[    {        "title": "2207.00641v1.Seismic_Response_of_Yielding_Structures_Coupled_to_Rocking_Walls_with_Supplemental_Damping.pdf",        "content": " \n Aghagholizadeh M, Makris N. Seismic Response of Yielding Structures Coupled to Rocking Walls with Supplemental Damping. Proceedings of the 12th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Salt Lake City, UT. 2022. \n Seismic Response of Yielding Structures Coupled to Rocking Walls with Supplemental Damping    M. Aghagholizadeh1  and N. Makris2   ABSTRACT  Given that the coupling of a framing structure to a strong, rocking wall enforces a first-mode response, this paper investigates the dynamic response of a yielding single-degree-of-freedom oscillator coupled to a rocking wall with supplemental damping (hysteretic or linear viscous) along its sides. The full nonlinear equations of motion are derived, and the study presents an earthquake response analysis in term of inelastic spectra. The study shows that for structures with preyielding period T1<1.0 s the effect of supplemental damping along the sides of the rocking wall is marginal even when large values of damping are used. The study uncovers that occasionally the damped response matches or exceeds the undamped response; however, when this happens, the exceedance is marginal. The paper concludes that for yielding structures with strength less than 10% of their weight the use of supplemental damping along the sides of a rocking wall coupled to a yielding structure is not recommended. The paper shows that supplemental damping along the sides of the rocking wall may have some limited beneficial effects for structures with longer preyielding periods (say T1>1.0 s). Nevertheless, no notable further response reduction is observed when larger values of hysteretic or viscous damping are used.  Introduction The concept of coupling the lateral response of a moment resisting frame with a rigid core system goes back to the early works of Paulay [1] and Fintel [2]. With this design, interstory drift demands are reduced at the expense of transferring appreciable shear-forces and bending moments at the foundation of the rigid core wall. In the early 1970s a new concept for seismic protection, by modifying the earthquake response of structures with specially designed supplemental devices, was brought forward in the seminal papers by Kelly et al. [3] and Skinner et al. [4] and was implemented in important structures that were under design at that time such as the South Rangitikei Rail Bridge, [5–7] the Union House Building in Auckland [8] and the Wellington Central Police Station in Wellington [9], New Zealand. Despite its remarkable originality and technical merit, the paper by Kelly et al. 1972 did not receive the attention it deserved, and it was some two decades later that the PRESSS Program [10,11] reintroduced the concept of uplifting and rocking of the joint shear wall system [12,13]. Given that damping during impact as the wall alternate pivot points is low, the idea of introducing supplemental energy dissipation devices in structural systems coupled with rocking walls received revived attention [14–18]. In view of the recent findings, this paper examines the contribution of viscous and hysteretic dampers to the response of a yielding frame coupled with a rocking wall shown in Figure 1.  1 Postdoc, Dept. of Civil Eng., Univ. of Southern California, Los Angeles, CA 90089 (email: aghaghol@usc.edu) 2 Professor, Dept. of Civil Eng., Southern Methodist University, Dallas, TX 75205  FIGURE 1 (a) Moment-resisting frame with a stepping rocking wall with dampers (b) A SDOF idealization of the yielding frame-rocking-wall system with a yielding oscillator coupled with a rocking wall with supplemental dampers. (c) Bilinear behavior of the yielding SDOF oscillator shown. (d) Geometric quantities pertinent to the dynamic analysis of a rocking wall with additional energy dissipators.  Dynamics of a Yielding Oscillator Coupled to a Rocking Wall with Supplemental Damping With reference to Fig. 1(b), this study examines the dynamic response of a yielding single-degree-of-freedom (SDOF) structure, with mass, ms, pre-yielding stiffness, k1, post-yielding stiffness, k2 and strength, Q that is coupled with a free-standing stepping rocking wall of size, 𝑅=√𝑏!+ℎ!, slenderness, tan𝛼=𝑏/ℎ, mass mw and moment of inertia about the pivoting (stepping) points O and O', 𝐼=4/3𝑚\"𝑅!. Vertical energy dissipation devices are mounted to the rocking wall at a distance, d, from the pivoting points of the wall as shown in Figures 1(b) and (d). In the interest of simplicity, it is assumed that the arm with length L, that couples the motion is articulated at the center mass of the rocking wall at a height, h from its foundation. With reference to Figure 1(b) dynamic equilibrium of the mass ms gives:   (1) Following a similar procedure as it described in [19,20], the equation of motion for positive and negative rotation (θ) of the stepping rocking wall can be written as:   (2)  in which 𝜎=𝑚#/𝑚\", 𝜉#= preyielding viscous damping ration of the SDOF oscillator (𝜉$=3%), 𝛼$\tis the () ( )sg ssmu u Ft c u T+= - -+!! !! !\n1222 2 211 1124cos ( ) cos( ) (sin sin( )) 2 cos( ) sin( ) (1 ) ( )3( 1) cos( ) sin( ) 0() [ ]ysss ss s s sddgwwuaa z tRFFurrgforRg m g R m g Rsa q q s a q waa q x w q a q q a q wsa q a q q+- + - - - + - + - + -éù=- + - + - + + ³êúëû!! ! !!!\n1222 2 211 1124cos ( ) cos( ) (sin sin( )) 2 cos( ) sin( ) (1 ) ( )3( 1) cos( ) sin( ) 0() [ ]ysss ss s s sddgwwuaa z tRFFurrgforRg m g R m g Rsa q q s a q waa q x w q a q q a q wsa q a q q++ - + - + - + + + - -éù=- + + - + + + <êúëû!! ! !!!\npost- to pre- yielding stiffness ration in hysteretic Bouc-Wen model [21,22], and 𝐹%is the damping force generated by the additional damping devices attached to the sides of the rocking-wall [19,23,24].  Earthquake Spectra of a Yielding Oscillator Coupled to a Rocking Wall with Supplemental Damping The effect of supplemental damping, either hysteretic or viscous along the sides of a stepping rocking wall coupled to a medium-to-high rise, yielding building is investigated with the generation of inelastic response spectra. Figures 2 and 3 (top and bottom) plots displacement response spectra of the yielding SDOF oscillator coupled to a rocking wall with vertical hysteretic dampers or similarly defined strength viscous dampers (for detailed calculations refer to [19]) appended to the pivot corners of the rocking wall with strength equal to 20% and 50% of the yielding strength of the structure 𝑄𝑠 = 0.08𝑚𝑠𝑔, postyield-to-preyield stiffness ratio equal to 𝑎𝑑 = 2.5% and the mass ratio 𝜎 = 𝑚𝑠\t⁄𝑚𝑤 = 10. Figure 2 shows that when the input ground motion is the 1994 Newhall record, the vertical hysteretic dampers further suppress the inelastic displacements. Figure 2 (top) also plots the peak angular velocity, 𝜃̇&'(\t, of the rocking wall with the scale shown on the right of the plots. Clearly, as the preyielding period, T1, of the frame structure increases, the peak angular velocity decreases. For each value of the preyielding period of the yielding oscillator appearing along the horizontal axis of the spectra, the value, 𝜃̇&'(, is used to calculate the equivalent viscous damping 𝑐)=𝜖𝑄#/<2𝑏𝜃̇&'(>that is needed to compute the corresponding spectra where the supplemental damping at the pivot corners of the rocking wall are linear viscous dampers. Figure 2 (bottom row) plots displacement response spectra of the yielding SDOF oscillator coupled to a rocking wall with vertical linear viscous dampers appended at the pivot corners of the rocking wall (d=0) with damping constant 𝑐)=𝜖𝑄#/<2𝑏𝜃̇&'(> in which 𝜃̇&'( is offered in Figure 2 (top row). Similarly, when the input ground motion is the 1994 Newhall record, the viscous dampers further suppress the inelastic displacements.  \n FIGURE 2 Peak response of SDOF yielding oscillator coupled with a stepping wall with zero-length supplemental hysteretic (top row) and viscous dampers (bottom row) appended at the pivoting points (𝑑 = 0) when excited by the Newhall/360 ground motion recorded during 1994 Northridge, California earthquake. Figures on the left correspond to a SDOF yielding oscillator with strength of 𝑄𝑠/𝑚𝑠 = 0.08𝑔,\twhereas for the figures on the right, 𝑄𝑠/𝑚𝑠 = 0.15𝑔.\t\n FIGURE 3 Peak response of SDOF yielding oscillator coupled with a stepping wall with zero-length supplemental hysteretic (top row) and viscous dampers (bottom row) appended at the pivoting points (𝑑 = 0) when excited by the Erzincan NS ground motion recorded during the 1992 Erzincan, Turkey earthquake. Figures on the left correspond to a SDOF yielding oscillator with strength of 𝑄𝑠/𝑚𝑠 = 0.08𝑔,\twhereas for the figures on the right, 𝑄𝑠/𝑚𝑠 = 0.15𝑔.\t There are situations where the structural response exceeds the structural response without dampers being appended at the pivot corners of the rocking wall. This “counter intuitive” finding should not be a surprise since it has been observed to also happen on the rocking response of solitary columns with supplemental damping [19,20] and results from the way that inertia, gravity, and damping forces combine. Similar trends can be observed in Figure 3 when the yielding SDOF system is coupled to the damped rocking wall when subjected to the Erzincan NS ground motion recorded during the 1992 Erzincan, Turkey earthquake.  Conclusions The need to ensure uniform interstory drift distribution in medium-to-high rise buildings when subjected to earthquake shaking has prompted a growing interest in coupling the lateral response of moment resisting frames to strong, heavy rocking walls. Given that the coupling with a strong rocking wall, enforces a deformation pattern of the yielding structural system that resembles a first mode, the analysis adopted a single-degree-of-freedom idealization to perform a response analysis. This paper investigates the dynamic response of a yielding SDOF oscillator coupled to a stepping rocking wall with supplemental damping (either hysteretic or viscous) along its sides. The full nonlinear equations of motion are derived, and the study presents a parametric analysis of the inelastic system in terms of inelastic response spectra and reaches the following conclusions: The participation of the stepping rocking wall suppresses invariably peak inelastic displacement; as has been shown in previous studies. In contrast, the effect of supplemental damping along the sides of the rocking wall is marginal for structures with preyielding periods lower that T1=1.0 s and occasionally the damped response exceeds the undamped response. Whenever the damped response exceeds the undamped response, the exceedance is marginal. The paper shows that supplemental damping along the sides of the rocking wall may have some limited beneficial effects for structures with longer preyielding periods (say T1>1.0 s). Nevertheless, no notable further response reduction is observed when larger values of hysteretic or viscous damping are used. \nReferences 1. Paulay T. The coupling of reinforced concrete shear walls. The Fourth World Conference on Earthquake Engineering, 1969. 2. Fintel M. Ductile Shear Walls in Earthquake-Resistant Multistory Buildings. Wind and Seismic Effects: Proceedings of the Seventh Joint Panel Conference of the US-Japan Cooperative Program in Natural Resources, vol. 470, US Department of Commerce, National Bureau of Standards; 1975. 3. Kelly JM, Skinner RI, Heine AJ. Mechanisms of energy absorption in special devices for use in earthquake resistant structures. Bulletin of NZ Society for Earthquake Engineering 1972; 5(3): 63–88. 4. Skinner RI, Beck JL, Bycroft GN. A practical system for isolating structures from earthquake attack. Earthquake Engineering & Structural Dynamics 1974; 3(3): 297–309. DOI: 10.1002/eqe.4290030308. 5. Beck JL, Skinner RI. The seismic response of a reinforced concrete bridge pier designed to step. Earthquake Engineering and Structural Dynamics 1974; 2(4): 343–358. 6. Skinner RI, Kelly JM, Heine AJ. Hysteretic dampers for earthquake‐resistant structures. Earthquake Engineering and Structural Dynamics 1974; 3(3): 287–296. 7. Kelly JM. Earthquake-Resistant Design with Rubber. Springer London; 1993. DOI: 10.1007/978-1-4471-3359-9. 8. Boardman P, Wood B, Carr A. Union House-A crossbraced structure with energy dissipators Seismic Vulnerability Assessment. Retrofit Strategies and Risk Reduction View Project Damage Avoidance Design (DAD) View project. Bulletin of the New Zealand National Society for Earthquake Engineering 1983; 16(2). 9. Charleson AW, Wright PD, Skinner RI. Wellington central police station, base isolation of an essential facility. Pacific Conference on Earthquake Engineering, 1987. 10. Priestley MJN. Overview of PRESSS research program. PCI Journal 1991; 36(4): 50–57. 11. Priestley MJN. The PRESSS Program—Current Status and Proposed Plans for Phase Ill. PCI Journal 1996; 4(2): 22–40. 12. Nakaki SD, Stanton JF, Sritharan S. An overview of the PRESSS five-story precast test building. PCI Journal 1999; 44(2). 13. Priestley MJN, Sritharan S, Conley JR, Pampanin S. Preliminary results and conclusions from the PRESSS five-story precast concrete test building. PCI Journal 1999; 44(6): 42–67. 14. Ajrab JJ, Pekcan G, Mander JB. Rocking wall-frame structures with supplemental tendon systems. Journal of Structural Engineering 2004; 130(6): 895–903. 15. Holden T, Restrepo J, Mander JB. Seismic performance of precast reinforced and prestressed concrete walls. Journal of Structural Engineering 2003; 129(3): 286–296. 16. Makris N, Aghagholizadeh M. The dynamics of an elastic structure coupled with a rocking wall. Earthquake Engineering & Structural Dynamics 2017; 46(6): 945–962. 17. Aghagholizadeh M, Makris N. Seismic Response of a Yielding Structure Coupled with a Rocking Wall. Journal of Structural Engineering 2018; 144(2): 04017196. DOI: 10.1061/(asce)st.1943-541x.0001894. 18. Aghagholizadeh M, Makris N. Earthquake response analysis of yielding structures coupled with vertically restrained rocking walls. Earthquake Engineering and Structural Dynamics 2018; 47(15): 2965–2984. DOI: 10.1002/eqe.3116. 19. Aghagholizadeh M, Makris N. Response analysis of yielding structures coupled to rocking walls with supplemental damping. Earthquake Engineering & Structural Dynamics 2021; 50(10): 2672–2689. DOI: 10.1002/EQE.3466. 20. Makris N, Aghagholizadeh M. Effect of Supplemental Hysteretic and Viscous Damping on Rocking Response of Free-Standing Columns. Journal of Engineering Mechanics 2019; 145(5): 4019028. 21. Bouc R. Forced vibration of mechanical systems with hysteresis. Fourth conference on non-linear oscillation, 1967. 22. Wen YK. Approximate method for nonlinear random vibration. Journal of Engineering Mechanics 1975; 101. 23. Aghagholizadeh M. Seismic Response of Moment Resisting Frames Coupled with Rocking Walls. Doctoral dissertation, University of Central Florida. 2018. 24. Aghagholizadeh M. A finite element model for seismic response analysis of vertically-damped rocking-columns. Engineering Structures 2020; 219: 110894. 25. Aghagholizadeh M, Makris N. Moment Resisting Frames Coupled with Rocking Walls Subjected to Earthquakes, PEER Report No. 2022/03. Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA (2022). https://doi.org/10.55461/MXXS2889. "    },    {        "title": "1001.2356v1.Multi_Error_Correcting_Amplitude_Damping_Codes.pdf",        "content": "arXiv:1001.2356v1  [quant-ph]  14 Jan 2010Multi-Error-Correcting Amplitude Damping Codes\nRunyao Duan∗†, Markus Grassl‡, Zhengfeng Ji§, and Bei Zeng¶\n∗Centre for Quantum Computation and Intelligent Systems (QC IS),\nFaculty of Engineering and Information Technology, Univer sity of Technology, Sydney, NSW 2007, Australia\n†State Key Laboratory of Intelligent Technology and Systems , Tsinghua National Laboratory for Information Science\nand Technology, Department of Computer Science and Technol ogy, Tsinghua University, Beijing 100084, China\n‡Centre for Quantum Technologies, National University of Si ngapore, Singapore 117543, Singapore\n§Perimeter Institute for Theoretical Physics, Waterloo, ON , N2L2Y5, Canada\n¶Institute for Quantum Computing and the Department of Combi natorics and Optimization,\nUniversity of Waterloo, Waterloo, ON, N2L3G1, Canada\nAbstract—We construct new families of multi-error-correcting\nquantum codes for the amplitude damping channel. Our key\nobservation is that, with proper encoding, two uses of the\namplitudedampingchannelsimulatea quantumerasure chann el.\nThis allows us to use concatenated codes with quantum erasur e-\ncorrecting codes as outer codes for correcting multiple amp litude\ndamping errors. Our new codes are degenerate stabilizer cod es\nand have parameters which are better than the amplitude\ndamping codes obtained by any previously known constructio n.\nIndex Terms —Amplitude damping channel, quantum error\ncorrection, concatenated quantum codes, quantum erasure c ode.\nI. INTRODUCTION\nIn most of works on quantumerror correction,it is assumed\nthat the errors to be corrected are completely random, with n o\nknowledge other than that they affect different qubits inde -\npendently [22], [9]. Or, equivalently, this is to assume tha t the\nPauli-type errors X= (0 1\n1 0),Y=/parenleftbig0−i\ni0/parenrightbig\n, andZ=/parenleftbig1 0\n0−1/parenrightbig\n,\nhappen with equal probability px=py=pz=p/3. The\nquantum channel described by this kind of noise is called\ndepolarizing channel EDP.\nThe most general physical operations (or quantum chan-\nnels) allowed by quantum mechanics are completely positive ,\ntrace preserving linear maps which can be represented in the\nfollowing Kraus decomposition form:\nN(ρ) =/summationdisplay\nkAkρA†\nk, (1)\nwhereAkare called Kraus operators of the quantum channel\nNand satisfy the completeness condition/summationtext\nkA†\nkAk= 1l. In\nthis language of quantum channels, the depolarizing channe l\nEDPwith error parameter pacting on any one-qubit quantum\nstateρ∈C2×2as\nEDP(ρ) = (1−p)ρ+p\n3(XρX+YρY+ZρZ),(2)\nso the Kraus operators for the depolarizing channel are the\nPauli matrices together with identity.\nHowever, if further information about an error process is\navailable, more efficient codes can be designed. Indeed in\nmany physical systems, the types of noise are likely to be\nunbalanced between amplitude ( X-type) errors and phase ( Z-\ntype) errors. Recently a lot of attention has been put intodesigning codes for this situation and in studying their fau lt\ntolerance properties [1], [7], [8], [15], [23]. All those wo rks\ndeal with error models which are still described by Kraus\noperators that are Pauli matrices (Pauli Kraus operators), but\ntheX- andY-errors happen with equal probability px=py,\nwhich might be different from the probability pzthat aZ-\nerror happens. The quantum channels described by this kind\nof noise are called asymmetric channels EASacting on any\none-qubit quantum state ρas\nEAS(ρ) = (1−(2px+pz))ρ\n+px(XρX+YρY)+pzZρZ. (3)\nThe choice px=pyis related to a physically realistic error\nmodel including amplitude damping (AD) noise and phase\ndamping noise [22]. The Kraus operators for AD noise with\ndamping rate γare\nA0=/parenleftbigg1 0\n0√1−γ/parenrightbigg\nandA1=/parenleftbigg0√γ\n0 0/parenrightbigg\n.(4)\nNote that\nA1=/parenleftbigg\n0√γ\n0 0/parenrightbigg\n=√γ\n2(X+iY)and\nA†\n1=√γ\n2(X−iY).\nHence the linear span of the operators A1andA†\n1equals the\nlinear span of XandY. If the system is at finite temperature,\nthe Kraus operator A†\n1will appear in the noise model [22].\nThus, if the code is capable of correcting tX- andtY-errors,\nit can also correct t A1- andt A†\n1-errors.\nIt was observed that when the temperature of a physical\nsystem is zero or very low, the error A†\n1is actually negligible\n[22]. For simplicity,we furtherignorethe phasedampinger ror\n(which is characterized by the Pauli operator Z). Then the\nerror model is fully characterized by A0andA1. In this work,\nwe will focus on this quantum channel with only amplitude\ndamping noise, i.e. the AD channel EAD, with only two Kraus\noperators given by Eq. (4). The AD channel is the simplest\nnonunital channel whose Kraus operators cannot be describe d\nbyPauli operations.The AD channelisa quantumanalogueof\nthe classical Z-channel which transmits 0faithfully, but maps1to either 0or1[26]. For the AD channel we only need\nto deal with the error A1(a quantum analogue of the error\n1→0), but not with A†\n1(a quantum analogue of the error\n0→1). So asking to be able to correct both X- andY-errors\nis a less efficient way for constructing quantum codes for the\nAD channel.\nSince the error model is not described by Pauli Kraus\noperators, the task of constructing good error-correcting codes\nbecomesverychallenging.Theknowntechniquesdealingwit h\nPauli errors cannot be applied or result in codes with bad\nparameters. Several new techniques for the construction of\ncodes which are adapted to this type of noise with non-\nPauli Kraus operators, and the AD channel in particular, hav e\nbeen developed [6], [8], [18], [19], [26]. After years’ effo rt,\nsystematic methods for constructing high performance sing le-\nerror-correcting codes have been found [18], [26]. However ,\nall these methods fail to construct good AD codes correcting\nmulti-errors.\nIn this paper we present a method for finding families of\ncodes correcting multi-amplitude-damping errors. Our con -\nstruction is based on the observation that with respect to a\nsimple encoding two uses of the amplitude damping channel\nsimulate a quantum erasure channel. This allows us to apply a\nconcatenated coding scheme with quantum erasure-correcti ng\ncodes as outer codes, resulting in codes correcting multi-\namplitude-damping errors. Our new codes are degenerate\nstabilizer codes which have better parameters than the code s\ngiven by any previously known construction.\nII. CORRECTING AMPLITUDE DAMPING ERRORS\nA quantum error-correcting code Qis a subspace of\n(C2)⊗n, the space of nqubits. For a K-dimensional code\nspace spanned by the orthonormal set |ψi/an}bracketri}ht,i= 1,...,Kand\na set of errors Ethere is a physical operation correcting all\nelementsEµ∈ Eif the error correction conditions [3], [16]\nare satisfied:\n∀ij,µν/an}bracketle{tψi|E†\nµEν|ψj/an}bracketri}ht=Cµνδij, (5)\nwhereCµνdepends only on µandν. If the matrix (Cµν)has\nfull rank the code is said to be nondegenerate, otherwise it i s\ndegenerate.\nFor the AD channel, if γis small, we would like to correct\nthe leading order errors that occur during amplitude dampin g.\nSettingA=X+iYandB=I−Z, we have\nA1=√γ\n2AandA0=I−γ\n4B+O(γ2).(6)\nIt has been shown that in order to improve the fidelity of\nthe transmission through an amplitude damping channel from\n1−γto1−γt, it is sufficient to satisfy the error-detection\nconditions for 2tA-errors andtB-errors [9, Section 8.7]. We\nwill say that such a code corrects tamplitude damping errors\nsince it improves the fidelity, to leading order, just as much\nas a truet-error-correcting code would for the same channel.\nStabilizer codes are a large kind of quantum codes which\ncontain many good quantum codes [9], [22]. A stabilizer code\nwithnqubits encoding kqubits is of distance dif all errors ofweight at most d−1(i.e., operators acting nontrivially on less\nthandindividual qubits) can be detected or have no effect on\nQ, and we denote the parameters of Qby[[n,k,d]]. We say\nan[[n,k]]stabilizer code is a t-code if it corrects tAD-errors.\nFor comparisonwith stabilizer codes, we say an [[n,k]]t-code\nis good if 2t+1>dfor the best possible [[n,k,d]]code; or,\nn<n′for the best possible [[n′,k,2t+1]]code; or,k >k′\nfor the best possible [[n,k′,2t+1]]code.\nThe first AD code given by Leung et al. [19] is a [[4,1]]\n1-code, i.e., correcting a single AD-error. Basis vectors of the\ncode are\n|0/an}bracketri}htL=1√\n2(|0000/an}bracketri}ht+|1111/an}bracketri}ht)\n|1/an}bracketri}htL=1√\n2(|0011/an}bracketri}ht+|1100/an}bracketri}ht). (7)\nUsing only 4 qubits, this 1-code is better than the [[5,1,3]]\ncode,a quantumcodecorrectingan arbitrarysingle-qubite rror\nand encoding one qubit using the minimal number of qubits\n[3], [17].\nFollowing the work by Leung et al. [19], several construc-\ntions for 1-codes have been proposed [8], [9], [18], [26],\nincludingsomehighperformance 1-codes.However,verylittle\nis known about good multi-error-correctingAD codes. It tur ns\nout that none of the methods known for constructing good 1-\ncodes can be directly generalized to t-codes with t>1.\nGottesman[9, Section 8.7] hasshown that Shor’s nine-qubit\ncode [25]\n|0/an}bracketri}htL=1\n2√\n2(|000/an}bracketri}ht+|111/an}bracketri}ht)⊗3\n|1/an}bracketri}htL=1\n2√\n2(|000/an}bracketri}ht−|111/an}bracketri}ht)⊗3(8)\ncan correct two AD-errors, despite the fact that it can corre ct\nonly a single general error. It is the best known 2-code and\nit is better than the [[11,1,5]]code [9], the best two-error-\ncorrecting stabilizer code encoding one qubit [10].\nIt is interesting to note that the 1-code given by Eq. (7) can\nbe rewritten in another basis as\n|+/an}bracketri}htL=1√\n2(|0/an}bracketri}htL+|1/an}bracketri}htL) =1\n2(|00/an}bracketri}ht+|11/an}bracketri}ht)⊗2\n|−/an}bracketri}htL=1√\n2(|0/an}bracketri}htL−|1/an}bracketri}htL) =1\n2(|00/an}bracketri}ht−|11/an}bracketri}ht)⊗2,(9)\nwhich is of a similar form as Eq. (8).\nTherefore, we can generalize the constructions of Eqs. (9)\nand (8) tot-codes with basis\n|0/an}bracketri}htL= 2−t+1\n2/parenleftBig\n|0...0/an}bracketri}ht/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nt+1+|1...1/an}bracketri}ht/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nt+1/parenrightBig⊗(t+1)\n|1/an}bracketri}htL= 2−t+1\n2/parenleftBig\n|0...0/an}bracketri}ht/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nt+1−|1...1/an}bracketri}ht/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nt+1/parenrightBig⊗(t+1)\n.(10)\nHowever,these [[n2,1,n]]so-called Bacon-Shorcode[2], [25]\ncorrectingt=n−1AD-errors scale badly when nis large.\nFor instance, there exists a [[25,1,9]]code and a [[29,1,11]]\ncode [10].Note that these [[n2,1]]codes are of Calderbank-Shor-\nSteane (CSS) type [5], [27]. They are also degenerate: for\ninstance, aZ-error acting on the first qubit or the second qubit\nhas the same effect on the code.\nIn general, CSS codes can be used to construct codes for\nthe AD channel [9, Section 8.7]:\nProposition 1 An[[n,k]]CSS code of X-distance 2t+1and\nZ-distancet+1is an[[n,k]]t-code.\nIn the first column of Table I we provide bounds on the\nlengthnof codes for the AD channel encoding one or two\nqubitsderivedfromCSS codeswith given Z- andX-distances\nt+ 1and2t+ 1, respectively. The lower bounds have been\nderived using linear programming techniques [24]. The uppe r\nbound is based on CSS codes constructed from the database\nof best known linear codes [4], [10].\nIn the fifth column we give upper and lower bounds on the\nlengthn′such that an [[n′,k,t+ 1]]code may exist. In the\nlast column, we list the bounds on the length of t-code from\nTheorem 1. The data for columns n′and2mis taken from\n[10]).\nn k t +1 2t+1 n′2m\n12–13 1 3 5 11 10\n19–20 1 4 7 17 20\n25–30 1 5 9 23–25 22\n33–41 1 6 11 29 32\n39–54 1 7 13 35–43 34\n47–70 1 8 15 41–53 44–48\n53–79 1 9 17 47–61 46–50\n–89 1 10 19 53–81 56\n–105 1 11 21 59–85 58\n14–17 2 3 5 14 16\n20–27 2 4 7 20–23 20\n27–37 2 5 9 26–27 28\n34–45 2 6 11 32–41 32\n41–62 2 7 13 38–51 40–46\n–71 2 8 15 44–59 44–52\n–87 2 9 17 50–78 52–54\n–102 2 10 19 56–83 56–56\n–110 2 11 21 62–104 64–82\nTABLE I\nBOUNDS ON THE LENGTH nOF AN[[n,1]]t-CODE DERIVED FROM CSS\nCODES,TOGETHER WITH THE BOUNDS ON THE LENGTH n′OF A\nSTABILIZER CODE [[n′,1,2t+1]]AND THE LENGTH 2mOF AN[[2m,1]]\nt-CODE FROM THEOREM 1.\nItcanbeseenfromTableIthattheconstructionofADcodes\nbased on CSS codes unlikely gives good AD codes. But as it\nis unknown whether these bounds for nandn′given in this\ntable can be achieved, we do not have the definite answer.\nThis problem will be addressed in future research.\nIII. AD CODE BASED ON QUANTUM ERASURE CODES\nAs discussed in Sec. II, no good method is known for\nconstructing good multi-error-correcting AD codes. In thi s\nsection we provide a construction which systematically giv es\nhigh performance t-codes with t >1. The construction\nuses concatenated quantum codes with an inner and an outer\nquantum code. After decoding the inner quantum code, theeffective channel is a quantum erasure channel. We start by\nproving the following lemma.\nLemma 1 Using the quantum dual-rail code Qiwhich en-\ncodes a single qubit into two qubits, given by\n|0/an}bracketri}htL=|01/an}bracketri}ht,|1/an}bracketri}htL=|10/an}bracketri}ht, (11)\ntwo uses of the AD channel simulate a quantum erasure\nchannel.\nProof:For any state ρof the code Q1, we observe that\nE⊗2\nAD(ρ) = (1−γ)ρ+γ(|00/an}bracketri}ht/an}bracketle{t00|). (12)\nThe state |00/an}bracketri}htis orthogonal to the code Q1. Using a measure-\nment that either projects on Q1or its orthogonal complement,\nit can be detected whether an AD error occurredor not. Hence\nwe obtain a quantum erasure channel with erasure symbol\n|00/an}bracketri}ht.\nRemark 1 It can easily be shown that with respect to the\ndual-rail code {01,10}, two uses of the Z-channel simulate\na classical erasure channel with erasure symbol 00(see, e.g.\n[21]). Lemma 1 is a quantumanalogueof this fact, yet Lemma\n1 is nontrivial due to the Kraus operator A0, which introduces\nsome relative phase error between |0/an}bracketri}htand|1/an}bracketri}htthat has no\nclassical analogue.\nLemma1 allowsustouse quantumerasure-correctingcodes\nas outer codes for correcting multiple amplitude damping\nerrors. It is known that an [[m,k,d]]quantum code corrects\nd−1erasure errors [9], [11], [22]. Our main result is given\nby the following theorem.\nTheorem 1 If there exists an [[m,k,d]]quantum code, then\nthere is a [[2m,k]]code correcting t=d−1amplitude\ndamping errors.\nProof:LetQbe the concatenated code with the inner\ncodeQ1givenEq.(11) andtheoutercode Q2with parameters\n[[m,k,d]]. The code Q2correctsd−1erasure errors. A\nsingle AD-error on each block of the inner code creates an\nerasure error for the outer code. The position of the error\nis indicated by the erasure state |00/an}bracketri}ht. Hence the outer codes\ntakes care of d−1AD-errors acting on different blocks. Two\nerrors acting on the same block annihilate the state, such th at\nthe quantum error correction condition given by Eq. (5) is\nnaturally satisfied. Hence Qis a[[2m,k]]AD code correcting\nt=d−1amplitude damping errors.\nRemark 2 It is interesting to compare our construction with\nthecorrespondingclassicalcase,whereconcatenationwit hthe\ndual-railcode {01,10}asinnercodeandan [m,k,d]erasure-\ncorrecting code as outer code yields an [2m,k] (d−1)-\ncode for the Z-channel. However, this (d−1)-code is in\ngeneral not good because simply repeating each codeword\nof an[m,k,d]classical code will straightforwardly give a\n[2m,k,2d]code correcting d−1arbitrary errors. In thequantumcase,however,theexistenceofan [[m,k,d]]stabilizer\ncode does not necessarily lead to a [[2m,k,2d]]stabilizer\ncode.\nIn Table II, we compare the t-codes from our construction\nwith the known upper and lower bounds on the minimum\ndistance of stabilizer codes from [10]. We fix the number of\nlogical qubits kand the number tof correctable AD-errors\nwithin the range k= 1,...,6andt= 1,...,10. The length\nn= 2mof the code is derived from the shortest known\nstabilizer code with parameters [[m,k,t+1]]from [10]. Hence\nthe first three columns gives the parameters of each line in\nthe table corresponds to an [[n,k]]t-code. The fourth column\nprovides 2t+1, which is the distance that is required for an\n[[n,k]]code to be capable to correct tarbitraryerrors. The last\ncolumn gives the lower and upper bounds on the distance dof\na[[n,k,d]]stabilizer code from [10]. Hence all t-codes with\n2t+1>dare better than the stabilizer codes with the same\nlength and dimension.With the exceptionof small parameter s,\nmany of our codes outperform the known—or even the best\npossible—correspondingstabilizer codes correcting tarbitrary\nerrors. Note that any improvement of the lower bound on the\ndistancedof a stabilizer code implies some improvement for\nt-codes as well.\nNote that all the t-codes listed in the table are degenerate\nstabilizer codes obtained by concatenation of a stabilizer code\nas outer code and the quantum dual-rail code Q1given by Eq.\n(11) as inner code. In order to compute the stabilizer of the\nconcatenatedcode, note that the inner code Q1is stabilized by\n−ZZ, and has logical operators ¯X=XXand¯Z=ZI. As\nan example, we compute the stabilizer for the [[10,1]] 2-code.\nExample 1 A[[10,1]] 2-code can be derived from the\n[[5,1,3]]code with stabilizer generated by:\ng1=X Z Z X I\ng2=I X Z Z X\ng3=X I X Z Z\ng4=Z X I X Z(13)\nThe stabilizer of the [[10,1]] 2-code is obtained by replacing\nthe operators in Eq. (13) by the logical operators of Q1and\nadding the stabilizer for each block of the inner code:\ng′\n1=XX Z I Z I XX I I\ng′\n2=I I XX Z I Z I XX\ng′\n3=XX I I XX Z I Z I\ng′\n4=Z I XX I I XX Z I\ng′\n5=−ZZ I I I I I I I I\ng′\n6=−I I Z Z I I I I I I\ng′\n7=−I I I I Z Z I I I I\ng′\n8=−I I I I I I Z Z I I\ng′\n9=−I I I I I I I I Z Z(14)\nAs a degenerate stabilizer code, this code has parameters\n[[10,1,4]].Asa2-code,thiscodeisnotasgoodasShor'snine-\nqubit code given in Eq. (8), but still better than the shortes tstabilizer code [[11,1,5]]encoding one qubit and correcting\ntwo arbitrary errors.\nHowever, the [[22,1]] 4-code given in Table II is better than\nthe[[25,1]] 4-codegivenin Eq. (10), the degenerate [[25,1,9]]\ncode constructed from concatenating two [[5,1,3]]codes, and\neven the putative stabilizer code [[22,1,8]].\nFrom the last column in Table I we see that, with the\nexception when both parameters tandkare small, the codes\nfrom our construction are better than the t-codes derived from\nCSS codes.\nIV. POSSIBLE GENERALIZATIONS\nOne possible generalization of our construction is to chose\na differentinnercode.For instance,we can take the innerco de\nas the following quantum code Q′\n1which encodes one qutrit\ninto three qubits:\n|0/an}bracketri}htL=|001/an}bracketri}ht,|1/an}bracketri}htL=|010/an}bracketri}ht,|2/an}bracketri}htL=|100/an}bracketri}ht.(15)\nFor any state ρof the code Q′\n1, we observe that\nE⊗3\nAD(ρ) = (1−γ)ρ+γ(|000/an}bracketri}ht/an}bracketle{t000|),(16)\nhencethe effectivechannelis a qutrit quantumerasurechan nel\nwhere the state |000/an}bracketri}htindicates an erasure.\nSince the inner code Q′\n1is of dimension 3, the outer code\nQ′\n2mustbechosenfromquantumcodesconstructedforqutrits\nrather than qubits, i.e. Q′\n2is a subspace of (C3)⊗m. Using a\n[[m,k,d]]3quantum code Q′\n2(where the subscript 3indicates\nthat this is a qutrit code), the concatenated code Qwith inner\ncodeQ1andoutercode Q2is anAD codecorrecting t=d−1\nAD errors, with length 3mand encodinga space of dimension\n3k. In general, quantum code of length nand dimension Kis\ndenoted by ((n,K)), so this construction yields a ((3m,3k))\n(d−1)-code.\nFor instance, an [[8,2,4]]3outer code (see [14], [20]) gives\na((24,9))AD code correcting 3AD errors. This is better\nthan the parameters [[24,3,7–8]]of a stabilizer code (cf. [10]),\nbut worse than the [[24,4]] 3-code given in Table II. It is\nnot yet clear whether this or other generalizations based on\nconcatenation using codes for the erasure channel yield bet ter\nAD codes than those obtained from the quantum dual-rail\ncodes.\nV. CONCLUSIONS\nWe have constructed families of good multi-error-correcti ng\nquantum codes for the amplitude damping channel based\non code concatenation and quantum erasure-correcting code s.\nAs the rate of our codes can never exceed the rate 1/2\nof the inner code, other methods—possibly generalized con-\ncatenation of quantum codes [12], [13]—have to be used in\norder to construct high-rate AD codes. However, our method\nprovidesthe first systematic constructionfor good multi-e rror-\ncorrectingAD codes. We hopethat ourmethodshade lightson\nconstructing good quantum codes adapted for other non-Paul i\nchannelsbeyondtheADchannel,andfurtherunderstandingo n\nthe role that degenerate codes play in quantum coding theory .n k t 2t+1 d\n8 1 1 3 3\n10 1 2 5 4\n20 1 3 7 7\n22 1 4 9 7–8\n32 1 5 11 11\n34 1 6 13 11–12\n48 1 7 15 13–17\n50 1 8 17 13–17\n56 1 9 19 15–19\n58 1 10 21 15–20\n8 2 1 3 3\n16 2 2 5 6\n20 2 3 7 6–7\n28 2 4 9 10\n32 2 5 11 10–11\n46 2 6 13 12–16\n52 2 7 15 14–18\n54 2 8 17 14–18\n56 2 9 19 14–19\n82 2 10 21 18–28n k t 2t+1 d\n12 3 1 3 4\n16 3 2 5 5\n24 3 3 7 7–8\n30 3 4 9 9–10\n40 3 5 11 10–13\n48 3 6 13 11–16\n52 3 7 15 13–17\n54 3 8 17 13–18\n72 3 9 19 15–24\n82 3 10 21 18–27\n12 4 1 3 4\n20 4 2 5 6\n24 4 3 7 6–8\n32 4 4 9 8–10\n40 4 5 11 10–13\n50 4 6 13 12–16\n52 4 7 15 12–17\n70 4 8 17 15–23\n80 4 9 19 16–26\n96 4 10 21 18–31n k t 2t+1 d\n16 5 1 3 4–5\n22 5 2 5 6–7\n28 5 3 7 7–9\n36 5 4 9 8–11\n42 5 5 11 9–13\n50 5 6 13 11–16\n60 5 7 15 13–19\n78 5 8 17 15–25\n86 5 9 19 18–28\n98 5 10 21 19–32\n16 6 1 3 4\n24 6 2 5 6–7\n28 6 3 7 6–8\n36 6 4 9 8–11\n48 6 5 11 10–15\n58 6 6 13 12–19\n64 6 7 15 14–21\n84 6 8 17 17–27\n92 6 9 19 18–29\n104 6 10 21 19–33\nTABLE II\nCOMPARISON OF OUR [[n,k]]t-CODES AND THE BOUNDS ON THE MINIMUM DISTANCE dOF A STABILIZER CODE [[n,k,d]].\nACKNOWLEDGMENTS\nWe thank Daniel Gottesman and Peter Shor for helpful\ndiscussions. RD is partly supported by QCIS, University of\nTechnology, Sydney, and the NSF of China (Grant Nos.\n60736011 and 60702080). BZ is supported by NSERC and\nQuantumWorks. Centre for Quantum Technologies is a Re-\nsearch Centre of Excellence funded by Ministry of Education\nand National Research Foundation of Singapore. Research at\nPerimeter Institute is supported by the Government of Canad a\nthrough Industry Canada and by the Province of Ontario\nthought the Ministry of Research & Innovation.\nREFERENCES\n[1] P. Aliferis and J. Preskill. Fault-tolerant quantum com putation against\nbiased noise. Physical Review A 78(5):052331, 2008.\n[2] D. Bacon. Operator quantum error-correcting subsystem s for self-\ncorrecting quantum memories. Physical Review A , 73(1):012340, 2006.\n[3] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Woo tters.\nMixed stateentanglement and quantum errorcorrection. PhysicalReview\nA, 54(5):3824–3851, 1996.\n[4] W. Bosma, J. J. Cannon, and C. Playoust. 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Shor. Nonadditive quantum error correc ting codes\nadapted to the amplitude damping channel. arXiv:0712.2586 , 2007.\n[19] D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto. Ap-\nproximate quantum error correction can lead to better codes .Physical\nReview A , 56(4):2567–2573, 1997.\n[20] S. Y. Looi, L. Yu, V. Gheorghiu, and R. B. Griffiths. Quant um\nerror correcting codes using qudit graph states. Physical Review A ,\n78(4):042303, 2008.\n[21] J. L. Massey. Zero Error. Lecture at Information Theory Winter\nSchool 2007 , La Colle sur Loup, France, 2007. Available on-line at\nhttp://itwinterschool07.eurecom.fr/Tutorials/Massey Zeroerror.pdf\n[22] M. Nielsen and I. Chuang. Quantum computation and quantum infor-\nmation. Cambridge University Press, Cambridge, England, 2000.\n[23] P. K. Sarvepalli, M. R¨ otteler, and A. Klappenecker. As ymmetric\nquantum LDPC codes. In Proceedings of the 2008 IEEE International\nSymposium on Information Theory , pp. 305–309, 2008.\n[24] P. K. Sarvepalli, A. Klappenecker, and M. R¨ otteler. As ymmetric\nquantum codes: constructions, bounds, and performance, Proceedings\nof the Royal Society London, Series A , 465(2105):1645–1672, 2009.\n[25] P. W. Shor. Scheme for reducing decoherence in quantum c omputer\nmemory. Physical Review A , 52(4):2493–2496, 1995.\n[26] P. W. Shor, G. Smith, J. A. Smolin, and B. Zeng. High perfo rmance\nsingle-error-correcting quantum codes for the amplitude d amping chan-\nnel.arXiv:0907.5149 , 2009.\n[27] A. Steane. Multiple particle interference and quantum error correction.\nProceedings of the Royal Society London, Series A , 452(1954):2551–\n2577, 1996."    },    {        "title": "1611.01870v1.Signals_for_Lorentz_and_CPT_Violation_in_Atomic_Spectroscopy_Experiments_and_Other_Systems.pdf",        "content": "arXiv:1611.01870v1  [hep-ph]  7 Nov 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nSignals for Lorentz and CPT Violation in\nAtomic Spectroscopy Experiments and Other Systems\nArnaldo J. Vargas\nPhysics Department, Indiana University, Bloomington, IN 4 7405, USA\nThe prospects of studying nonminimal operators for Lorentz violation using\nspectroscopy experiments with light atoms and muon spin-pr ecession experi-\nments are presented. Possible improvements on bounds on min imal and non-\nminimal operators for Lorentz violation are discussed.\n1. Motivation and introduction\nThe Standard-Model Extension (SME) has facilitated a worldwide sy s-\ntematic search for Lorentz violation. Promptly after the introduc tion of\nthe SME1models for Lorentz violation in spectroscopy experiments with\nlight atoms and muon spin-precessionexperiments were introduced .2These\nmodels triggered experimental searches for Lorentz violation with hydrogen\nmasers,3muonium spectroscopy,4and muon spin-precession experiments.5\nRecently the effective Lorentz-violating hamiltonians used to obtain the\nmodels for the systems aforementioned were extended to include c ontri-\nbutions from Lorentz-violating operators of arbitrary mass dimen sions.6\nThese new hamiltonians motivated two publications. The first publicat ion\ndiscusses the changes, due to the introduction of nonminimal term s, to the\nwell-known phenomenology for Lorentz violation in spectroscopy ex peri-\nments with muonic atoms and muonaspin-precession experiments.8The\nother publication concentrates on the instance of spectroscopy experiments\nwith light atoms that are composed of first-generationparticles.9Note that\nmost of the results presented in Ref. 9 can readily be applied to any t wo-\nfermion atom, including muonium and muonic hydrogen, and some of th e\ndiscussion here is based on this fact.\naSee Ref. 7 for a similar work with first-generation particles instead of muons.Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\n2. Sidereal variations\nA signalfor Lorentz violationis a siderealvariation of frequencies m easured\nin a laboratory on the surface of the Earth. In this section we will limit\nattention only to effects produced by a breaking of the rotation sy mmetry\nof the laboratory frame as observed in the Sun-centered frame.\n2.1.Bounds from previous studies\nExperimental studies that are sensitive to minimal operators might also be\nsensitive to nonminimal operators. Using the results from sidereal varia-\ntion studies in muonium spectroscopy4and muon spin-precession exper-\niments,5bounds on muon nonminimal coefficients for Lorentz violation\nwere reported.8In the future some of these bounds might be improved8by\nthe planned new measurements of the hyperfine structure of muo nium at\nJ-PARC, and of the antimuon anomalous frequency at J-PARC and F er-\nmilab. Bounds on proton and electron nonminimal coefficients for Lor entz\nviolation were obtained9from the results of sidereal variation studies with\nhydrogen masers.3\n2.2.Prospects and new signals\nNot all the harmonics of the sidereal frequency can contribute to the side-\nreal variation of the energy level of an atom. For two-fermion ato ms such\nas hydrogen the maximum harmonic of the sidereal frequency that can con-\ntribute to the variation of an energy level is given by the expression 2K−1,\nwhereKis the maximum the total angular momentum Jof the lighter\nfermion and the total angular momentum Fof the atom.\nTheminimal operatorsforLorentzviolationcanonlyproducevariat ions\nup to the second harmonic of the sidereal frequency and contribu tions to\nvariations of the energy levels with higher harmonics of the sidereal fre-\nquency are strictly due to the presence of nonminimal operators. To study\nsystematically the nonminimal operators, it is necessary to be sens itive to\nthese higher harmonics by performing sidereal variation studies of transi-\ntions involving energy levels with J >3/2 orF >1. This suggests that\nsome of the most promising experimental studies sensitive to these nonmin-\nimal terms in spectroscopy experiments with light atoms, including mu o-\nnium and muonic hydrogen, are sidereal variation studies of two-ph oton\ntransitions such as the 2 S-nDand 2S-nPtransitions.9\nThesensitiveofaspectroscopyexperimenttosomeofthe coefficie ntsfor\nLorentzviolationcandependonthe particularlightatomusedinthe e xper-Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\niment. For example, the contributions due to the nonminimal operat ors de-\npendonthemomentumofthefermionrelativetothezero-momentu mframe\nof the atom. Realizing the same experimental studies with systems w ith\nhigher momentum such as deuterium and muonic hydrogen can subst an-\ntially improve bounds on some of the nonminimal coefficients for Loren tz\nviolation.8,9\nThe nonminimal terms allow contributions to the sidereal variation of\nthe muon or antimuon anomalous frequency from all possible harmon ics of\nthe sidereal frequency.8The harmonics of the sidereal frequency that can\ncontribute to the sidereal variation can be limited by restraining the mass\ndimensions of the operators that contribute to the energy shift t o be equal\nto or smaller than d. The maximum harmonic that can contribute in that\ncase will be obtained by d−2 for even values of dandd−3 for odd values\nofd.\nThe new measurement of the antimuon anomalous frequency at Fer -\nmilab will use more energetic antimuons compared to the experiment a t\nJ-PARC10and for that reason it will be more sensitive to nonminimal\nLorentz-violating operators.8This implies that the signals for Lorentz vio-\nlation that the Fermilab experiment could target would include variatio ns\nwith harmonics higher than the second harmonic of the sidereal fre quency.\nThe J-PARC experiments would be more sensitive to the minimal coef-\nficients for Lorentz violation8and the targeted signals would be sidereal\nvariations with the first and second harmonic of the sidereal frequ ency.\n3. Boost corrections\nFrequenciesmeasuredinalaboratoryonthesurfaceoftheEarth canexhibit\nannual and sidereal variations due to the change of the velocity of the\nlaboratory frame relative to the Sun-centered frame. Some of th e signals\nfor Lorentz violation presented in this section might overlap with sign als\npresented in the previous section, however they are produced by a different\nset of coefficients for Lorentz violation.\nThe corrections due to the velocity of the laboratory frame to the 1S-\n2S transition and the hyperfine splitting of the ground state in hydr ogen\nand deuterium were obtained including contributions from coefficient s for\nLorentz violation up to mass dimension eight.9,11The results obtained for\nhydrogen can be adapted for other light atoms such as muonium, po sitro-\nnium, and muonic hydrogen. The signals for Lorentz violation in this ca se\nare annual variation and sidereal variation of the transition frequ encies.9Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n4\nCorrections to the antimuon anomalous frequency due to the motio n of\nthe laboratory were obtained including contributions from coefficien ts for\nLorentz violation up to mass dimension four.8The signals are annual vari-\nation and variations up to the second harmonic of the sidereal freq uency.8\n4. Antihydrogen\nAllthesignalsforLorentzviolationthatcanbestudiedinhydrogenc analso\nbe studied in antihydrogen.9The planned measurements of the hyperfine\nsplitting of the ground state of antihydrogen by the ASACUSA collab o-\nration and the 1S-2S transitions by ALPHA and ATRAP collaborations\ncould in the future be among the most sensitive tests discriminating b e-\ntween the CPT even and the CPT odd coefficients for Lorentz violatio n in\nspectroscopy experiments. The study of these transitions in ant ihydrogen\nshould only be the beginning. These transitions are insensitive to CPT -\nviolating operators that could only be studied by using transitions th at\ninvolve energy levels with higher values of FandJ.\nAcknowledgments\nThis work was supported by Department of Energy grant DE-SC00 10120\nand by the Indiana University Center for Spacetime Symmetries.\nReferences\n1. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998); V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n2. R.Bluhm, V.A.Kosteleck´ y, andN.Russell, Phys.Rev.Let t.82, 2254 (1999);\nR.Bluhm, V.A.Kosteleck´ y, andC.D.Lane, Phys.Rev.Lett. 84, 1098 (2000).\n3. D.F. Phillips, M.A. Humphrey, E.M. Mattison, R.E. Stoner , R.F.C. Vessot,\nand R.L. Walsworth, Phys. Rev. D 63, 111101(R) (2001).\n4. V.W. Hughes, M. Perdekamp, D. Kawall, W. Liu, K. Jungmann, and G. zu\nPutlitz, Phys. Rev. Lett. 87, 111804 (2001).\n5. G.W. Bennett et al., Phys. Rev. Lett. 100, 091602 (2008).\n6. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 88, 096006 (2013).\n7. Y. Ding and V.A. Kosteleck´ y, Phys. Rev. D 94, 056008 (2016); Y. Ding,\nthese proceedings.\n8. A.H. Gomes, V.A. Kosteleck´ y, and A.J. Vargas, Phys. Rev. D90, 076009\n(2014).\n9. V.A. Kosteleck´ y and A.J. Vargas, Phys. Rev. D 92, 056002 (2015).\n10. R.M. Carey et al., Fermilab proposal 0989, 2009; M. Aoki et al., J-PARC\nproposal J-PARC-PAC2009-12, 2009.\n11. A. Matveev et al., Phys. Rev. Lett. 110, 230801 (2013)."    },    {        "title": "1006.1817v1.Scale_interactions_in_magnetohydrodynamic_turbulence.pdf",        "content": "arXiv:1006.1817v1  [physics.flu-dyn]  9 Jun 2010Scale Interactions in MHD Turbulence 1\nScale Interactions in Magnetohydrodynamic\nTurbulence\nPablo D. Mininni\nDepartamento de F´ ısica, Facultad de Ciencias Exactas y Nat urales, Universidad\nde Buenos Aires and CONICET, Ciudad Universitaria, 1428 Bue nos Aires,\nArgentina,\nand\nNCAR, P.O. Box 3000, Boulder, Colorado 80307-3000, USA.\nKey Words magnetohydrodynamics, modeling, simulation, isotropy, u niversal-\nity\nAbstract This article reviews recent studies of scale interactions i n magnetohydrodynamic\nturbulence. The present day increase of computing power, wh ich allows for the exploration\nof different configurations of turbulence in conducting flows , and the development of shell-to-\nshell transfer functions, has led to detailed studies of int eractions between the velocity and the\nmagnetic field and between scales. In particular, processes such as induction and dynamoaction,\nthe damping of velocity fluctuations by the Lorentz force, or the development of anisotropies,\ncan be characterized at different scales. In this context we c onsider three different configurations\noften studied in the literature: mechanically forced turbu lence, freely decaying turbulence, and\nturbulence in the presence of a uniform magnetic field. Each c onfiguration is of interest for\ndifferent geophysical and astrophysical applications. Loc al and non-local transfers are discussedAnnual Review of Fluid Mechanics 2011 1056-8700/97/0610-0 0\nfor each case. While the transfer between scales of solely ki netic or solely magnetic energy\nis local, transfers between kinetic and magnetic fields are o bserved to be local or non-local\ndepending on the configuration. Scale interactions in the ca scade of magnetic helicity are also\nreviewed. Based on the results, the validity of several usua l assumptions in hydrodynamic\nturbulence, such as isotropy of the small scales or universa lity, is discussed.\nCONTENTS\nIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\nIndirect evidence of non-locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\nThe shell-to-shell transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\nDirect studies of multi-scale interactions . . . . . . . . . . . . . . . . . . . . . . . . 14\nForced isotropic and homogeneous turbulence . . . . . . . . . . . . . . . . . . . . . . . 15\nFreely decaying turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\nAnisotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\nMagnetic helicity and the inverse cascade . . . . . . . . . . . . . . . . . . . . . . . . . 21\nNon-local interactions and universality of MHD turbulence . . . . . . . . . . . . . 22\nConcluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26\nSummary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\nAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n1 Introduction\nTurbulence is a multiscale phenomenon ubiquitous in geophy sical and astrophys-\nical flows. In many of these flows, the coupling of a conducting fluid with elec-\ntromagnetic fields requires consideration of the magnetohy drodynamic (MHD)\n2Scale Interactions in MHD Turbulence 3\nequations (see, e.g., Moffatt 1978). The equations describe t he dynamics of non-\nrelativistic conducting fluids as, e.g., in the Earth’s core or in industrial appli-\ncations, and under some approximations they can also descri be the large-scale\nbehavior of magnetospheric, space, and astrophysical plas mas. In these latter\ncases, care must be taken to consider only the scales where a o ne-fluid approxi-\nmation holds, as scales small enough may require considerat ion of kinetic plasma\neffects such as ambipolar diffusion in weakly ionized plasmas as the interstellar\nmedium, or the Hall current for highly ionized media such as s mall scales in the\nsolar wind. However, in those cases the MHD equations still g ive a good de-\nscription of large scales, and the approximation gives a use ful approach to get\nlowest-order physical insight into the fate of the flows.\nIn the simplest case, that of an incompressible flow with cons tant mass density,\nthe equations give the evolution of the bulk fluid velocity uand of the magnetic\nfieldb:\n∂tu+u·∇u=−∇p+b·∇b+ν∇2u, (1)\n∂tb+u·∇b=b·∇u+η∇2b, (2)\nwhere the magnetic field is written in Alfvenic units, the den sity is set to unity,\nandpis the (fluid plus magnetic) pressure. The kinematic viscosi tyνand mag-\nnetic diffusivity ηcontrol respectively the viscous and Ohmic dissipation. Th ese\nequations areconstrainedbytheincompressibilitycondit ion andbythesolenoidal\ncharacter of the magnetic field,\n∇·u= 0,∇·b= 0. (3)\nTwo different Reynolds numberscan bedefinedin MHD flows. Theme chanical4 Mininni\nReynolds number\nRe=UL\nν, (4)\nwhich is the ratio of convective to viscous forces (where Uis the rms velocity and\nLa characteristic lengthscale of the flow), and the magnetic R eynolds number\nRm=UL\nη, (5)\nthat can be interpreted as the ratio of induction to Ohmic dis sipation. In many\nflows these Reynolds numbers are very large, and the flows are i n a turbulent\nregime.\nWhile in the hydrodynamic case the phenomenological theory of Kolmogorov\n(K41) gives to a good approximation (albeit intermittency c orrections) the power\nlaw of the energy spectrum, no clearly established phenomen ological counterpart\nexists in MHD. This has many implications as the energy dissi pation rate (re-\nquired to predict, e.g., heating rates in solar and space phy sics) depends on the\nslope of the energy spectrum. Also, subgrid models, require d to do numerical\nmodeling in astrophysics and geophysics given the large sca le separation involved\nin such flows, are less developed in MHD as a result of the lack o f detailed knowl-\nedge of its energy spectrum.\nIn the Kolmogorov description of hydrodynamic turbulence, the interactions of\nsimilar size eddies play the basic role of cascading the ener gy to smaller scales on\na scale-dependent time scale τℓ∼ℓ/uℓ, whereℓis the examined length scale and\nuℓthe characteristic velocity at that scale. This time scale, which is proportional\nto the eddy turnover time at the scale ℓ, is the only time scale available on di-\nmensional grounds in the inertial range, provided enough sc ale separation exists\nbetween forcing and dissipation. In this context, interact ions between scales are\nlocal (in spectral space) as dominant interactions are betw een eddies of similarScale Interactions in MHD Turbulence 5\nsizes. One then expects the statistical properties of suffici ently small scales to\nbe independent of the way turbulence is generated, and to hav e therefore uni-\nversal character. Recent experiments showed deviations fr om this behavior even\nfor simple hydrodynamic flows (e.g., slower than expected re covery of isotropy,\nor presence of long-time correlations in the small scales; s ee Carlier et al. 2001,\nPoulain et al. 2006, Shen & Warhaft 2000, Wiltse & Glezer 1993 , 1998). Numer-\nical simulations also gave evidence of the presence of non-l ocal interactions with\nthe large scale flow playing a role in the cascade of energy (Al exakis et al. 2005a,\nDomaradzki 1988, Domaradzki & Rogallo 1990, Zhou 1993). In n umerical sim-\nulations with Reynolds numbers as high as Rλ≈800, it was observed that\n20% of the energy flux in the small scales was due to interactio ns with the large\nscale flow (Mininni et al. 2006). However, more recent simula tions with Reynolds\nnumbers up to Rλ≈1300 using spatial resolutions of 20483grid points showed\nthat as the Reynolds number is increased, the percentage of t he non-local flux\ndecreases as a power law of the Reynolds number, suggesting t hat the flux in hy-\ndrodynamicturbulencemay be predominantly local for very l arge Reynolds num-\nber (Mininni et al. 2008). Recent theoretical results put th is in firmer grounds\n(Aluie & Eyink 2009a, Eyink & Aluie 2009), showing that the en ergy flux in hy-\ndrodynamic turbulence is local in the limit of infinite Reyno lds number and\nobtaining bounds on the scaling of the non-local contributi on to the flux with\nReynolds number which are in agreement with the numerical re sults.\nThe case for MHD turbulence is less clear and has given rise to more controver-\nsies. Several attempts have been done to extend the phenomen ological arguments\nofKolmogorov toconductingflows(see, e.g., Boldyrev 2006, Goldreich & Sridhar 1995,\nIroshnikov 1963, Kraichnan 1965, Matthaeus & Zhou 1989). Ho wever, the MHD6 Mininni\nequivalent of the 4/5 law in hydrodynamic turbulence (the Po litano-Pouquet re-\nlations, see Politano & Pouquet 1998a,b) couple the velocit y and the magnetic\nfield in a way that can be compatible with several power law beh aviors; in three\ndimensions they read\n/angbracketleftBig\nδz∓\n/bardbl(l)/vextendsingle/vextendsingleδz±(l)/vextendsingle/vextendsingle2/angbracketrightBig\n=−4\n3ǫ±l, (6)\nwhereǫ±are the dissipation rates of the Els¨ asser variables z±=u±b, and the\nsubindex /ba∇dbldenotes the increment of the field along the displacement vec torl.\nMoreover, even in the simplest incompressible case, at leas t two time scales\ncan be identified in the inertial range of MHD turbulence. Bes ides the eddy\nturnover time, incompressible MHD flows are also characteri zed by the period\nof Alfv´ en waves τ∼(B0L)−1, where B0is the amplitude of the large scale\nmagnetic field in Alfvenic units. In a first attempt to derive a phenomenologi-\ncal theory, Iroshnikov (1963) and Kraichnan (1965) (IK) ass umed that the large\nscale magnetic field acts as a uniform field for the small scale fluctuations, which\nthen behave as Alfv´ en waves. In that case, small scales can i nteract not only\nthrough the eddies but also through Alfv´ en packages, which reduce the energy\nflux to small scales by increasing its transfer time. This int roduces in prac-\ntice a non-local interaction as the waves propagate along th e large scale field\n(see Gomez et al. 1999 for a discussion). From dimensional an alysis, Iroshnikov\nand Kraichnan then derived an isotropic energy spectrum pro portional to k−3/2.\nLater, extensions whereconsidered to take into account the anisotropy inducedat\nsmall scales by the large scale magnetic field (Boldyrev 2006 , Galtier et al. 2000,\n2005, Goldreich & Sridhar 1995). Some of these extensions, a fter accounting for\nthe anisotropy, rely on some form of a balance between the two fields that leaves\nonly the turnover time as the relevant time scale, and can be t herefore consideredScale Interactions in MHD Turbulence 7\nlocal or non-local depending on the authors.\nAt the core of the early disquisitions is the fact that in MHD t he roles of a\nlarge scale flow and of a large scale magnetic field are different . While a (uni-\nform) large scale flow can be removed by a Galilean transforma tion, a large scale\nmagnetic field cannot. As a remarkable coincidence, lack of G alilean invariance\nis at the basis of the ∼k−3/2spectrum for hydrodynamic turbulence within the\nframework of the Direct Interaction Approximation (DIA) by Kraichnan (1959),\na flaw later corrected by the development of the Test Field Mod el (TFM) and\nthe Lagrangian History DIA (LHDIA). However, in MHD magneti c fields are not\nGalileaninvariantandforthisreasontheassociatedAlfv´ enwaveshavetobetaken\ninto account in phenomenological theories, andare also con sidered when studying\nnon-local effects in the Eddy-Damped Quasi-Normal Markovian (EDQNM) clo-\nsure (Pouquet et al. 1976) or in weak turbulence theory (Naza renko et al. 2001).\nHowever, although phenomenological descriptions assume t hat a large scale field\nhas the effect of reducing the energy cascade rate, the transfe r of energy (and the\ncascade) in many cases still takes place between eddies of si milar size, presumably\nallowing for recovery of universal statistical properties at small scales.\nIn recent years, this universal behavior has been questione d by different au-\nthors. Because energy can be injected in MHD by a mechanical f orcing or by\nan electromotive forcing, MHD turbulence is characterized by a larger number\nof regimes than hydrodynamic turbulence even in its simples t configurations.\nMagnetic fields in planets and stars are believed to be genera ted by dynamo\naction, where turbulent motions sustain magnetic fields aga inst Ohmic dissipa-\ntion (Brandenburg & Subramanian 2005, Krause & Raedler 1980 , Moffatt 1978,\nPouquet et al. 1976). This regime is often studied in numeric al simulations (and8 Mininni\nrecently in experiments, see Monchaux et al. 2007) by mechan ically stirring the\nflow. Depending on the amount of mechanical helicity in the flo w (the alignment\nbetween the velocity and the vorticity), or in the presence o f large-scale shear,\nthe magnetic field generated may have large or small-scale co rrelation (compared\nwith the integral scale of the flow), giving a steady state tha t may be dominated\nbymechanical ormagneticenergy. Ontheotherhand,plasmas inthesolarcorona\nand in the solar wind are dominated by magnetic energy, and ar e often studied\nnumerically by stirring the flow with electromotive forces o r using simulations\nof freely decaying turbulence. Finally, the amount of cross -correlation between\nthe velocity and the magnetic field depends on the flow (e.g., o n the heliocentric\ndistance in the solar wind) and can also be varied in the simul ations.\nThe questioning of universality was accompanied by recent d etailed studies of\nscale interactions in MHD turbulence. Many of the studies co nsidered the so\ncalled shell-to-shell transfer functions and partial ener gy fluxes, either in numer-\nical simulations, observations, closures, or from the theo retical point of view. In\nthe following sections we give a review of the results in this area, considering the\nseveral regimes studied by different authors, and also some ex amples of possible\nsources of non-locality in MHD. Finally, we discuss the resu lts in the context\nof universality and of phenomenological theories for MHD. T o briefly summarize\nthe results, several authors have shown that the locality of energy transfer is in\nquestion in MHD flows. In particular, it was shown from simula tions that the\ntransfer of energy in MHD has two components: a local one that shares similar\nproperties with hydrodynamic turbulence, and a component c oupling the veloc-\nity and magnetic fields for which energy from the large scales can be under some\ncircumstances injected directly into the small scales with out the intervention ofScale Interactions in MHD Turbulence 9\nintermediate scales.\n2 Indirect evidence of non-locality\nSome theoretical, phenomenological, and (more recently) n umerical results in-\ndicate scale interactions in MHD can be, under some conditio ns, of a different\nnature than in hydrodynamic turbulence. In this section we r eview early theoret-\nical indications of non-locality in MHD turbulence, as well as numerical results\nthat support the theoretical arguments without directly me asuring scale interac-\ntions.\nEarly studies of dynamo action, and of magnetic field evoluti on under flows\nwith simple strain, show that a large scale flow can excite, th rough field line\nstretching, magnetic fieldsat widely separated scales. One of thefirstworks along\nthese lines is the work of Batchelor (1950) where he consider ed the similarity\nbetween the induction equation and the vorticity equation ( ω=∇×u):\n∂ω\n∂t+u·∇ω=ω·∇u+ν∇2ω. (7)\nWhile the second term on the l.h.s. advects the vorticity, th e first term on\nthe right (in three dimensions) produces vorticity by vorte x stretching. For\nPM=η/ν >1 (the magnetic Prandtl number) Batchelor then concluded th at\nthe magnetic field would grow as magnetic field line stretchin g overcomes Ohmic\ndissipation. Later works considering stretching by unifor m straining motion\n(Moffatt & Saffman 1964, Zel’Dovich et al. 1984) showed that a lar ge scale mag-\nneticfieldcandirectlycreatesmall-scalemagneticfields. TheworkofKazanstev (1968)\nconsidered a similar process under a random velocity field an d described a non-\nlocal coupling that sustains the so-called small-scale dyn amo, where magnetic\nfields are amplified at scales smaller than the integral scale of the flow. Several10 Mininni\nnumerical simulations support these results, and show that smooth motions at\nthe viscous scale give exponential growth of magnetic fields that can peak at the\nmagnetic diffusion scale (Schekochihin et al. 2002a, 2004, 20 02b).\nThe opposite limit, when the magnetic Prandtl number is much smaller than\nunity(a case ofinterest for industrialflows), issometimes studiedusingthequasi-\nstatic approximation (see Knaepen & Moreau 2008 for a review ). In this case, an\nexternal uniform magnetic field is applied, and the magnetic Reynolds number\nis chosen small enough that magnetic field fluctuations are ra pidly damped. In\nthat limit the Lorentz force in the momentum equation reduce s to linear Joule\ndamping\n∂u\n∂t+u·∇u=−∇p+σB2\n0∇−2∂2u\n∂z2+ν∇2u, (8)\nwhereσis the conductivity of the medium and the uniform magnetic fie ldB0is\nin thezdirection. The Joule damping, although anisotropic in spec tral space, is\nroughly independent of the wavenumber, and unlike viscous d amping is not con-\ncentrated at small scales but rather acts at all scales. As a r esult, the large-scale\nmagnetic field in this approximation exerts work over all sca les in the velocity\nfield (damping turbulent fluctuations) in a non-local way. We will see that the\nshell-to-shell transfers indicate in some cases similar be havior of the Lorentz force\neven in cases far from this approximation.\nAnother important example concerns Alfv´ en waves, which ar e also non-linear\nsolutions of the ideal MHD equations. Alfv´ enic states with u=±bmake the\nnon-linear terms in Eqs. (1) and (2) zero, leaving only inter actions with the large\nscale fields to transport energy across scales. Finally, it i s worth mentioning\nhere some recent attempts to build shell models of MHD turbul ence (see e.g.,\nLessinnes et al. 2009, Plunian & Stepanov 2007, Stepanov & Pl unian 2008). InScale Interactions in MHD Turbulence 11\nthese models, it was found that many features of steady state MHD turbulence\ncan be reproduced using local coupling between shells, but t hat to reproduce the\nsmall-scale dynamo and turbulence at PM≫1, non-local transfers have to be\nconsidered (Stepanov & Plunian 2008).\n3 The shell-to-shell transfer\nIn recent years, the increase in computing power allowed num erical exploration\nof MHD turbulence in different regimes. The development of she ll-to-shell trans-\nfers (see Alexakis et al. 2005b, Dar et al. 2001, Debliquy et a l. 2005) allowed for\nexplicit computation of detailed scale interactions in MHD turbulence using the\noutputstemmingfrom thesimulations andwithouttheneedto computethemore\nexpensive triadic interactions. In this section we briefly i ntroduce the isotropic\nshell-to-shell energy transfer functions, and describe ho w fluxes can be obtained\nfrom them.\nA shell filter decomposition of the two fields is introduced as\nu(x) =/summationdisplay\nKuK(x), (9)\nb(x) =/summationdisplay\nKbK(x), (10)\nwhere\nuK(x) =/summationdisplay\nK1<|k|≤K2˜ u(k)eik·x(11)\nand\nbK(x) =/summationdisplay\nK1<|k|≤K2˜b(k)eik·x. (12)\nHere˜u(k) and˜b(k) are respectively the Fourier transforms of the velocity an d\nmagnetic fields with wavenumber k. The shell filtered fields uKandbKare\ntherefore defined as the field components whose Fourier trans forms contain only12 Mininni\nwavenumbers in a given shell K. These shells can be defined with linear binning\nusingK1=KandK2=K+1, or alternatively with logarithmic binning using\nK1=γnK0andK2=γn+1K0for some positive γ >1 and integer n(γ= 2 is\noften used). The latter definition has the advantage of being conceptually closer\nto the idea of “scale” of eddies in turbulence, which in gener al implies logarith-\nmic division of wavenumbers. The former has the advantage of having a direct\nassociation with Alfv´ en waves, which are of the form u=±b∼ei(k·x±ωt)in\nperiodic boxes or in infinite domains, and which are more akin to linear treat-\nment of spectral space. Note that the transfer among logarit hmic shells can be\nreconstructed by summing over the linearly spaced shells.\nAnother variant when defining the shell filter decomposition has to do with\nthe choice of using sharp filters (as in the equations above) o r smooth filters\n(Eyink 1994, 2005). This issue has raised some controversy i n the hydrodynamic\ncase, with claims that non-localities observed in simulati ons may be due to the\ncommonlyusedsharpfilters. Recentnumericalcomparisons( Domaradzki & Carati 2007a,b)\nhave shown that results are only weakly dependent on the shap eof the filter used,\nexcept in the case where a very broad smooth filter is consider ed. Moreover, re-\ncent theoretical results were able to show locality of hydro dynamic turbulence\nin Fourier space in the limit of infinite Reynolds numbers for both smooth and\nsharp filters (Aluie & Eyink 2009a, Eyink & Aluie 2009).\nBased on the shell filter decomposition, the evolution of the kinetic energy in\na shellK,Eu(K) =/integraltextu2\nK/2dx3, can be derived from Eq. (1) as\n∂Eu(K)\n∂t=/summationdisplay\nQ[Tuu(Q,K)+Tbu(Q,K)]−νDu(K), (13)Scale Interactions in MHD Turbulence 13\nand for the magnetic energy, Eb(K) =/integraltextb2\nK/2dx3, from Eq. (2)\n∂Eb(K)\n∂t=/summationdisplay\nQ[Tbb(Q,K)+Tub(Q,K)]−ηDb(K), (14)\nwhere the functions Du(K) andDb(K) express respectively the kinetic and\nmagnetic energy dissipation in the shell K. The transfer functions Tuu(Q,K),\nTub(Q,K),Tbb(Q,K), andTbu(Q,K), that express the energy transfer between\ndifferent fields and shells are given by\nTuu(Q,K) =−/integraldisplay\nuK(u·∇)uQdx3, (15)\nTbu(Q,K) =/integraldisplay\nuK(b·∇)bQdx3, (16)\nTbb(Q,K) =−/integraldisplay\nbK(u·∇)bQdx3, (17)\nTub(Q,K) =/integraldisplay\nbK(b·∇)uQdx3. (18)\nThe function Tuu(Q,K) measures the transfer rate of kinetic energy in the shell\nQto kinetic energy in the shell Kdue to the advection term in the momentum\nequation (1). This is the non-linear transfer that is also pr esent in hydrodynamic\nturbulence. Similarly, Tbb(Q,K) expressestherateofmagnetic energytransferred\nfrom the shell Qto magnetic energy in the shell Kdue to the magnetic advection\nterm. TheLorentzforceisresponsibleforthetransferofen ergyfromthemagnetic\nfield in the shell Qto the velocity field in the shell K, as measured by Tbu(Q,K).\nFinally, the term responsible for the stretching of magneti c field lines, the first\nterm on the r.h.s. of Eq. (2), results in the transfer of kinet ic energy from shell\nQto magnetic energy in shell Kand is expressed by Tub(Q,K). This is the term\nthat describes magnetic induction and dynamo action.\nAll these transfer functions satisfy\nTvw(Q,K) =−Twv(K,Q). (19)14 Mininni\n(wherethesubindices vandwstandfor uorb). Thisexpressionindicates thatthe\nrate at which the shell Qgives energy to the shell Kis equal to the rate the shell\nKreceives energy from the shell Q, and is a necessary condition to define shell-\nto-shell transfers that satisfy a detailed energy balance b etween shells. Then, the\ncontribution of these transfers to the total energy flux can b e computed as:\nΠvw(k) =−k/summationdisplay\nK=0/summationdisplay\nQTvw(K,Q). (20)\nBesides the total energy, the MHD equations have two more ide al invariants:\nthe cross-helicity C=/integraltextu·bdx3, and the magnetic helicity H=/integraltexta·bdx3where\nais the vector potential such as b=∇×a. These quantities also satisfy detailed\nbalance equations equivalent to Eqs. (13) and (14). Shell-t o-shell transfer func-\ntions for the magnetic helicity have been defined in Alexakis et al. (2006). Its\ntransfer from shell Qto shellKis measured by\nTH(K,Q) =/integraldisplay\nbK·(uK×bQ)dx3. (21)\nThe energy transfer functions were also generalized in rece nt works to consider\nthe flux of energy in terms of the Els¨ asser variables (Alexak is et al. 2007a, 2005b,\nCarati et al. 2006), anisotropictransfers(Alexakis et al. 2007a,Teaca et al. 2009),\nforward and backward transfers in an attempt to quantify bac kscatter required\nfor subgrid models (Carati et al. 2006, Debliquy et al. 2005) , and extensions to\nconsider compressibility effects Graham et al. (2010), and ki netic plasma effects\nas in two-fluid MHD approximations (Mininni et al. 2007).\n4 Direct studies of multi-scale interactions\nTheshell-to-shell energytransfershavebeenstudiedexte nsively(Alexakis et al. 2005b,\nCarati et al. 2006,Dar et al. 2001,Debliquy et al. 2005,Mini nni et al. 2005a,Verma 2004)Scale Interactions in MHD Turbulence 15\nfor a variety of mechanically forced and decaying MHD flows in two and three\ndimensions. Depending on the configuration, different degree s of non-locality\nwere reported. In the following subsections we present a sho rt summary of\nthe results discriminating by the forcing configuration. Ov erall, we can say\nthat in all cases examined in the literature the transfers TuuandTbbhave a\nlocal behavior: energy is transferred forward between near by shells, in a fashion\nsimilar to what is observed in hydrodynamic turbulence (Ale xakis et al. 2005a,\nDomaradzki & Rogallo 1990,Mininni et al. 2006,Ohkitani & Ki da 1992,Yeung et al. 1995,\nZhou 1993). On the other hand, the TbuandTubtransfers that express the en-\nergy exchange between the velocity and the magnetic field hav e a rather different\nbehavior.\n4.1 Forced isotropic and homogeneous turbulence\nAs mentioned before, forced simulations of MHD turbulence c an be attained by\nforcing both fields, or by forcing only the velocity field (in w hich case magnetic\nfields are sustained by dynamo action; a distinction must be d one then between\nthe kinematic regime, where the magnetic field has no backrea ction on the flow,\nand the turbulent steady state wherethe magnetic field modifi es the flow through\nthe Lorentz force). The mechanically forced case is of more i nterest as it is closer\nto astrophysical and geophysical configurations, and as it i s consistent with the\nconstraintofmagneticfluxconservation. Thefirststudieso fshell-to-shell transfer\nfrom simulations in such configuration were presented in Ale xakis et al. (2005b).\nIn the simulations with a resolution of 2563grid points, the velocity field was\nforced with time independent mechanical forcing until a hyd rodynamic turbulent\nsteady state was reached. Two different forcing functions wer e studied: one16 Mininni\nnon-helical and one helical. Magnetic Prandtl numbers of un ity and smaller\nwere considered. Once a hydrodynamic steady state was reach ed for each forcing\nfunction, a small magnetic field was introduced and, after th e transient kinematic\ndynamo amplification, the system reached a steady state MHD t urbulent regime.\nIn such a state the transfer functions described in the previ ous section were\ncomputed. Typical results are illustrated in Fig. 1.\nTheTuuandTbbtransfers were observed to behave in a similar fashion, givi ng\ndirect and local transfer of energy. In Fig. 1, this is indica ted by the negative and\npositive peaks, which show energy is removed by these transf er functions from\nsmaller wavenumbers and given to slightly larger wavenumbe rs. However, for\nTuba distinct behavior appeared. The large scale flow injected e nergy (through\nstretching) directly into the magnetic field at all scales. T his manifests itself as\na peak at the mechanical forcing scale for all receiving shel ls, and as an extended\npositive plateau (note positive Tubindicates energy given by the velocity field at\nshellQto magnetic field at shell K). In other words, at a given shell K, the\nmagnetic field receives energy from the velocity field in all s hellsQ < K, and\ngives energy to the velocity field in shells Q > K. This result, reminiscent of\nthe theoretical arguments by Batchelor (1950) and Zel’Dovi ch et al. (1984), was\ninterpreted as the sustainment of the magnetic field against Ohmic dissipation\nby dynamo action: to maintain the magnetic field when only the velocity field\nis stirred, a nonzero flux from the velocity field to the magnet ic field is required\nat all times. It is worth pointing out here that in the steady s tate this non-local\ntransferis small compared withthelocal transfers(approx imately 10 −20% at the\nresolutions studied). When considering Els¨ asser variabl es, the transfer functions\nwere observed to become more local.Scale Interactions in MHD Turbulence 17\nThecase of random forcingwith magnetic Prandtl numberof un ity was studied\nin Carati et al. (2006) using 5123simulations. The analysis, which used logarith-\nmic binning, confirmed the previous results, showing local t ransfer in Tuuand\nTbb, and non-local coupling between the velocity and the magnet ic field. This\nindicates that the phenomenon may be independent of the type of forcing, and\nassociated to the stretching process that sustains the magn etic field. The work\nalso discussed the possibility of splitting the transfer fu nctions to discriminate\nbetween forward and backward contributions, which were use d to discuss im-\nplications of the shell-to-shell transfers for LES models. Similar results were\nobtained for forced two dimensional MHD turbulence (Dar et a l. 2001).\nA different approach was considered by Yousef et al. (2007), wh o studied the\nsteady state of small-scale dynamo action for PM≤1. Instead of using transfer\nfunctions, to measure the different components of the energy fl ux they considered\nthe Politano-Pouquet relations (6) in terms of the velocity and the magnetic field\n/angbracketleftBig\nδu/bardbl/parenleftBig\n|δu|2+|δb|2/parenrightBig/angbracketrightBig\n∓/angbracketleftBig\nδb/bardbl/parenleftBig\n|δu|2+|δb|2/parenrightBig/angbracketrightBig\n±2/angbracketleftBig\nδu·δb/parenleftBig\nδu/bardbl∓δb/bardbl/parenrightBig/angbracketrightBig\n=−4\n3ǫ±l,\n(22)\ntogether with Chandrasekhar (1951) law\n/angbracketleftBig\nδu3\n/bardbl/angbracketrightBig\n−6/angbracketleftBig\nb2\n/bardblδu/bardbl/angbracketrightBig\n=−4\n5ǫl, (23)\nwhereǫis the total energy flux. The authors discriminated between t he different\nterms to look how they balanced to give rise to the direct flux. Each of the terms\nin these expressions can indeed be associated to the counter part in real space of\nthe Πuu, Πbb, and Π ub+Πbufluxes in Fourier space.\nThe dominant balance was identified between (4 /5)ǫland 6/angbracketleftBig\nb2\n/bardblδu/bardbl/angbracketrightBig\n, and they\nconcluded that at their available resolution, the local dir ect cascade of energy was18 Mininni\n“short-circuited” by the transfer of kinetic energy into ma gnetic energy. They\nalso associated this non-local coupling with the folded str ucture of the small-scale\nmagnetic field. Using the shell-to-shell transfer approach , Alexakis et al. (2007a)\nfurther showed that the non-local effects disappear if phases are randomized for\nthe two fields, which also make the current sheet and folded st ructures disappear.\nThe non-local effects play a more important role in the kinemat ic dynamo\nregime (Mininni et al. 2005a), as in that case the turbulence is not in a steady\nregime and Tubaccounts for all mechanisms that amplify the magnetic field. In\nthat case, the Tubtransfer has been shown to be useful to identify and quantify\nscale-by-scalesourcesofdynamoaction(Alexakis et al. 20 07b,Mininni et al. 2005a).\n4.2 Freely decaying turbulence\nThe non-local effects observed in forced turbulence are eithe r absent or negligible\nin the freely decaying case. In Debliquy et al. (2005), 5123simulations of freely\ndecaying MHD turbulence were considered. The TuuandTbbtransfers are similar\nto the forced case (see Fig. 2) and indicate local direct tran sfer. However, the\nTubandTbutransfer functions were also observed to be local, with most of the\ntransfer between the velocity and the magnetic field taking p lace between the\nsame shell. The remaining transfer (for non-neighboring sh ells) was observed to\ndecay more slowly than in the TuuandTbbfunctions, but except for this detail\nno other indications of non-locality were reported.\nSimilarresultswereobtainedfromanalysisofsolarwindtu rbulence(Strumik & Macek 2008a,b).\nSolar wind turbulence is often considered the MHD equivalen t of hydrodynamic\nfreely decaying wind tunnel turbulence(see Bruno & Carbone 2005 for a review).\nFrom 1996 Ulysses magnetometers time series and using a Mark ov process ap-Scale Interactions in MHD Turbulence 19\nproach, Strumik & Macek (2008b) concluded that the transfer of magnetic to\nmagnetic energy was local. Then, using velocity and magneti c field time series\nfrom ACE spacecraft from 1999 to 2006 and performing the same analysis on the\nremaining transfers they concluded that all transfers were local.\nThedifferences between the forced and freely decaying cases c an beunderstood\nas in the mechanically forced runs, the velocity field has to c ontinuously supply\nenergy to themagnetic field in order to sustain it against Ohm icdissipation. This\nis not necessarily the case for freely decaying runswherebo th fields are dissipated\nin time.\n4.3 Anisotropic turbulence\nRecently, the shell-to-shell transfers were extended to co nsider anisotropies when\nan external uniform magnetic field is imposed. This case is of interest as in\nmany astrophysical problems a strong large-scale magnetic field is present cre-\nating small-scale anisotropy. Unlike hydrodynamic turbul ence, MHD turbulence\ndoes not recover isotropy at small scales, and theoretical a nd numerical results\nindicate anisotropy becomes stronger at smaller scales.\nTo study anisotropic transfers, different foldings of the she lls in Fourier space\ncan beimplemented. Fig 3 shows the possibleoptions. In Alex akis et al. (2007a),\nanisotropic shell-to-shell transfer functions were intro duced folding Fourier shells\nin cylinders (associated to wavenumbers k⊥perpendicular to the mean mag-\nnetic field) and in planes (associated to parallel wavenumbe rsk/bardbl). Shell-to-shell\ntransfers were only considered for the Els¨ asser variables , but the fluxes were\nreconstructed from these functions to measure the relative contribution of non-\nlocality to the total flux. Freely decaying simulations with spatial resolution of20 Mininni\n2563grid points were analyzed, and the amplitude of the imposed m agnetic field\nwas varied from 0 to 15 (in units of the initial small scale fluc tuations). The\ntransfer functions of the two Els¨ asser energies were found local in both paral-\nlel and perpendicular directions, irrespectively of the am plitude of the external\nfield. However, interactions between the counterpropagati ng Alfv´ en waves were\nreported to become non-local. For strong magnetic fields, mo st of the energy flux\nin the perpendicular direction was found to be due to interac tions with modes\nwithk/bardbl= 0 (see Fig. 4). In the parallel direction, however, k/bardbl= 0 modes\ncannot transfer energy and most of the interactions were obs erved to take place\nwith modes near k/bardbl≈0. The results are in qualitative agreement with predic-\ntions from weak turbulence theory (Galtier et al. 2000) and w ith recent non-local\nphenomenological models (Alexakis 2007).\nAdifferentapproachtostudyanisotropictransferswasprese ntedbyTeaca et al. (2009),\nwho decomposed the spectral space into rings, studying then transfers along ra-\ndial and angular directions in spectral space (which they te rmed “ring-to-ring”\ntransfers). They considered forced simulations of MHD turb ulence with an im-\nposed magnetic field, with a spatial resolution of 5123grid points and varying\nthe imposed magnetic field between 0 to√\n10 (in units of the small-scale mag-\nnetic field fluctuations). They also observed dominance of en ergy transfer in the\ndirection perpendicular to the uniform magnetic field, and s uppression of the\ntransfer in the parallel direction. Their approach is usefu l to understand how\nenergy is angularly distributed in spectral space to create anisotropy. Non-local\neffects with the forcing shell were observed in the shell-to-s hell transfers, but in\nthe angular ring-to-ring transfers were too weak to be notic ed.Scale Interactions in MHD Turbulence 21\n4.4 Magnetic helicity and the inverse cascade\nNon-local transfers were also reported in investigations o f the cascade of mag-\nnetic helicity. Magnetic helicity is an ideal invariant in M HD, that is known to\ncascade inversely (to the large scales) in a turbulent flow (A lexakis et al. 2006,\nBrandenburg 2001, Brandenburg & Subramanian 2005, G´ omez & Mininni 2004,\nMeneguzzi et al. 1981, Pouquet et al. 1976). The generation o f large scale mag-\nnetic fields in galaxies and other astrophysical bodies is so metimes attributed\nto this inverse cascade. In Alexakis et al. (2006) magnetica lly and mechanically\nforced simulations were considered. In both cases, both loc al and non-local trans-\nfers were observed. At early times, magnetic helicity was ob served to cascade\ninversely and locally from the closest neighbor shells, and non-locally from the\nforced shells. When the correlation length became of the siz e of the box, the\ndirect input from the forced scales became dominant, and a lo cal direct trans-\nfer of helicity from large to small scales also developed. Th is latter effect was\nspeculated to be dependent on boundary conditions and there fore non-universal.\nIn the mechanically forced case the inverse cascade of helic ity was associated to\nthelarge-scale dynamo α-effect(Brandenburg 2001,Brandenburg & Subramanian 2005,\nKrause & Raedler 1980,Pouquet et al. 1976,Steenbeck et al. 1 966). Inthatcase,\nthe mechanical forcing creates equal amounts of magnetic he licity of opposite\nsigns at large and small scales. The process can be understoo d using the concep-\ntual“stretch, twist, andfold”(STF)dynamomechanism(Chi ldress & Gilbert 1995,\nVainshtein & Zeldovich 1972). Each time a closed magnetic flu x tube is twisted\nby the helical velocity field, magnetic helicity is created a t large scales, while\nsmall scale magnetic field lines are twisted in the opposite d irection thus creating\nequal amount of magnetic helicity of opposite sign in the sma ll scales. As the22 Mininni\nSTF process is repeated, the large-scale helicity is transf erred inversely both lo-\ncally and non-locally (with constant flux), while the small- scale helicity is pushed\ntowards smaller scales (see Fig 5). This latter process remo ves magnetic helicity\nfrom the large scales and allows the magnetic field to “disent angle” through re-\nconnection events, destroying in that way magnetic helicit y (Alexakis et al. 2006,\n2007b). At this moment it is unclear whether this processes s hould be associated\nto a cascade (i.e., if the process takes place with constant fl ux), although results\nin Alexakis et al. (2007b) and Mininni & Pouquet (2009) sugge st this may not\nbe the case.\n5 Non-local interactions and universality of MHD turbulenc e\nThe above considerations led several authors to consider wh ether some of the\nusual assumptions in hydrodynamic turbulence hold in the MH D case. From the\nshell-to-shell transfer, thescenariopicturedinFig. 6se emstoarisefortheenergy:\ninteractions between the same fields are mostly local, and in teractions between\nthe velocity and the magnetic field can have different degrees o f non-locality\ndepending on whether the turbulence is forced or freely deca ying, depending on\nhow the velocity and the magnetic fields are maintained again st dissipation in\nthe forced case, and depending on the presence of an external magnetic field.\nIt is unclear for the moment whether the varying degree of non -locality with\nthe configuration will converge to a universal solution for v ery large Reynolds\nnumbers.\nTheoretical arguments considering interactions in MHD tur bulence also ob-\ntained conflicting results. Using the EDQNM closure, Pouque t et al. (1976) re-\nportednon-localinteractionswhichwereassociatedtoAlf v´ enwaves. InVerma (2003,Scale Interactions in MHD Turbulence 23\n2004) and Verma et al. (2005), field-theoretic calculations were used to compute\nthe shell-to-shell transfers and it was concluded that they were local except for\nthe transfer between the velocity and the magnetic field, whi ch was found to\nbe somewhat non-local. The helicity transfer was also found to be non-local.\nRecently, Aluie & Eyink (2009b) gave strict bounds for fluxes in MHD turbu-\nlence under the assumptions that both the velocity and the ma gnetic energy\nfollow power laws in the inertial range between k−1andk−3. The velocity-to-\nvelocity and magnetic-to-magnetic fluxes were found to be lo cal in the limit of\ninfinite Reynolds number, and the fluxes coupling velocity an d magnetic fields\nwere found to be local although counterexamples to their pro of as the ones men-\ntioned in Sect. 2 were acknowledged. However, these results shed light into why\nsome simulations were found to be more local than others, as m echanisms as\nthe small-scale dynamo can be expected to be less relevant in freely decaying\nturbulence in approximate equipartition between the two fie lds.\nAt presently attainable spatial resolutions, other indica tions of possible non-\nuniversalbehaviorhasbeenreportedinnumericalsimulati ons. InDmitruk et al. (2003),\nsimulations of forced reduced magnetohydrodynamics (RMHD ) where presented\nwhere the energy spectrum changed its power law depending on the timescale of\ntheexternalforcing. SpectracompatiblewithKolmogorov, Iroshnikov-Kraichnan,\nweak turbulence theory, or even steeper laws were observed. The RMHD equa-\ntions correspondto an approximation of the MHD equations wh ena strong exter-\nnalmagneticfieldisimposed. Similarresultswerereported byMason et al. (2008),\nwho considered forced MHD with an imposed magnetic field. Oth er numeri-\ncal simulations of forced MHD turbulence (see e.g., Beresny ak & Lazarian 2009,\nHaugen et al. 2003,M¨ uller et al. 2003,M¨ uller & Grappin 200 5)alsoreportedcon-24 Mininni\nflicting results. In freely decaying isotropic turbulence, some simulations were\nobserved to develop Iroshnikov-Kraichnan scaling while ot hers Kolmogorov-like\nscaling(Mininni & Pouquet 2007,2009,M¨ uller & Grappin 200 5). Recently, large\nresolution simulations of freely decaying MHD flows showed t hat depending on\nthe amplitude of the dynamically consistent large-scale ma gnetic field, different\npower laws can be realized (Lee et al. 2009). Finally, recent studies of spectral\nlaws in solar wind data (Podesta et al. 2007) indicate that ma ny of these power\nlaws can also be identified in space plasmas.\nAlthough the main aim of this review is to consider studies of scale inter-\nactions in MHD, in this context it is worth mentioning some of the existing\nphenomenological theories for MHD turbulence. While Irosh nikov and Kraich-\nnan considered small scale fluctuations as isotropic, it is c lear now that MHD\nturbulence does not recover isotropy at small scales (Goldr eich & Sridhar 1995,\nMilano et al. 2001, Shebalin et al. 1983) and may become even m ore anisotropic\nas the scales are decreased. To take this into account, a differ ent MHD spec-\ntrum has been advocated in Goldreich & Sridhar (1995), where by the anisotropy\nof the flow induces a Kolmogorov-like spectrum in the perpend icular direction.\nA balance between linear and non-linear timescales (the Alf v´ en and turnover\ntimes) is assumed which leads to a “critical balance” of the f ormk/bardblB0∼k⊥bl.\nAnother anisotropic model based on dynamic alignment of the velocity and mag-\nnetic fields (Boldyrev 2006) gives IK-like scaling in the per pendicular direction.\nIn this case, the angle between the two fields decreases (and t herefore the fields\nbecome more aligned) with the scale as ∼l1/4. Consideration of this alignment in\nthe Politano-Pouquet relations leads to the aforementione d scaling for the energy\nspectrum. Earlyextensionstoflowswithsizablecross-corr elations canbefoundinScale Interactions in MHD Turbulence 25\nGaltier et al. (2000), Grappin et al. (1983). Other models ha ve considered tran-\nsitions fromKolmogorov toIroshnikov-Kraichnan scalingb ytakingdifferent com-\nbinations of the non-linear and Alfv´ en timescales (Mattha eus & Zhou 1989), or\nconsider non-locality (Alexakis 2007).\nTherefore, while the assumptions of locality and of isotrop ization of the small\nscales common in hydrodynamic turbulence allow for a simple r phenomenologi-\ncal treatment of MHD, the development of local anisotropies , the variety of time\nscales in the problem (see Zhou et al. 2004 for a review), and t he different sim-\nulations showing scaling consistent with different phenomen ological theories, led\nsomeauthorstodiscusssomeoftheseassumptions. InScheko chihin et al. (2008),\nnon-locality, anisotropy, and non-universality were cons idered as defining prop-\nerties of MHD turbulence. The authors argued that the small- scale dynamo, a\nfundamental process in MHD turbulence, shows clear signatu res of non-locality\n(Haugen et al. 2003, 2004, Mininni et al. 2005a, Schekochihi n et al. 2002a, 2004,\n2002b). They also argued that anisotropy is intrinsic to MHD , and that non-\nuniversality manifests itself just from the needed distinc tion between MHD tur-\nbulence in the presence and in the absence of a strong mean fiel d. Similar\nconcerns about universal behavior in MHD were discussed in L ee et al. (2009)\nfor the case of freely decaying turbulence. Beresnyak & Laza rian (2009) con-\nsider the lack of a “bottleneck” in MHD (an accumulation of en ergy at the\nbeginning of the viscous range observed in hydrodynamic tur bulence) as evi-\ndence of non-locality (see also Graham et al. 2009). In some s ense, some of\nthese discussions can be tracked back to early consideratio ns of freely decay-\ning MHD turbulence and the processes of selective decay (Kin ney et al. 1995,\nMatthaeus & Montgomery 1980, Mininni et al. 2005b, Ting et al . 1986) and dy-26 Mininni\nnamic alignment (Ghosh et al. 1988, Grappin et al. 1983, Mini nni et al. 2005c,\nPouquet et al. 1986). MHD, having three ideal invariants, is known to decay for\nvery long times into different attractors depending on the ini tial ratio of these\ninvariants (Stribling & Matthaeus 1991, Ting et al. 1986). A lthough these solu-\ntions involve final stages of the decay, recent numerical sim ulations showed that\nthese relaxed states can be realized locally in the flow in ver y short times scales\n(Mason et al. 2006,Matthaeus et al. 2008,Perez & Boldyrev 20 09,Servidio et al. 2008),\ngiving rise to different regimes.\n6 Concluding remarks\nSincethesuccessof thephenomenological theory ofKolmogo rov inhydrodynamic\nturbulence, several attempts have been made to apply simila r considerations to\ntackle MHD turbulence. The presence of waves and of several t ime scales, and\nof several ideal invariants limited these approaches givin g rise to many possible\nmodels. Solar wind observations and numerical simulations later showed that\nassumptions like isotropy of the small scales, or equiparti tion between the fields,\nmay not hold in the MHD case. More recently, the increase in co mputing power\nallowed for some exploration of the parameter space giving r ise to conflicting\nresults for scaling laws in the energy spectrum.\nThe recent introduction of shell-to-shell transfers allow ed for detailed studies\nof scale interactions in MHD turbulence and opened the door t o discuss another\nhypothesis: that of locality of interactions between scale s. The results, at in-\ntermediate spatial resolutions and Reynolds numbers, show different degrees of\nnon-locality depending on the configuration studied: force d or freely decaying\nturbulence, in presence or in absence of an external magneti c field, etc. Non-Scale Interactions in MHD Turbulence 27\nlocal transfers, when observed, involve the coupling betwe en the velocity and the\nmagnetic field, or the transfer of magnetic helicity. In the f ormer case, the non-\nlocal transfers were not larger than 10 −20% of the total, although they played\nfundamental roles, e.g., sustaining the magnetic field by dy namo action against\nOhmic dissipation.\nIn spite of some conflicting results in the simulations and th eory, there is grow-\ning consensus that MHD turbulence is less local than hydrody namic turbulence.\nTo what extent it is a matter of debate. It is unclear for the mo ment whether\nthese effects will go away for larger Reynolds numbers, or if th ey stay how much\nimpact they will have in the flow dynamics, and under what cond itions. How-\never, the different degrees of non-locality observed at prese nt resolutions, and the\nexistence of non-local processes in MHD (as, e.g., the small scale dynamo), call\nfor a discussion about the validity of the hypothesis of loca lity of interactions,\nand of whether there is only one kind of MHD turbulence or many . This raises\nthe question of what is the definition of MHD turbulence in phe nomenological\nor theoretical approaches. If only configurations as the one s in the solar wind\n(with an imposed magnetic field) are to be considered, then a u niversal scaling\n(or several classes of universality) may be possibly identi fied. However, if pro-\ncesses like the small-scale dynamo, the large-scale dynamo , and inverse cascades\nare to be considered as manifestations of MHD turbulence, no n-local interactions\nand non-universal behavior may persist even for very large R eynolds numbers.\nIn this context, many of the works reviewed here may have to be revisited in the\nfollowing years, as experiments and increased computing po wer will allow us to\nexplore new regions of the parameter space of MHD turbulence .28 Mininni\n7 Summary points\n1. Simply applying properties of hydrodynamic turbulence t o the MHD case\nmay not be possible. In particular, assumptions of scale loc ality of MHD\nturbulence must be tested in experiments and simulations.\n2. Shell-to-shell transfer functions allow for detailed st udies of coupling be-\ntween fields and scales in numerical simulations. The shell- to-shell trans-\nfers can also be associated with physical processes such as a s Alfv´ en wave\ninteractions, Joule damping, and dynamo action.\n3. The degree of non-locality observed at the presently atta inable spatial res-\nolutions depends on the configuration.\n4. Mechanically forced turbulence shows local transfer of m agnetic and ki-\nnetic energy, but the coupling between the velocity and magn etic field that\nsustains the latter against Ohmic dissipation is non-local .\n5. In freely decaying MHD turbulence, non-local effects seem t o be negligible.\n6. Studies of the energy transfer in the presence of an impose d magnetic field\nshow most of the transfer takes place in the direction perpen dicular to the\nexternal field, with strong non-local interactions with mod es withk/bardbl= 0.\n7. The transfer of energy for the Els¨ asser variables is more local than the\ntransfer in terms of the velocity and magnetic fields.\n8. The shell-to-shell transfer of magnetic helicity is more complex, with su-\nperimposed direct and inverse transfers. The inverse trans fer has a local\ncomponent, and a non-local one that moves energy from the for ced scale\ndirectly to the largest scales in the system.Scale Interactions in MHD Turbulence 29\n8 Acknowledgments\nThe National Center for Atmospheric Research is sponsored b y the National\nScience Foundation. The author acknowledges fruitful disc ussions with Alexan-\ndros Alexakis, Daniele Carati, Gregory Eyink, Annick Pouqu et, and Alexander\nSchekochihin, and figures gently provided by A. Alexakis and D. Carati. The\nauthor also expresses his gratitude to A. Pouquet for her car eful reading of the\nmanuscript. PDM is a member of the Carrera del Investigador C ient´ ıfico of\nCONICET,and acknowledges supportfrom grants UBACYT X468/ 08 and PICT\n2007-02211.\nReferences\n1. Alexakis A. 2007. Nonlocal phenomenology for anisotropi c magnetohydrody-\nnamic turbulence. Astrophys. J. 667:L93–L96\n2. Alexakis A,Bigot B,PolitanoH,Galtier S.2007a. Anisotr opicfluxesandnonlocal\ninteractions in magnetohydrodynamic turbulence. Phys. Rev. E 76:056313\n3. Alexakis A, Mininni PD, Pouquet A. 2005a. Imprint of large -scale flows on\nturbulence. Phys. Rev. Lett. 95:264503\n4. Alexakis A, Mininni PD, Pouquet A. 2005b. Shell to shell en ergy transfer in\nMHD. I. Steady state turbulence. Phys. Rev. E 72:046301\n5. Alexakis A, Mininni PD, Pouquet A. 2006. On the inverse cas cade of magnetic\nhelicity. Astrophys. J. 640:335–343\n6. Alexakis A, Mininni PD, Pouquet A. 2007b. 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The TuuandTbbfunctions are local, with a negative peak for K < Qand\na positive peak for K > Q, which indicate energy is removed by these transfer\nfunctions from smaller wavenumbers and given to slightly la rger wavenumbers.\nThe transfer between magnetic to kinetic energy is of smalle r amplitude and also\nseems local. Right: The Tubtransfer, for different values of K. This function is\nnon-local, with a strong peak at the forcing scale and with a c onstant positive\nplateau that extends up to K≈Q. Adapted from Alexakis et al. (2005b).Scale Interactions in MHD Turbulence 41\nFigure 2: Left: Tuutransfer function in freely decaying MHD turbulence, for\ndifferent shells. The Tbbtransfer function is similar but has twice the amplitude.\nRight:Tbutransfer functionin thesame simulation. Note thepeak for K−Q= 0,\nindicating most interchange of energy between the velocity and the magnetic\nfield takes place between similar scales. The shells are loga rithmically binned.\nAdapted from Debliquy et al. (2005).42 Mininni\nFigure 3: Isotropic (spherical) shells in the left, and anis otropic foldings of shells\nin Fourier space. The uniform magnetic field is assumed to be i n thezdirection.\nCylindricalandplanar shellsareshownin themiddle, andri ngshells areshownin\nthe right. Transfer of energy across planes is denoted by T(Q/bardbl,K/bardbl), and transfer\nacross cylindersis denoted by T(Q⊥,K⊥). For ring-to-ring transfers, thenotation\nT{Q,α}\n{K,β}denotes transfer can be measured between KandQspherical shells, as\nwell as between two azimuthal angles αandβ.Scale Interactions in MHD Turbulence 43\nFigure4: Left: totalenergyflux(solid)acrosscylindersan dpartialfluxassociated\nto interactions with modes with k/bardbl= 0 (dashed), with four different values of the\nexternal magnetic field B0from 0 to 15 (from top to bottom). Right: same but\nwith the total flux and partial flux associated to interaction s with modes with\nk/bardbl= 1 across planes. From Alexakis et al. (2007a).44 Mininni\nFigure 5: Left: the stretch, twist, and fold dynamo mechanis m. Each time a\nclosed magnetic flux tube is twisted, magnetic helicity of op posite sign is created\nat large and small scales. The folding creates regions where helical magnetic\nfields can reconnect. Right: (a) The helicity spectrum in a si mulation with\n(positive) helical mechanical forcing at k= 10. Magnetic helicity is negative at\nlarger scales and positive at smaller scales. (b,c,d): The t ransfer of helicity for\nQ= 2, 10, and 20. The red arrows indicate transfer of negative h elicity and the\ngreen arrows transfer of positive helicity. At large scales (b), negative magnetic\nhelicity inversely cascades locally between neighbor shel ls and non-locally from\nthe forced shell and to the small scale shells. At the forced s hell (c), the forcing\ninjects opposite signs of helicity at large and small scales . At small scales (d),\npositive magnetic helicity has a local direct transfer of he licity, while the small\nscales also remove negative magnetic helicity from the larg e scales. Note direct\ntransfer of negative helicity is equivalent to inverse tran sfer of positive helicity.Scale Interactions in MHD Turbulence 45\nFigure 6: Sketch of the several shell-to-shell energy trans fers identified in sim-\nulations of isotropic and homogeneous MHD turbulence. The Tuutransfers are\nshown in red, Tbbtransfers are shown in blue, and TubandTbuin green. The\nthickness of the arrows roughly indicates the strength of th e transfers. Above:\nmechanically forced simulations. At the shell K, the magnetic field receives en-\nergy from the velocity field at all larger scales, and gives en ergy to the velocity\nfield at slightly smaller scales. Below: Freely decaying tur bulence. The Tuband\nTbutransfers only interchange energy between similar scales. In both cases, the\nTuuandTbbtransfers are local and give the largest contribution to the flux."    },    {        "title": "1303.5843v2.Stringent_constraint_on_neutrino_Lorentz_invariance_violation_from_the_two_IceCube_PeV_neutrinos.pdf",        "content": "arXiv:1303.5843v2  [astro-ph.HE]  25 Jun 2013Stringent constraint on neutrino Lorentz-invariance viol ation\nfrom the two IceCube PeV neutrinos\nEnrico Borriello, Sovan Chakraborty, and Alessandro Mirizzi\nII Institut f¨ ur Theoretische Physik, Universit¨ at Hambur g,\nLuruper Chaussee 149, 22761 Hamburg, Germany\nPasquale Dario Serpico\nLAPTh, Univ. de Savoie, CNRS, B.P.110, Annecy-le-Vieux F-7 4941, France\nIt has been speculated that Lorentz-invariance violation ( LIV) might be generated by quantum-\ngravity (QG) effects. As a consequence, particles may not tra vel at the universal speed of light. In\nparticular, superluminal extragalactic neutrinos would r apidly lose energy via the bremssthralung of\nelectron-positron pairs ( ν→νe+e−), damping their initial energy into electromagnetic casca des, a\nfigure constrained by Fermi-LAT data. We show that the two cas cade neutrino events with energies\naround 1 PeV recently detected by IceCube—if attributed to e xtragalactic diffuse events, as it\nappears likely—can place the strongest bound on LIV in the neutrino sector, namely δ= (v2−1)<\nO(10−18), corresponding to a QG scale MQG>∼105MPl(MQG>∼10−4MPl) for a linear (quadratic)\nLIV, at least for models inducing superluminal neutrino effe cts (δ >0).\nPACS numbers: 11.30.Cp, 95.85.Ry LAPTH-015/13\nI. INTRODUCTION\nIt is conceivable that Lorentz-invariance might be vi-\nolated in a candidate theory of Quantum Gravity (QG),\nsee [1] for a review. In this context, the space-time\nfoam due to QG fluctuations might cause e.g. highly\nboosted energetic particles to propagate at speed vdif-\nferent from the velocity of light c. The resulting Lorentz-\ninvariance violation (LIV) effect can be phenomenologi-\ncally parametrized in terms of δdefined as (we set c= 1)\nδ=/parenleftbiggv\nv0/parenrightbigg2\n−1, v=∂E\n∂p,v0=p/radicalbig\np2+m2,(1)\nwhere, assuming that there is at least one frame in which\nspace and time translations and spatial rotations are ex-\nact symmetries (typically the lab one), one writes\nE2=p2+m2+f(p,...). (2)\nThefunction fisoftenexpressedinpowersofmomentum\noverthe phenomenologicalscale MQGwhenLIVbecomes\nimportant, of the type ∝pℓM2−ℓ\nQG. Hence one can trans-\nlate a constraint on δinto a constraint on MQGvia a\nrelation of the kind\nδ=/parenleftbiggv\nv0/parenrightbigg2\n−1≃v0\nE∂f\n∂p≃ ±/parenleftbiggE\nMQG/parenrightbiggn\n,(3)\nwith e.g. cubic ( ℓ= 3) or quartic ( ℓ= 4) terms in powers\nofpadded to the dispersion relation of Eq. (3), inducing\n“linear” ( n= 1) or “quadratic” ( n= 2) deviations from\nLI occurring at a mass scale MQG. The + ( −) implies\nsuperluminal (subluminal) velocity.\nThe observation of LIV effects of this nature, if sup-\npressed by the Planck scale MPl= 1.22×1028eV, repre-\nsents a very challenging task for any Earth-based experi-\nment (see [2] for a rewiev of the current bounds). In thisrespect, high-energy astrophysics experiments represent\na promising, complementary tool able to probe the con-\nsequencesof tiny LIV[1, 3]. Tests ofLIV in photon prop-\nagation from objects like Gamma Ray Bursts (GRB) [4],\nActive Galactic Nuclei (AGN) [5] and Pulsars [6] look-\ning into delays in arrival times of photons of different\nenergies can probe a quantum gravity mass scale up to\nMQG∼1026eV for the linear case, and MQG∼1019eV\nfor the quadratic case.\nExperimental probes of LIV for neutrinos are limited\nby the scarcity of neutrino data from distant astrophysi-\ncalsources. Inparticular,fromSN1987Adata, exploiting\ntheflightdelayofthe γwithrespectto νofafewhours,it\nhas been obtained a bound of δ<∼10−9[7], correspond-\ning toMQG>∼1015eV forn= 1, or MQG>∼1011eV\nforn= 2, taking E≃10 MeV as typical SN νenergy.\nThe supernova limit on LIV can reach MQG∼1021eV\n(2×1014eV) for the linear (quadratic) energy depen-\ndence, observing narrow time structure in the neutrino\nemission, like the ones associated with the neutroniza-\ntion burst [8]. The bounds on LIV could be further\nimproved in case of detection of high-energy neutrinos\n(E>∼O(GeV)) from astrophysical sources, like Gamma\nRay Bursts or Active Galactic Nuclei [9]. Indeed, from\ntheenergydependenttimeofflightdelayofneutrinosand\nthe corresponding γrays, induced by LIV, the limit that\nwould be placed is MQG>∼1026eV for the n= 1 case,\nandMQG>∼1019eV for the n= 2 case, respectively.\nFor the specific case of superluminal neutrinos, LIV\nwould also allow processes that would be otherwise\nkinematically forbidden in vacuum, namely neutrino\nCherenkov radiation ( ν→νγ), neutrino splitting ( ν→\nνν¯ν) and bremssthralung of electron-positron pairs\n(ν→νe+e−) (see, e.g. [10]). All these processes would\nproduce a depletion of the high-energy neutrino fluxes\nduring their propagation. In particular, among the pre-2\nvious processes, the neutrino pair production has been\nrecognized as the fastest energy-loss process for LIV neu-\ntrinos and has been used in [11–13] to invalidate the su-\nperluminal velocity claim made by the OPERA Collabo-\nration [14]. A similar analysis has also been able to put\nstrongconstraintsonsuperluminalvelocitiesofhigheren-\nergy neutrinos. The observation of upward going atmo-\nspheric neutrino showers(with path-length L≃500 km),\nmeasured at E>∼100 TeV at the IceCube experiment,\nhas allowed to put the strongest bound on LIV, namely\nδ<∼10−13[15]. That bound was derived by comparing\ntheobservedspectrumwiththetheoreticalpredictedone.\nIt was noted before that a conceptually similar method\ncould lead to improvements if it could be applied to the\nspectrum of an astrophysical source such as a supernova\nremnant, as noted in [13]. In Sec. II, following from re-\ncentobservations of IceCube, we explore a new stringent\nbound based on this particle physics mechanism but fol-\nlowing from a different, calorimetric (as opposed to spec-\ntral) diagnostic tool, based on multi-messenger consider-\nations involving diffuse gamma-ray data. In Sec. III we\ndiscuss these results and conclude.\nII. NEW STRINGENT BOUND\nSince the effects of LIV processes typically grow with\nenergy and distance, it is worthwhile to exploit high-\nenergy neutrinos traveling along a sufficiently long base-\nline. It has been argued for example that observations of\nthe cosmogenic neutrino background (produced by ultra-\nhigh energy cosmic ray losses) might lead to stringent\nconstraints, see [16]. In this context, the IceCube exper-\niment has recently reported the detection of two cascade\nneutrino events with energies around E≃1 PeV in the\nperiod ofdatataking 2011-2012[17]. At the moment, the\noriginofthese eventsis notsettled. It seemsthat conven-\ntionalatmosphericneutrinosorcosmogenicneutrinos[18]\nare unlikely to be the sources of these events, and that\nthe chance that these events are associated to prompt at-\nmospheric neutrinos from heavy quark decays is at most\na few % [19]. There is also another intriguing experimen-\ntal indication: an IceCube analysis of neutrino-induced\nmuon track events shows that, although the sample is\ndominated by conventional atmospheric neutrinos, data\ndo prefer marginally (at the 1.8 σlevel) an extra com-\nponent at E >100TeV [20]. Albeit of little significance\nat present, it is intriguing that the extrapolation of this\nbest-fit flux to PeV scale is in perfect agreement with\nthe observed two PeV cascade events mentioned above.\nAlthough the significance of an additional component is\nstilllow, the hypothesisthatthesetwoneutrinoscouldbe\nthe first indication ofa diffuse extragalacticastrophysical\nflux at the PeV scale appears at the same time the mostlikely and the most exciting one1.\nIn this section, we deduce an important improvement\nover the current bounds on superluminal neutrino veloc-\nity, following from the above-mentioned assumption. In\nSec. III we shall comment further on this point.\nFirst, let us remind that for δ >0 the process ν→\nνe+e−is kinematically allowed provided that\nEν>∼2me/√\nδ≃PeV/radicalbig\n10−18/δ (4)\nand assuming LI conservation in the electron sector [11].\nThis process implies a pair production decay rate of [11]:\nΓe±=1\n14G2\nFE5δ3\n192π3= 2.55×1053δ3E5\nPeVMpc−1.(5)\nAs in [11], we neglect here the process ( ν→νν¯ν), which\nis only expected to bring minor modification to our ar-\ngument. On the other hand, if the process associated\nwith the rate of Eq. (5) is forbidden because of threshold\neffects, the ν→νγis anyway operational and a channel\nfor energy losses, although it is two to three orders of\nmagnitude less efficient than ν→νe+e−.\nRemarkably, without any significant additional infor-\nmation on astrophysical sources of neutrinos (neither on\nthe specific mechanism of neutrino production), this is\nsufficient to derive a strong bound. Let us assume that\nthe PeV neutrinos are generated somehow in extragalac-\ntic sources. If the above processes are effective, the neu-\ntrinofluxwillbesoondepleted attheexpenseofinjecting\nelectromagnetic energy while propagating to the Earth.\nThe interaction length of e±onto CMB photons is ex-\ntremely short, of the order of few kpc. Even a gamma\nray will not propagate much more than O(10) kpc at\nthese energies before pair-producing on the CMB (see for\nexample [23]). The inverse-Compton scattering photons\ninduced by e±will populate a gamma-ray flux at whose\nenergy is roughly stored between E1∼ O(1) GeV and\nE2O(100) GeV, see for example Fig. 5 in [24]. This flux\nis constrained by Fermi data [25] to have an integrated\nenergy density\nωγ=4π\nc/integraldisplayE2\nE1Edϕγ\ndEdE<∼5.7×10−7eV/cm3.(6)\nWe are taking the normalization for the extragalactic\nflux from Fermi and the spectral shape for the energy\ndensitydϕγ/dE∼E−2.41, just like in previous pub-\nlications on this subject such as [26]. Note also that\nthe unaccounted flux (not reasonably attributed to unre-\nsolved astrophysical sources) is probably only a fraction\nof the above one, see e.g. [27] for a recent modeling of\n1Although the referred analysis is based on the more advanced\nstage/statistics of IceCube-59, it is worth reminding that differ-\nent analysisofIceCube-40 publicdata show contradicting r esults.\nIn particular, an excess can be seen in νedata [21], but not in\nνµdata [22].3\nthat. Now, the two events detected in IceCube in two\nyears of data taking would imply a diffuse energy flux\nE2\nνdϕE/dE≃3.6×10−8GeV cm−2s−1sr−1[17], actu-\nally quite close to the so-called Waxman-Bahcall bench-\nmark[28]. Thesenumbersimply that onecannottolerate\nan energy density ωin\nνin theinitialneutrino flux larger\nby about two orders of magnitude than the observed one\nωobs\nν=4π\nc1.2PeV/integraldisplay\n1PeVEdϕE\ndEdE≃2.7×10−9eV/cm3,(7)\notherwise the electromagnetic energy injected via the\nprocess would basically saturate the bound. Note that\nwe did not extrapolate the observed flux beyond the en-\nergy window at which the two events have been mea-\nsured. Once accounting for the fact that the process in\nquestion transfers a large fraction of the initial neutrino\nenergy into the e±pair [11], a simple back-of-the enve-\nlopecalculationleadstoanapproximateconstraintofthe\ntype:\ne−Γd>∼ωobs\nν\nωγ∼10−2, (8)\nwheredis the characteristic distance of the sources. Pro-\nvided that the channel ν→νe+e−is open, we get\nδ3\ne±dMpc<1.8×10−53. (9)\nFor the simplest scenario of cosmologically distant\nsources, a reasonable value is d∼ O(103)Mpc. In fact\nthe emissivity of a diffuse neutrino flux from cosmologi-\ncallydistributedsources(suchasGRBs)ispeakedatred-\nshiftz∼1 [29], that corresponds to a comoving distance\nof more than 3 Gpc. Equation (9) nominally implies\nδ<∼O(10−19), which means that the pair-production\nmechanism is not operational and that the actual bound\nis thusδ <O(10−18) (see the condition of Eq. (4)). A\npriori, the ν→νγiskinematicallyaccessibletoPeVneu-\ntrinos even for δ <O(10−18) and depending on its rate\nit could put a more stringent bound, but it is easy to\ncheck that it is not the case: it leads instead to a slightly\nweaker(albeit of comparableorderof magnitude) bound.\nHowever, the latter channel allows to infer that the de-\nrived bound probably holds under more general hypothe-\nses than the ones we assumed: for example it should not\ndepend crucially on the Ansatz that the LIV in the elec-\ntron sector is much smaller than in the neutrino one.\nIII. DISCUSSION AND CONCLUSION\nWe have derived a very stringent bound on LIV in the\nneutrino sector, δ <O(10−18), from the observations of\ntwo PeV scale IceCube events and remarkably few as-\nsumptions on the underlying astrophysical sources.\nIt is important to emphasize that our argument is nei-\nther based on a study of the neutrino flux shape distor-\ntion, like in [15] nor on the assumption of any putativeGalactic source, like in [13], as none is currentlyplausible\nto explain the observed flux we refer to. What we pro-\npose here is instead a novel “calorimetric” bound, based\non the observed extragalactic gamma ray diffuse flux.\nLet us now discuss the model dependence in more de-\ntail. We just assumed that the events are due to an\nextragalactic flux, hence with characteristic distance of\nsource(s) of the order of a Gpc. Of course, once addi-\ntional information will be available (e.g. on the number\ndensityand redshift distribution ofthe sourcescontribut-\ning to the flux) an improved calculation will be possible.\nWe do not expect however that the ignorance of these\ndetails affects the bound very much. In fact, within our\napproximations the bound on δscales as\nδ<∼10−18×Max/bracketleftBigg\n1.6/parenleftbiggln(ωγ/ωobs\nν)\nE5\nPeVdMpc/parenrightbigg1\n3\n, E−2\nPeV/bracketrightBigg\n,(10)\nwith a very weak dependence on the initial assumption\non the fluxes, as well as a dependence only on the cubic\nroot of the typical distance. For example, it is worth\nnotingthateveninthe extreme(andunlikely)casewhere\nthe observed neutrino flux were dominated by the closest\nextragalactic source candidates, at distances of the order\nof several Mpc—Centaurus A, which is the closest one,\nat about 4 Mpc from us—one would still deduce δ<∼\nO(10−18). Again, this is basically suggesting that the\nenergy loss channel via pair emission must be a closed\nchannel. On the other hand, detecting higher energy\nevents would improve and make the bounds more robust.\nThe extragalactic nature is essential in making the\nelectromagnetic “cascade”argument operational, but ac-\ntuallyitisalsothemostlikelyexplanation,giventhatthe\nflux level required to match observations appear close to\npredicted diffuse fluxes (see e.g. [30]). Not only that: an\nindirect argument against a closer astrophysical origin is\nthat the Galactic diffuse flux at PeV energy can be com-\nputed quite reliably and is expected to be much smaller,\nsee e.g. [31]. This is true provided that the source is rela-\ntively“thin” tocosmicrays, sothat the flux isdominated\nby cosmic ray production in interstellar medium. This\nappears almost unavoidable, especially since we would\nneed accelerators capable of maximal energies up to 1016\neV/nucleon at least, which would be hard to impossi-\nble to achieve with large energy losses within the accel-\nerator. Anyway, some bound should exist also if one\nor more hypothetical Galactic sources contribute to this\nflux. Lackingpromising candidates of this type in the lit-\nerature, we do not consider this further. In general, the\nbound should be then obtained by estimating an input\nflux upper bound by some other sort of multi-messenger\nconstraint: Typically, other by-products (or tertiaries of\ntheirenergydegradation)ofthe neutrinoproductionpro-\ncesses at the source.\nFinally, what if the events should be eventually at-\ntributed to atmospheric background? We briefly men-\ntioned that this appears unlikely at present. Yet, it\nis worth noting that even in that case, the current4\nbound [15] could be still improved by a more modest\namount (perhaps one order of magnitude), although a\nmore detailed analysis of the type of [15]—properly ac-\ncounting for prompt contribution and its uncertainty—\nwould be needed to draw a more quantitative conclusion.\nIn summary, we have argued that a confirmation of\nthe extragalactic astrophysical nature of the PeV events\ndetected by IceCube would not only open a new win-\ndow to the high-energy universe, but also allow a signif-\nicant jump in tests of fundamental physics. To obtain\nthis result, as more and more customary in astroparticle\nphysics, wereliedonamulti-messengerstrategy(herethe\nlink with diffuse gamma ray background), which proves\nonce again the power of this approach.Acknowledgments\nThe authors thank Basudeb Dasgupta, Tom Gaisser,\nTeresa Montaruli, and Anne Schukraft for useful con-\nversations, Jorge S. Diaz for feedback on the manuscript,\nand particularlyLuca Maccione for comments. The work\nof E.B. and S.C. and A.M. was supported by the Ger-\nman Science Foundation (DFG) within the Collaborative\nResearch Center 676 “Particles, Strings and the Early\nUniverse”. At LAPTh, this activity was developed co-\nherently with the research axes supported by the Labex\ngrant ENIGMASS.\n[1] S. Liberati and L. Maccione, Annu. Rev. Nucl. Part. Sci.\n59(2009) 245–267.\n[2] V. A. Kostelecky and N. Russell, Rev. Mod. Phys. 83\n(2011) 11.\n[3] D. Mattingly, Living Rev. Rel. 8, 5 (2005).\n[4] J. R. Ellis, N. E. Mavromatos, D. V. Nanopoulos,\nA. S. Sakharov and E. K. G. Sarkisyan, Astropart. Phys.\n25, 402 (2006) [Astropart. Phys. 29, 158 (2008)]\n[5] J. Albert et al.[MAGIC and Other Contributors Collab-\norations], Phys. Lett. B 668, 253 (2008).\n[6] P. Kaaret, astro-ph/9903464.\n[7] L. Stodolsky, Phys. Lett. B 201, 353 (1988).\n[8] S. Chakraborty, A. Mirizzi and G. Sigl, Phys. Rev. D 87,\n017302 (2013).\n[9] U. Jacob and T. Piran, Nature Phys. 3, 87 (2007).\n[10] L. Maccione, S. Liberati and D. M. Mattingly,\narXiv:1110.0783 [hep-ph].\n[11] A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. 107,\n181803 (2011).\n[12] L. Gonzalez-Mestres, arXiv:1109.6630 [physics.gen- ph].\n[13] X. -J. Bi, P. -F. Yin, Z. -H. Yu and Q. Yuan, Phys. Rev.\nLett.107(2011) 241802.\n[14] T. Adam et al.[OPERA Collaboration], JHEP 1210\n(2012) 093.\n[15] R. Cowsik, T. Madziwa-Nussinov, S. Nussinov and\nU. Sarkar, Phys. Rev. D 86, 045024 (2012).\n[16] D. M. Mattingly, L. Maccione, M. Galaverni, S. Liberati\nand G. Sigl, JCAP 1002, 007 (2010).\n[17] M. G. Aartsen et al. [IceCube Collaboration],arXiv:1304.5356 [astro-ph.HE].\n[18] E. Roulet, G. Sigl, A. van Vliet and S. Mollerach, JCAP\n1301, 028 (2013).\n[19] I. Cholis and D. Hooper, arXiv:1211.1974 [astro-ph.HE ].\n[20] Anne Schukraft, talk at NOW 2012, to appear\nNucl. Phys. B Proc. Supplements. Slides available at\nwww.ba.infn.it/ now/\n[21] R. Abbasi et al.[IceCube Collaboration], Phys. Rev. D\n84(2011) 082001.\n[22] R. Abbasi et al. [IceCube Collaboration],\narXiv:1111.2736 [astro-ph.HE].\n[23] S. Lee, Phys. Rev. D 58, 043004 (1998).\n[24] D. V. Semikoz and G. Sigl, JCAP 0404, 003 (2004)\n[hep-ph/0309328].\n[25] A. A. Abdo et al.[Fermi-LAT Collaboration], Phys. Rev.\nLett.104, 101101 (2010)\n[26] V. Berezinsky, A. Gazizov, M. Kachelriess and\nS. Ostapchenko, Phys. Lett. B 695, 13 (2011).\n[27] F. Calore, M. Di Mauro and F. Donato, arXiv:1303.3284\n[astro-ph.CO].\n[28] E. Waxman and J. N. Bahcall, Phys. Rev. D 59, 023002\n(1999).\n[29] F. Iocco, K. Murase, S. Nagataki and P. D. Serpico, As-\ntrophys. J. 675(2008) 937.\n[30] J. K. Becker, Phys. Rept. 458, 173 (2008).\n[31] C. Evoli, D. Grasso and L. Maccione, JCAP 0706, 003\n(2007)."    },    {        "title": "2108.07395v2.Asymptotic_behaviour_of_the_wave_equation_with_nonlocal_weak_damping__anti_damping_and_critical_nonlinearity.pdf",        "content": "arXiv:2108.07395v2  [math.AP]  6 May 2022Asymptotic behavior of the wave equation with nonlocal weak\ndamping, anti-damping and critical nonlinearity\nChunyan Zhaoa, Chengkui Zhonga,∗, Zhijun Tanga\naDepartment of Mathematics, Nanjing University, Nanjing, 2 10093, China\nAbstract\nIn this paper, we first prove an abstract theorem on the existenc e of polynomial attractors\nand the concrete estimate of their attractive velocity for infinite- dimensional dynamical\nsystems, then apply this theorem to a class of wave equations with n onlocal weak damping\nand anti-damping in case that the nonlinear term fis of subcritical growth.\nKeywords: Wave equation, nonlocal weak damping, nonlocal weak anti-damping , critical\nnonlinearity, global attractor.\n2010 MSC: 35B40, 35B41, 35L05\n1. Introduction\nIn this paper, we investigate the existence of the global attracto r for the wave equation\nwithnonlocal weak damping, nonlocal weak anti-damping andcritical nonlinear source term\nutt−∆u+k||ut||p\nL2(Ω)ut+f(u) =/integraldisplay\nΩK(x,y)ut(y)dy+h(x) in [0,∞)×Ω,(1.1)\nu= 0 on [0,∞)×∂Ω, (1.2)\nu(x,0) =u0(x),ut(x,0) =u1(x), x∈Ω, (1.3)\nwhere Ω is a bounded domain in RN(N≥3) with smooth boundary ∂Ω and the following\nassumption holds:\nAssumption 1.1. (i)kandpare positive constants, K∈L2(Ω×Ω),h∈L2(Ω);\n(ii)f∈C1(R)satisfies\n|f′(s)| ≤M(|s|2\nN−2+1), (1.4)\nliminf\n|s|→∞f′(s)≡µ>−λ1, (1.5)\nwhereM≥0andλ1is the first eigenvalue of the operator −∆equipped with Dirichlet\nboundary condition.\n∗Corresponding author.\nEmail addresses: emmanuelz@163.com (Chunyan Zhao), ckzhong@nju.edu.cn (Chengkui Zhong),\ntzj960629@163.com (Zhijun Tang)\nPreprint submitted to Elsevier May 9, 2022Since the pioneering work of J.K. Hale et al. [26] on the dynamical beha vior of dissipa-\ntive wave equations in the 1970s, there has been a wealth of literatu re on the asymptotic\nstate (with particular reference to existence of the global attra ctor, estimate of its fractal\ndimension and existence of exponential attractors) of solutions o f wave equations with vari-\nous damping. Among them, we refer to [1, 2, 4, 7, 9, 24, 25, 27, 34, 45, 49, 51] for the wave\nequation with weak damping kutwhich models the oscillation process occurring in many\nphysical systems, including electrodynamics, quantum mechanics, nonlinear elasticity, etc.\nWave equations with strong damping −k∆ut(see [42] for their physical background) were\nstudied in [5, 6, 23, 24, 41, 42]. Literatures [20–22, 28, 30, 33, 44, 50] were devoted to wave\nequations with nonlinear damping g(ut). The damping ( −∆)αut(α∈(0,1)) is called frac-\ntional damping. In particular, it is referred to as structural damp ing when 1/2≤α<1 and\nas moderate damping when 0 ≤α<1/2. Studies related to wave equations with fractional\ndamping can be found in [6, 52, 53] and references therein.\nOn the other hand, the long-time behavior of hyperbolic equations w ith nonlocal damp-\ning also has received great attention. For example, we refer to [11 , 19] for the study of the\nKirchhoff equation with the damping M(||∇u||2\nL2(Ω))∆ut, to [8, 35] for the case of nonlocal\nweak damping M(||∇u||2\nL2(Ω))ut, to [17] for the case of the damping M(||∇u||2\nL2(Ω))g(ut),\nand to [10, 16] for the case of the damping M(||∇u||2\nL2(Ω))(−△)θut. The damping terms\ninvolved in the references listed above all have Kirchhoff type coeffic ientsM(||∇u||2\nL2(Ω)).\nIn addition, Lazo [38] proved the existence of a global solution to th e equation\nutt+M(|A1\n2u|2)Au+N(|Aαu|2)Aαut=f,\nwhereAis a positive self-adjoint operator defined in Hilbert space H,α∈(0,1] and the\nfunctionsM,Nsatisfy the nondegenerate condition.\nWhile, to the best of our knowledge, only very few results are availab le for damped\nhyperbolic equations whose nonlocal damping coefficient depends on ut. Among them we\nhighlight that in 1989 Balakrishnan and Taylor [3] presented some ext ensible beam equa-\ntions with nonlocal energy damping/bracketleftbigg/integraldisplay\nΩ(|∆u|2+|ut|2)dx/bracketrightbiggq\n∆utto model the damping phe-\nnomena in flight structures. Recently Silva, Narciso and Vicente [31 ] have proved the global\nwell-posedness, polynomial stability of the following beam model with t he nonlocal energy\ndamping\nutt−κ∆u+∆2u−γ/bracketleftbigg/integraldisplay\nΩ(|∆u|2+|ut|2)dx/bracketrightbiggq\n∆ut+f(u) = 0.\nLazo [37] considered the local solvability of the wave equation\nutt−M(/ba∇dbl∇u/ba∇dbl2\nL2(Ω))△u+N(/ba∇dblut/ba∇dbl2\nL2(Ω))ut=b|u|p−1u.\nWe are motivated by the literature mentioned above to study proble m (1.1)-(1.3) in\nour last work [55]. As far as we know, this constituted the first resu lt on the long-time\nbehavior of wave equation with nonlocal damping k||ut||p\nL2(Ω)ut. In [55], we have proved\nvia the method of Condition (C) that the system possesses a global attractor in the case\nthatfsatisfies the subcritical growth condition. However, we did not solv e the problem of\n2the existence of the global attractor for the critical case, which is the aim of the present\npaper.\nProblem(1.1)-(1.3)isaweaklydampedmodel,inwhichthenonlocalcoefficient k||ut||pre-\nflects the effect of kinetic energy on damping in physics. The term/integraltext\nΩK(x,y)ut(y)dyis an\nanti-damping because it may provide energy. The difficulty of this pro blem lies first in the\nnondegenerate, nonlocal coefficient of damping and the arbitrarin ess of the exponent p>0.\nDue to the influence of nonlocal coefficient k||ut||p, when the velocity utis very small, the\nnonlocal damping is weaker than the linear damping. Furthermore, a s the velocity utis\nsmaller and pis larger, the damping is weaker and thus energy dissipation is slower. In\naddition, the presence of the anti-damping term leads to the energ y not decreasing along\nthe orbit, and moreover, the effect of energy supplement brough t by the anti-damping term\nneeds to be overcome by the damping. All these factors cause diffic ulties in studying the\nlong-term behavior of this model. At the same time, since fis of critical growth, the cor-\nresponding Sobolev embedding is no longer compact, which makes all t he methods based\non compactness, including Condition (C), no longer available to prove the existence of the\nglobal attractor.\nIn this paper, to overcome the difficulty of lack of compactness in th e critical case, we\nemploy the criterion of asymptotic smoothness relying on the repea ted inferior limit (see\nLemma 2.5 below) to prove the existence of the global attractor. Chuesho v and Lasiecka\n[14, 15] proposed this criterion based on the idea of Khanmamdov [32 ]. To handle the\ndifficulty that nonlocal damping coefficient /ba∇dblut/ba∇dblpbrings in energy estimate, we use the\nstrong monotone inequality for the general inner product space ( see Lemma 2 .6 below).\nThe proof of compactness borrows many ideas from [14].\nAs for the dissipativity, A. Haraux [29] obtained via barrier’s method the uniform bound\noftheenergyintermsoftheinitialenergyforadissipative waveequ ationwithanti-damping.\nThe key element of the method is that the dissipative term in the inequ ality for Lyapunov’s\nfunction has a coefficient sublinearly dependent on the energy. I. C hueshov and I. Lasiecka\n[14] further proved that systems whose Lyapunov’s functions sa tisfy such inequalities are ul-\ntimately dissipative. Their strategy was to select the perturbation parameterǫinthe energy\ninequality as a suitable function of the initial energy according to the sublinear dependence\nof the coefficient of the dissipation term on the energy; and thus th ey deduced that the\nenergy is ultimately bounded by a constant independent on the initial data. Following the\nmethod in [14], we prove the dissipativity for problem (1 .1)-(1.3).\nThe establishment of the global well-posedness follows the idea in [13, 14].\nThis paper is organized as follows. In Section 2, we present some not ations and lemmas\nwhich will be needed later. In Section 3 and Section 4, we prove the glo bal well-posedness\nand dissipativity of the dynamical system generated by problem (1 .1)-(1.3), respectively. In\nSection 5, we establish the existence of the global attractor for t his system.\n2. Preliminaries\nThroughout this paper, we will denote the inner product and the no rm onL2(Ω) by (·,·)\nand/ba∇dbl·/ba∇dbl, respectively, and the norm on Lp(Ω) by/ba∇dbl·/ba∇dblp. The symbol Adenotes the strictly\npositive operator on L2(Ω) defined by A=−△with domain D(A) =H2(Ω)∩H1\n0(Ω). The\nsymbols֒→and֒→֒→standforcontinuousembeddingandcompactembedding, respect ively.\n3The capital letter “C” with a (possibly empty) set of subscripts will d enote a positive\nconstant depending only on its subscripts and may vary from one oc currence to another.\nAnd we write\nΨ(ut(t,x)) =/integraldisplay\nΩK(x,y)ut(t,y)dy.\nIn later sections, we will use the following Sobolev embeddings:\nH1\n0(Ω)֒→L2N\nN−2(Ω), Hs(Ω)֒→L2N\nN−2s(Ω)(s∈(0.1)).\nWe then present some preliminaries.\nDefinition 2.1. [14] Let{S(t)}t≥0be a semigroup on a complete metric space ( X,d). A\nclosed set B ⊆Xis said to be absorbing for {S(t)}t≥0iff for any bounded set B⊆Xthere\nexistst0(B) such that S(t)B⊆ Bfor allt > t0(B). The semigroup {S(t)}t≥0is said to be\ndissipative iff it possesses a bounded absorbing set.\nDefinition 2.2. [14] A compact set A ⊆Xis said to be a global attractor of the dynamical\nsystem (X,{S(t)}t≥0) iff\n(i)A ⊆Xis an invariant set, i.e., S(t)A=Afor allt≥0;\n(ii)A ⊆Xis uniformly attracting, i.e., for all bounded set B⊆Xwe have\nlim\nt→+∞dist(S(t)B,A) = 0.\nHere and below dist( A,B) := supx∈AdistX(x,B) is the Hausdorff\nsemi-distance.\nDefinition 2.3. [14]Adynamical system ( X,{S(t)}t≥0)issaidtobeasymptoticallysmooth\niff for any bounded set Bsuch thatS(t)B⊆Bfort >0 there exists a compact set Kin\nthe closure BofB, such that\nlim\nt→+∞dist(S(t)B,K) = 0.\nLemma 2.4. [14] Let/parenleftbig\nX,{S(t)}t≥0/parenrightbig\nbe a dissipative dynamical system, where the phase\nspaceXis a complete metric space. Then/parenleftbig\nX,{S(t)}t≥0/parenrightbig\npossesses a global attractor if and\nonly if/parenleftbig\nX,{S(t)}t≥0/parenrightbig\nis asymptotically smooth.\nLemma 2.5. [15] Let/parenleftbig\nX,{S(t)}t≥0/parenrightbig\nbe a dynamical system, where the phase space Xis a\ncomplete metric space. Assume that for any bounded positive ly invariant set BinXand\nanyǫ>0there exists T≡T(ǫ,B)such that\nliminf\nm→∞liminf\nn→∞dist(S(T)yn,S(T)ym)≤ǫfor every sequence {yn} ⊆B.(2.1)\nThen/parenleftbig\nX,{S(t)}t≥0/parenrightbig\nis asymptotically smooth.\n4Lemma 2.6. [54] Let(H,(·,·)H)be an inner product space with the induced norm /ba∇dbl · /ba∇dblH\nand constant p>1. Then there exists some positive constant Cpsuch that for any x,y∈H\nsatisfying (x,y)/ne}ationslash= (0,0), we have\n/parenleftbig\n/ba∇dblx/ba∇dblp−2\nHx−/ba∇dbly/ba∇dblp−2\nHy,x−y/parenrightbig\nH≥\n\nCp/ba∇dblx−y/ba∇dblp\nH, p≥2;\nCp/ba∇dblx−y/ba∇dbl2\nH\n(/ba∇dblx/ba∇dblH+/ba∇dbly/ba∇dblH)2−p,1<p<2.(2.2)\nInequality (2 .2), whichwasverifiedfor RNin[43,47]andthenforageneralinnerproduct\nspace in [54], will play a crucial role in our estimate.\nLemma 2.7. [48] Assume X ֒→֒→B ֒→YwhereX, B, Y are Banach spaces. The\nfollowing statements hold.\n(i) LetFbe bounded in Lp(0,T;X)where1≤p<∞, and∂F/∂t={∂f/∂t:f∈F}be\nbounded in L1(0,T;Y), where∂/∂tis the weak time derivative. Then Fis relatively\ncompact in Lp(0,T;B).\n(ii) LetFbe bounded in L∞(0,T;X)and∂F/∂tbe bounded in Lr(0,T;Y)wherer >1.\nThenFis relatively compact in C(0,T;B).\n3. Global well-posedness\nIn this section we discuss the global well-posedness of problem (1 .1)-(1.3). We will use\nthe following definitions of solutions.\nDefinition 3.1. A function u(t)∈C([0,T];H1\n0(Ω))∩C1([0,T];L2(Ω)) withu(0) =u0and\nut(0) =u1is said to be\n(i) strong solution to problem (1 .1)-(1.3) on the interval [0 ,T], iff\n•u∈W1,1(a,b;H1\n0(Ω)) andut∈W1,1(a,b;L2(Ω)) for any 0 <a<b<T ;\n• −∆u(t)+k||ut(t)||p\nL2(Ω)ut(t)∈L2(Ω) for almost all t∈[0,T];\n•equation (1 .1) is satisfied in L2(Ω) for almost all t∈[0,T];\n(ii) generalized solution to problem (1 .1)-(1.3) on the interval [0 ,T], iff there exists a se-\nquence of strong solutions {uj(t)}to problem (1 .1)-(1.3) with initial data ( uj\n0,uj\n1) instead\nof (u0,u1) such that ( uj,uj\nt)→(u,ut) inC([0,T];H1\n0(Ω)×L2(Ω)) asj→+∞;\n(iii) weak solution to problem (1 .1)-(1.3) on the interval [0 ,T], iff\n/integraldisplay\nΩut(t,x)ψ(x)dx=/integraldisplay\nΩu1ψdx+/integraldisplayt\n0/bracketleftbigg/integraldisplay\nΩ×ΩK(x,y)ut(τ,y)ψ(x)dxdy\n+/integraldisplay\nΩh(x)ψ(x)dx−/integraldisplay\nΩ∇u(τ,x)∇ψ(x)dx\n−k||ut(τ)||p/integraldisplay\nΩut(τ,x)ψ(x)dx−/integraldisplay\nΩf(u(τ,x))ψ(x)dx/bracketrightbigg\ndτ(3.1)\nholds for every ψ∈H1\n0(Ω) and for almost all t∈[0,T].\n5In order to prove the global well-posedness for (1 .1)-(1.3), we need the following defini-\ntions and lemmas.\nDefinition 3.2. [46] LetXbe a real Banach space with dual space X∗,F:X→X∗is\nsaid to be monotone if /an}b∇acketle{tF(u)−F(v),u−v/an}b∇acket∇i}ht ≥0,∀u,v∈X, and hemicontinuous if for each\nu,v∈Xthe real-valued function λ→ /an}b∇acketle{tF(u+λv),v/an}b∇acket∇i}htis continuous.\nDefinition 3.3. [46] LetXandYbe Banach spaces. F:X→Yis called demicontinuous\nif it is continuous from Xwith norm convergence to Ywith weak convergence.\nLemma 3.4. [18] LetXbe a real reflexive Banach space, F:X→X∗hemicontinuous,\nmonotone and coercive, i.e./angbracketleftFx,x/angbracketright\n||x||→ ∞as||x|| → ∞. ThenFis ontoX∗.\nLemma 3.5. [46] LetXbe a reflexive Banach space. If F:X→X∗hemicontinuous,\nmonotone and bounded, then it is demicontinuous.\nLemma 3.6. [13] LetA:D(A)⊆H→Hbe a maximal accretive operator on a Hilbert\nspaceH, i.e.,(Ax1−Ax2,x1−x2)H≥0for anyx1,x2∈D(A)andRg(I+A) =H; besides,\nassume that 0∈A0. LetB:H→Hbe a locally Lipschitz. If u0∈D(A),f∈W1,1(0,t;H)\nfor allt>0, then there exists tmax≤+∞such that the initial value problem\nut+Au+Bu∋fandu=u0∈H (3.2)\nhas a unique strong solution uon the interval [0,tmax).\nWhereas, if u0∈D(A),f∈L1(0,t;H)for allt >0, then problem (3.2)has a unique\ngeneralized solution u∈C([0,tmax);H).\nMoreover, in both cases we have lim\nt→tmax/ba∇dblu(t)/ba∇dblH=∞providedtmax<∞.\nWe are now ready to establish the global well-posedness for (1 .1)-(1.3).\nTheorem 3.7. LetT >0be arbitrary. Under Assumption 1.1, we have the following\nstatements.\n(i) For any (u0,u1)∈H1\n0(Ω)×H1\n0(Ω)such that −∆u0+k||u1||pu1∈L2(Ω), there exists\na unique strong solution u(t)to problem (1.1)-(1.3)on[0,T].\n(ii) For every (u0,u1)∈H1\n0(Ω)×L2(Ω)there exists a unique generalized solution, which\nis also the weak solution to problem (1.1)-(1.3).\nProof.We divide our proof into three steps.\nStep1.We first prove local well-posedness of problem (1 .1)-(1.3).\nLetU= (u,v)Twithv=ut. We rewrite (1 .1)-(1.3) as\n/braceleftBigg\nUt+A(U) =B(U), t>0,\nU(0) =U0,(3.3)\n6whereU0= (u0,u1)T,A:D(A)⊆H1\n0(Ω)×L2(Ω)→H1\n0(Ω)×L2(Ω) is given by\nA(U) =/parenleftbigg−v\n−∆u+k||v||pv/parenrightbigg\n,\nU∈D(A) =/braceleftbig\n(u,v)T∈H1\n0(Ω)×H1\n0(Ω)|−∆u+k||v||pv∈L2(Ω)/bracerightbig\nandB:H1\n0(Ω)×L2(Ω)→H1\n0(Ω)×L2(Ω) is given by\nB(U) =/parenleftbigg\n0\nΨ(v)+h(x)−f(u)/parenrightbigg\n,U= (u,v)T∈H1\n0(Ω)×L2(Ω).\nWe note that\nD(A) =H1\n0(Ω)×L2(Ω), (3.4)\nbecause\n/parenleftBig\nH1\n0(Ω)/intersectiondisplay\nH2(Ω)/parenrightBig\n×/parenleftBig\nH1\n0(Ω)/intersectiondisplay\nH2(Ω)/parenrightBig\n⊆D(A)⊆H1\n0(Ω)×L2(Ω)\nand (H1\n0(Ω)/intersectiontextH2(Ω))×(H1\n0(Ω)/intersectiontextH2(Ω)) is dense in H1\n0(Ω)×L2(Ω).\nBy Lemma 2 .6, for every v1,v2inL2(Ω) we have\n(/ba∇dblv1/ba∇dblpv1−/ba∇dblv2/ba∇dblpv2,v1−v2)≥0. (3.5)\nConsequently, we have\n/parenleftbig\nA(U1)−A(U2),U1−U2/parenrightbig\nH1\n0(Ω)×L2(Ω)\n=/parenleftbig\n∇(v2−v1),∇(u1−u2)/parenrightbig\n+/parenleftbig\n∇(u1−u2),∇(v1−v2)/parenrightbig\n+/parenleftbig\nk/ba∇dblv1/ba∇dblpv1−k/ba∇dblv2/ba∇dblpv2,v1−v2/parenrightbig\n≥0,(3.6)\nfor allU1,U2∈D(A), whereU1= (u1,v1)T,U2= (u2,v2)T.\nWe proceed to show that\nRg(I+A) =H1\n0(Ω)×L2(Ω), (3.7)\ni.e., for∀(f0,f1)T∈H1\n0(Ω)×L2(Ω), the equation\n(A+I)(U) =/parenleftbigg\n−v+u\n−∆u+k||v||pv+v/parenrightbigg\n=/parenleftbigg\nf0\nf1/parenrightbigg\n(3.8)\nhas a solution.\nEliminating ufrom (3.8) gives\n−∆v+k/ba∇dblv/ba∇dblpv+v=f1+∆f0∈H−1(Ω). (3.9)\nDefineG:H1\n0(Ω)→H−1(Ω) byG(v) =−∆v+k/ba∇dblv/ba∇dblpv+vfor eachv∈H1\n0(Ω). Obviously,\n7for eachv1,v2∈H1\n0(Ω),\n/angbracketleftbig\nG(v1+λv2),v2/angbracketrightbig\n=/parenleftbig\n∇(v1+λv2),∇v2/parenrightbig\n+(1+k||v1+λv2||p)/parenleftbig\nv1+λv2,v2/parenrightbig\nis a continuous function of real variable λ.\nIt follows from (3 .5) that\n/angbracketleftbig\nG(v1)−G(v2),v1−v2/angbracketrightbig\n=/ba∇dbl∇(v1−v2)/ba∇dbl2+/ba∇dblv1−v2/ba∇dbl2+k/parenleftbig\n/ba∇dblv1/ba∇dblpv1−/ba∇dblv2/ba∇dblpv2,v1−v2/parenrightbig\n≥0\nfor allv1,v2∈H1\n0(Ω).\nMoreover we have\n/angbracketleftbig\nG(v),v/angbracketrightbig\n/ba∇dbl∇v/ba∇dbl=/ba∇dbl∇v/ba∇dbl2+/ba∇dblv/ba∇dbl2+k/ba∇dblv/ba∇dblp+2\n/ba∇dbl∇v/ba∇dbl−→+∞ (3.10)\nas/ba∇dbl∇v/ba∇dbl → ∞.\nIn summary, Gis hemicontinuous, monotone and coercive. Thus, by Lemma 3 .4,G\nis ontoH−1(Ω) and (3.7) follows immediately. Combining (3 .6) and (3.7) meansAis m-\naccretive.\nFor anyu1,u2∈H1\n0(Ω), by (1.4) we have\n/ba∇dblf(u1)−f(u2)/ba∇dbl\n=/braceleftBigg/integraldisplay\nΩ/bracketleftbigg/integraldisplay1\n0f′/parenleftbig\nu2+θ(u1−u2)/parenrightbig/parenleftbig\nu1−u2/parenrightbig\ndθ/bracketrightbigg2\ndx/bracerightBigg1\n2\n≤C/braceleftbigg/integraldisplay\nΩ/parenleftbig\n|u1|4\nN−2+|u2|4\nN−2+1/parenrightbig\n|u1−u2|2dx/bracerightbigg1\n2\n≤C/parenleftbig\n/ba∇dblu1/ba∇dbl2\nN−2\n2N\nN−2+/ba∇dblu2/ba∇dbl2\nN−2\n2N\nN−2+1/parenrightbig\n/ba∇dblu1−u2/ba∇dbl2N\nN−2\n≤C/parenleftbig\n/ba∇dbl∇u1/ba∇dbl2\nN−2+/ba∇dbl∇u2/ba∇dbl2\nN−2+1/parenrightbig\n/ba∇dbl∇(u1−u2)/ba∇dbl.(3.11)\nMoreover, using the H¨ older inequality, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nΩK(x,y)/parenleftbig\nv1(y)−v2(y)/parenrightbig\ndy/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤ /ba∇dblK/ba∇dblL2(Ω×Ω)||v1−v2|| (3.12)\nfor anyv1,v2∈L2(Ω).\nIt follows from (3 .11) and (3.12) thatBis locally Lipschitz.\nNow that we have proved that Ais m-accretive, Bis locally Lipschitz and D(A) =\nH1\n0(Ω)×L2(Ω), by Lemma 3 .6, we conclude that there exists tmax≤+∞such that problem\n(1.1)-(1.3) has a unique strong solution uon [0,tmax) for every ( u0,u1)∈D(A) and it has\na unique generalized solution uon [0,tmax) for every ( u0,u1)∈H1\n0(Ω)×L2(Ω), moreover\n[0,tmax) is the maximal interval on which the solution exists. Furthermore, for both the\n8strong solution and the generalized solution we have\nlim\nt→tmax/ba∇dbl(u,ut)/ba∇dblH1\n0(Ω)×L2(Ω)=∞,providedtmax<+∞. (3.13)\nStep2.Next, we will prove the global well-posedness of problem (1 .1)-(1.3).\nWe denote\nE(u(t),ut(t)) =1\n2/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2/parenrightBig\n+/integraldisplay\nΩF(u)dx−/integraldisplay\nΩhudx.\nChooseµ0∈R+∩(−µ,λ1). By (1.5), there exists M >0 such that\nf′(s)>−µ0,|s|>M.\nIt follows that \n\nF(s)≥ −λ1+µ0\n4s2−C,|s|>M,\n|F(s)| ≤C, |s| ≤M.\nConsequently,\n/integraldisplay\nΩF(u)dx≥/integraldisplay\nΩ1/parenleftBig\n−λ1+µ0\n4u2−C/parenrightBig\ndx+/integraldisplay\nΩ2F(u)dx\n≥ −λ1+µ0\n4/integraldisplay\nΩu2dx−C1,(3.14)\nwhere Ω 1=/braceleftbig\nx∈Ω :|u(x)|> M/bracerightbig\n, Ω2=/braceleftbig\nx∈Ω :|u(x)| ≤M/bracerightbig\nandC1is some positive\nconstant.\nIt is easy to get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩhudx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n16/parenleftbigg\n1−µ0\nλ1/parenrightbigg\n/ba∇dbl∇u/ba∇dbl2+C. (3.15)\nBy Poincar´ e’s inequality we have\n/ba∇dbl∇u/ba∇dbl2≥λ1/ba∇dblu/ba∇dbl2. (3.16)\nBy (3.14)-(3.16) we have\nE(u(t),ut(t))\n≥1\n2/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2/parenrightBig\n−λ1+µ0\n4/integraldisplay\nΩu2dx−C1−1\n8/parenleftbigg\n1−µ0\nλ1/parenrightbigg\n/ba∇dbl∇u/ba∇dbl2−C\n≥1\n8/parenleftbigg\n1−µ0\nλ1/parenrightbigg/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2/parenrightBig\n−C.(3.17)\n9We deduce from (1 .4) that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF(u)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\nΩC(1+|u|2N−2\nN−2)dx\n≤C/parenleftBig\n/ba∇dbl∇u/ba∇dbl2N−2\nN−2+1/parenrightBig\n.(3.18)\nUsing (3.15) and (3.18) we obtain\nE(u(t),ut(t))≤C/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2N−2\nN−2+1/parenrightBig\n. (3.19)\nMultiplying (1.1) by utand integrating on Ω yields\nd\ndtE(u(t),ut(t)) =−k/ba∇dblut/ba∇dblp+2+/integraldisplay\nΩΨ(ut)utdx (3.20)\nfort∈[0,tmax).\nUsing Young’s inequality with ǫ, we deduce from (3 .20) that\nd\ndtE(u(t),ut(t))≤ −k/ba∇dblut/ba∇dblp+2+/ba∇dblK/ba∇dblL2(Ω×Ω)/ba∇dblut/ba∇dbl2\n≤ −k/ba∇dblut/ba∇dblp+2+k\n2/ba∇dblut/ba∇dblp+2+C\n≤C(3.21)\nfort∈[0,tmax).\nIntegrating (3 .21), we have\nE(u(t),ut(t))≤ E(u0,u1)+Ct. (3.22)\nIftmax<+∞, we duduce from (3 .17),(3.19) and (3.22) that\n/ba∇dbl(u(t),ut(t))/ba∇dbl2\nH1\n0(Ω)×L2(Ω)≤C/parenleftBig\n/ba∇dblu1/ba∇dbl2+/ba∇dbl∇u0/ba∇dbl2+/ba∇dbl∇u0/ba∇dbl2N−2\nN−2+1+tmax/parenrightBig\n<+∞.(3.23)\nBythedefinition ofthegeneralized solution, it iseasy toverify that ( 3.23) isalso trueforthe\ngeneralized solution. Thus by (3 .13), we have proved the global existence and uniqueness\nof the strong solution as well as the generalized solution.\nStep3.Finally, we will verify that the generalized solution to problem (1 .1)-(1.3) is also\nweak.\nObviously, the strong solution to problem (1 .1)-(1.3) is also weak.\nLetu(t) bethe generalized solution to problem (1 .1)-(1.3), then by definition there exists\nasequence ofstrongsolutions {uj(t)}to problem(1 .1)-(1.3)withinitial data( uj\n0,uj\n1) instead\nof (u0,u1) such that\n(uj,uj\nt)→(u,ut) inC([0,T];H1\n0(Ω)×L2(Ω)) asj→+∞. (3.24)\n10We have\n/integraldisplay\nΩuj\nt(t,x)ψ(x)dx=/integraldisplay\nΩuj\n1ψdx+/integraldisplayt\n0/bracketleftbigg/integraldisplay\nΩ×ΩK(x,y)uj\nt(τ,y)ψ(x)dxdy\n+/integraldisplay\nΩh(x)ψ(x)dx−/integraldisplay\nΩ∇uj(τ,x)∇ψ(x)dx\n−k||uj\nt(τ)||p/integraldisplay\nΩuj\nt(τ,x)ψ(x)dx−/integraldisplay\nΩf(uj(τ,x))ψ(x)dx/bracketrightbigg\ndτ(3.25)\nholds for every ψ∈H1\n0(Ω) and for almost all t∈[0,T].\nDefineD:L2(Ω)→L2(Ω) byG(v) =/ba∇dblv/ba∇dblpvfor eachv∈L2(Ω). Inequality (3 .5)\nindicates that Dis accretive. Besides, it is apparent that Dis hemicontinuous and bounded.\nConsequently, due to Lemma 3 .5,Dis demicontinuous. Thus, we have\n||uj\nt(τ)||p/integraldisplay\nΩuj\nt(τ,x)ψ(x)dx→ ||ut(τ)||p/integraldisplay\nΩut(τ,x)ψ(x)dxasj→+∞.(3.26)\nSince by (3 .24) there exists J∈Nsuch that max\nτ∈[0,T]/ba∇dbluj\nt(τ)/ba∇dbl ≤max\nτ∈[0,T]/ba∇dblut(τ)/ba∇dbl+ 1 for every\nj≥J, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle||uj\nt(τ)||p/integraldisplay\nΩuj\nt(τ,x)ψ(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbigg\nmax\nτ∈[0,T]/ba∇dblut(τ)/ba∇dbl+1/parenrightbiggp+1\n/ba∇dblψ/ba∇dbl ≤C. (3.27)\nBy the Lebesgue convergence theorem, it follows from (3 .26) and (3.27) that\nlim\nj→+∞/integraldisplayt\n0/bracketleftbigg\n||uj\nt(τ)||p/integraldisplay\nΩuj\nt(τ,x)ψ(x)dx/bracketrightbigg\ndτ=/integraldisplayt\n0/bracketleftbigg\n||ut(τ)||p/integraldisplay\nΩuj\nt(τ,x)ψ(x)dx/bracketrightbigg\ndτ.(3.28)\nLettingj→+∞in (3.25) and using (3 .24) and (3.28), we see that u(t) satisfies (3 .1), which\ncompletes the proof.\nBy Theorem 3 .7, problem (1 .1)-(1.3) generates an evolution semigroup {S(t)}t≥0in the\nspaceH1\n0(Ω)×L2(Ω) by the formula S(t)(u0,u1) = (u(t),ut(t)), where (u0,u1)∈H1\n0(Ω)×\nL2(Ω) andu(t) is the weak solution to problem (1 .1)-(1.3).\n4. Dissipativity\nIn this section, we will prove the dissipativity of the dynamical syste m generated by the\nweak solution to problem (1 .1)-(1.3), which is a necessary condition for the existence of the\nglobal attractor.\nTheorem 4.1. Under Assumption 1.1, the dynamical system/parenleftbig\nH1\n0(Ω)×L2(Ω),\n{S(t)}t≥0/parenrightbig\ngenerated by the weak solution of problem (1.1)-(1.3)is dissipative, i.e., there\nexistsR>0satisfying the property: for any bounded set BinH1\n0(Ω)×L2(Ω), there exists\nt0(B)such that /ba∇dblS(t)y/ba∇dblH1\n0(Ω)×L2(Ω)≤Rfor ally∈Bandt≥t0(B).\n11Proof.Chooseµ0∈R+∩(−µ,λ1). By (1.5), there exists M >0 such that\nf′(s)>−µ0,|s|>M. (4.1)\nIt follows that \n\nF(s)≥ −λ1+µ0\n4s2−C,|s|>M;\n|F(s)| ≤C, |s| ≤M.\nConsequently,\n/integraldisplay\nΩF(u)dx≥/integraldisplay\nΩ1/parenleftBig\n−λ1+µ0\n4u2−C/parenrightBig\ndx+/integraldisplay\nΩ2F(u)dx\n≥ −λ1+µ0\n4/integraldisplay\nΩu2dx−C1,(4.2)\nwhere Ω 1=/braceleftbig\nx∈Ω :|u(x)|> M/bracerightbig\n, Ω2=/braceleftbig\nx∈Ω :|u(x)| ≤M/bracerightbig\nandC1is some positive\nconstant.\nLet\nVǫ(t) =1\n2/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2/parenrightBig\n+/integraldisplay\nΩF(u)dx−/integraldisplay\nΩhudx+ǫ/integraldisplay\nΩutudx.\nSince by Poincar´ e’s inequality we have\n/ba∇dbl∇u/ba∇dbl2≥λ1/ba∇dblu/ba∇dbl2, (4.3)\nthere exists ǫ0>0 such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleǫ/integraldisplay\nΩutudx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n16/parenleftbigg\n1−µ0\nλ1/parenrightbigg/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2/parenrightBig\n(4.4)\nholds for all ǫ≤ǫ0.\nHereafter we assume ǫ∈(0,ǫ0).\nWe also have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩhudx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n16/parenleftbigg\n1−µ0\nλ1/parenrightbigg\n/ba∇dbl∇u/ba∇dbl2+C. (4.5)\nWe deduce from (1 .4) that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF(u)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/parenleftBig\n/ba∇dbl∇u/ba∇dbl2N−2\nN−2+1/parenrightBig\n. (4.6)\nWe deduce from (4 .2)-(4.6) that\n1\n8/parenleftbigg\n1−µ0\nλ1/parenrightbigg/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2/parenrightBig\n−C≤Vǫ(t)≤C/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2N−2\nN−2+1/parenrightBig\n.(4.7)\n12Multiplying (1.1) by ut+ǫuand integrating on Ω yields\nd\ndtVǫ(t) =−k/ba∇dblut/ba∇dblp+2+/integraldisplay\nΩΨ(ut)utdx+ǫ/bracketleftbigg\n−/ba∇dbl∇u/ba∇dbl2+/ba∇dblut/ba∇dbl2\n−k/ba∇dblut/ba∇dblp/integraldisplay\nΩutudx−/integraldisplay\nΩf(u)udx+/integraldisplay\nΩΨ(ut)udx+/integraldisplay\nΩhudx/bracketrightbigg\n.(4.8)\nWe estimate the terms on the right hand side of identity (4 .8) as follows:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−k/ba∇dblut/ba∇dblp/integraldisplay\nΩutudx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤kC/ba∇dblut/ba∇dblp+1/ba∇dbl∇u/ba∇dbl\n≤Ck/parenleftBig\n/ba∇dblut/ba∇dblp+1/ba∇dbl∇u/ba∇dblp\np+2/parenrightBigp+2\np+1+1\n12/parenleftbigg\n1−µ0\nλ1/parenrightbigg\n/ba∇dbl∇u/ba∇dbl2\np+2(p+2)\n=Ck/ba∇dblut/ba∇dblp+2/ba∇dbl∇u/ba∇dblp\np+1+1\n12/parenleftbigg\n1−µ0\nλ1/parenrightbigg\n/ba∇dbl∇u/ba∇dbl2;(4.9)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩΨ(ut)udx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n12/parenleftbigg\n1−µ0\nλ1/parenrightbigg\n/ba∇dbl∇u/ba∇dbl2+C/ba∇dblut/ba∇dbl2; (4.10)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩΨ(ut)utdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dblK/ba∇dblL2(Ω×Ω)/ba∇dblut/ba∇dbl2. (4.11)\nWe infer from (4 .1) that\nF(s)≤f(s)s+µ0\n2s2+C,|s|>M. (4.12)\nBy (4.3) and (4.12) we have\n−/integraldisplay\nΩf(u)udx≤ −/parenleftBig/integraldisplay\nΩF(u)dx+λ1+µ0\n4/integraldisplay\nΩu2dx+C1/parenrightBig\n+1\n4/parenleftbigg3µ0\nλ1+1/parenrightbigg\n/ba∇dbl∇u/ba∇dbl2+C.(4.13)\nUsing (4.2), (4.5), (4.7), (4.9), (4.10), (4.11), (4.13) and Young’s inequality with ǫ, we\ndeduce from (4 .8) that\nd\ndtVǫ(t)\n≤−k/ba∇dblut/ba∇dblp+2/parenleftBig\n1−Cǫ/ba∇dbl∇u/ba∇dblp\np+1/parenrightBig\n+k\n2/ba∇dblut/ba∇dblp+2+C\n−ǫ/bracketleftbigg1\n2/parenleftbigg\n1−µ0\nλ1/parenrightbigg/parenleftBig\n/ba∇dblut/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2/parenrightBig\n+/parenleftBig/integraldisplay\nΩF(u)dx+λ1+µ0\n4/integraldisplay\nΩu2dx+C1/parenrightBig/bracketrightbigg\n≤−k/ba∇dblut/ba∇dblp+2/bracketleftbigg1\n2−Cǫ/parenleftBig\nVǫ(t)+C/parenrightBigp\n2(p+1)/bracketrightbigg\n−2\n3/parenleftbigg\n1−µ0\nλ1/parenrightbigg\nǫVǫ(t)+C.(4.14)\n13Integrating (4 .14) fromstotand rescaling ǫ, we have\nVǫ(t)≤e−ǫ(t−s)Vǫ(s)+C\nǫ−/integraldisplayt\nse−ǫ(t−τ)k/ba∇dblut/ba∇dblp+2/bracketleftbigg1\n2−Cǫ/parenleftBig\nVǫ(τ)+C/parenrightBigp\n2(p+1)/bracketrightbigg\ndτ(4.15)\nfor allt≥s≥0.\nInequality (4 .15) is exactly formula (3 .44) in Theorem 3.15 in [14] with b(·) =Cand\nγ=p\n2(p+1), and thus Theorem 3.15 in [14] can be directly applied to gain the ultimat e\ndissipativity of the dynamical system generated by the problem (1 .1)-(1.3).\n5. The existence of the global attractor\nHaving verified the dissipativity, by Lemma 2 .4, in order to establish the existence of the\nglobal attractor, we only need to prove that the system is asympt otically smooth. Further,\nby Lemma 2 .5, it is sufficient to verify inequality (2 .1). This is exactly what we do when\nproving the following theorem.\nTheorem 5.1. Under Assumption 1.1, the dynamical system generated by the weak solution\nof problem (1.1)-(1.3)possesses a global attractor.\nProof.LetBbe a positively invariant bounded set in H1\n0(Ω)×L2(Ω).\nFor any sequence/braceleftbig\n(u(n)\n0,u(n)\n1)/bracerightbig∞\nn=1inB, we set S(t)(u(n)\n0,u(n)\n1) =/parenleftbig\nu(n)(t),\nu(n)\nt(t)/parenrightbig\n. It follows from the positive invariance property of Bthat\n/vextenddouble/vextenddouble/parenleftbig\nu(n)(t),u(n)\nt(t)/parenrightbig/vextenddouble/vextenddouble\nH1\n0(Ω)×L2(Ω)≤CB,∀t>0,n∈N. (5.1)\nWrite\nEn,m(t) =1\n2/bracketleftBig\n/ba∇dbl∇(u(n)(t)−u(m)(t))/ba∇dbl2+/ba∇dblu(n)\nt(t)−u(m)\nt(t)/ba∇dbl2/bracketrightBig\n.\nStep 1. We first estimate En,m(T).\nThe difference u(n)−u(m)satisfies\nu(n)\ntt−u(m)\ntt−△(u(n)−u(m))+k/ba∇dblu(n)\nt/ba∇dblpu(n)\nt−k/ba∇dblu(m)\nt/ba∇dblpu(m)\nt\n=−f(u(n))+f(u(m))+Ψ(u(n)\nt−u(m)\nt).(5.2)\nMultiplying (5 .2) by (u(n)\nt(t)−u(m)\nt(t)) inL2(Ω) and then integrating from ttoT, we obtain\nEn,m(T)\n=En,m(t)+/integraldisplayT\nt/integraldisplay\nΩ/bracketleftBig/parenleftbig\nΨ(u(n)\nt(τ)−u(m)\nt(τ))/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig\n−/parenleftbig\nf(u(n)(τ))−f(u(m)(τ))/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig\n−/parenleftbig\nk/ba∇dblu(n)\nt(τ)/ba∇dblpu(n)\nt(τ)−k/ba∇dblu(m)\nt(τ)/ba∇dblpu(m)\nt(τ)/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig/bracketrightBig\ndxdτ.(5.3)\n14Integrating (5 .3) with respect to tbetween 0 and Tgives\nT·En,m(T)\n=/integraldisplayT\n0En,m(t)dt+/integraldisplayT\n0/integraldisplayT\nt/integraldisplay\nΩ/bracketleftBig/parenleftbig\nΨ(u(n)\nt(τ)−u(m)\nt(τ))/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig(5.4)\n−/parenleftbig\nf(u(n)(τ))−f(u(m)(τ))/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig\n−/parenleftbig\nk/ba∇dblu(n)\nτ(τ)/ba∇dblpu(n)\nt(τ)−k/ba∇dblu(m)\nt(τ)/ba∇dblpu(m)\nt(τ)/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig/bracketrightBig\ndxdτdt.\nMultiplying (5 .2) by (u(n)(t)−u(m)(t)) inL2(Ω) and then integrating from 0 to T, we obtain\n/integraldisplayT\n0En,m(t)dt\n=−1\n2/bracketleftbigg/integraldisplay\nΩ/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig\ndx/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleT\n0\n+/integraldisplayT\n0/ba∇dblu(n)\nt(t)−u(m)\nt(t)/ba∇dbl2dt\n+1\n2/integraldisplayT\n0/integraldisplay\nΩ/bracketleftBig/parenleftbig\nΨ(u(n)\nt(t)−u(m)\nt(t))/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig\n−/parenleftbig\nf(u(n)(t))−f(u(m)(t))/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig\n−/parenleftbig\nk/ba∇dblu(n)\nt(t)/ba∇dblpu(n)\nt(t)−k/ba∇dblu(m)\nt(t)/ba∇dblpu(m)\nt(t)/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig/bracketrightBig\ndxdt.(5.5)\nBy Lemma 2 .6,\n/integraldisplayT\n0/integraldisplayT\nt/integraldisplay\nΩ/parenleftbig\nk/ba∇dblu(n)\nτ(τ)/ba∇dblpu(n)\nt(τ)−k/ba∇dblu(m)\nt(τ)/ba∇dblpu(m)\nt(τ)/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig\ndxdτdt≥0.(5.6)\nLet 0<s<1. We infer from (5 .1) that\n−1\n2/bracketleftbigg/integraldisplay\nΩ/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig\ndx/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleT\n0≤CB (5.7)\nand\n/integraldisplayT\n0/integraldisplay\nΩ−/parenleftbig\nk/ba∇dblu(n)\nt(t)/ba∇dblpu(n)\nt(t)−k/ba∇dblu(m)\nt(t)/ba∇dblpu(m)\nt(t)/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig\ndxdt\n≤k/integraldisplayT\n0/parenleftbig\n/ba∇dblu(n)\nt(t)/ba∇dblp+1+/ba∇dblu(m)\nt(t)/ba∇dblp+1/parenrightbig\n·/ba∇dblu(n)(t)−u(m)(t)/ba∇dbldt\n≤TCBsup\nt∈[0,T]/ba∇dblu(n)(t)−u(m)(t)/ba∇dbl\n≤TCBsup\nt∈[0,T]/ba∇dblu(n)(t)−u(m)(t)/ba∇dblHs(Ω).(5.8)\n15By (1.4) and (5.1), we have\n/ba∇dblf(u(n)(t))−f(u(m)(t))/ba∇dbl\n=/braceleftBigg/integraldisplay\nΩ/bracketleftbigg/integraldisplay1\n0f′/parenleftbig\nu(m)(t)+θ(u(n)(t)−u(m)(t)/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig\ndθ/bracketrightbigg2\ndx/bracerightBigg1\n2\n≤C/braceleftbigg/integraldisplay\nΩ/parenleftbig\n|u(n)(t)|4\nN−2+|u(m)(t)|4\nN−2+1/parenrightbig\n|u(n)(t)−u(m)(t)|2dx/bracerightbigg1\n2\n≤C/parenleftbig\n/ba∇dblu(n)(t)/ba∇dbl2\nN−2\n2N\nN−2+/ba∇dblu(m)(t)/ba∇dbl2\nN−2\n2N\nN−2+1/parenrightbig\n/ba∇dblu(n)(t)−u(m)(t))/ba∇dbl2N\nN−2(5.9)\n≤C/parenleftbig\n/ba∇dbl∇u(n)(t)/ba∇dbl2\nN−2+/ba∇dbl∇u(m)(t)/ba∇dbl2\nN−2+1/parenrightbig\n/ba∇dbl∇(u(n)(t)−u(m)(t)/ba∇dbl\n≤CB.\nConsequently,\n/integraldisplayT\n0/integraldisplay\nΩ−/parenleftbig\nf(u(n)(t))−f(u(m)(t))/parenrightbig/parenleftbig\nu(n)(t)−u(m)(t)/parenrightbig\ndxdt\n≤/integraldisplayT\n0/ba∇dblf(u(n)(t))−f(u(m)(t))/ba∇dbl·/ba∇dblu(n)(t)−u(m)(t)/ba∇dbldt\n≤TCBsup\nt∈[0,T]/ba∇dblu(n)(t)−u(m)(t)/ba∇dbl\n≤TCBsup\nt∈[0,T]/ba∇dblu(n)(t)−u(m)(t)/ba∇dblHs(Ω).(5.10)\nBy Lemma 2 .6, for anyǫ>0, we have\n/ba∇dblu(n)\nt(t)−u(m)\nt(t)/ba∇dbl2\n≤ǫ\n2+Cǫ/ba∇dblu(n)\nt(t)−u(m)\nt(t)/ba∇dblp+2\n≤ǫ\n2+Cǫk/integraldisplay\nΩ/parenleftbig\n/ba∇dblu(n)\nt(t)/ba∇dblpu(n)\nt(t)−/ba∇dblu(m)\nt(t)/ba∇dblpu(m)\nt(t)/parenrightbig/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig\ndx.(5.11)\n16We deduce from (5 .1), (5.3) and (5.11) that\n/integraldisplayT\n0/ba∇dblu(n)\nt(t)−u(m)\nt(t)/ba∇dbl2dt\n≤ǫ\n2T+Cǫ/braceleftbigg\nEn,m(0)−En,m(T)\n+/integraldisplayT\n0/integraldisplay\nΩ/bracketleftBig/parenleftbig\nΨ(u(n)\nt(t)−u(m)\nt(t))/parenrightbig/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig\n−/parenleftbig\nf(u(n)(t))−f(u(m)(t))/parenrightbig/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig/bracketrightBig\ndxdt/bracerightbigg\n≤ǫ\n2T+Cǫ,B+Cǫ/integraldisplayT\n0/integraldisplay\nΩ/bracketleftBig/parenleftbig\nΨ(u(n)\nt(t)−u(m)\nt(t))/parenrightbig/parenleftbig\nu(n)\nt(t)−\nu(m)\nt(t)/parenrightbig\n−/parenleftbig\nf(u(n)(t))−f(u(m)(t))/parenrightbig/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig/bracketrightBig\ndxdt.(5.12)\nPlugging (5 .5), (5.6), (5.7), (5.8), (5.10) and (5.12) into (5 .4), we obtain\nEn,m(T)\n≤Cǫ,B\nT+ǫ\n2+CBsup\nt∈[0,T]/ba∇dblu(n)(t)−u(m)(t)/ba∇dblHs(Ω)\n+Cǫ,B1+T\nT/integraldisplayT\n0/ba∇dblΨ(u(n)\nt(t)−u(m)\nt(t))/ba∇dbldt\n+Cǫ\nT/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nΩ/parenleftbig\nf(u(n)(t))−f(u(m)(t))/parenrightbig/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig\ndxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplayT\nt/integraldisplay\nΩ/parenleftBig\nf(u(n)(τ))−f(u(m)(τ))/parenrightbig/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig\ndxdτdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightBigg\n.(5.13)\nStep 2. Next, we will investigate some convergence properties of the term s on the right in\n(5.13).\nBy Alaoglu’s theorem and Lemma 2 .7, we deduce from (5 .1) andH1\n0(Ω)֒→֒→Hs(Ω)֒→\nL2(Ω) that there exists a subsequence of/braceleftbig\n(u(n),u(n)\nt)/bracerightbig∞\nn=1, still denoted by/braceleftbig\n(u(n),u(n)\nt)/bracerightbig∞\nn=1,\nsuch that\n/braceleftBigg\n(u(n),u(n)\nt)∗⇀(u,v) inL∞(0,T;H1\n0(Ω)×L2(Ω)),\nu(n)→winC([0,T];Hs(Ω)),asn→ ∞. (5.14)\nMoreover, we can verify that v=utandw=u. Indeed, by (5 .14), for any φ(s)∈\n17C∞\nc[0,t] and anyψ0(x)∈H2(Ω)∩H1\n0(Ω), we have\n/integraldisplayt\n0/parenleftbig\nu(n)\nt(s),φ(s)△ψ0(x)/parenrightbig\nds\n=/integraldisplayt\n0φ(s)d\ndt/parenleftbig\nu(n)(s),△ψ0(x)/parenrightbig\nds\n=−/integraldisplayt\n0φ′(s)/parenleftbig\nu(n)(s),△ψ0(x)/parenrightbig\nds\n=/integraldisplayt\n0/parenleftbig\n∇u(n)(s),φ′(s)∇ψ0(x)/parenrightbig\nds\n−→/integraldisplayt\n0/parenleftbig\n∇u(s),φ′(s)∇ψ0(x)/parenrightbig\nds\n=/integraldisplayt\n0/parenleftbig\nut(s),φ(s)△ψ0(x)/parenrightbig\nds\nand /integraldisplayt\n0/parenleftbig\nu(n)\nt(s),φ(s)△ψ0(x)/parenrightbig\nds−→/integraldisplayt\n0/parenleftbig\nv(s),φ(s)△ψ0(x)/parenrightbig\nds\nasn→ ∞. It follows that v=ut.\nSince\n/integraldisplayT\n0/parenleftbig\n∇(u(n)(t)−w),∇ϕ/parenrightbig\ndt=/integraldisplayT\n0/parenleftbig\nAs\n2(u(n)(t)−w),A1−s\n2ϕ/parenrightbig\ndt\n≤sup\nt∈[0,T]/ba∇dblu(n)(t)−w/ba∇dblHs(Ω)/integraldisplayT\n0/ba∇dblϕ/ba∇dblH2−s(Ω)dt\nholds for any ϕ∈L1(0,T;H2−s(Ω)), by (5 .14), we have/integraltextT\n0/parenleftbig\n∇(u(n)(t)−w),∇ϕ/parenrightbig\ndt\n→0 asn→ ∞, which together with (5 .14) givesw=u.\nLetVbe the completion of L2(Ω) with respect to the norm /ba∇dbl · /ba∇dblVgiven by /ba∇dbl · /ba∇dblV=\n/ba∇dblΨ(·)/ba∇dbl+/ba∇dblA−1\n2·/ba∇dblandWbe the completion of L2(Ω) with respect to the norm /ba∇dbl·/ba∇dblWgiven\nby/ba∇dbl·/ba∇dblW=/ba∇dblA−1\n2·/ba∇dbl. SinceK∈L2(Ω×Ω), we have\nL2(Ω)֒→֒→V ֒→W. (5.15)\nReplacingu(m)(t) by 0 in (5 .9) gives/ba∇dblf(u(n)(t))−f(0)/ba∇dbl ≤CB, i.e.,/ba∇dblf(u(n)(t))/ba∇dbl ≤CB. In\naddition, it is easy to get\n/ba∇dblΨ(u(n)\nt(t))/ba∇dbl ≤ /ba∇dblK/ba∇dblL2(Ω×Ω)/ba∇dblu(n)\nt(t)/ba∇dbl ≤CB.\n18Therefore, from (1 .1) we get\n/ba∇dblA−1\n2u(n)\ntt(t)/ba∇dbl\n≤/ba∇dbl∇u(n)(t)/ba∇dbl+k/ba∇dblu(n)\nt(t)/ba∇dblp/ba∇dblA−1\n2u(n)\nt(t)/ba∇dbl+/ba∇dblA−1\n2/parenleftbig\nΨ(u(n)\nt(t))+h−f(u(n)(t))/parenrightbig\n/ba∇dbl\n≤CB.\nConsequently,/integraldisplayT\n0/ba∇dblA−1\n2u(n)\ntt(t)/ba∇dbldt≤CB,T. (5.16)\nBesides, we have/integraldisplayT\n0/ba∇dblu(n)\nt(t)/ba∇dbldt≤CB,T. (5.17)\nBy Lemma 2 .7, (5.15)-(5.17) imply that/braceleftbig\nu(n)\nt(t)/bracerightbig∞\nn=1is relatively compact in L1(0,\nT;V). Thus there exists a subsequence of/braceleftbig\n(u(n),u(n)\nt)/bracerightbig∞\nn=1(still denoted by itself) such\nthat\nlim\nn,m→∞/integraldisplayT\n0/ba∇dblΨ/parenleftbig\nu(n)\nt(t)−u(m)\nt(t)/parenrightbig\n/ba∇dbldt= 0. (5.18)\nIn addition, it follows from (5 .14) that\nlim\nn,m→∞sup\nt∈[0,T]/ba∇dblu(n)(t)−u(m)(t)/ba∇dblHs(Ω)= 0, (5.19)\nwhich together with (5 .18) gives\nI1≡liminf\nn→∞liminf\nm→∞/bracketleftbig\nCBsup\nt∈[0,T]/ba∇dblu(n)(t)−u(m)(t)/ba∇dblHs(Ω)\n+Cǫ,B1+T\nT/integraldisplayT\n0/ba∇dblΨ(u(n)\nt(t)−u(m)\nt(t))/ba∇dbldt/bracketrightbig\n=0.(5.20)\n19LetF(µ) =/integraldisplayµ\n0f(τ)dτ. By (1.4) and (5.1),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF(u(n)(t))dx−/integraldisplay\nΩF(u(t))dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay\nΩ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0f/parenleftbig\nu(t)+θ(u(n)(t)−u(t))/parenrightbig\n·(u(n)(t)−u(t))dθ/vextendsingle/vextendsingle/vextendsingle/vextendsingledx\n≤C/integraldisplay\nΩ(|u(n)(t)|N\nN−2+|u(t)|N\nN−2+1)·|u(n)(t)−u(t)|dx\n≤C/ba∇dblu(n)(t)−u(t)/ba∇dbl·/parenleftbig\n1+/ba∇dblu(n)(t)/ba∇dblN\nN−2\n2N\nN−2+/ba∇dblu(t)/ba∇dblN\nN−2\n2N\nN−2/parenrightbig\n≤C/ba∇dblu(n)(t)−u(t)/ba∇dblHs(Ω)/parenleftbig\n1+/ba∇dbl∇u(n)(t)/ba∇dblN\nN−2+/ba∇dbl∇u(t)/ba∇dblN\nN−2/parenrightbig\n≤CB/ba∇dblu(n)(t)−u(t)/ba∇dblHs(Ω)(5.21)\nholds for all t≥0.\nCombining (5 .14) and (5.21) gives\n/integraldisplay\nΩF(u(n)(t))dx⇒/integraldisplay\nΩF(u(t))dxasn→ ∞. (5.22)\nIt follows from HN(Ω)֒→L∞(Ω) thatL1(Ω)֒→(L∞(Ω))∗֒→H−N(Ω). Hence we deduce\nfrom (1.4) and (5.1) that\n/ba∇dblA−N\n2f(u(n)(t))−A−N\n2f(u(t))/ba∇dbl\n=/ba∇dblf(u(n)(t))−f(u(t))/ba∇dblH−N(Ω)\n≤C/ba∇dblf(u(n)(t))−f(u(t))/ba∇dbl1\n≤C/integraldisplay\nΩ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0f′/parenleftbig\nθu(n)(t)+(1−θ)u(t)/parenrightbig/parenleftbig\nu(n)(t)−u(t)/parenrightbig\ndθ/vextendsingle/vextendsingle/vextendsingle/vextendsingledx\n≤C/integraldisplay\nΩ(|u(n)(t)|2\nN−2+|u(t)|2\nN−2+1)·|u(n)(t)−u(t)|dx\n≤C/ba∇dblu(n)(t)−u(t)/ba∇dbl·(1+/ba∇dblu(n)(t)/ba∇dbl2\nN−2\n4\nN−2+/ba∇dblu(t)/ba∇dbl2\nN−2\n4\nN−2)\n≤C/ba∇dblu(n)(t)−u(t)/ba∇dblHs(Ω)(1+/ba∇dbl∇u(n)(t)/ba∇dbl2\nN−2+/ba∇dbl∇u(t)/ba∇dbl2\nN−2)\n≤CB/ba∇dblu(n)(t)−u(t)/ba∇dblHs(Ω)(5.23)\nholds for all t≥0.\nCombining (5 .14) and (5.23) gives\nsup\nt∈[0,T]/ba∇dblA−N\n2/parenleftbig\nf(u(n)(t))−f(u(t))/parenrightbig\n/ba∇dbl −→0 asn→ ∞. (5.24)\n20For each fixed t∈[0,T] and eachϕ∈L1/parenleftbig\n0,T;HN(Ω)∩H1\n0(Ω)/parenrightbig\n, we have\n/integraldisplayT\nt/parenleftbig\nf(u(n)(τ))−f(u(τ)),ϕ/parenrightbig\ndτ\n=/integraldisplayT\nt/parenleftbig\nA−N\n2(f(u(n)(τ))−f(u(τ))),AN\n2ϕ/parenrightbig\ndτ\n≤sup\nτ∈[0,T]/ba∇dblA−N\n2/parenleftbig\nf(u(n)(τ))−f(u(τ))/parenrightbig\n/ba∇dbl/integraldisplayT\n0/ba∇dblϕ/ba∇dblHN(Ω)dτ,\nwhich, together with (5 .24), gives\n/integraldisplayT\nt/parenleftbig\nf(u(n)(τ))−f(u(τ)),ϕ/parenrightbig\ndτ−→0 asn→ ∞. (5.25)\nSinceL1/parenleftbig\nt,T;HN(Ω)∩H1\n0(Ω)/parenrightbig\nis dense in L1/parenleftbig\nt,T;L2(Ω)/parenrightbig\n, (5.25) implies\nf(u(n))∗⇀f(u) inL∞/parenleftbig\nt,T;L2(Ω)/parenrightbig\nasn→ ∞. (5.26)\nBy (5.14), we have\n(u(n),u(n)\nt)∗⇀(u,ut) inL∞/parenleftbig\nt,T;H1\n0(Ω)×L2(Ω)/parenrightbig\nasn→ ∞. (5.27)\nFrom (5.26) and (5.27), we obtain\nlim\nn→∞lim\nm→∞/integraldisplayT\nt/integraldisplay\nΩf(u(n)(τ))u(m)\nt(τ)dxdτ (5.28)\n= lim\nn→∞/integraldisplayT\nt/integraldisplay\nΩf(u(n)(τ))ut(τ)dxdτ\n=/integraldisplayT\nt/integraldisplay\nΩf(u(τ))ut(τ)dxdτ\n=/integraldisplay\nΩF(u(T))dx−/integraldisplay\nΩF(u(t))dx\nand\nlim\nn→∞lim\nm→∞/integraldisplayT\nt/integraldisplay\nΩf(u(m)(τ))u(n)\nt(τ)dxdτ=/integraldisplay\nΩF(u(T))dx−/integraldisplay\nΩF(u(t))dx. (5.29)\n21We deduce from (5 .22), (5.28) and (5 .29) that\nlim\nn→∞lim\nm→∞/integraldisplayT\nt/integraldisplay\nΩ/parenleftbig\nf(u(n)(τ))−f(u(m)(τ))/parenrightbig\n·/parenleftbig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightbig\ndxdτ\n= lim\nn→∞lim\nm→∞/bracketleftBig/integraldisplay\nΩF(u(n)(T))dx−/integraldisplay\nΩF(u(n)(t))dx+/integraldisplay\nΩF(u(m)(T))dx\n−/integraldisplay\nΩF(u(m)(t))dx−/integraldisplayT\nt/integraldisplay\nΩf(u(m)(τ))u(n)\nt(τ)dxdτ\n−/integraldisplayT\nt/integraldisplay\nΩf(u(n)(τ))u(m)\nt(τ)dxdτ/bracketrightBig\n=0(5.30)\nfor allt∈[0,T].\nDue to (5.1) and (5.9),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\nt/integraldisplay\nΩ/parenleftBig\nf(u(n)(τ))−f(u(m)(τ))/parenrightBig\n·/parenleftBig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightBig\ndxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CB,T.(5.31)\nBy Lebesgue’s dominated convergence theorem, combining (5 .30) and (5.31) yields\nlim\nn→∞lim\nm→∞/integraldisplayT\n0/integraldisplayT\nt/integraldisplay\nΩ/parenleftBig\nf(u(n)(τ))−f(u(m)(τ))/parenrightBig\n·/parenleftBig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightBig\ndxdτdt= 0.(5.32)\nIt follows from (5 .30) and (5.32) that\nI2≡lim\nn→∞lim\nm→∞/braceleftBigg\nCǫ\nT/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nΩ/parenleftBig\nf(u(n)(t))−f(u(m)(t))/parenrightBig/parenleftBig\nu(n)\nt(t)−u(m)\nt(t)/parenrightBig\ndxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplayT\nt/integraldisplay\nΩ/parenleftBig\nf(u(n)(τ))−f(u(m)(τ))/parenrightBig/parenleftBig\nu(n)\nt(τ)−u(m)\nt(τ)/parenrightBig\ndxdτdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightBigg\n+Cǫ,B\nT+ǫ\n2/bracerightBigg\n=Cǫ,B\nT+ǫ\n2.(5.33)\nWe deduce from (5 .13), (5.20) and (5.33) that\nliminf\nm→∞liminf\nn→∞En,m(T)≤I1+I2=Cǫ,B\nT+ǫ\n2≤ǫ\nforT≥2Cǫ,B\nǫ, which implies\nliminf\nm→∞liminf\nn→∞/vextenddouble/vextenddouble/vextenddouble/parenleftbig\nu(n)(T),u(n)\nt(T)/parenrightbig\n−/parenleftbig\nu(m)(T),u(m)\nt(T)/parenrightbig/vextenddouble/vextenddouble/vextenddouble\nH1\n0(Ω)×L2(Ω)≤√\n2ǫ.\n22Consequently, by Lemma 2 .5, the dynamical system generated by problem (1 .1)-(1.3) is\nasymptotically smooth. 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Weinerc)\na)Department of Applied Analysis and Computational Mathemat ics, E¨ otv¨ os Lor´ and University,\nP´ azm´ any P. stny. 1/C, H-1117 Budapest, Hungary\nb)Department of Energy Engineering, Budapest University of T echnology and Economics,\nBertalan L. u. 4-6, H-1111 Budapest, Hungary\nc)Department of Analysis, Budapest University of Technology and Economics,\nEgry J. u. 1, H-1111 Budapest, Hungary\nAbstract\nWetaketheviewpointthatthephysicallyacceptable soluti ons oftheLorentz–Dirac equation for radiation\nback-reaction are actually determined by a second order equ ation of motion, the self-force being given as\na function of spacetime location and velocity. We propose th ree different methods to obtain this self-force\nfunction. For two example systems, we determine the second o rder equation of motion exactly in the\nnonrelativistic regime via each of these three methods, the three methods leading to the same result. We\nreveal that, for both systems considered, back-reaction in duces a damping proportional to velocity and, in\naddition, it decreases the effect of the external force.\n1 Introduction\nThe Maxwell equations of classical electrodynamics describe how giv en charged particles determine the elec-\ntromagnetic field, while the Newton equation with the Lorentz force tells how a given electromagnetic field\ndetermines the motion of charged particles. As a first step toward s describing interaction between matter and\nfield, the radiation back-reaction(self-force) of a point charge ehavingthe special relativistic world line function\nris deduced to be[1, 2, 3, 4, 5, 6]\nfj\nself=η/parenleftbig\ngj\nk−˙rj˙rk/c2/parenrightbig...rk, (1)\nwhere indices run from 0 to 3, gis the spacetime metric, cis the speed of light, η= (2/3)e2/c3, and overdot\ndenotes differentiation with respect to proper time.1This force is added to the external force f, which may\ndepend on both the spacetime location and the four-velocity of the particle, to obtain\nm¨rj=fj(r,˙r)+η/parenleftbig\ngj\nk−˙rj˙rk/c2/parenrightbig...rk, (2)\ncalled the Lorentz–Dirac equation, for the motion of the point part icle with mass m.\nThe problems with this equation are well-known. First, it is of third ord er so the initial values of spacetime\nposition, velocity and acceleration are necessary to obtain the mot ion, and there is no apparent reasoning how\nto prescribe acceleration. Second, the equation admits ’runaway’ solutions—motions accelerating exponentially\nin time—, and, third, it exhibits acausal behavior.\nThere are a number of attempts to treat these problems (see, e.g ., [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17],\nresearch along the lines of [18], etc.), trying to find the physically acc eptable solutions of the Lorentz–Dirac\nequation and giving further and further insight into the situation.\nIt is important to note here that, irrespective of whether the Lor entz–Dirac equation is considered an exact\none for exactlypointlike chargesor anapproximateone fordistribu ted ones, it is legitimate, useful and insightful\nto investigate whether an equation of the form (2) finds physically m eaningful use.\nOne of the proposed and most frequently applied approach is as follo ws[19]. As a zeroth approximation, the\nequation without radiation is considered:\nm¨xj=fj(x,˙x). (3)\nThe third derivative is computed from this equation,\n...xj=1\nm/parenleftbigg∂fj\n∂xk˙xk+∂fj\n∂˙xk¨xk/parenrightbigg\n. (4)\n∗Corresponding author. E-mail: fulop@energia.bme.hu\n1Note that, upon ˙ rj˙rj= 1, we also have −˙rj˙rk...rk= ˙rj/bracketleftBig\n¨rk¨rk−(1/2)/parenleftbig\n˙rk˙rk/parenrightbig¨/bracketrightBig\n= ˙rj¨rk¨rk. Frequently, the self-force is written using\nthis latter form.\n1Then the second derivative here is replaced by the rhs of (3) (divide d bym), an expression of lower order\nderivatives, leading to the following approximation of the self-force (1):\nbj\n{1}(x,˙x) :=η/parenleftbigg\ngj\nk−˙xj˙xk\nc2/parenrightbigg1\nm/parenleftbigg∂fk\n∂xl˙xl+1\nm∂fk\n∂˙xlfl(x,˙x)/parenrightbigg\n. (5)\nThis is added to the external force to derive an approximate secon d order equation of motion:\nm¨xj=fj(x,˙x)+bj\n{1}(x,˙x). (6)\nIn known examples, bj\n{1}(x,˙x) is found to be a damping dissipative term.\nAnother idea [10] is that the initial values of spacetime position, velo city and acceleration cannot be given\nindependently, and one has to find a ‘critical manifold’ formed by tho se initial values which do not result in\nrunaway solutions. In [10], it is stated that the critical manifold admit s a second order differential equation for\nthe physically acceptable motions but the actual form of such an eq uation is not given. Instead, by a singular\nperturbation, only a first approximation is provided, which results in (6).\nHere, ourobject ofinterest isthe exactformofthe secondorde rdifferentialequationforthe criticalmanifold.\nBefore proceeding, we mention that other results [20] also sugge st, implicitly, the existence of a second order\nequation of motion in the background.\nIn what follows, we explicitly assume the existence of a second order equation of motion,2\nm¨xj=fj(x,˙x)+bj(x,˙x), (7)\nwith the self-force as a function of spacetime position and velocity,\nfj\nself=bj(x,˙x), (8)\nand derive the condition on it that ensures that all its solutions are s olutions of (2) as well. This condition\nturns out to take the form of a partial differential equation for (8 ).\nIn parallel, based on physically plausible ideas, we also propose two iter ative methods for obtaining the\nself-force function. Actually, the first step of one of these itera tive methods corresponds to (6).\nFor demonstration, we investigate the partial differential equatio n and the iterative methods in quantitative\ndetail, in the nonrelativistic regime,3treating two special cases. For both systems, we calculate the se lf-force\nfunction exactly via each of the three methods, and find that the t hree approaches provide the same result.\n2 The self-force function\nFor convenience, we introduce the shorthands\nˆf:=f/m, ˆη:=η/m, ˆb:=b/m. (9)\n2.1 Differential equation for the self-force function\nAs said, we assume that the equation of motion of a radiating particle is of the form\n¨xj=ˆfj(x,˙x)+ˆbj(x,˙x), (10)\nwhere the second term on the rhs is the self-force as a function of spacetime position and four-velocity. In order\nto obtain its actual expression, we use a fixed-point-like property , as follows. Computing the third derivative\nfrom the expected equation of motion (10), substituting ¨ xin the obtained expression by the rhs of (10), and\napplying ˆ η/parenleftbig\ngj\nk−˙xj˙xk/c2/parenrightbig\nto the result, we have to recover the self-force:\nˆbj= ˆη/parenleftbigg\ngj\nk−˙xj˙xk\nc2/parenrightbigg/bracketleftBigg\n∂/parenleftbigˆfk+ˆbk/parenrightbig\n∂xl˙xl+∂/parenleftbigˆfk+ˆbk/parenrightbig\n∂˙xl/parenleftbigˆfl+ˆbl/parenrightbig/bracketrightBigg\n. (11)\n2Apparently, this hypothesis is considerably stronger than assuming dependence on the extended past, which phenomenon could\nalso be plausible by physical expectations about how intera ction takes place between matter and field. Here, driven by th e above\nmotivations, we investigate whether one can succeed with th is stronger assumption.\n3Namely, we take only j= 1,2,3 in (2) and omit terms of the order of 1 /c2or higher.\n2Naturally, one has to keep in mind that xand ˙xare to be understood as independent variables. We can display\nthis fact in a more self-explaining way, writing (11) as\nˆbj(x,u) = ˆη/parenleftbigg\ngj\nk−ujuk\nc2/parenrightbigg/braceleftBigg\n∂/bracketleftbigˆfk+ˆbk/bracketrightbig\n(x,u)\n∂xlul+∂/bracketleftbigˆfk+ˆbk/bracketrightbig\n(x,u)\n∂ul/bracketleftbigˆfl+ˆbl/bracketrightbig\n(x,u)/bracerightBigg\n. (12)\nThis is a first order partial differential equation (system) for the t wo-variable function(s) ˆbj. Its solution\nis expected to contain an arbitrary free function; on physical gro unds, one can impose some requirements, via\nwhich one can obtain the sought self-force function uniquely.\nTo formulate the fundamental condition, let us draw attention to t hat the self-force depends on the external\nforce,b(x,u) =bf(x,u). It is evident that there is no self-force without external action ; this fact can be taken\ninto account in two ways.\nFirst, we require that if fis zero in a neighborhood of a spacetime point x0and four-velocity u0then\nbf(x0,u0) = 0. (13)\nSecond, it is plausible to expect that less action generates less reac tion; consequently, we demand that if\nthe external action tends to zero then the self-force must tend to zero, too. Specifically, we will consider the\nself-force function bκffor every 0 ≤κ≤1 and will impose, in the pointwise sense,\nlim\nκ→0bκf(x,u) = 0. (14)\nA third natural assumption is that if the external field has a spacet ime symmetry then the self-force function\nhas the same symmetry. Namely, if fis invariant under a Poincar´ e transformation P(with the underlying\nLorentz transformation L), i.e.,Lf/parenleftbig\nP−1x,L−1u/parenrightbig\n=f(x,u), then the same invariance must hold for bf, too.\nThough not utilized in the present considerations, a natural gener alization of this criterion to include non-\ninvariant cases would be that the transformation of ftofP(x,u) =Lf/parenleftbig\nP−1x,L−1u/parenrightbig\nis accompanied by the\ncorresponding transformation of bftobfP(x,u) =Lbf/parenleftbig\nP−1x,L−1u/parenrightbig\n. Similarly, our two other conditions could\nalso be generalized/weakened for future needs.\n2.2 Iteration of the radiation term\nEquation (11) is, in general, a rather complicated system of partial differential equations so it is not easy to\nfind its solutions. Hence, we look for other methods as well, to deter mine the self-force function.\nAn idea is suggested by (6) which, as said, cannot be an exact equat ion. We can consider it, however, as a\nfirst approximation. Then it is a straightforward idea that we take a n analogous second approximation:...xis\ncomputed as the derivative of\n¨xj=ˆfj(x,˙x)+ˆbj\n{1}(x,˙x), (15)\nand then ¨ xis substituted by the rhs of (15). Thus, we obtain an expression ˆb{2}(x,˙x), with which the second\napproximation for the equation of motion is\n¨xj=ˆfj(x,˙x)+ˆbj\n{2}(x,˙x). (16)\nThesameprocedurecanberepeatediterativelyforallhigherorde rs. Ifthesequenceofterms ˆb{n}(x,˙x)converges\nto aˆb(x,˙x)—in some appropriate sense, e.g., pointwise—then we arrive at a sec ond order equation of motion\nof the form (10). Naturally, it is a tough problem is whether the sequ ence converges or not.\n2.3 Iteration of the solution\nThe above iteration method suggests another one, which would not directly result in an equation of motion but\nin the motion corresponding to initial values x0of spacetime position and u0of four-velocity.\nLet the solution of the zeroth approximation (3) corresponding to initial values x0andu0be denoted by\nr{0}. Taking its first and third derivatives, we establish the differential e quation\n¨xj=ˆfj(x,˙x)+ ˆη/parenleftBig\ngj\nk−˙rj\n{0}˙r{0}k/slashbig\nc2/parenrightBig...rk\n{0}(17)\n3as the first approximation for the equation of motion. Let r{1}be its solution for initial values x0andu0.\nTaking its first and third derivative, we establish the second approx imation, and so on; at step n+1 we solve\n¨xj=ˆfj(x,˙x)+ ˆη/parenleftBig\ngj\nk−˙rj\n{n}˙r{n}k/c2/parenrightBig...rk\n{n}. (18)\nIf the sequence of solutions r{n}converges to an rthen we succeeded in finding a motion satisfying equality (2)\nwithout the need for the initial value of acceleration. This motion is, t herefore, a good candidate for the sought\nphysical solution.\nNaturally, here, too, convergence is a tough problem.\nWe can observe that, although this iterative method provides solut ions rather than the equation of motion,\na corresponding self-force function b(x,u) can be read off from the solutions. Namely, the value of the corre-\nsponding ˆbat any spacetime point x0and four-velocity value u0can be calculated from the third derivative of\nthe solution rbelonging to initial values x0andu0, at the initial proper time value:\nˆbj(x0,u0) = ˆη/parenleftbig\ngj\nk−(u0)j(u0)k/c2/parenrightbig...rk(0). (19)\n3 Applying the three approaches in the nonrelativistic regi me: Con-\nstant field\nLet us consider a constant external electromagnetic field, which a cts on the charged particle via the Lorentz\nforce\nf=eE+ev×B=eE+eFv, (20)\nwhereEis the electric field three-vector, Bis the magnetic axial vector field, F= (−B×) is the corresponding\nantisymmetric three-tensor—now each assumed space and time ind ependent—, and vis the velocity of the\nparticle. With the shorthands\n˜E:=e\nmE,˜B:=e\nmB,˜F:=e\nmF, (21)\nwe can simply write\nˆf=ˆf(v) =˜E+˜Fv. (22)\nThe electromagnetic nature of the field will not play any role here so t he subsequent considerations will be\napplicable for any force of the form (22), including a constant grav itational attraction, for example.\nWe will need some technical remarks regarding ˜F. Its kernel is spanned by ˜B, its range is the plane\northogonal to ˜B, and, with Pdenoting the orthogonal projection onto this plane, we find\n˜F=P˜F=˜FP=P˜FP,(I−P)˜F=0,˜F2=−˜B2P, (23)\nwhereIstands for the three-identity tensor (and ˜Bis the magnitude of ˜B).\nTo keep the formulae shorter and more easily accessible, we first tr eat˜E=0. In this case, the equation of\nmotion without radiation is\n˙v=˜Fv. (24)\nThe results for the general case ˜E/negationslash=0are presented in section 3.4.\n3.1 Differential equation for the self-force function\nWe can start with ruling out the space and time dependence of ˆb, based on the requirement of section 2.1 that\na spacetime symmetry of the external force should be respected by the self-force, too. In the present case, the\nsymmetry in question is spacetime translation invariance. Hence, th e sought equation of motion is of the form\n˙v=˜Fv+ˆb(v). (25)\nIn addition, the external field is invariant for space inversion, −˜F(−v) =˜F(v), and thus −ˆb(−v) =ˆb(v) is\nrequired, too.\nAccording to our assumption described in section 2.1, ˆbmust obey the differential equation\nˆb(v) = ˆη/bracketleftbig˜F+ˆb′(v)/bracketrightbig/bracketleftbig˜Fv+ˆb(v)/bracketrightbig\n, (26)\n4with′denoting the derivative map. Let us assume that we can expand ˆbin a series. Because of the space\ninversion symmetry, the even powers are zero, so\nˆb(v) =L1v+L3(v,v,v)+L5(v,v,v,v,v)+··· (27)\nwhereL1is a linear map, L3is a symmetric trilinear map etc.; keep in mind that they depend on ˜F. Using\nthe notation L3/parenleftbig\nv3/parenrightbig\n:=L3(v,v,v) etc., we obtain\nL1v+L3/parenleftbig\nv3/parenrightbig\n+L5/parenleftbig\nv5/parenrightbig\n+···= ˆη/bracketleftbig˜F+L1+3L3/parenleftbig\nv2,·/parenrightbig\n+5L5/parenleftbig\nv4,·/parenrightbig\n+···/bracketrightbig\n×/bracketleftbig˜Fv+L1v+L3/parenleftbig\nv3/parenrightbig\n+L5/parenleftbig\nv5/parenrightbig\n+···/bracketrightbig\n, (28)\nfrom which it follows, order by order, that\nL1= ˆη/parenleftbig˜F+L1/parenrightbig2, (29)\nL3/parenleftbig\nv3/parenrightbig\n= ˆη/bracketleftbig\n3L3/parenleftbig\nv2,/parenleftbig˜F+L1/parenrightbig\nv/parenrightbig\n+/parenleftbig˜F+L1/parenrightbig\nL3/parenleftbig\nv3/parenrightbig/bracketrightbig\n, (30)\nL5/parenleftbig\nv5/parenrightbig\n= ˆη/bracketleftbig\n3L3/parenleftbig\nv2,L3/parenleftbig\nv2/parenrightbig/parenrightbig\n+5L5/parenleftbig\nv4,/parenleftbig˜F+L1/parenrightbig\nv/parenrightbig\n+/parenleftbig˜F+L1/parenrightbig\nL5/parenleftbig\nv5/parenrightbig/bracketrightbig\n, (31)\netc. Multiplying (29) from the left and from the right by ˜F+L1, we find that L1˜F=˜FL1. Then it is a simple\nalgebraic fact that\nL1=−α˜F−γP+β(I−P), (32)\nwhereα,γandβare scalar coefficients depending on ˜F. Applying the second condition put in section 2.1, γ\nandβmust tend to zero if ˜Ftends to zero.\nHaving (32), (29) yields\n−α˜F−γP+β(I−P) = ˆη/bracketleftbig\n−(1−α)2˜B2P+γ2P+β2(I−P)−2γ(1−α)˜F/bracketrightbig\n, (33)\nwhich tells\nα= 2ˆηγ(1−α), γ= ˆη(1−α)2˜B2−ˆηγ2(34)\nas well as β= ˆηβ2, according to which βis either zero or equals 1 /ˆη. The second possibility is excluded by the\ndemand that βmust be zero for zero external field so\nβ= 0. (35)\nThe first equation in (34) rules out α= 1, and then another equivalent pair of equations is\nˆηγ=α\n2(1−α),4(ˆη˜B)2(1−α)4+(1−α)2−1 = 0. (36)\nThe latter condition is a quadratic equation for ̺:= (1−α)2≥0, with the only non-negative root\n̺=/parenleftBig\n−1+/radicalBig\n1+16/parenleftbig\nˆη˜B/parenrightbig2/parenrightBig/slashBig/parenleftBig\n8/parenleftbig\nˆη˜B/parenrightbig2/parenrightBig\n. (37)\nAgain, the condition that γmust be zero for zero external field, gives the result:\nα= 1−√̺, γ = (1/√̺−1)/(2ˆη), (38)\nthe latter obtained from the first equation in (36). As it is proved in t he Appendix, (30) yields L3=0, and,\nsimilarly, all the higher order terms are found to be zero. Hence, th e self-force is\nˆb(v) =−(α˜F+γP)v, (39)\nand the equation of motion is\n˙v=˜Fv+ˆb(v),i.e.,˙v=/bracketleftbig\n(1−α)˜F−γP/bracketrightbig\nv. (40)\nIt is informative to inspect the solution of this equation of motion, wh ich is\nw(t) = (I−P)v0+e−γte(1−α)˜FtPv0 (41)\nfor initial velocity v0at zero initial time. For nonzero ˜B, we have 0 <1−α <1 andγ >0 so radiation causes\nthat\n•the effect of the external magnetic field is reduced by a certain fac tor, and\n•the component of velocity perpendicular to the magnetic field tends to zero as time passes.\nNote that, in this simple case, two conditions given at the end of sect ion 2.1 suffice to determine the self-force\nfunction completely.\n53.2 Iteration of the radiation term\nItfollowsfromtheequationwithoutradiation—thezerothapproxim ation(24)—that ¨v=˜F˙v=˜F2v=−˜B2Pv.\nAccordingly, the first approximation of the radiation term is\nˆb{1}(v) =L{1}v,L{1}=−ˆη˜B2P, (42)\nwith which the first approximation of the equation of motion becomes ˙v=/parenleftbig˜F+L{1}/parenrightbig\nv. The radiation term\nderived from this equation equals\nˆb{2}(v) =L{2}v,L{2}= ˆη/parenleftbig˜F+L{1}/parenrightbig2. (43)\nRepeating this again and again, we recognize the generic recursion f ormula\nˆb{n}(v) =L{n}v,L{n}= ˆη/parenleftbig˜F+L{n−1}/parenrightbig2. (44)\nSince\nL{2}= ˆη/parenleftbig˜F+L{1}/parenrightbig2= ˆη/parenleftbig\n−˜B2P−2ˆη˜B2˜F+ ˆη2˜B4P/parenrightbig\n, (45)\neveryL{n}is a linear combination of ˜FandP. Supposing that the sequence ˆb{n}(v) converges for all v,L{n}\nshould converge to an L, for which we have\nL= ˆη(˜F+L)2, (46)\nwhereLis a linear combination of ˜FandP:L=−α˜F−γP.Consequently,\n−α˜F−γP= ˆη/bracketleftBig\n−(1−α)2˜B2P−2γ(1−α)˜F+γ2P/bracketrightBig\n, (47)\nwhich imposes the pair of equations in (34), i.e. we arrive at the same r esult as previously.\nThe ambiguity α= 1±√̺arises here, too. If convergence holds then, naturally, only one o f the possibilities\ncan be the limit. To select the correct one, we can consider the simple case when there is no external force.\nThen all the iteration terms are zero, and the iteration converges trivially. Hence, the coefficients must be zero\nfor zero magnetic field, as previously.\nA problem with the method of iterating the radiation term is that it is ha rd to obtain conditions for the\nconvergence of the iteration. Unfortunately, convergence may not hold. Indeed, for ˆ η˜B= 1—i.e., for such\nspecial magnetic fields—convergence does not occur, since then L{2}=−2˜F, soL{3}=L{1}. Thus, for all\nhigherns,\nL{n}=/braceleftBigg\nL{2}(nis even),\nL{1}(nis odd).(48)\nIt is interesting, however, that the final result (38) does not exc lude the case ˆ η˜B= 1, and provides solution for\nˆη˜B >1 as well (which is expected to be outside the domain of convergence ).\n3.3 Iteration of the solution\nConsidering an initial value v0at zero initial time, the zeroth equation (24) has the solution\nw{0}(t) = e˜Ftv0. (49)\nThen¨w{0}(t) =−˜B2e˜Ftv0, so the first approximation satisfies the equation\n˙v=˜Fv−ˆη˜B2e˜Ftv0, (50)\nwhose solution with initial value v0is\nw{1}(t) =/parenleftBig\n1−ˆη˜B2t/parenrightBig\ne˜Ftv0. (51)\nThen¨w{1}(t) =−/bracketleftBig\n˜B2/parenleftBig\n1−ˆη˜B2t/parenrightBig\n+2ˆη˜B2˜F/bracketrightBig\ne˜Ftv0. The second approximation satisfies the equation\n˙v=˜Fv−ˆη/bracketleftBig\n˜B2/parenleftbig\n1−ˆη˜B2t/parenrightbig\n+2ˆη˜B2˜F/bracketrightBig\ne˜Ftv0, (52)\n6with the solution\nw{2}(t) =/bracketleftbigg\n1−ˆη˜B2/parenleftbigg\nt−1\n2ˆη˜B2t/parenrightbigg\n−2ˆη2˜B2t˜F/bracketrightbigg\ne˜Ftv0. (53)\nAt thenth step of this iteration scheme, we find\nw{n}(t) =/bracketleftBig\np{n}(t)+q{n}(t)˜F/bracketrightBig\ne˜Ftv0, (54)\nwherep{n}andq{n}are polynomials of tsatisfying the recursive formulae\n˙p{n}= ˆη/parenleftBig\n¨p{n−1}−2˜B2˙q{n−1}−˜B2pn−1/parenrightBig\n, (55)\n˙q{n}= ˆη/parenleftBig\n¨q{n−1}+2˙p{n−1}−˜B2q{n−1}/parenrightBig\n(56)\nand the initial conditions p{n}(0) = 1, q{n}(0) = 0.\nLet us suppose that p:= lim np{n}andq:= lim nq{n}exist and, moreover, that the limit procedure and\ndifferentiation can be interchanged. Then, for w:= lim nw{n},\nw(t) =/bracketleftbig\np(t)+q(t)˜F/bracketrightbig\ne˜Ftv0, (57)\n˙p= ˆη/parenleftBig\n¨p−2˜B2˙q−˜B2p/parenrightBig\n, (58)\n˙q= ˆη/parenleftBig\n¨q+2˙p−˜B2q/parenrightBig\n(59)\ntogether with the conditions p(0) = 1, q(0) = 0. With the notation s:=p+i˜Bq, (58) and (59) can be comprised\nas\n¨s−/parenleftBig\n1/ˆη−2i˜B/parenrightBig\n˙s−˜B2s= 0. (60)\nThe roots of the corresponding characteristic polynomial are\n1/2ˆη−i˜B±/radicalBig\n1/(4ˆη2)−i˜B/ˆη. (61)\nThe emerging ambiguity can be resolved as previously: if the magnetic force is sent to zero then p{n}(t) = 1\nfor alln, therefore, we choose the root that vanishes for ˜B→0 so as to obtain the solution p(t) = 1, i.e., the\nfree motion (no self-force).\nTherefore, if ( −γ) andαare the real and the imaginary part of the root, respectively, the n the solution of\nour system of differential equations is\np(t) = e−γtcos(α˜Bt), q(t) =−e−γtsin(α˜Bt)/˜B, (62)\nwhereαandγsatisfy the relations in (34).\nWe have obtained the same result as previously. In particular, utilizin g (23), one can demonstrate that (57)\nis the same as (41). At last, it is straightforward to find that the se lf-force function corresponding to (57)—see\n(19)—proves to be the same as (39).\nOn the other side, one can also observe that, though leading to the same result, the two iterations themselves\nare different: the solutions of the iterated equations ˙v=/parenleftbig˜F+L{n}/parenrightbig\nvdo not equal the functions (54).\nConvergence is a nontrivial question in this approach, too. Nevert heless, it is interesting that, with this\nmethod, there is no evidence for excluding the case ˆ η˜B= 1.\n3.4 Nonzero electric and magnetic field\nAs anticipated in section 3, now we turn towards the case of a nonze ro constant electric field in addition to the\nnonzero constant magnetic field.\nIt is beneficial to decompose the electric field into components para llel to and orthogonal to the magnetic\nfield, respectively, and to observe that this decomposition can be w ritten in the form\n˜E= (I−P)˜E+˜F/parenleftbigg\n−1\n˜B2˜F˜E/parenrightbigg\n. (63)\n7This induces a decomposition of the equation of motion without self-f orce [i.e., containing only the external\nLorentz force (22)]. The ˜B-parallel component,\n[(I−P)v]˙= (I−P)˜E, (64)\ngoverns only the ˜B-parallel component of v, while the ˜B-orthogonal part can be written as\n˙u=˜Fuwith u:=Pv−1\n˜B2˜F˜E,u⊥˜B, (65)\nand determines the time evolution of the ˜B-orthogonal component of v.\nNow we add the self-force term. Both iteration methods, namely, t hat of the radiation term and that of the\nsolution, can be found to preserve this decomposition, where the ˜B-orthogonal part can actually be treated the\nsame way for uas we proceeded in the ˜E=0case forv. Both approaches provide\nˆb(v) =−αP˜E+γ1\n˜B2˜F˜E−/bracketleftbig\nα˜F+γP/bracketrightbig\nv (66)\nfor the self-force and\nv(t) = (I−P)˜Et+(I−P)v0+1\n˜B2˜F˜E+e−γte(1−α)˜Ft/parenleftbigg\nPv0−1\n˜B2˜F˜E/parenrightbigg\n(67)\nfor the solution, after putting the two decomposed parts togeth er. The found self-force function proves to be\na solution of the partial differential equation of the third approach as well, and satisfies the two additional\nrequirements—vanishing for vanishing external field and symmetry preservation.\n4 Applying the three approaches in the nonrelativistic regi me: Har-\nmonic force\nAs our second physical system considered as example for the thre e proposed methods, we next investigate the\none-dimensional nonrelativistic motion due to a harmonic elastic forc e. Without radiation back-reaction, the\nequation is\n¨x=−ω2x, (68)\nwhereωis a non-negative constant.\n4.1 Differential equation for the self-force function\nAccording to our assumption described in section 2.1, the radiation s elf-force function ˆb(x,˙x) in the anticipated\nequation of motion\n¨x=−ω2x+ˆb(x,˙x) (69)\nis to satisfy the quasi-linear partial differential equation\nˆb(x,v) = ˆη/bracketleftBigg\n−ω2v+∂ˆb(x,v)\n∂xv+∂ˆb(x,v)\n∂v/parenleftBig\n−ω2x+ˆb/parenrightBig/bracketrightBigg\n. (70)\nNote that, although not denoted explicitly, the sought ˆbalso depends on ω, i.e., on the external force.\nThe characteristic ordinary differential equation corresponding t o (70) reads\ndx\ndξ=v,dv\ndξ=−ω2x+ˆb,dˆb\ndξ=1\nˆηˆb+ω2v. (71)\nThis is a simple linear differential equation whose characteristic roots λfulfill the equation\nˆηλ3−λ2−ω2= 0. (72)\nWe can find its solutions (roots) by the Cardano formula. With the no tation\nρ±:=3/radicalBig\nˆη2ω2/2+1/27±/radicalbig\nˆη4ω4/4+ ˆη2ω2/27, (73)\n8the roots are\nˆηλ1=1\n3−ρ++ρ−\n2+i√\n3(ρ+−ρ−)\n2, (74)\nˆηλ2=1\n3−ρ++ρ−\n2−i√\n3(ρ+−ρ−)\n2, (75)\nˆηλ3=1\n3+ρ++ρ−. (76)\nWith the three roots, the solutions of equation (71) are of the for m\nx(ξ) =3/summationdisplay\ni=1cieλiξ, v (ξ) =3/summationdisplay\ni=1λicieλiξ, ˆb(ξ) =3/summationdisplay\ni=1λ2\nicieλiξ+ω2x(ξ).(77)\nAccording to the method of characteristics, via eliminating the auxilia ry variable ξ, one obtains ˆbas a function\nofxandv. Now we use the condition of section 2.1 that ˆbmust be zero for zero external force i.e. for ω= 0.\nSinceλ1andλ2are zero for ω= 0 and λ3is not zero, we deduce from the last equality above that c3= 0 is\nnecessary.\nThenξcan be eliminated easily; with the notations z1:=c1eλ1ξandz2:=c2eλ2ξ, we have\nx=z1+z2, v =λ1z1+λ2z2, ˆb=λ2\n1z1+λ2\n2z2+ω2x. (78)\nHere, the first two equations enable one to express z1andz2as a linear function of xandv. Substituting them\ninto the third equation provides ˆbalso as a linear function of xandv:ˆb(x,v) = (ω2−λ1λ2)x+(λ1+λ2)v.\nA convenient way to proceed is to write the coefficients in another fo rm:\nˆb(x,v) =αω2x−γv. (79)\nEvidently,\nˆηγ=−2\n3+ρ++ρ−, (80)\nand, substituting (79) into (70), we obtain\nα=ˆηγ\n1+ ˆηγand ˆηγ(1+ ˆηγ)2= ˆη2ω2. (81)\nWe can see that γ >0 and 0< α <1 are necessary for ω/negationslash= 0. Hence, the equation of motion with the radiation\nterm (79) reads\n¨x=−(1−α)ω2x−γ˙x. (82)\nAccording to this equation, radiation causes that\n•the effect of the harmonic force is reduced by a certain factor, an d\n•the motion is damped by a term proportional to velocity.\nWe can see in this case, too, that the condition that the self-force must be zero if the external force is zero\nsuffices to determine the self-force function completely.\n4.2 Iteration of the radiation term\nIt follows from the equation without radiation (the zeroth approxim ation) (68) that...x=−ω2˙x, so the the first\napproximation of the radiation term is\nˆb{1}(x,˙x) =−ˆηω2˙x, (83)\nand the first approximate equation of motion becomes ¨ x=−ω2x+ˆb{1}(x,˙x). Computing...xfrom this equation\nand then replacing ¨ xwith−ω2x+ˆb{1}(x,˙x), we obtain the second approximation\nˆb{2}(x,˙x) = ˆη2ω4x−ˆηω2/parenleftbig\n1−ˆη2ω2/parenrightbig\n˙x (84)\nand the second equation ¨ x=−ω2x+ˆb{2}(x,˙x).\n9It is straightforward then that the nth approximation is of the form\nˆb{n}(x,˙x) =αnω2x−γn˙x, (85)\nwhereαnand ˆηγnare functions (polynomials) of ˆ η2ω2and satisfy the recursive formulae\nαn+1= ˆηγn(1−αn), ˆηγn+1= ˆη2ω2(1−αn)−(ˆηγn)2. (86)\nSupposing that the sequence ˆb{n}converges to a ˆb, i.e., the limits α:= lim nαnandγ:= lim nγnexist, then\nˆb(x,˙x) =αω2x−γ˙x (87)\nand\nα=ˆηγ\n1+ ˆηγ,ˆηγ(1+ ˆηγ)2= ˆη2ω2, (88)\nwhich coincide with (81). Putting ˆ ηλ:= 1 + ˆηγ, the second equation above is transformed into the equation\n(72). Naturally, now we are interested in the real roots. We find th at only one root is real, namely, with the\nnotation (73),\nˆηγ=−2\n3+ρ++ρ−. (89)\nHence, we arrive at the same result as previously.\nIt is a problem, though, that it is difficult to obtain conditions for the c onvergence of the iteration. Unfor-\ntunately, convergence does not hold necessarily. Indeed, if ˆ ηω= 1 then the sequence does not converge because\nthenˆb{2}(x,˙x) =ωx2, and thus ˆb{3}=ˆb{1}. Consequently,\nˆb{n}=/braceleftBiggˆb{2}(nis even),\nˆb{1}(nis odd).(90)\nIt is interesting, however, that, with (89), (88) provides a solutio n for all values of ˆ ηω.\n4.3 Iteration of the solution\nConsidering some initial values x0andv0for position and velocity, the zeroth equation (68) has the solution\nr{0}(t) =z0eiωt+z∗\n0e−iωt=z0eiωt+c.c. (91)\nwith\nz0:= (x0−iv0/ω)/2. (92)\nSince...r{0}(t) =−iz0ω3eiωt+c.c., the equation for the first iterated solution reads\n¨x=−ω2x+ ˆη/parenleftbig\n−iz0ω3eiωt+c.c./parenrightbig\n, (93)\nwhose solution with the chosen initial values is\nr{1}(t) =/bracketleftbig\n−(1/2)ˆηω3t+z0/bracketrightbig\neiωt+c.c. (94)\nThen we find that the nth solution is of the form\nr{n}(t) =pn(t)eiωt+c.c., (95)\nwherepn(t) is a polynomial of tofnth degree; and we have\n¨r{n}= (¨pn+2iω˙pn−ω2pn)eiωt+c.c. (96)\nand...r{n}= (...pn+3iω¨pn−3ω2˙pn−iω3pn)eiωt+c.c., (97)\nand the resulting recursive formula (arising both from the coefficien t of eiωtand from that of e−iωt) is\n¨pn+1+2iω˙pn+1−ω2pn+1=−ω2pn+1+ ˆη(...pn+3iω¨pn−3ω2˙pn−iω3pn). (98)\n10Thus, supposing convergence (and that differentiation can be inte rchanged with taking the limit), we find for\nthe limit p:= lim npn\n¨p+2iω˙p−ω2p=−ω2p+ ˆη(...p+3iω¨p−3ω2˙p−iω3p). (99)\nThis is a linear differential equation of third order. Its solutions are o f the form eλtwith\n(λ+iω)2+ω2= ˆη(λ+iω)3. (100)\nPuttingλ+iω=µ+iν, whereµandνare real, we can rewrite this as\nµ2+2iµν−ν2+ω2= ˆη(µ3+3iµ2ν−3µν2−iν3), (101)\nwhich is equivalent to the pair of real equations\nν2= 3µ2−2µ/ˆη, (102)\n8(ˆηµ)3−8(ˆηµ)2+2(ˆηµ)−(ˆηω)2= 0. (103)\nWithγ:= 2µ, (103) reduces to the second equation of (88). In parallel, puttin gν2=: (1−α)ω2−γ2/4 (i.e.,\ndefining αin this way), we find ˆ ηγ(ˆηγ+ 1) = (1 −α)ω2, which, together with the second equation of (88),\nresults in its first one.\nAs a consequence, whenever convergence holds, the iteration of solutions gives the same solutions and same\nself-force function as the iteration of the radiation term.\n5 Discussion\nWe looked for a second order equation of motion whose solutions sat isfy the Lorentz–Dirac equality and, at the\nsame time, are physically acceptable. Our aim was to give back-react ion as a function of spacetime position and\nvelocity. A simple argument showed that this self-force function is d etermined by a first order partial differential\nequation. Two iterative methods, too, were proposed for finding t he self-force function.\nIn the nonrelativistic approximation we could exactly calculate the se lf-force function for two systems:\na constant external electromagnetic field and a one-dimensional e lastic external force. The three suggested\nmethods turned out to lead to the same result. As concerns the ph ysical picture, for both systems, radiation\nback-reaction has two manifestations: inducing a damping linear in ve locity and reducing the strength of the\nexternal force.\nThe latter effect could also allow the—quantum field theory motivated —interpretation that back-reaction\ncauses a positive renormalizationof the mass of the particle. Howev er, for the constant external electromagnetic\nfield, this renormalization turns out to differ from a simple scalar multip lying of the mass. Rather, renormal-\nization is a direction dependent, tensorial multiplication.\nThis system also proves to show a limitation of the criterion by Dirac an d Haag [7, 8], which would choose\nthat solution for initial position and velocity for which acceleration te nds to zero for asymptotically large\ntimes. In fact, this system is found to decouple into two independen t subsystems, one parallel to and the\nother orthogonal to the magnetic field. In the former subsystem , acceleration remains time independent and\ncorresponds to unrenormalized mass, while damping and renormalize d mass (or renormalized external force)\nemerges in the latter subsystem.\nIt is therefore an important open question on what grounds the de crease of the external force can be\ninterpreted as mass renormalization, and whether in other system s this renormalization is not only direction\ndependent but, forexample, also (spacetime) position and velocity dependent (assuggested by somepreliminary\nconsiderations not detailed here).\nThe three methods we proposed and investigated led to the same re sult for the two systems considered.\nSome differences among the three approaches were found, thoug h. First, the problematic aspect is that there\nis an encoded amount of ambiguity in the partial differential equation to solve. This ambiguity was easy to\nrule out for the two systems we considered but may be a harder tas k for other systems. Second, in the two\niteration approaches, convergence remained a tough open mathe matical problem; moreover, not all coefficients\nof the model ensured the existence of a solution in one of the iterat ion methods.\nFurtherstudyisneeded, accordingly,abouteachmethodsepara telyandalsoaboutsomepossibleconnections\namong them.\n11A The proof utilized in section 3.1\nThe claim that (30) yields L3=0can be proved as follows.\nWe start with deriving a simple algebraic fact. Let Tbe a symmetric trilinear map. Then, with the\nsimplifying notation used earlier,\nT/parenleftbig\n(w+v)3/parenrightbig\n=T/parenleftbig\nw3/parenrightbig\n+3T/parenleftbig\nw2,v/parenrightbig\n+3T/parenleftbig\nw,v2/parenrightbig\n+T/parenleftbig\nv3/parenrightbig\n(104)\nfor allvandw. Thus, if T/parenleftbig\n(·)3/parenrightbig\n=0then 3T/parenleftbig\nw2,v/parenrightbig\n+ 3T/parenleftbig\nw,v2/parenrightbig\n=0for allvandw. Here, for a fixed\nv, the first term is bilinear, the second term is linear in w; their sum can be zero only if both are zero. As a\nconsequence, if T/parenleftbig\nv3/parenrightbig\n=0for allvthenT(w2,v) =0for allwandv. Further, for a fixed v, we have\nT/parenleftbig\n(w1+w2)2,v/parenrightbig\n=T/parenleftbig\nw2\n1,v/parenrightbig\n+2T(w1,w2,v)+T/parenleftbig\nw2\n2,v/parenrightbig\n, (105)\nwhich shows that, if T/parenleftbig\nw2,v/parenrightbig\n=0for allwandv, then\nT(w1,w2,v) =0for allw1,w2,v. (106)\nHence, we have the result: if T(v3) =0for allvthen (106) holds.\nThe previous consideration and (30) yield that\n/bracketleftbig\nI−ˆη(˜F+L1)/bracketrightbig\nL3(w1,w2,v) = 3ˆηL3/parenleftbig\nw1,w2,(˜F+L1)v/parenrightbig\n(107)\nfor allw1,w2andv. For fixed w1andw2,W:=L3(w1,w2,·) is a linear map satisfying\n/bracketleftbig\nI−ˆη(˜F+L1)/bracketrightbig\nW= 3ˆηW(˜F+L1), (108)\nwhich implies\nW/bracketleftbig\nI−3ˆη(˜F+L1)/bracketrightbig\n= (˜F+L1)ˆηW, (109)\ntoo. Recall that ˜F+L1= (1−α)˜F−γP. Then it is a simple fact that the linear maps multiplying Won the\nleft hand side in (108) and (109), respectively, are nondegenerat e. Therefore, applying I−Pfrom the right to\n(108) and from the left to (109), we find WP=W=PW.\nAs a consequence, Wcommutes with ˜F, too, so it must be of the form W=β˜F+ζP,and either (108) or\n(109) implies\n(1+4ˆηγ)W= 4ˆη(1−α)W˜F. (110)\nThis equation and W=β˜F+ζPresult in\n(1+4ˆηγ)β= 4ˆη(1−α)ζ, (1+4ˆηγ)ζ=−4ˆη(1−α)˜B2β, (111)\nimplying ˜B2β2=−ζ2, which is possible only if β= 0 and ζ= 0, i.e., W=0.\nSince we had W=L3(w1,w2,·) for arbitrary w1andw2, we arrive at L3=0.\nAcknowledgments\nSupport from the Hungarian Scientific Research Fund (OTKA, Gran t No. K116375) is appreciated.\nReferences\n[1] J.D. Jackson, Classical electrodynamics (Wiley, New York, 1998) 3rd ed.\n[2] S.R. de Groot, L.G. Suttorp, Foundations of electrodynamics (North-Holland, Amsterdam, 1972).\n[3] J.G. Taylor, Classical electrodynamics as a distribution theory, Math. Proc. Camb. Phil. Soc. 52(1956),\n119–134.\n[4] T. Matolcsi, Classical electrodynamics , Extracts from the Scientific Works of the Department of Applied\nAnalysis, E¨ otv¨ os University, Budapest, Hungary, 1977/4.\n12[5] E.G.P. Rowe, Structure of the energy tensor in the classical elec trodynamics of point particles, Phys. Rev.\nD18, 3639–3654 (1978).\n[6] A. Gsponer, The self-interaction force on an arbitrarily moving p oint-charge and its energy-momentum\nradiation rate: A mathematically rigorous derivation of the Lorentz –Dirac equation of motion,\narXiv:0812.3493v2; ISRI-07-01, 2008.\n[7] P.A.M. Dirac, Classical theory of radiating electrons, Proc. Royal Soc. London A 167, 148–169 (1938).\n[8] R. Haag, Die Selbstwechselwirkung des Elektrons, Z. Naturforsch. 10a, 752–761 (1955).\n[9] M.M.de Souza, The Lorentz–Dirac equation and the structures o f spacetime, Bras. J. Phys. 28, 250–256\n(1998).\n[10] H. Spohn, The critical manifold of the Lorentz-Dirac equation, Europhys. Lett. 50, 287–292 (2000).\n[11] R.F. O’Connell, The equation of motion of an electron, Phys. Lett. A313, 491–497 (2003).\n[12] H. Spohn, Dynamics of charged particles and their radiation field (Cambridge University Press, Cambridge,\n2004).\n[13] A.D. Yaghjian, Relativistic dynamics of a charged sphere (Lect. Notes Phys.686, Springer, New York, 2006)\n2nd ed.\n[14] F. Rohrlich, Classical charged particles (World Scientific, Singapore, 2007) 3rd ed.\n[15] R. Mares,P.I.Ram´ ırez-Baca,G.Aresde Parga,Lorentz–Dir acandLandau-Lifshitzequationswithout mass\nrenormalization: ansatz of Pauli and renormalization of the force, J. Vectorial Relativity 5, 1–8 (2010).\n[16] G.Ares-de-Parga,R.Mares,M.Ortiz-Dom´ ınguez,Astudyo fthecentralfieldproblembyusingtheLandau–\nLifshitz equation of motion for a charged particle in classical electro dynamics, J. Vectorial Relativity 5,\n26–33 (2010).\n[17] A. Kar, S.G. Rajeev, On the relativistic classical motion of a radia ting spinning particle in a magnetic field,\nAnnals of Physics 326, 958–967 (2011).\n[18] P. Forg´ acs, T. Herpay, P. Kov´ acs, Comment on “Finite Size C orrections to the Radiation Reaction Force\nin Classical Electrodynamics”, Phys. Rev. Lett. 109, 029501 (2012).\n[19] L.D. Landau, E.M. Lifshitz: Classical theory of fields (Butterworth-Heinemann, Oxford, 1982).\n[20] J. Polonyi, Effective dynamics of a classical point charge, Annals of Physics 342, 239–263 (2014).\n13"    },    {        "title": "2108.02090v1.Nonlinear_fluid_damping_of_elastically_mounted_pitching_wings_in_quiescent_water.pdf",        "content": "This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1\nNonlinear fluid damping of elastically mounted\npitching wings in quiescent water\nYuanhang Zhu1y, Varghese Mathai2and Kenneth Breuer1\n1Center for Fluid Mechanics, School of Engineering, Brown University, Providence, RI 02912, USA\n2Department of Physics, University of Massachusetts, Amherst, MA 01003, USA\n(Received xx; revised xx; accepted xx)\nWe experimentally study the nonlinear fluid damping of a rigid but elastically mounted\npitching wing in the absence of a freestream flow. The dynamics of the elastic mount\nare simulated using a cyber-physical system. We perturb the wing and measure the fluid\ndamping coefficient from damped oscillations over a large range of pitching frequencies,\npitching amplitudes, pivot locations and sweep angles. A universal fluid damping scaling\nis proposed to incorporate all these parameters. Flow fields obtained using particle image\nvelocimetry are analyzed to explain the nonlinear behaviors of the fluid damping.\n1. Introduction\nThe interaction between elastically mounted pitching wings and unsteady flows is central\nto many applications. With a free-stream flow, this interaction can lead to self-sustained,\nflow-induced oscillations, which have been studied for understanding classic aeroelastic\nbehaviour (Dowell et al.1989; Dugundji 2008), as well as in developing oscillating foil\nenergy harvesting devices (Xiao & Zhu 2014; Young et al.2014). Without a free stream,\nbut with prescribed heaving or flapping (i.e. hovering), the passive flow-induced pitching\nmotionsareusedinmodellingthethrustgenerationandmaneuveringinanimalflight(Wang\n2005; Bergou et al.2007; Shinde & Arakeri 2013; Kang & Shyy 2014; Beatus & Cohen\n2015).\nOne of the critical parameters that govern the flow-structure interactions of passively\npitching wings is the fluid damping. According to the semi-empirical Morison equation\n(Morison etal.1950),thetotalfluidforceexertedonawingsubmergedinunsteadyviscous\nfluid can be divided into two parts – the force associated with fluid inertia (i.e. the added\nmass force), which is in phase with acceleration (Brennen 1982; Corkery et al.2019), and\nthe force induced by vortices in the flow (i.e. the fluid damping force), which is in phase\nwith velocity (Shih & Buchanan 1971; Kang & Shyy 2014; Su & Breuer 2019). While the\nstructuraldampingforceistypicallyproportionaltovelocitybecauseoftheconstantstructural\ndampingcoefficient,thefluiddampingforceisexpectedtoscalequadraticallywithvelocity\n(Morison et al.1950; Keulegan & Carpenter 1958), and due to this nonlinearity, the fluid\ndamping coefficient is usually obtained empirically as a function of the reduced frequency,\nthe Reynolds number, the oscillation amplitude, etc (Shih & Buchanan 1971). For pitching\nflexible wings (Alben 2008) and heaving membrane wings (Tzezana & Breuer 2019), the\nfluid damping coefficient is found to scale inversely with the oscillation frequency.\nyEmail address for correspondence: yuanhang_zhu@brown.eduarXiv:2108.02090v1  [physics.flu-dyn]  4 Aug 20212 Y. Zhu, V. Mathai and K. Breuer\nFor elastically mounted pitching wings with a free stream, the interplay between the fluid\ndamping and the structural damping governs the flow-induced oscillation. By mapping out\nthe cycle-averaged energy transfer between the elastic system and the ambient fluid using\nprescribedkinematics,Menon&Mittal(2019)andZhu etal.(2020)showedthattheenergy\ninjectedbythenegativefluiddampingmustbeequaltotheenergydissipatedbythepositive\nstructural damping in order for the flow-induced oscillations to sustain. In other words,\nthe total damping of the system must be zero (Dugundji 2008). The negative fluid damping\narisesprimarilyfromtheformationandsheddingofdynamicstallvortices(McCroskey1982;\nCorke&Thomas2015).Intheabsenceofafreestream,however,thefluiddampingbecomes\npositive and counteracts the pitching motion because of the drag effect. With both the fluid\ndamping and the structural damping being positive, any perturbations to the system will\nbe damped out. However, little is known about how the fluid damping shapes the damped\noscillations, and understanding this is of critical importance for understanding the fluid-\nstructure interactions of elastically mounted pitching wings under external perturbations\nsuch as gusts.\nIn the present study, we use laboratory experiments to characterise the fluid damping\nof elastically mounted pitching wings in quiescent water, with the elastic mount simulated\nusingacyber-physicalsystem(§2).Weperform‘ringdown’experimentstoextractthefluid\ndamping(§3.1).Theeffectsofmanyparametersareinvestigated,includingtheeffectsofthe\npitching frequency, the pitching amplitude, the pivot location and the sweep angle (§3.2).\nWe propose a universal fluid damping scaling to incorporate these parameters (§3.3), and\ncorrelate the nonlinear behaviour of the fluid damping with the dynamics of the vortical\nstructures measured using particle image velocimetry (§3.4). Finally, the key findings are\nsummarised in §4.\n2. Experimental set-up\nFigure 1(a) shows a schematic of the experimental set-up. We conduct all the experiments\nin the Brown University free-surface water tunnel (test section width\u0002depth\u0002length=\n08 m\u000206 m\u000240 m), with the flow speed kept at zero ( 𝑈1=0m/s). A NACA 0012\nwing, made of clear acrylic, is mounted vertically in the tunnel, with an endplate on the\ntop to skim surface waves and eliminate wingtip vortices at the root. The wing is connected\nto a six-axis force/torque transducer (ATI 9105-TIF-Delta-IP65), which measures the fluid\ntorque𝜏𝑓exerted on the wing. This 𝜏𝑓is then fed into the cyber-physical system (CPS).\nDepending on the input virtual structural parameters, specifically the torsional stiffness 𝑘𝑣,\ndamping𝑏𝑣andinertia𝐼𝑣,theCPScalculatesthepitchingpositionofthewingandoutputs\nthe signal to the servo motor (Parker SM233AE). An optical encoder (US Digital E3-2500)\nwhich is independent of the CPS is used to measure the pitching position 𝜃. The CPS is\noperated at 4000 Hz to minimise any phase delay between the input 𝜏𝑓and the output 𝜃. A\ndetailed explanation of the CPS can be found in Zhu et al.(2020).\nWeusetwo-dimensionalparticleimagevelocimetry(PIV)tomeasuretheflowfieldaround\nthewing.Theflowisseededusing50 𝜇mdiameterhollowceramicspheresandilluminated\nbyalasersheetatthemid-spanplane.Thelasersheetisgeneratedbyadouble-pulseNd:YAG\nlaser(532nm,QuantelEverGreen)withLaVisionsheetoptics.Thetransparentwingenables\nflow field measurements on both sides of the wing. Due to the limitation of space beneath\nthe tunnel, a 45\u000emirror is used to reflect the images into two co-planar sCMOS cameras\n(LaVision).WeusetheDaVissoftware(LaVision)tocalculate(twopassesat 64\u000264pixels,\ntwo passes at 32\u000232pixels, both with 50% overlap) and stitch the velocity fields from the\ntwo cameras to form a field of view of 32𝑐\u000232𝑐, where𝑐is the chord length of the wing.\nFigure 1(b) sketches the two types of wings we use in the present study. For the unsweptNonlinear fluid damping of pitching wings 3\nServo motor\n& gearbox\nForce \ntransducer\nLaser sheet\n@ mid-spanEndplate\n2×sCMOS\nwith 35mm lensMirrorOptical encoder\nU   = 0 m/sNACA 0012 \n(transparent)CPS\nUnswept wing\nΛx/c0 10.5\nSwept wing(a) (b)\ncs\ncsLE \nLE TE \nTE k\nb\nIv\nv\nv\n∞\nF/i.pc/g.pc/u.pc/r.pc/e.pc 1. (a) A schematic of the experimental set-up. The structural dynamics of the wing is simulated by\na cyber-physical system (CPS). ( b) Sketches of unswept and swept wings. The leading edge (LE) and the\ntrailing edge (TE) are parallel. Dashed lines represent the pivot axis.\nwing, a wing holder mechanism (not shown) enables the pivot axis to be adjusted between\n𝑥𝑐=0and 1 with a step size of 0.125. For the swept wings, the sweep angle Λis defined\nastheanglebetweentheleadingedgeandtheverticalaxis.Foursweptwingswith Λ=10\u000e,\n15\u000e,20\u000eand25\u000eareused.Asshowninthefigure,thepivotaxisofsweptwingsisavertical\nline passing through the mid-chord point ( 𝑥𝑐=05) of the mid-span plane. All the wings\nhaveaspanof 𝑠=03mandachordlengthof 𝑐=01m,whichresultsinanaspectratioof\n𝐴𝑅=3.\nThe governing equation of the system is\n𝐼¥𝜃¸𝑏¤𝜃¸𝑘𝜃=𝜏𝑓 (2.1)\nwhere𝜃,¤𝜃,and¥𝜃aretheangularposition,velocityandacceleration,respectively. 𝐼,𝑏and𝑘\naretheeffectiveinertia,dampingandstiffnessofthesystem.Theeffectiveinertia 𝐼isthesum\nof the virtual inertia 𝐼𝑣, which we prescribe with the CPS, and the physical inertia 𝐼𝑝of the\nwing (i.e.𝐼=𝐼𝑣¸𝐼𝑝). The effective damping 𝑏equals the virtual damping 𝑏𝑣(i.e.𝑏=𝑏𝑣)\nbecause the friction in the system is negligible. The effective stiffness 𝑘equals the virtual\nstiffness(i.e. 𝑘=𝑘𝑣).𝜏𝑓isthenonlinearfluidtorqueexperiencedbythewing,whichcanbe\ndivided into the added mass torque, 𝜏𝑎=\u0000𝐼𝑎¥𝜃, where𝐼𝑎is the added fluid inertia, and the\nfluid damping torque, for simplicity 𝜏𝑏=\u0000𝑏𝑓¤𝜃, where𝑏𝑓is the fluid damping coefficient\n(see §1). Note that 𝑏𝑓is expected to be a function of ¤𝜃(Mathaiet al.2019). Equation 2.1\ncan thus be rearranged as\n¹𝐼¸𝐼𝑎º¥𝜃¸¹𝑏¸𝑏𝑓º¤𝜃¸𝑘𝜃=0 (2.2)\nAfter a perturbation of amplitude 𝐴0is applied at time 𝑡0, the damped oscillations of the\nsystem can be described as\n𝜃=𝐴0𝑒\u0000𝛾¹𝑡\u0000𝑡0ºcos»2𝜋𝑓𝑝¹𝑡\u0000𝑡0º¼ (2.3)\nwhere\n𝛾=𝑏¸𝑏𝑓\n2¹𝐼¸𝐼𝑎ºand𝑓𝑝=1\n2𝜋√︂\n𝑘\n𝐼¸𝐼𝑎\u0000𝛾2¹24abº4 Y. Zhu, V. Mathai and K. Breuer\n0 50 100 150-2-1012\n0 0.5 1 1.5 2 2.5024610-3\n51 5501\n0 1 20.380.400.42(a) (b)\nn+1 n\nF/i.pc/g.pc/u.pc/r.pc/e.pc 2. (a) System response and amplitude decay in a typical ‘ring down’ test, where an elastically\nmounted unswept wing ( Λ=0\u000e) pivots around the mid-chord ( 𝑥𝑐=05) at a frequency of 𝑓𝑝=040Hz.\nThe inset shows the measurements of the pitching amplitude 𝐴𝑛and the pitching frequency 𝑓𝑝of the𝑛-th\npeak. The fluid damping 𝑏𝑓at𝐴𝑛is extracted by fitting an exponential curve (i.e. the red solid line) to the\nadjacent three peaks. ( b) Extracted𝑏𝑓in air and in water. The zero value is indicated by the black dashed\nline. The inset compares the measured pitching frequency 𝑓𝑝in air and in water.\n3. Results and Discussion\n3.1.Extracting the fluid damping from ‘ring down’ experiments\nWeconduct‘ringdown’experimentstomeasurethefluiddampingexperiencedbyelastically\nmountedpitchingwings.Inthe‘ringdown’experiment,ashort-timeconstant-torqueimpulse\nisappliedtotheCPSastheperturbation,afterwhichthesystemresponseandtheamplitude\ndecay of the wing is recorded and analysed. Figure 2( a) shows the results from a typical\n‘ringdown’experiment.Inthisspecificcase,weuseanunsweptwing( Λ=0\u000e)whichpivots\naround the mid-chord ( 𝑥𝑐=05) at a frequency of 𝑓𝑝=040Hz. We conduct the ‘ring\ndown’experimenttwice–onceinairandonceinwater.Thepitchingamplitudeofthewing\ndecays faster in water than in air, indicating a higher total damping in water.\nToquantifythisamplitudedecay,thepositivepeaksofthesystemresponseareidentified.\nAsshownintheinset,theamplitudeofthe 𝑛-thpeakisdenotedby 𝐴𝑛,andthecorresponding\npitchingfrequencyismeasuredas 𝑓𝑝=2¹𝑡𝑛¸1\u0000𝑡𝑛\u00001º.Tomeasurethetotaldamping 𝑏¸𝑏𝑓\nat amplitude 𝐴𝑛, we fit an exponential, 𝑦=𝛼𝑒\u0000𝛾𝑡, to the three adjacent peaks, 𝑛\u00001,𝑛and\n𝑛¸1,andextractthecorresponding 𝛾(seeequation2.3).Nowtheonlyunknowninequation\n2.4bistheaddedmass, 𝐼𝑎.Afterobtaining 𝐼𝑎,thefluiddamping, 𝑏𝑓,isthencalculatedusing\nequation 2.4 a(Rao 1995). Since 𝑓𝑝and𝛾are both measured, 𝐼𝑎and𝑏𝑓are alsomeasured\nquantities. Moreover, both 𝐼𝑎and𝑏𝑓arecycle-averaged , meaning they cannot reflect the\ninstantaneousvariationofthefluidinertiaanddamping.Themeasuredfluiddamping, 𝑏𝑓,in\nbothairandwaterarecomparedinfigure2( b).Since𝜏𝑓inequation2.1isnegligibleinairas\ncomparedtootherforcesintheequation, 𝑏𝑓staysnearzero,whichisindicatedbythegood\nagreementbetweentheredcirclesandtheblackdashedline.Asshownbythegreensquares,\n𝑏𝑓in water is significant because of the existence of the fluid damping torque, 𝜏𝑏. It is also\nobserved that 𝑏𝑓in water increases non-monotonically with 𝐴. This nonlinear behaviour\nwillberevisitedlaterin§3.4.Theinsetoffigure2( b)showsthemeasuredpitchingfrequency,\n𝑓𝑝, in both air and water. Due to the combined effect of the fluid inertia and damping, we\nsee that𝑓𝑝is slightly lower in water than in air.Nonlinear fluid damping of pitching wings 5\n0 0.5 1 1.5 2 2.5051015\n0 0.5 1 1.5 2 2.504812(a) (b) 10-310-3\nF/i.pc/g.pc/u.pc/r.pc/e.pc 3. (a) Extracted fluid damping 𝑏𝑓at different pitching frequencies for 𝑥𝑐=05. (b) A frequency\nscaling for the fluid damping which collapses 𝑏𝑓at different𝑓𝑝into one curve. Note that ( a) and (b) share\nthe same legend.\n3.2.Frequency scaling of the fluid damping\nWe repeat the ‘ring down’ experiment for the unswept wing ( Λ = 0\u000e) pivoting at the mid-\nchord,𝑥𝑐=05, and change the pitching frequency by tuning the virtual inertia, 𝐼𝑣, and\nthe virtual stiffness, 𝑘𝑣, while keeping the virtual damping 𝑏𝑣constant (Onoue & Breuer\n2016, 2017). The extracted fluid damping, 𝑏𝑓, are shown in figure 3( a). Note that figure\n3(a) and (b) share the same legend. We observe that 𝑏𝑓increases monotonically with the\npitching frequency, 𝑓𝑝, and that the trend of 𝑏𝑓remains consistent for all frequencies.\nThis observation agrees with those observed in heaving rigid plates (Keulegan & Carpenter\n1958; Shih & Buchanan 1971), where the fluid damping coefficient scales inversely with\nthe oscillation period. As we discussed earlier, 𝑏𝑓derives from the fluid damping torque\n𝜏𝑏, which depends strongly on the vortex-induced forces on the wing (Kang & Shyy 2014).\nOnoue & Breuer (2016, 2017) have shown that the circulation of LEVs scales with the\nstrength of the feeding shear-layer velocity. In our case without a free-stream flow, the\nfeedingshear-layer velocityequals theleading-/trailing-edgevelocity, whichisproportional\nto𝑓𝑝. Based on this, we divide 𝑏𝑓by𝑓𝑝(figure 3b). It is seen that with this scaling, all of\nthe fluid damping curves collapse nicely.\nWe extend this frequency scaling to unswept wings with different pivot axes (figure 4 a)\nand to swept wings with different sweep angles (figure 4 b). For comparison, we include the\nprevious results (figure 3 b) using purple circles in both figure 4( a) and (b). Note that each\nsymbol shape in figure 4 contains fivedifferent pitching frequencies, 𝑓𝑝=020, 0.28, 0.40,\n0.56 and 0.78 Hz.\nFor the unswept wing ( Λ = 0\u000e), we change the pivot axis from 𝑥𝑐=0to 1 with a step\nsizeof0.125(seetheinsetoffigure4 a).Weobservethat 𝑏𝑓𝑓𝑝increasesasthepivotaxisis\nmoved away from the mid-chord, 𝑥𝑐=05. For pivot axes that are symmetric with respect\nto the mid-chord (i.e. 𝑥𝑐=0375& 0.625, 0.25 & 0.75, 0.125 & 0.875 and 0 & 1), 𝑏𝑓𝑓𝑝\nroughlyoverlap.Theslightinconsistencybetween 𝑏𝑓𝑓𝑝for𝑥𝑐¡05and𝑥𝑐05comes\nfromtheasymmetryoftheNACA0012winggeometrywithrespecttothemid-chord;wesee\nthat the scaled damping, 𝑏𝑓𝑓𝑝, is always slightly higher for 𝑥𝑐 05. In these cases, the\ndamping at the trailing edge dominates due to the higher velocity and longer moment arm,\nand is stronger than the cases when 𝑥𝑐 ¡05, where the leading-edge damping dominates.\nWe will show in §3.4 that this is due to differences in the vortex structures generated by the\nsharp and rounded geometries.\nThisfrequencyscaling, 𝑏𝑓𝑓𝑝,alsoholdsforthree-dimensional(3D)sweptwings(figure6 Y. Zhu, V. Mathai and K. Breuer\n0 0.5 1 1.5 2 2.500.030.060.09\n0 0.5 1 1.5 2 2.500.050.100.15\n0 0.5 1(a) ( b)\nNACA 0012 \nΛPivot axis\nF/i.pc/g.pc/u.pc/r.pc/e.pc 4. (a)𝑏𝑓𝑓𝑝for an unswept wing ( Λ=0\u000e) pivoting at 𝑥𝑐=0to 1 with a step size of 0.125. The\npivotlocationforeachdatasetisshownbytheinset.( b)𝑏𝑓𝑓𝑝forsweptwingswith Λ=0\u000e,10\u000e,15\u000e,20\u000e\nand25\u000e.Theinsetshowssideviewsofthefivesweptwingsandthedashedlineindicatesthepivotaxis.The\ncolours of the wings correspond to the colours of 𝑏𝑓𝑓𝑝curves in the figure. The purple circles in ( a) and\n(b) are replotted from figure 3( b). Note that each dataset in ( a) and (b) includes fivedifferent𝑓𝑝.\n4b). Again, each curve includes data from five pitching frequencies. Here, the pivot axes\nof swept wings are kept as a vertical line passing through the mid-chord of the mid-span\nplane (see the inset of figure 4 b). AsΛincreases, the average pivot axes of the top and the\nbottom portion of the swept wing move away from the mid-chord, leading to the increase\nof the scaled damping, 𝑏𝑓𝑓𝑝, in a manner similar to that observed for unswept wings with\ndifferent pivot locations (figure 4 a). This argument will be revisited in the next section.\n3.3.Universal fluid damping scaling for unswept and swept wings\nFigure 4( a) indicates that the pivot axis plays an important role in determining the fluid\ndamping of unswept wings. We extend the frequency scaling of 𝑏𝑓to take into account\nthis effect. First, we divide the wing into two parts, the fore part from LE to the pivot axis\nwith a chord length of 𝑐𝐿𝐸, and the aft part from the pivot axis to TE with a chord length\nof𝑐𝑇𝐸(see the inset of figure 5 for an example when the wing pivots at 𝑥𝑐=05). The\nMorisonequation(Morison etal.1950)indicatesthatthefluiddampingforce 𝐹scaleswith\n05𝜌𝑈2𝑠𝑐, where𝜌is the fluid density, 𝑈\u0018¤𝜃𝑐is the characteristic velocity and 𝑠𝑐is the\nwingarea.Wecanexpressthetotalfluiddampingtorqueasthesumofthetorqueexertedon\nthe fore and aft portions of the wing,\n𝜏𝑏\u0018𝐾𝐿𝐸𝐹𝐿𝐸𝑐𝐿𝐸¸𝐾𝑇𝐸𝐹𝑇𝐸𝑐𝑇𝐸 (3.1)\nwhere the subscripts 𝐿𝐸and𝑇𝐸refer to the leading- and trailing-edge contributions, and\n𝐾𝐿𝐸and𝐾𝑇𝐸are empirical factors that account for the subtle differences in the damping\nassociated with the specific geometries of the leading and trailing edges (figure 4 a). Since\nthe differences are small, 𝐾𝐿𝐸and𝐾𝑇𝐸should be close to one, and for consistency, their\naverage value must equal one ( ¹𝐾𝐿𝐸¸𝐾𝑇𝐸º2=1).\nSince the damped oscillations are observed to be near-sinusoidal (figure 2 a), the average\nangular velocity is given by 4𝑓𝑝𝐴. Simplifying, we arrive at an expression for the fluid\ndamping:\n𝑏𝑓\u00182𝜌𝑓𝑝𝐴𝑠¹𝐾𝐿𝐸𝑐4\n𝐿𝐸¸𝐾𝑇𝐸𝑐4\n𝑇𝐸º (3.2)Nonlinear fluid damping of pitching wings 7\n0 0.5 1 1.5 2 2.50123\ncLE cTE s\ncLE,botcTE,bots/2 s/2 ΛcLE,topcTE,topFLE FTE \nF/i.pc/g.pc/u.pc/r.pc/e.pc 5. Non-dimensional fluid damping coefficient 𝐵\u0003\n𝑓versus pitching amplitude 𝐴for unswept wings\npivotingat𝑥𝑐=0to1andsweptwingswithsweepangles Λ=0\u000eto25\u000e.Theinsetshowsthedefinitionof\nthe leading-edge chord 𝑐𝐿𝐸and the trailing-edge chord 𝑐𝑇𝐸, with black dashed lines indicating the pivot\naxes. The black dotted line indicates the small amplitude prediction for a drag coefficient of 𝐶𝐷=28.\nor, in non-dimensional form,\n𝐵\u0003\n𝑓\u0011𝑏𝑓\n2𝜌𝑓𝑝𝑠¹𝐾𝐿𝐸𝑐4\n𝐿𝐸¸𝐾𝑇𝐸𝑐4\n𝑇𝐸º/𝐴 (3.3)\nForsweptwings,becausethepivotaxispassesthrough 𝑥𝑐=05atthemid-span,thetop\nhalf of the wing has an average pivot axis 𝑥𝑐 ¡05, while the bottom half has an average\npivot axis𝑥𝑐 05. Ignoring three-dimensional effects, we approximate the swept wing\nby two ‘equivalent’ unswept wing segments. We choose not to divide the wing into a large\nnumberofnarrow‘bladeelements’(Glauert1983),becausethepivotaxisofsomeelements\nnear the wing root/tip for large sweep angles may lie outside the range 𝑥𝑐=»01¼, where\nour scaling has not been tested. The inset of figure 5 shows how these two unswept wing\nsegmentsareconfigured(rectangleswithreddottedlines).Basedonthewinggeometry,we\nsee that\n𝑐𝐿𝐸𝑡𝑜𝑝=𝑐𝑇𝐸𝑏𝑜𝑡=𝑐\n2¸𝑠\n4tanΛ\n𝑐𝑇𝐸𝑡𝑜𝑝=𝑐𝐿𝐸𝑏𝑜𝑡=𝑐\n2\u0000𝑠\n4tanΛ(3.4)\nFollowing the same analysis as for the unswept wing, and adding the fluid damping of the\ntopandthebottomwingsegmentstogether,wefindthatthefluiddampingforthefullswept\nwing is given by\n𝑏𝑓\u0018𝜌𝑓𝑝𝐴𝑠¹𝐾𝐿𝐸𝑐4\n𝐿𝐸𝑡𝑜𝑝¸𝐾𝑇𝐸𝑐4\n𝑇𝐸𝑡𝑜𝑝¸𝐾𝐿𝐸𝑐4\n𝐿𝐸𝑏𝑜𝑡¸𝐾𝑇𝐸𝑐4\n𝑇𝐸𝑏𝑜𝑡º(3.5)\nIf we define an effective leading-edge chord 𝑐𝐿𝐸=𝑐𝐿𝐸𝑡𝑜𝑝=𝑐𝑇𝐸𝑏𝑜𝑡and an effective\ntrailing-edgechord 𝑐𝑇𝐸=𝑐𝑇𝐸𝑡𝑜𝑝=𝑐𝐿𝐸𝑏𝑜𝑡,thisscalingreducestoequation3.2with 𝐾𝐿𝐸\nand𝐾𝑇𝐸cancelled out. This cancellation results because the effective pivot axes of the top\nand the bottom segments are symmetric about 𝑥𝑐=05at the mid-span, which averages\nout the slight differences in fluid damping experienced by the top and the bottom segments.\nFor the same reason, 𝐾𝐿𝐸and𝐾𝑇𝐸also cancel out in equation 3.3 for swept wings.8 Y. Zhu, V. Mathai and K. Breuer\n(a)\n-1 0 1 (b) (c) (d)\n(e)\n-1 0 1 -1 0 1 (f)\n-1 0 1 (g)\n-1 0 1 (h)\n-1 0 1 \n-50-25025 50 \nLEVTEV\nLEVTEVLEVTEVLEVTEV\nLEVTEV\nLEV TEV LEVTEVLEV\nTEV= 0.52 = 1.05 = 1.57 = 2.09\nF/i.pc/g.pc/u.pc/r.pc/e.pc 6. PIV flow field measurements for an unswept wing undergoing prescribed sinusoidal pitching\nmotions in quiescent water. ( a–d) Pivot axis (shown by green dots) 𝑥𝑐=05, pitching frequency 𝑓𝑝=05\nHz, pitching amplitude 𝐴=052¹30\u000eº105¹60\u000eº157¹90\u000eºand209¹120\u000eº. (e–h) Same as ( a–d),\nexcept that the pivot axis is at 𝑥𝑐=025. All the velocity fields are phase-averaged over 20 cycles. Only\neveryfifthvelocityvectorisshown.Spanwisevorticity 𝜔:positive(red),counterclockwise;negative(blue),\nclockwise. See supplementary materials for the full video.\nFigure 5 shows the non-dimensional fluid damping, 𝐵\u0003\n𝑓, as a function of the pitching\namplitude,𝐴,forunsweptandsweptwings.Here,wehaveused 𝐾𝐿𝐸=095and𝐾𝑇𝐸=105.\nWe see that all of our measurements collapse remarkably well under the proposed scaling,\nespecially for 𝐴  157¹90\u000eº, despite the wide range of pitching frequencies ( 𝑓𝑝=020\nto 0.78 Hz), pivot axes ( 𝑥𝑐=0to 1) and sweep angles ( Λ = 0\u000eto25\u000e) tested in the\nexperiments.Inthesmall-amplitudelimit( 𝐴05),𝐵\u0003\n𝑓scaleslinearlywith 𝐴,withaslope\nthat corresponds to the drag coefficient, 𝐶𝐷. We note that 𝐶𝐷\u001928, which is comparable\nto that of an accelerated normal flat plate (Ringuette et al.2007). At higher pitching angles\n(𝐴¡05),however,thelinearapproximationnolongerholdsandweseeadecreasingslope\nof𝐵\u0003\n𝑓as a function of 𝐴. This is presumably because the shed vortices no longer follow\nthe rotating wing and the fluid force becomes non-perpendicular to the wing surface as 𝐴\nincreases.For 𝐴¡157¹90\u000eº,thescalingworksreasonablywellexceptforthecase Λ=0\u000e,\n𝑥𝑐=05,whereadecreasing 𝐵\u0003\n𝑓isobserved.Inthenextsection,wewilluseinsightsfrom\nthe velocity fields to explain this non-monotonic behaviour.\n3.4.Insights obtained from velocity fields\nTo gain more insight regarding the nonlinear behaviour of 𝐵\u0003\n𝑓, we conduct 2D PIV\nexperiments to measure the surrounding flow fields of an unswept wing ( Λ = 0\u000e) with\na prescribed pitching motion: 𝜃=𝐴sin¹2𝜋𝑓𝑝𝑡º. The results are shown in figure 6. The\npitching frequency is kept at 𝑓𝑝=05Hz for all the cases and the pitching amplitude is\nvariedfrom𝐴=052¹30\u000eºto209¹120\u000eºwithastepsizeof 052¹30\u000eº.Twopivotaxesare\ntested,𝑥𝑐=05(figure6a–d)and𝑥𝑐=025(figure6e–h).Notethattheflowfieldsshown\nin figure 6 are notsequential. Instead, all the snapshots are taken right before 𝑡𝑇=025\nfor different pitching amplitudes, where 𝑇is the pitching period. This specific time instant\nis chosen because it best reflects the difference in dynamics associated with the different\npitching amplitudes and pivot axes.\nFor both pivot locations (figure 6 a–d:𝑥𝑐=05ande–h:𝑥𝑐=025), the spanwise\nvorticity of the pitch-generated leading-edge vortex (LEV) and trailing-edge vortex (TEV)Nonlinear fluid damping of pitching wings 9\nincreases with the pitching amplitude, 𝐴. This can be explained by the increase in the\nfeeding shear-layer velocities associated with the higher pitching amplitudes (Onoue &\nBreuer 2016). The boundary vortices near the wing surface, which are related to the added\nmass effect (Corkery et al.2019), also become more prominent due to the increase of the\nangular acceleration. When the wing pivots at 𝑥𝑐=05(figure 6a–d), the leading-edge\nvelocityequalsthetrailing-edgevelocity.Asaresult,theLEVandTEVarefairlysymmetric\nabout the pivot axis, with some subtle differences caused by the rounded and sharp edges,\nrespectively. This confirms the arguments given earlier for the differences between 𝑏𝑓𝑓𝑝\nfor𝑥𝑐 ¡05and𝑥𝑐 05(figure 4a). For𝑥𝑐=025(figure 6e–h), however, the TEV is\nmuchmoreprominentthantheLEVbecauseofthehighertrailing-edgevelocity.Duetothe\nlow leading-edge velocity and the pitch-induced rotational flow, the sign of the LEV even\nreverses and becomes negative for 𝐴=105to209(figure 6f–h).\nForbothpivotlocations,duetotheabsenceofaconvectivefreestream,andtheexistence\nof the pitch-induced rotational flow, the LEV and TEV (only the TEV for 𝑥𝑐=025) are\nentrainedclosertothewingsurfaceas 𝐴increases.For 𝑥𝑐=05,asshowninfigure6( c–d),\nthe LEV moves towards the aft portion of the wing and the TEV moves towards the fore\nportionofthewingwhen 𝐴>157¹90\u000eº.Thetorquegeneratedbythesetwovortices,which\ncounteracts the wing rotation for small 𝐴, now assists the rotation as the wing pitches up\ntowardshigherangularpositions.Thisassistreducesthefluiddragexperiencedbythewing\nandthuslowersthefluiddamping.Thiseffectcanaccountforthenon-monotonicbehaviour\nof𝐵\u0003\n𝑓for𝑥𝑐=05(figure 5). For 𝑥𝑐=025(figure 6g–h), a similar scenario is observed,\nin which the TEV moves towards the fore portion of the wing and gets closer to the wing\nsurfaceas𝐴increases.However,becauseoftheexistenceofacounter-rotatingLEV,theTEV\nisnotabletoapproachthewingsurfaceascloselyasinthe 𝑥𝑐=05case.Thisexplainswhy\na flattening behaviour, rather than a non-monotonic trend of 𝐵\u0003\n𝑓, is observed for 𝑥𝑐=025\nand presumably for other pivot locations at high pitching amplitudes.\n4. Conclusions\nBy utilising a cyber-physical control system to create an elastically mounted pitching wing,\nwehaveexperimentallymeasuredthenonlinearfluiddampingassociatedwithvorticesshed\nfrom a bluff body. A theoretical scaling has been proposed and validated, based on the\nMorison equation, which incorporates the frequency, amplitude, pivot location and sweep\nangle. The nonlinear behaviour of the scaled fluid damping has been correlated with the\nvelocity fields measured using particle image velocimetry.\nOne should note that our scaling may not be applicable for instantaneous fluid damping,\nbecausethedampingcharacterisedinthepresentstudyiscycle-averagedovernear-sinusoidal\noscillations.Inaddition,wehavenotconsideredthree-dimensionaleffects,whicharepresent\ndue to the wing tip flows. Incorporating these may further improve the collapse of the fluid\ndamping coefficient, 𝐵\u0003\n𝑓(figure 5).Lastly, in §3.4, onlyqualitative analysis ofthe flow field\nhasbeenperformedthusfar.Inordertogetmoreaccuratecorrespondencebetweenthefluid\ndampingandtheflowdynamics,quantitativeanalysisofthevortextrajectoryandcirculation\nis needed, which will be the focus of future study.\nDespitetheselimitations,theproposedscalinghasbeenshowntocollapsethedataovera\nwiderangeofoperatingconditions( 𝑓𝑝=020to0.78Hzand 𝐴=0to2.5)forbothunswept\n(𝑥𝑐=0to1)andsweptwings( Λ=0\u000eto25\u000e).Itcanbeusedtopredictdampingassociated\nwithshedvortices,andthusbenefitthefuturemodellingofawidevarietyofflows,including\nunswept and swept wings in unsteady flows as well as other bluff body geometries. The\nuniversality of this scaling reinforces the underlying connection between swept wings and10 Y. Zhu, V. Mathai and K. Breuer\nunswept wings with different pivot locations. 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Sci. 67, 2–28.\nZ/h.pc/u.pc, Y., S/u.pc, Y. & B/r.pc/e.pc/u.pc/e.pc/r.pc, K. S. 2020Nonlinearflow-inducedinstabilityofanelasticallymountedpitching\nwing.J. Fluid Mech. 899, A35."    },    {        "title": "1902.05344v1.Lorentz_violation_and_Gravitoelectromagnetism__Casimir_effect_and_Stefan_Boltzmann_law_at_Finite_temperature.pdf",        "content": "arXiv:1902.05344v1  [hep-th]  14 Feb 2019Lorentz violation and Gravitoelectromagnetism: Casimir e ffect and\nStefan-Boltzmann law at Finite temperature\nA. F. Santos1,∗and Faqir C. Khanna†2,‡\n1Instituto de Física, Universidade Federal de Mato Grosso,\n78060-900, Cuiabá, Mato Grosso, Brazil\n2Department of Physics and Astronomy, University of Victori a,\n3800 Finnerty Road Victoria, BC, Canada\nAbstract\nThe standard model and general relativity are local Lorentz invariants. However it is possible that at\nPlanck scale there may be a breakdown of Lorentz symmetry. Mo dels with Lorentz violation are constructed\nusing Standard Model Extension (SME). Here gravitational s ector of the SME is considered to analyze the\nLorentz violation in Gravitoelectromagnetism (GEM). Usin g the energy-momentum tensor, the Stefan-\nBoltzmann law and Casimir effect are calculated at finite temp erature to ascertain the level of local Lorentz\nviolation. Thermo Field Dynamics (TFD) formalism is used to introduce temperature effects.\n†Professor Emeritus - Physics Department, Theoretical Phys ics Institute, University of Alberta\nEdmonton, Alberta, Canada\n∗alesandroferreira@fisica.ufmt.br\n‡khannaf@uvic.ca\n1I. INTRODUCTION\nLorentz and CPT symmetries play a central role in the Standar d Model (SM) and Einstein\nGeneral Relativity (GR). GR is a classical theory that descr ibes the gravitational force. The SM\ndescribes other three fundamental forces that are defined in a quantum version. There are models\nthat seek to unify the two fundamental theories into a single one. Such a theory is expected to\nemerge at the Planck scale, ∼1019GeV, where some new physics may emerge. The new physics\nmay involve different properties, such as the appearance of L orentz violation effects [1, 2]. Studies\nof Lorentz violation, both theoretical and experimental, a re described by an effective field theory\ncalled the Standard Model Extension (SME) [3]. The SME inclu des the SM, GR and all possible\noperators that break the Lorentz symmetry. A complete descr iption of GR in the framework of\nthe SME has been considered [4–6]. In the gravitational sect or of the SME [7, 8] there are 19\ncoefficients for Lorentz violation in addition to an unobserv able scalar parameter. A similarity\nbetween the gravitational sector and the electromagnetic s ector of the SME, specifically CPT-even\ncoefficients, has been developed [9]. This would suggest a clo se relationship between gravitational\nand CPT-even electromagnetic sectors.\nThe search for analogies between electromagnetism and grav ity, for Lorentz invariant theories,\nstarted with Faraday [10] and Maxwell [11] and has a long hist ory [12–20]. For a review of Grav-\nitoelectromagnetism (GEM) follow references [21]. Experi mental efforts to test GEM have been\ndeveloped [22]. There are three different ways to construct G EM theory: (i) using the similarity\nbetween the linearized Einstein and Maxwell equations [21] ; (ii) based on an approach using tidal\ntensors [23] and (iii) the decomposition of the Weyl tensor i nto gravitomagnetic ( Bij=1\n2ǫiklCkl0j)\nand gravitoelectric ( Eij=−C0i0j) approach [24]. Here Weyl approach is considered. The Weyl\ntensor is connected with the curvature tensor and it is the tr ace-less part of the Riemann ten-\nsor. The analogy between electromagnetism and General Rela tivity is based on the correspondence\nCασµν↔Fασ, where the Weyl tensor is the free gravitational field and Fασis the electromagnetic\ntensor. The Weyl tensor gives contributions due to nonlocal sources. In the Weyl tensor approach,\na Lagrangian formulation for GEM has been developed [25]. In this formalism a symmetric gravito-\nelectromagnetic tensor potential, Aµν, which describes the gravitational interaction, is defined . For\nexample, the GEM theory at finite temperature has been analyz ed [26]. The gravitational Bhabha\nscattering has been calculated [27]. Using the Lagrangian f ormalism for GEM, our main objective\nin this paper is to calculate contributions of the Lorentz vi olation to the Casimir effect and the\nStefan-Boltzmann law of the GEM theory.\n2The Casimir effect is the interaction between two parallel co nducting plates [28]. The attraction\nbetween plates is the result of electromagnetic modes due to boundary conditions or topological\neffects. Initially this effect was predicted for the electrom agnetic field. However now it has been\ndefined for any quantum field. The Casimir effect was confirmed e xperimentally first by Sparnaay\n[29]. Now high degree of accuracy has been achieved experime ntally [30], [31]. If the gravitational\nfield has a quantum nature, this effect would be expected for gr avitational waves. Using the GEM\nformulation and considering plates that are made of superco nducting material, the gravitational\nCasimir effect has been analyzed [32]. The Casimir effect for G EM at finite temperature has been\ncalculated [33]. In the present study the Casimir energy and pressure and the Stefan-Boltzmann\nlaw for the GEM field with Lorentz-violating corrections at fi nite temperature are calculated. The\nThermo Field Dynamics (TFD) formalism is used to introduce t he finite temperature effects.\nTFD is a real-time finite temperature formalism [34, 35]. Thi s formalism leads to an interpre-\ntation of the statistical average of an arbitrary operator O, as the expectation value in a thermal\nvacuum, i.e., /an}b∇acketle{tO/an}b∇acket∇i}ht=/an}b∇acketle{t0(β)|O|0(β)/an}b∇acket∇i}ht. The thermal vacuum |0(β)/an}b∇acket∇i}htdescribes the thermal equilibrium of\nthe system, where β=1\nkBT,Tis the temperature and kBis the Boltzmann constant. To construct\nthis thermal state two basic elements are necessary: (i) the doubling of the original Fock space and\n(ii) the Bogoliubov transformation. This doubling consists of Fock space composed of the original,\nS, and a fictitious space (tilde space), ˜S. The map between the tilde and non-tilde operators is\ndefined by the tilde (or dual) conjugation rules. The Bogoliub ov transformation is a rotation among\noperators involving these two spaces. Here we use natural un its, i.e.,kB=/planckover2pi1=c= 1.\nThis paper is organized as follows. In section II, a Lagrangi an formulation for GEM is intro-\nduced. In section III, the GEM theory with Lorentz-violatin g parameter is analyzed. The vacuum\nexpectation value of the energy-momentum tensor is calcula ted. In section IV, TFD and some char-\nacteristics of the finite temperature formalism are present ed. In section V, some applications are\ndeveloped. The Stefan-Boltzmann law and the Casimir effect wi th Lorentz-violating corrections at\nzero and finite temperature are calculated. In section VI, so me concluding remarks are presented.\nII. AN INTRODUCTION TO GEM FIELD\nA brief introduction to the lagrangian formulation of GEM is presented in this section. The GEM\ndescribes the dynamics of the gravitational field in a manner similar to that of the electromagnetic\nfield. Here the GEM approach will be used with the Weyl tensor c omponents ( Cijkl) being: Eij=\n−C0i0j(gravitoelectric field) and Bij=1\n2ǫiklCkl\n0j(gravitomagnetic field). The field equations for the\n3components of the Weyl tensor have a structure similar to tho se of Maxwell equations. The GEM\nequations are given as\n∂iEij=−4πGρj, (1)\n∂iBij= 0, (2)\nǫ(i|kl∂kBl|j)+∂Eij\n∂t=−4πGJij, (3)\nǫ(i|kl∂kEl|j)+∂Bij\n∂t= 0, (4)\nwhereGis the gravitational constant, ǫiklis the Levi-Civita symbol, ρjis the vector mass density\nandJijis the mass current density. The symbol (i|···|j)denotes symmetrization of the first and\nlast indices, i.e., iandj.\nA lagrangian formulation for the GEM equations has been cons tructed [25]. In such a construc-\ntion, the fields EijandBijare defined as\nE=−gradϕ−∂˜A\n∂t, B= curl˜A, (5)\nwhere˜Awith components Aµνis a symmetric rank-2 tensor field, gravitoelectromagnetic tensor\npotential, and ϕis the GEM vector counterpart of the electromagnetic scalar potential φ. A\ngravitoelectromagnetic tensor Fµναis defined as\nFµνα=∂µAνα−∂νAµα, (6)\nwhereµ,ν,α= 0,1,2,3. Then GEM equations are written as\n∂µFµνα= 4πGJνα, ∂ µGµ/an}bracketle{tνα/an}bracketri}ht= 0, (7)\nwhereJναdepends on the mass density ( ρi) and the current density ( Jij) andGµναis the dual\nGEM tensor defined as Gµνα=1\n2ǫµνγσηαβFγσβ.Then the GEM lagrangian is\nLG=−1\n16πFµναFµνα−GJναAνα. (8)\nSince the nature of Aµνis different from hµν, we use a different approach. The tensor potential\nis not related to the perturbation of the spacetime. It is con nected directly with the description\nof the gravitational field in flat spacetime. The gauge transf ormation for the tensor potential is\nA′\nµν=Aµν+∂µθν, whereθνis 4-vector. The gravitoelectromagnetic tensor Fµναis invariant under\nthis transformation. Then the GEM Lagrangian is gauge invar iant. For more details see [36].\nTherefore, the gauge transformation in GEM is similar to tha t of electromagnetism.\n4III. LORENTZ-VIOLATING CONTRIBUTIONS TO THE GEM FIELD\nThe Lagrangian that includes the Lorentz-violating contri butions to the GEM field is\nL=−1\n16πFρσθFρσθ−1\n4/parenleftbig\nk(4)/parenrightbig\nκλξρηγθFκλγFξρθ, (9)\nwhere/parenleftbig\nk(4)/parenrightbig\nκλξρis a dimensionless coefficient field that belongs to minimal se ctor of SME gravity\n[4, 37]. This tensor has the same symmetries as the Riemann te nsor and can be decomposed into\n20 coefficients, i.e., sµνwith 9 independent quantities, tµναγthat have symmetries of the Riemann\ncurvature tensor, implying 10 independent quantities and uis a scalar. In the weak field approxi-\nmation, coefficients for Lorentz violation are taken as const ants in a special coordinate system and\nare donated by ¯sµν,¯tµναγand¯u. The¯ucoefficient is not observable. Then the gravitational sector\nhas 19 coefficients. The CPT-even part of the electromagnetic (EM) field sector has 19 Lorentz\nviolation coefficients which are decomposed into two parts: 1 0 birefringent and 9 non-birefringent\ncomponents. In addition, coefficients of the gravity sector a re reminiscent of those for the coefficient\n(kF)µναβin the electromagnetic part of the SME [4]. Therefore, there is a correspondence between\nLorentz violation effects for the EM field and for the weak field gravitational field, i.e. GEM field\n[9]. Here, for simplicity, the calculations are developed c onsidering all components of the tensor\n/parenleftbig\nk(4)/parenrightbig\nκλξρ.\nIn order to calculate the Casimir effect, first the energy-mom entum tensor for the Lagrangian\n(9) is defined as\nTµν=∂L\n∂(∂µAλξ)∂νAλξ−gµνL. (10)\nHere, this tensor is divided into two parts,\nTµν=Tµν\nGEM+Tµν\nLV, (11)\nwhere\nTµν\nGEM=−1\n4πFµλξ∂νAλξ+1\n16πgµνFρσθFρσθ(12)\nis the part that corresponds to the GEM field and\nTµν\nLV=−/parenleftbig\nk(4)/parenrightbigκλµξFκλΛ∂νAξΛ+1\n4gµν/parenleftbig\nk(4)/parenrightbig\nκλξρFκλθFξρθ(13)\nis the Lorentz-violating part. It is to be noted that this ten sor (11) is not symmetric. The Belinfante\nmethod [38] is used to define the Lorentz invariant part. Then the symmetric energy-momentum is\nTµν\nGEM=1\n4π/bracketleftbigg\n−FµλξFνλξ+1\n4gµνFρσθFρσθ/bracketrightbigg\n. (14)\n5The same method is not applicable for the Lorentz violating p art. However this is written as\nTµν\nLV=−/parenleftbig\nk(4)/parenrightbigκλµρFνρΛFκλΛ+1\n4gµν/parenleftbig\nk(4)/parenrightbig\nκλξρFκλθFξρθ. (15)\nThus the total energy-momentum tensor becomes\nTµν=1\n4π/bracketleftbigg\n−FµλξFνλξ+1\n4gµνFρσθFρσθ/bracketrightbigg\n−/parenleftbig\nk(4)/parenrightbigκλµρFνργFκλγ+1\n4gµν/parenleftbig\nk(4)/parenrightbig\nκλξρFκλθFξρθ.(16)\nThis tensor is not completely symmetric. This is a feature of theories which exhibit Lorentz viola-\ntion.\nThe canonical conjugate momentum related to the tensor Aκλis given as\nπκλ=∂L\n∂(∂0Aκλ)=−1\n4πF0κλ. (17)\nAdopting the Coulomb gauge, where A0i= 0anddiv˜A=∂iAij= 0, the covariant quantization is\ncarried out and the commutation relation is\n/bracketleftbig\nAij(x,t),πkl(x′,t)/bracketrightbig\n=i\n2/bracketleftBig\nδikδjl−δilδjk−1\n∇2/parenleftBig\nδjl∂i∂k−δjk∂i∂l−δil∂j∂k+δik∂j∂l/parenrightBig/bracketrightBig\nδ3(x−x′).(18)\nOther commutation relations are zero.\nTo avoid divergences, the energy-momentum tensor is writte n at different space-time points as\nTµν(x) =Tµν\nGEM(x)+Tµν\nLV(x), (19)\nwhere\nTµν\nGEM(x) =1\n4πlim\nx′→x/bracketleftBig\n−Fµλξ,νλξ(x,x′)+1\n4gµνFρσθ,ρσθ(x,x′)/bracketrightBig\n, (20)\nand\nTµν\nLV(x) = lim\nx′→x/bracketleftBig\n−/parenleftbig\nk(4)/parenrightbigκλµρFνργ,κλγ(x,x′)+1\n4gµν/parenleftbig\nk(4)/parenrightbig\nκλξρFκλθ,ξρθ(x,x′)/bracketrightBig\n, (21)\nwhere\nFξκγ,µνρ(x,x′)≡τ/bracketleftBig\nFξκγ(x)Fµνρ(x′)/bracketrightBig\n. (22)\nwithτbeing the time order operator. Using the τoperator explicity\nFξκγ,µνρ(x,x′) =Fξκγ(x)Fµνρ(x′)θ(x0−x′\n0)+Fµνρ(x′)Fξκγ(x)θ(x′\n0−x0), (23)\nwithθ(x0−x′\n0)being the step function. In calculations, the commutation r elation eq. (18) and\n∂µθ(x0−x′\n0) =nµ\n0δ(x0−x′\n0), (24)\n6are used where nµ\n0= (1,0,0,0)is a time-like vector. Then we get\nFξκγ,µνρ(x,x′) = Γξκγ,µνρ,λǫωυ(x,x′)τ/bracketleftbig\nAλǫ(x)Aωυ(x′)/bracketrightbig\n+Iκγ,µνρ(x,x′)nξ\n0δ(x0−x′\n0)−Iξγ,µνρ(x,x′)nκ\n0δ(x0−x′\n0), (25)\nwhere\nΓξκγ,µνρ,λǫωυ(x,x′) =/parenleftBig\ngκλgǫγ∂ξ−gξλgǫγ∂κ/parenrightBig/parenleftbig\ngνωgρυ∂′µ−gµωgρυ∂′ν/parenrightbig\n(26)\nand\nIκγ,µνρ(x,x′) =/bracketleftbig\nAκγ(x),Fµνρ(x′)/bracketrightbig\n(27)\nThen the complete energy-momentum tensor is\nTµν(x) =−lim\nx′→x/braceleftBig/parenleftBig1\n4π∆µν,λǫωυ\nGEM(x,x′)+∆µν,λǫωυ\nLV(x,x′)/parenrightBig\nτ/bracketleftbig\nAλǫ(x)Aωυ(x′)/bracketrightbig/bracerightBig\n, (28)\nwith\n∆µν,λǫωυ\nGEM(x,x′) = Γµρξ,νρξ,λǫωυ(x,x′)−1\n4gµνΓρσθ,ρσθ,λǫωυ(x,x′) (29)\nand\n∆µν,λǫωυ\nLV(x,x′) =/parenleftbig\nk(4)/parenrightbigκλµρΓνργ,κλγ,λǫωυ(x,x′)−1\n4gµν/parenleftbig\nk(4)/parenrightbigκλγρΓκλθ,γρθ,λǫωυ(x,x′).(30)\nThe vacuum expectation value of Tµνis\n/an}b∇acketle{tTµν(x)/an}b∇acket∇i}ht=−lim\nx′→x/braceleftBig/parenleftBig1\n4π∆µν,λǫωυ\nGEM(x,x′)+∆µν,λǫωυ\nLV(x,x′)/parenrightBig\n/an}b∇acketle{t0|τ/bracketleftbig\nAλǫ(x)Aωυ(x′)/bracketrightbig\n|0/an}b∇acket∇i}ht/bracerightBig\n.\nUsing the graviton propagator,\nDλǫωυ(x−x′) =i\n2NλωǫυG0(x−x′), (31)\nwhereNλωǫυ≡ηλωηǫυ+ηλυηǫω−ηλǫηωυandG0(x−x′)is the massless scalar field propagator. An\nimportant note, Lorentz-violating coefficients are small an d hence can be treated perturbatively.\nThus, to obtain first-order corrections in Lorentz-violati ng coefficients, an expansion of the prop-\nagator is considered. By taking the zeroth-order term in Lore ntz-violating parameter the vacuum\nexpectation value of Tµνis\n/an}b∇acketle{tTµν(x)/an}b∇acket∇i}ht=−i\n2lim\nx′→x/braceleftBig/parenleftBig1\n4πΓµν\nGEM+Γµν\nLV/parenrightBig\nG0(x−x′)/bracerightBig\n, (32)\n7where\nΓµν\nGEM(x,x′) = 8/parenleftbigg\n∂µ∂′ν−1\n4gµν∂ρ∂′\nρ/parenrightbigg\n. (33)\nand\nΓµν\nLV(x,x′) = 8/bracketleftBig/parenleftbig\nk(4)/parenrightbigκλµ\nλ∂ν∂′\nκ+/parenleftbig\nk(4)/parenrightbigνλµρ∂ρ∂′\nλ\n−1\n4gµν/parenleftBig/parenleftbig\nk(4)/parenrightbigκλγ\nλ∂κ∂′\nγ+/parenleftbig\nk(4)/parenrightbigλκ\nλρ∂κ∂′\nρ/parenrightBig/bracketrightBig\n. (34)\nUsing the tilde conjugation rules, the vacuum average of Tµνin terms of the α-dependent fields\nis\n/an}b∇acketle{tTµν(ab)(x;α)/an}b∇acket∇i}ht=−i\n2lim\nx′→x/braceleftBig/parenleftBig1\n4πΓµν\nGEM+Γµν\nLV/parenrightBig\nG(ab)\n0(x−x′;α)/bracerightBig\n, (35)\nwith the α-parameter being a compactification parameter defined by α= (α0,α1,···αD−1). Here\na field theory on the topology Γd\nD= (S1)d×RD−dwith1≤d≤Dis considered. Then any set\nof dimensions of the manifold RDcan be compactified, where the circumference of the nthS1is\nspecified by αn.Dare the space-time dimensions and dis the number of compactified dimensions.\nThe physical energy-momentum tensor is defined as\nTµν(ab)(x;α) =/an}b∇acketle{tTµν(ab)(x;α)/an}b∇acket∇i}ht−/an}b∇acketle{tTµν(ab)(x)/an}b∇acket∇i}ht. (36)\nThis definition describes a renormaliation procedure to obt ain measurable physical quantities at\nfinite temperature. Both the energy-momentum tensor at finite and zero temperature are divergent.\nThen by subtracting the energy-momentum tensor at zero temp erature non-divergent results are\nobtained at finite temperature. With this procedure a measur able physical quantity is given by\nTµν(ab)(x;α) =−i\n2lim\nx′→x/braceleftBig/parenleftBig1\n4πΓµν\nGEM+Γµν\nLV/parenrightBig\nG(ab)\n0(x−x′;α)/bracerightBig\n, (37)\nwith\nG(ab)\n0(x−x′;α) =G(ab)\n0(x−x′;α)−G(ab)\n0(x−x′). (38)\nThe relevant component of the Fourier representation is G0(x−x′;α)≡G(11)\n0(x−x′;α)that is\ngiven by\nG0(x−x′;α) =/integraldisplayd4k\n(2π)4e−ik(x−x′)v2(kα;α)[G0(k)−G∗\n0(k)]. (39)\nwherev2(kα;α)is the generalized Bogoliubov transformation [39] that is gi ven as\nv2(kα;α) =d/summationdisplay\ns=1/summationdisplay\n{σs}2s−1∞/summationdisplay\nlσ1,...,lσs=1(−η)s+/summationtexts\nr=1lσrexp/bracketleftBig\n−s/summationdisplay\nj=1ασjlσjkσj/bracketrightBig\n, (40)\n8wheredis the number of compactified dimensions, η= 1(−1)for fermions (bosons) and {σs}\ndenotes the set of all combinations with selements. In order to obtain physical conditions at finite\ntemperature and spatial confinement, α0has to be taken as a positive real number, while αnfor\nn= 1,2,···,d−1must be pure imaginary of the form iLn.\nIV. THERMO FIELD DYNAMICS\nA brief introduction to Thermo Field Dynamics (TFD) is prese nted. TFD is a real-time finite\ntemperature field theory. In this formalism the usual Fock sp aceSof the system is doubled, such\nthat the expanded space is ST=S ⊗˜S, which is applicable to systems in a thermal equilibrium\nstate. This doubling is defined by the tilde (∼) conjugation rules, associating each operator in Sto\ntwo operators in ST.\nThermal effects are introduced through a Bogoliubov transfor mation that corresponds to a\nrotation in the tilde and non-tilde variables. For bosons th is is given as\nd(α) =u(α)d(k)−v(α)˜d†(k), (41)\n˜d†(α) =u(α)˜d†(k)−v(α)d(k), (42)\nwhere(d†,˜d†)are creation operators, (d,˜d)are destruction operators, and the algebraic rules for\nthermal operators are\n/bracketleftBig\nd(k,α),d†(p,α)/bracketrightBig\n=δ3(k−p),\n/bracketleftBig\n˜d(k,α),˜d†(p,α)/bracketrightBig\n=δ3(k−p), (43)\nand other commutation relations are null. The quantities u(α)andv(α)are related to the Bose\ndistribution function as v2(α) = (eαω−1)−1andu2(α) = 1+v2(α). Hereω=ω(k)andα=β.\nA doublet notation is defined by\n\nd(α)\n˜d†(α)\n=B(α)\nd(k)\n˜d†(k)\n, (44)\nwhereB(α)is the Bogoliubov transformation given as\nB(α) =\nu(α)−v(α)\n−v(α)u(α)\n. (45)\nAs an example, let us consider a free scalar field in Minkowski space-time specified by diag(gµν) =\n(+1,−1,−1,−1). The scalar field propagator is given as\nG(ab)\n0(x−x′;α) =i/an}b∇acketle{t0,˜0|τ[φa(x;α)φb(x′;α)]|0,˜0/an}b∇acket∇i}ht, (46)\n9whereφ(x;α) =B(α)φ(x)B−1(α)anda,b= 1,2. Then\nG(ab)\n0(x−x′;α) =i/integraldisplayd4k\n(2π)4e−ik(x−x′)G(ab)\n0(k;α), (47)\nwhere\nG0(k;α) =G0(k)+v2(k;α)[G0(k)−G∗\n0(k)], (48)\nwithG0(k) = (k2−m2+iǫ)−1and[G0(k)−G∗\n0(k)] = 2πiδ(k2−m2). As the physical information\nis given by the non-tilde components, i.e. G(11)\n0(k;α), hereG(11)\n0(k;α)≡G0(k;α)is used.\nV. SOME APPLICATIONS\nHere three different applications which depend on the choice of theαparameter are considered.\nA. Stefan-Boltzmann law\nConsider the thermal effect for the choice α= (β,0,0,0). Then the generalized Bogoliubov\ntransformation (40) becomes\nv2(β) =∞/summationdisplay\nj0=1e−βk0j0. (49)\nThen the Green function, eq. (39), is given as\nG(11)\n0(x−x′;α) = 2∞/summationdisplay\nj0=1G0/parenleftbig\nx−x′−iβj0n0/parenrightbig\n, (50)\nwherenµ\n0= (1,0,0,0). The vacuum expectation value of the energy-momentum tenso r, eq. (37),\nbecomes\nTµν(11)(x;α) =−ilim\nx′→x/braceleftBig/parenleftBig1\n4πΓµν\nGEM+Γµν\nLV/parenrightBig∞/summationdisplay\nj0=1G0/parenleftbig\nx−x′−iβj0n0/parenrightbig/bracerightBig\n. (51)\nForµ=ν= 0, we obtain\nT00(11)(T) =π\n30(1+4πκ0)T4, (52)\nthe Stefan-Boltzmann law for the GEM field with corrections du e to Lorentz-violating parameters,\nwith\nκ0≡1\n2/parenleftbig\nk(4)/parenrightbig0λ0\nλ+/parenleftbig\nk(4)/parenrightbig0000−1\n6/parenleftBig/parenleftbig\nk(4)/parenrightbig1λ1\nλ\n+/parenleftbig\nk(4)/parenrightbig2λ2\nλ+/parenleftbig\nk(4)/parenrightbig3λ3\nλ−2/parenleftbig\nk(4)/parenrightbig0101−2/parenleftbig\nk(4)/parenrightbig0202−2/parenleftbig\nk(4)/parenrightbig0303/parenrightBig\n. (53)\n10Whenκ0= 0, the Lorentz invariant result obtained in [33] is recovered . More results in statistical\nmechanics in the presence of Lorentz-violating background fields have been studied [40].\nThe component µ=ν= 3is given as\nT33(11)(β) =π\n90β4(1+4πκ1), (54)\nwhere\nκ1≡3/parenleftbig\nk(4)/parenrightbig0303+3\n2/parenleftbig\nk(4)/parenrightbig0λ0\nλ−/parenleftbig\nk(4)/parenrightbig3λ3\nλ+/parenleftbig\nk(4)/parenrightbig3333\n+5\n4/parenleftBig/parenleftbig\nk(4)/parenrightbig1λ1\nλ+/parenleftbig\nk(4)/parenrightbig2λ2\nλ+/parenleftbig\nk(4)/parenrightbig3λ3\nλ/parenrightBig\n. (55)\nIt is important to observe that the lowest order of the Lorent z violation leads to a modification\nin the Stefan-Boltzmann law. However, while small, Lorentz v iolating terms do not contradict any\nexperimental measurements of the Stefan-Boltzmann law. In a ddition, constraints on the Lorentz-\nviolating parameters can be obtained if the precision of the measurements will improve significantly.\nB. Casimir effect at zero temperature\nThe Casimir effect for the GEM field of the SME with Lorentz symm etry violation at zero\ntemperature is calculated. For parallel plates perpendicu lar to the z direction and separated by a\ndistance dtheαparameter is chosen as α= (0,0,0,i2d). In this case, the Bogoliubov transformation\nis\nv2(d) =∞/summationdisplay\nl3=1e−i2dk3l3(56)\nand the Green function is\nG(11)\n0(x−x′;d) = 2∞/summationdisplay\nl3=1G0/parenleftbig\nx−x′−2dl3z/parenrightbig\n. (57)\nThen the energy-momentum tensor becomes\nTµν(11)(x;d) =−ilim\nx′→x/braceleftBig/parenleftbigg1\n4πΓµν\nGEM+Γµν\nLV/parenrightbigg∞/summationdisplay\nl3=1G0/parenleftbig\nx−x′−2dl3z/parenrightbig/bracerightBig\n. (58)\nThus the Casimir energy and pressure are obtained\nE(d) =T00(11)(d) =−π\n1440d4(1+4πκ2) (59)\nP(d) =T33(11)(d) =−π\n480d4(1+4πκ3), (60)\n11where\nκ2≡1\n2/parenleftbig\nk(4)/parenrightbig0λ0\nλ+/parenleftbig\nk(4)/parenrightbig0000+1\n2/parenleftbig\nk(4)/parenrightbig1λ1\nλ\n−/parenleftbig\nk(4)/parenrightbig0101+1\n2/parenleftbig\nk(4)/parenrightbig2λ2\nλ−/parenleftbig\nk(4)/parenrightbig0202−3\n2/parenleftbig\nk(4)/parenrightbig3λ3\nλ+3/parenleftbig\nk(4)/parenrightbig0303(61)\nand\nκ3≡3/parenleftbig\nk(4)/parenrightbig0303+3\n2/parenleftbig\nk(4)/parenrightbig0λ0\nλ−/parenleftbig\nk(4)/parenrightbig3λ3\nλ\n+/parenleftbig\nk(4)/parenrightbig3333−5\n12/parenleftBig/parenleftbig\nk(4)/parenrightbig\n)1λ1λ+/parenleftbig\nk(4)/parenrightbig2λ2\nλ−3/parenleftbig\nk(4)/parenrightbig3λ3\nλ/parenrightBig\n. (62)\nThese expressions are consequences of the periodic conditi ons introduced by the topology Γ1\n4=\nS1×R3whereS1stands for the compactification of x3-axis in a circumference of length L= 2d.\nBy taking L= 2din the Green function is equivalent to the contributions of e ven images used\nin [41], for Dirichlet boundary condition. Then the the toro idal topology method can be used for\ncalculating the Casimir effect for Dirichlet boundary condi tion.\nThis result shows that the Lorentz-violating term modifies t he Casimir effect for the GEM field.\nSince the Lorentz-violating parameter is small (κ≪1), our result shows that the Casimir force\nbetween the plates is attractive, similar to the case of the e lectromagnetic field.\nC. Casimir effect at finite temperature\nThe effect of finite temperature is introduced by taking α= (β,0,0,i2d)and then the Bogoliubov\ntransformation, eq. (40), becomes\nv2(k0,k3;β,d) =∞/summationdisplay\nj0=1e−βk0j0+∞/summationdisplay\nl3=1e−iLk3l3+2∞/summationdisplay\nj0,l3=1e−βk0j0−iLk3l3. (63)\nThe Green function, corresponding to the first two terms, is g iven in eq. (50) and in eq. (57),\nrespectively. For the third term the Green function is\nG(11)\n0(x−x′;β,d) = 4∞/summationdisplay\nj0,l3=1G0(x−x′−iβj0n−2dl3z). (64)\nThen the energy-momentum tensor becomes\nTµν(11)(β,d) =−2ilim\nx′→x/braceleftBig/parenleftbigg1\n4πΓµν\nGEM+Γµν\nLV/parenrightbigg∞/summationdisplay\nj0,l3=1G0(x−x′−iβj0n−2dl3z)/bracerightBig\n. (65)\nThe complete expression for the Casimir energy at finite temp erature is obtained as\nE(β,d) =π\n30β4(1+4πκ0)−π\n1440d4(1+4πκ2)\n+2\nπ3∞/summationdisplay\nj0,l3=13(βj0)2−(2dl3)2\n[(βj0)2+(2dl3)2]3(1+4πκ4), (66)\n12whereE(β,d) =T00(11)(β,d). Hereκ0,κ2,κ4represent the contribution of Lorentz violation into\nthe Casimir Energy. The Casimir pressure at finite temperatu re is\nP(β,d) =π\n90β4(1+4πκ1)−π\n480d4(1+4πκ3)\n+2\nπ3∞/summationdisplay\nj0,l3=1(βj0)2−3(2dl3)2\n[(βj0)2+(2dl3)2]3(1+4πκ5), (67)\nwhereP(β,d) =T33(11)(β,d). In this case κ1,κ3,κ5give the Lorentz violation effects to the Casimir\npressure. The first term is the Stefan-Boltzmann law, the seco nd and third term are Casimir effect\nat zero and finite temperature, respectively. The Lorentz-v iolating parameters κ4andκ5are defined\nas\nκ4≡1\n2/parenleftbig\nk(4)/parenrightbig0λ0\nλ+/parenleftbig\nk(4)/parenrightbig0000+3(2dl3)2−(βj0)2\n[(2dl3)2−3(βj0)2]/parenleftbigg\n−1\n2/parenleftbig\nk(4)/parenrightbig3λ3\nλ+/parenleftbig\nk(4)/parenrightbig0303/parenrightbigg\n−(βj0)2+(2dl3)2\n[(2dl3)2−3(βj0)2]/parenleftBig\n−1\n2/parenleftbig\nk(4)/parenrightbig1λ1\nλ+/parenleftbig\nk(4)/parenrightbig0101−1\n2/parenleftbig\nk(4)/parenrightbig2λ2\nλ+/parenleftbig\nk(4)/parenrightbig0202/parenrightBig\n(68)\nand\nκ5≡(2dl3)2−3(βj0)2\n[3(2dl3)2−(βj0)2]/parenleftbigg/parenleftbig\nk(4)/parenrightbig0303+1\n2/parenleftbig\nk(4)/parenrightbig0λ0\nλ/parenrightbigg\n+1\n4/parenleftbig\nk(4)/parenrightbig3λ3\nλ+/parenleftbig\nk(4)/parenrightbig3333−5\n4(2dl3)2+(βj0)2\n[3(2dl3)2−(βj0)2]/parenleftBig/parenleftbig\nk(4)/parenrightbig1λ1\nλ+/parenleftbig\nk(4)/parenrightbig2λ2\nλ/parenrightBig\n.\nThe modifications due to the Lorentz-violating terms at zero and finite temperature are similar.\nThese results are similar to the case of electromagnetic fiel d. It is important to point out\nthat although these results are similar there are important difference between two theories. For\nexample, electromagnetic fields are vectors whereas GEM fiel ds are tensors. The electromagnetic\nfield propagates on a given space-time, whereas the gravitat ional field itself generates the space-time.\nVI. CONCLUSIONS\nThe SME is an effective theory that includes all Lorentz-viol ating parameter besides the known\nphysics of the SM and GR. In this paper Lorentz-violating cor rections to the GEM theory are\nconsidered. GEM is a gravitational theory based on an analog y with electromagnetism. A La-\ngrangian formalism of Gravitoelectromagnetism (GEM) is us ed. Using this formalism, the energy\nmomentum tensor for the GEM field with Lorentz violation is ca lculated. Our main objective is\nto calculate the Lorentz- violating contributions to the St efan-Boltzmann law and Casimir effect\nat finite temperature. The TFD formalism is used to introduce finite temperature effects. Our\n13results show that contributions due to the Lorentz-violati ng term are linearly proportional to all\ncomponents of the tensor/parenleftbig\nk(4)/parenrightbig\nκλξρ. Here the gravitational Casimir force is found to be propor-\ntional to ∼(1 +κ)FG, whereFGis the gravitational Casimir effect and κis the correction due\nto the Lorentz violation. The gravitational Casimir effect f or conventional plates is very small.\nHowever plates of special material, to measure the gravitat ional Casimir effect, using the GEM\nfield have been developed [32]. Thus, while small, Lorentz vi olating terms do not contradict any\nexperimental measurements of the gravitational Casimir fo rce and the Stefan-Boltzmann law. 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Matsumoto, Thermofield D ynamics and Condensed States, North-\nHolland, Amsterdan, (1982); F. C. Khanna, A. P. C. Malbouiss on, J. M. C. Malboiusson and A. E.\nSantana, Themal quantum field theory: Algebraic aspects and applications, World Scientific, Singapore,\n(2009).\n[36] J. Ramos, M. Montigny and F. C. Khanna, Gen. Rel. Grav. 50, 83 (2018).\n[37] Q. G. Bailey, V. A. Kostelecky and R. Xu, Phys. Rev. D 91, 022006 (2015).\n[38] F. J. Belinfante, Physics 7, 449 (1940).\n[39] F. C. Khanna, A. P. C Malbouisson, J. M. C. Malbouisson an d A. E. Santana, Ann. Phys. 326, 2634\n(2011).\n[40] D. Colladay and P. McDonald, Phys. Rev. D 70, 125007 (2004); Phys. Rev. D 73, 105006 (2006).\n[41] L. S. Brown, G. J. Maclay, Phys. Rev. 184, 1272 (1969).\n15"    },    {        "title": "1706.08477v1.Landau_Damping_of_Beam_Instabilities_by_Electron_Lenses.pdf",        "content": "Landau Damping  of Beam Instabilities by Electron \nLenses  \nV. Shiltsev, Y. Alexahin , A.Burov, and A. Valishev  \nFermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA  \n \nAbstract  \nModern and future particle accelerators employ increasingly higher intensity \nand brighter beams of charged particles and become operationally limited by  \ncoherent beam instabilities. Usual methods to control the instabilities, such as \noctupole  magn ets, beam feedback dampers and use of chromatic effects , become less \neffective and insufficient . We show that, i n contrast,  Lorentz  forces of a low-energy, \na magnetically stabilized electron beam , or “electron lens”,  easily introduce s \ntransverse nonlinear focusing sufficient  for Landau damping of transverse beam \ninstabilities in accelerators . It is also important that, unlike  other nonlinear elements, \nthe electron lens provides the frequency spread mainly at the beam core, thus \nallowing much higher frequenc y spread without lifetime degradation.  For the \nparameters of the Future Circular Collider, a single conventional electron lens a few meters long would provide stabilization superior to tens of  thousands of \nsuperconducting octupole magnets.    \n   PACS numbers: 29.27.Bd, 29.20.dk, 52.35.Qz, 41.85.Ew  \n \nIntroduction. - Collective instabilit ies of charged particle beams set important  \nlimitations on the beam intensity  [1, 2, 3]. In general, the instability is always driven \nby a certain agent that, first, respon ds to the beam collective perturbation,  and, \nsecond,  acts back  on it. Such responses can occur through beam -induced \nelectromagnetic wake -fields [4], interaction with accumulated residual ions or  \nelectron clouds  [3, 5]. \nSuppression  of the collective  instabilities is typically ac hieved by a joint action \nof feedback  systems and Landau damping [6,  7, 8]. For multi-bunch beams, such \nfeedback s usually suppress the most unstable coupled- bunch and beam -beam modes. \nHowever, having limited bandwidths, these dampers are normally inefficient for the \nintra-bunch modes and Landau damping is needed for their suppression. To make it \npossible, the spectrum of  incoherent, or individual particle  frequencies  must overlap \nwith frequencies of the unstable collective modes, thus allowing absorption o f the \ncollective energy by the resonant particle s. The frequency spread can be generated by  \nnon-linear focusing forces, such as those due to the space charge of an opposite \ncolliding beam  in colliders, or by non-linear - usually , octupole - magnets. The first \noption is not available at one -beam facilities, but even in the colliders, it does not exist at injection and during the acceleration ramp , where the beams do not yet \ncollide . Thus far, commonly  used are octupole  magnets with the transverse magnetic \nfields on beam’s axis of 𝐵𝐵 𝑥𝑥+𝑖𝑖𝐵𝐵𝑦𝑦=𝑂𝑂3(𝑥𝑥+𝑖𝑖𝑖𝑖)3, which generate the transverse, or \nbetatron,  frequency shifts proportional to the square of particles ’ amplitude s [7]. For \nhigher energy E  of the accelerated particles , the octupoles become less and less \neffective: the corresponding frequency spread scales  as 1/E2 due to incr easing rigidity \nand smaller size  of the beam, while the instability growth rates scale only as 1/E , \nsince the  transverse  beam size is not important for them. As a consequence, one \nneeds to increase the strength of these magnets accordingly. For example, in the \nTevatron  proton -antiproton collider , with 𝐸𝐸≈1 𝑇𝑇𝑇𝑇𝑇𝑇 , there were 35  superconducting \noctupole magnets installed in  1 m long package cryostats and operated with up to \n50 A current  [9], while in the  7 TeV LHC, 336 superconducting octupole magnets, \neach about 0.32 m long,  operate at the maximum current of 500 A [10] – and even \nthat is not always sufficient to maintain the beam stability  above certain proton \nbunch intensities. The anticipated 50 𝑇𝑇𝑇𝑇𝑇𝑇 beam energy  in the proton -proton Future \nCircular Collider (FCC-pp , [11]) would require a further  factor of more than 60 in \nintegrated octupole strength [12], which makes stabilization by  octupoles greatly \nimpractical.  \nAnother very serious concern is that at their maximum strength, the octupoles  \ninduce  significant non-linear fields and dangerous betatron frequency shifts for the larger amplitude particles , destabilizing their dynamics.  This leads to increased rate \nof particle loss es, and therefore, higher  radiation load [1 3].  \nTo provide a sufficient spread of the betatron frequencies without  beam  \nlifetime degradatio n, we  propose the use of  an electron lens  – a high brightness low \nenergy electron beam system [1 4, 15]. In this Letter, we calculate the accelerator \nbeam coherent stability diagrams for  various sizes of the electron  beam , simulate \nnumerically the effect of the electron lenses on incoherent particle dynamics and \ncompare it with the case of octupoles. Major parameters of the electron lens devices \nfor effective suppression of coherent insta bilities are presented as examples for the \nLHC  and for the FCC.   \n Stability diagrams with electron lenses.  - The Lorenz force  acting on an ultra-\nrelativistic  proton from a  low energy electr on beam  with velocity \n and current \ndensity distribution \n , \n \n  , (1)  \nis diminishing at large rad ius r as ~1/r; therefore,  outside of the electron beam, the \ncorresponding betatron frequency shifts  \n  drop quadratically  with the  proton’s \ntransverse amplitude s \n . For a  round Gaussian -profile  electron beam of rms transverse size σ e, the amplitude dependent tune shift \n , where \n  is the \nproton revolution frequency,  equal to [16 ]:  \n \n   (2) \nHere \n  are the modified Bessel functions , Le is the length of the electron beam , \nIe is the electron current,  \n  is the Alfven current, m e and mp are \nelectron and proton masses, εn is the normalized rms emittance, or the action average, \nof the proton beam, \n  is the beam rms size, where \n  is the ring beta -\nfunction at the lens location and \n  is the relativistic  factor . The two transverse \nemittances and beam sizes at the lens position are assumed to be identical. The tune \nshift versus amplitude parameters \n  is shown in Fig ure 1. \n \n  \nFIG. 1: The incoherent tune shift by the round electron lens , \n , versus the \nparticle transverse amplitudes, Eq. (2).   \n \nWhen the coherent tune shift \n  is much smaller than the longitudinal, or the \nsynchrotron,  tune,  \n , which is typical for high -energy colliders with feedbacks \non, the beam stability is  conventionally  quantified by means of the stability diagram  \n[7]: \n \n   (3) \nHere  F is the normalized phase space density  as a function of actions  \n, so that  \n; the symbol \n  stands for an infinitesimally small positive \nvalue in accordance with the Landau rule [6].  The function \n  maps the real axis \nin the complex plane \n  onto a complex plane D , showing the stability thresholds \nfor the coherent tune shifts \n ; the beam is unstable if and only if there is a \ncollective mode whose tune shift stays above the stability diagram D .  \nIn case of octupoles, the incoherent tune shifts are linear functions of the actions:  \n \n   (4) \nFor the LHC at 7  TeV with  \n , its 168 Landau octupoles per beam, fed with \nthe maximal current of 500 A, provide the nonlinearity matrix  with \n \n  [8]. The corresponding stab ility diagram is \nshown in Figure 2.  \n \nFIG. 2: Stability diagram for the 7 TeV proton beams in LHC  at the maximal \nstrength of the Landau octupoles.    \nFor the electron lens, the stability diagram, Eq. (3), with the tune shift \n  given by \nEq.(2) , is presented in Fig. 3 for various electron beam sizes and the same current \ndensity at the center; both real and imaginary parts of the diagram are in the units of \nδνmax. \n \n \nFIG. 3: Electron lens stability diagrams are presented for various electron beam sizes \n(noted in units of  the proton beam rms size), assuming the same current density at the \ncenter . \n \nTable I  lists main  parameters of the electron lens required to generate a tune \nspread δνmax = 0.01 in the LHC . For the LHC parameters, such a lens provide s \napproximately an order  of magnitude  larger  stability diagram than the existing \nLandau octupoles all operating at their maximum current of 500 A.. In the  50 TeV proton-proton  Future Circular Collider,  the same  single lens w ould introduce the \nsame tune spread δνmax = 0.01, provided that the normalized emittance is the same \nand the beta -function scales as the energy, i.e. \n  at the lens location in the \nFCC . To make similar stability diagram for the FCC, ~20000 LHC -type octupoles \nwould be needed. The electron sy stem parameters listed in T able I are either modest \nor comparable to the electron lenses already commissioned and operational for beam -\nbeam compensati on in the Tevatron proton- antiproton collider [17 , 18] and in the \nRelativistic Heavy Ion Collider (RHIC) [ 19]. Given the flexibility of the electron \nlenses [14], they can be effectively used for proton beam stabilization at all stages of \ncollider operation  – at injection, on the energy ramps, during the low -beta squeeze , \nadjustment to collisions,  and, if necessary, in collisions.  Moreover, the electron \ncurrent can be easily regulated over short time intervals and the electron lenses can \nbe set to operate on  a subset of least stable bunches in the accelerator or even on \nindividual bunches,  as was  demonstrated in the Tevatron [20 ]. The increased betatron \nfrequency spread δν of about 0.004- 0.01 induced by the electron lenses  has been \ndemonstrated in the 980 GeV proton beam in the Tevatron [21] and in the RHIC 100 \nGeV polarized proton beams [2 2].  \n \nTABLE I : Electron beam requirements to generate the  tune shift δνmax =0.01   \nin the 7 TeV LHC  proton beams with \n   \nParameter  Symbol           Value  Unit \n     \nLength  Le 2.0       m \nBeta-functions at the e -lens  βx,y 240       m \nElectron c urrent  Ie 0.8  A  \nElectron energy  Ue 10  kV  \ne-beam radius in main solenoid  σe   0.28  mm  \nFields  in main/gun  solenoids  Bm / Bg  6.5/0.2  T \nMax. t une spread by e -lens  δνmax  0.01   \n  \nLong -term single particle stability .- To compare the effects of  Landau \ndamping by octupole magnets with  that by the electron lenses on the long- term single \nparticle stability , we have applied frequency map analysis (FMA)  and Dynamic \nAperture calculations –  methods  widely used to explore dynamics of Hamiltonian \nsystems [13, 23, 24].  The phase space plot of such systems is usually  a complicated \nmixtur e of periodic, quasiperiodic,  and chaotic trajectories arranged in stable  and \nunstable areas.  Analysis of these trajectories and distinction between regular \n(periodic or quasiperiodic) and chaotic ones provides useful information on the \nmotion features,  such as working resonances, their widths, and locations in the planes \nof the betatro n tunes and amplitudes. The FMA method is a quick tool widely used in \nthe accelerator community for studies of particle motion stability [25 , 26]. The \nDynamic Aperture (DA – the area of stable long- term particle dynamics) calculation \nemploys more computer -intensive simulations (normally hundreds of thousand or millions of turns) and is used as a figure of merit in the accelerator design and \noperations [2 7].   \nFigure 4 presents the simulated FMA and DA plots for the illustrative case of \n7 TeV protons circulating without collisions in a focusing optics model ( HL-LHC \noptics Version 1.0 [28]) in the presence of realistic multipole magnetic field errors in \nthe LHC with machine c hromaticity , i.e. tune de rivative  on the relative momentum \ndeviatio n, \n . Two Landau damping mechanisms  are examined : with  \nexisting octupole magnets set to create tune spread of δ ν = 0.01 within  the \namplitude s Ax=Ay=3.5 σp (Fig.  4 a) and with a single electron lens, placed  at the \nlocation IR4 of the ring such that it generates the maximal tune shift δν max = 0.01 \nwith the e lectron beam size matched to the proton  beam size of σ p =0.28  mm \n(Fig.  4 b). The colors progressively changing from blue to red  indicate the range  of \nthe betatron frequency (tune) modulation for protons from 10-7 to 10-3, respectively . \nThe initial amplitudes A x and Ay vary from 0  σp (core) to 8  σp (halo). Each point on \nthe plots indicate s the result of 8000  turns of tracking . The DA calculation data are \nshown on the same plots – the cyan lines depict the range of initial parameters \nbeyond which particles are lost after 100,000 turns One can see a significant \nadvantage of the dynamics with the electron lens: FMA in Fig.  4 a shows large tune \nvariations – a clear indication of enhanced diffusion in the FMA methods  – for \nparticles with A x,y > 4 σp in the case of the octupole magnets. Moreover, the particles \nwith initial horizontal amplitude above 5 σp are lost during the tracking  over 8000 \nturns.  The dynamic aperture in the case of the electron lens is significantly larger and \nexceeds 8 σp. That makes  the electron lens the method of choice to provide strong \nLandau damping in accelerators without instigation of dangerous halo diffusion.   \nFIG. 4. Frequency Map Analysis ( FMA) and Dynamic Aperture modeling of  LHC \nproton dynamics with comparable strength Landau damping provided by octupole \nmagnets ( a) and by  the electron lens ( b).  Horizontal and vertical axes –  initial \nparticle amplitudes A x, Ay in units of the rms beam size varying from 0 σp (core) to 8  \nσp (halo).  Brighter colors indicate exponentially stronger tune modulation indicating \nresonances (see color palette ). 100,000 turns DA is shown in cyan lines.   \n \nIn conclusion, we are stressing  that electron lenses are the proper Landau \noptical elements, since they can efficiently provide required nonlinearity where it is \nneeded for  beam stabilization, i.e. at the beam core, and do not introduce \nnonlinearity where it is detrimental for the lifetime, i.e. far outside the beam. \nFlexibility in the control of transverse electron charge distribution and fast current \nmodulation allows the generation of the required spread of betatron frequencies by \nvery short electron lenses with modest parameters, which have been demonstrated in \nthe devices built so far. Landau damping by electron lenses is free of many \ndrawback s of other methods present ly used or proposed – the lenses do not reduce \nthe dynamic aperture and do not require numerous superconducting octupole \nmagnets; they suppress all the u nstable beam modes in contrast to available feedback \nsystems which act only on the modes with non- zero dipole moment  [8]; their \nefficiency will not be dependent on the bunch length as in an RF qua drupole  based \nsystem [29], and corresponding single particle stability concerns due to synchro-\nbetatron resonances will be avoided.  All of this  makes the Landau damping by \nelectron lenses a unique instrument for the next generation high- current accelerators, \nincluding hadron supercolliders.  Electron lenses may also be helpful in low -energy \nhigh-brightness  accelerators, where Landau damping  is intrinsically suppressed by a \nshift of single particle  tunes away from the frequency of coherent oscillations [ 30]; a \npreliminary study of this issue is suggested in Ref.  [31].  \nThe technology  of the electron lenses is well established and well up to the \nrequirements of Landau damping in particle acceler ators , as discussed above. T wo \nelectron lenses were built and installe d in the Tevatron ring [ 17] at Fermilab, and \ntwo similar ones in the BNL’s RHIC [2 2]. They employed some 10 kV Ampere -\nclass electron beams of millimeter to submillimeter sizes with a variety of the \ntransverse current distributions  je(r) generated at the thermionic electron gun, \nincluding Gaussian ones.  The electron beam s in the lenses are very stable \ntransversely being u sually immers ed in a strong magnetic field - about Bg=0.1-0.3 T at the electron gun cathode and some Bm=1.0-6.5 T insid e a few meters  long main \nsuperconducting solenoi ds. The electron beam  transverse alignment on the high-\nenergy  beam is done by trajectory  correctors to better than a small fraction of the \nrms beam siz e σe. The electron lens magnetic system adiabatically compresses the \nelectron- beam cross- section area in the interaction region by the factor of  Bm/Bg≈10 \n(variable from 2 to 60), proportionally increasing the current density j e of the \nelectron beam in the interaction region compared to its value on the gun cathode, \nusually of about 2 -10 A/cm2. In-depth experimental studies of Landau damping with \nelectron lenses are being planne d at Fermilab’s IOTA ring [ 32].  \nWe would like to thank B.Brown, E.Gianfelice -Wendt , V.Lebedev  and \nE.Prebys for many useful comments. 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Antipov , et al,  JINST  12, T03002  (2017)  \n "    },    {        "title": "2104.11974v1.Random_Euclidean_embeddings_in_finite_dimensional_Lorentz_spaces.pdf",        "content": "arXiv:2104.11974v1  [math.FA]  24 Apr 2021Random Euclidean embeddings in finite dimensional\nLorentz spaces\nDaniel J. Fresen∗\nApril 27, 2021\nAbstract\nQuantitative bounds for random embeddings of Rkinto Lorentz sequence spaces\nare given, with improved dependence on ε.\n1 Introduction\nOur starting point is Milman’s general Dvoretzky theorem [8]. The dep endence on εis\ndue to Schechtman [14] following Gordon [5], and the Gaussian formula tion due to Pisier\n[13]. We refer the reader to [9, 11, 15] for more details.\nTheorem 1 There exists a universal constant c >0such that the following is true.\nConsider any\n(n,k,ε)∈N×N×(0,1)\nLet/ba∇dbl·/ba∇dblbe a norm on Rnand set\nM= (2π)−n\n2/integraldisplay\nRn/ba∇dblx/ba∇dblexp/parenleftbigg\n−1\n2|x|2/parenrightbigg\ndx b = sup/braceleftbig\n/ba∇dblθ/ba∇dbl:θ∈Sn−1/bracerightbig\nwhere|·|denotes the standard Euclidean norm. Assume that k≤dwhere\nd=c/parenleftbiggM\nb/parenrightbigg2\nε2\nand letGbe ann×krandom matrix with i.i.d. standard normal random variables as\nentries. With probability at least 1−2e−d, the following event occurs: for all x∈Rk,\n(1−ε)M|x| ≤ /ba∇dblGx/ba∇dbl ≤(1+ε)M|x| (1)\n∗UniversityofPretoria,DepartmentofMathematicsandApplied Mat hematics, daniel.fresen@up.ac.za\nMSC: 46B06, 46B07, 46B09, 52A21, 52A23.\n1If (X,/ba∇dbl·/ba∇dbl) is a general n-dimensional normed space over R, one can put coordinates\nonXby identifying it with Rnin such a way so that\nb= 1 M≥c√\nlnn\nand therefore if nis sufficiently large we can ensure that dis large. Logarithmic depen-\ndence onnis the worst case scenario (for the best possible choice of coordina tes), which\nis the correct behavior in ℓn\n∞but can be improved significantly for other spaces, such as\nℓn\npfor fixedp∈[1,∞), where one gets power dependence on n.\nBy rotational invariance of the normal distribution, Range( G) is a uniformly dis-\ntributed random subspace in the Grassmannian Gn,kof allk-dimensional linear subspaces\nofRn. The bounds in (1) mean that on Range( G),/ba∇dbl·/ba∇dblapproximates the pushforward\nnorm/ba∇dbly/ba∇dbl♯:=M|G−1y|, whereG−1: Range(G)/ma√sto→Rkdenotes the inverse of the linear\nmap associated to G. Since the sub-level sets of this pushforward norm are ellipsoids,\nMilman’s general Dvoretzky theorem can be interpreted as follows: Assumingnis suf-\nficiently large and we have chosen an appropriate coordinate syste m through which to\nidentify a given normed space XwithRn,\n•Mostk-dimensional subspaces of Xare almost isometric to Hilbert spaces\n•Mostk-dimensional cross-sections of the unit ball B={x:/ba∇dblx/ba∇dbl ≤1}are approximately\nellipsoidal.\nForε=ε0, for any universal constant ε0∈(0,1), Theorem 1 is in a particular sense\nsharp (see [6, 10]), and in this sense dependence on nis understood. However the question\nof optimal dependence on εis open. We refer the reader to [12, 16] for the best existing\nbounds of the form ρ(ε)lnnin the existential Dvoretzky theorem, where one is satisfied\nwith a single subspace of this dimension, and bounds for general clas ses of spaces with\nsymmetries are contained in [2, 17].\nPaouris, Valettas and Zinn [11] studied dependence on εin the randomized Dvoretzky\ntheorem for the ℓn\npspaces, both in the range 1 ≤p≤Clnnandp>Clnn, improving on\nthe bounds in Theorem 1.\nThe results we now present extend results in [11] to the class of Lo rentz spaces. These\nspaces have a structure that is more complicated than that of the ℓn\npspaces, and the\nLorentz norm of a Gaussian random vector is typically not written in t erms of the sum of\ni.i.d. random variables. Our approach is different to that in [11] and in th e special case\nofℓn\npit allows for a simpler proof without the use of Talagrand’s L1−L2inequality, and\nwith improved dependence on p(removing a factor of p−p).\n2 What’s new?\nConsider the finite dimensional Lorentz spaces, i.e. Rnendowed with the norm\n|x|ω,p=/parenleftBiggn/summationdisplay\ni=1ωixp\n[i]/parenrightBigg1/p\n2where 1 ≤p <∞, (ωi)n\n1is any non-increasing sequence in [0 ,1] withω1= 1, and/parenleftbig\nx[i]/parenrightbign\n1denotes the non-increasing rearrangement of ( |xi|)n\n1. These spaces are the finite\ndimensional counterparts to the infinite dimensional Lorentz spac es (see [7]) which play\na classical role in analysis.\nBefore studying the general case in Section 9, we study the specia l case where ωi=i−r\nfor 0≤r<∞, using the notation\n|x|r,p=/parenleftBiggn/summationdisplay\ni=1i−rxp\n[i]/parenrightBigg1/p\nIn this special case, using Lemmas 13 and 14, Theorem 1 applies with\nd=\n\ncr,pnε2: 0≤r≤1/2, p<2(1−r)\ncr,pn(lnn)1−2\npε2: 0≤r≤1/2, p= 2(1−r)\ncr,pn2(1−r)\npε2: 0≤r≤1/2, p>2(1−r)\ncr,pn2(1−r)\npε2: 1/2<r<1\ncr,p(lnn)1+2\npε2:r= 1\nwhere the coefficients cr,p>0 do not depend on anything except randpand can be\nwritten explicitly. Our first main result, Theorem 17, is a little hard on t he eye, and we\nhave hidden it near the end of the paper. Its two main corollaries, ho wever, are simpler\n(Corollaries 2 and 4). Both improve the dependence on ε.\nCorollary 2 In the case /ba∇dbl·/ba∇dbl=|·|r,p, where0≤r≤1and1≤p<∞, Theorem 1 holds\nwith the sufficient condition k≤dreplaced with k≤d′, where\nd′≥\n\ncr,pnε2: 0≤r<1/2,p<2−2r\ncr,pmin/braceleftBig\nnε2,n(lnn)1−2\npε2\np/bracerightBig\n: 0≤r<1/2,p= 2−2r\ncr,pmin/braceleftBig\nnε2,n2(1−r)\npε2\np/bracerightBig\n: 0≤r<1/2,p>2−2r\ncr,pn(lnn)−1ε2:r= 1/2,p= 2−2r= 1\ncr,pmin/braceleftBig\nn(lnn)−pε2,n1\npε2\np/bracerightBig\n:r= 1/2,p>2−2r= 1\ncr,pmin/braceleftBig\nn2(1−r)(lnn)−(p−1)ε2,n2(1−r)\npε2\np/bracerightBig\n: 1/2<r<1\ncr,pmin/braceleftBig\n(lnn)3ε2,(lnn)1+2\npε2\np/bracerightBig\n: r= 1\nFor the spaces under consideration, Corollary 2 improves on Theor em 1 in all cases\nexcept when either p <2−2ror whenp= 1, and in those cases it reduces to the old\nbound.\nWe have excluded the case r >1 from Corollary 2 because as n→ ∞, the resulting\nLorentz space is isomorphic to ℓn\n∞, and Theorem 3 below gives better estimates. More\ngenerally, when\np>cln/parenleftbigg\n1+1+n1−r\n1+|1−r|lnnlnn/parenrightbigg\n3for arbitrarily small universal constant c>0, Lemma 10 (later in the paper) implies\n|x|∞≤ |x|r,p≤C|x|∞/parenleftbigg\n1+1+n1−r\n1+|1−r|lnnlnn/parenrightbigg1/p\n≤C′|x|∞\nIn this case the estimates in Theorem 17 start to break down. The f ollowing result, which\ngeneralizes the case p>clnn(andr= 0) in [11, Theorem 1.2] can then be used instead\n(with slightly improved probability bound from 1 −Cn−cε/ln(1/ε)in [11]).\nTheorem 3 For all0<c1<C1there exists c2>0such that the following is true: let n∈\nNand let|·|♯be a norm on Rnthat is invariant under coordinate permutations and satisfi es\nc1|x|∞≤ |x|♯≤C1|x|∞(for allx∈Rn). Letε∈(0,1)and0< k≤c2ε(lnε−1)−1lnn.\nLetGbe ann×kstandard Gaussian random matrix. Then with probability at l east\n1−Cn−c2ε, for allx∈Rk,(1−ε)M|Ge1|♯|x| ≤ |Gx|♯≤(1+ε)M|Ge1|♯|x|.\nProof.LetT:Rn→Rnbe the map that arranges the coordinates of a vector in non-\ndecreasing order. Let XandYbe independent standard normal random vectors in Rn.\nFrom estimates for the normal distribution, see e.g. (5) in Lemma 12 , with probability at\nleast 1−Cexp(−ct2),\n/vextendsingle/vextendsingle/vextendsingle|X|♯−|Y|♯/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle|TX|♯−|TY|♯/vextendsingle/vextendsingle/vextendsingle≤ |TX−TY|♯≤C1|TX−TY|∞≤CC1t2\n√\nlnn\nThis can be converted to a deviation of |X|♯aboutM|X|♯and is the same deviation\nestimate satisfied by |X|∞. The usual proof of the randomized Dvoretzky theorem (using\nthe (ε/4)-net argument, see e.g. [9, 15]) then transfers to |·|♯.\nThe second corollary of Theorem 17 applies to the classical ℓn\npspaces, with improved\ndependence on pcompared to [11, Theorem 1.2].\nCorollary 4 For allC1>0there exists c2>0such that the following statement is true.\nIn the case /ba∇dbl·/ba∇dbl=|·|0,p=|·|p, and under the added assumption that p<C 1lnn, Theorem\n1 holds with the sufficient condition k≤dreplaced with k≤d′, whered′is defined as\nfollows,\nd′=/braceleftBigg\ncnε2: 1≤p≤2\nc2min/braceleftBig\ncpnε2,pn2\npε2\np/bracerightBig\n: 2<p<C 1lnn\nandc>0is a universal constant.\nWe end the paper with a result for general Lorentz norms.\nTheorem 5 (refer to Theorem 18 for a more detailed statement) A random e mbedding\nof/parenleftbig\nRk,|·|/parenrightbig\ninto/parenleftBig\nRn,|·|ω,p/parenrightBig\nusing a standard Gaussian matrix is, with probability at lea st\n1−2e−d, anεalmost isometry in a sense similar to (1) provided k≤d, wheredis defined\nas follows: If 1<p<∞set\nd=/parenleftbigg\n1+1\np−1/parenrightbigg−1\nmin\n\ncp/parenleftBig/summationtextn\ni=1ωi/parenleftbig\nlnn\ni/parenrightbigp/2/parenrightBig2\nε2\n/summationtextn\ni=1ω2\ni/parenleftbig\nlnn\ni/parenrightbigp−1,cB−1/p/parenleftBiggn/summationdisplay\ni=1ωi/parenleftBig\nlnn\ni/parenrightBigp/2/parenrightBigg2/p\nε2/p\n\n\n4where\nB=\n\n/summationtextn\ni=1ω2\nii−(p−1): 1<p<3/2/parenleftbigg/summationtextn\ni=1ω2\n2−p\ni/parenrightbigg2−p\n: 3/2≤p<2\n1 : 2 ≤p<∞\nand ifp= 1set\nd=c/parenleftBig/summationtextn\ni=1ωi/parenleftbig\nlnn\ni/parenrightbig1/2/parenrightBig2\nε2\n/summationtextn\ni=1ω2\ni\n3 The main engine: Gaussian concentration\nWe make extensive use of the classical Gaussian concentration ineq uality. The simple\nproof of Maurey and Pisier is contained in [13].\nTheorem 6 Letf:Rn→Rand letXbe a random vector in Rnwith the standard\nnormal distribution. Then for all t≥0,\nP{|X−Mf(X)|>CtLip(f)} ≤2exp/parenleftbig\n−t2/parenrightbig\nAssuming Lip (f)<∞, the same holds true with Mreplaced by E.\nIf/ba∇dbl·/ba∇dbl:Rn→[0,∞) is a norm, then\nLip(/ba∇dbl·/ba∇dbl) = sup/braceleftbig\n/ba∇dblθ/ba∇dbl:θ∈Sn−1/bracerightbig\nwhich can be seen by applying the triangle inequality. Denoting this sup remum asb(/ba∇dbl·/ba∇dbl),\nGaussian concentration implies that with probability at least 1 −2exp(−t2),\nM/ba∇dblX/ba∇dbl−Ctb(/ba∇dbl·/ba∇dbl)≤ /ba∇dblX/ba∇dbl ≤M/ba∇dblX/ba∇dbl+Ctb(/ba∇dbl·/ba∇dbl)\nFor our purposes the right hand inequality /ba∇dblX/ba∇dbl ≤M/ba∇dblX/ba∇dbl+Ctb(/ba∇dbl·/ba∇dbl) will be sufficient.\nThe following result due to Schechtman[14] provides a uniform bound over a sphere rather\nthan just at a point and reduces to Theorem 6 in the case k= 1.\nTheorem 7 Letf:Rn→Rand letGbe ann×krandom matrix with i.i.d standard\nnormal random variables as entries. Let t≥0and assume that k≤ct2. Then with\nprobability at least 1−2exp(−ct2)the following event occurs: For all θ∈Sk−1,\n|f(Gθ)−Mf(Gθ)| ≤tLip(f)\n4 Key methods\nWe will deal with functions that may not be Lipschitz, or whose Lipsch itz constant is not\nrepresentative of the typical behaviour of the function. For suc h functions we will need\nto prove deviation inequalities for f(X) aboutMf(X), whereXis a standard normal\n5random vector as in Section 3. In order to do so, it will be useful to r estrict the function\nfto a setKwith the following two properties:\n•Lip(f|K) is nicely bounded\n•P{X /∈K}is small (here Xis normally distributed as in Theorem 6)\nExactly how one interprets ‘nicely bounded’ and ‘small’ may depend on the situation,\nand the reader will see the details in the proofs of Theorems 17 and 1 8. It follows from\nelementary metric space theory that f|Kcan be extended to a function F:Rn→Rsuch\nthat Lip(F) = Lip(f|K). One can then apply Gaussian concentration to Fand transfer\nthe result back to fsincef(X) =F(X) with high probability. This procedure appears\nin Bobkov, Nayar and Tetali [1] and is further explored in [3].\nThe original way of applying concentration of measure to prove Dvo retzky’s theorem,\nas in the classical works of Milman and Schechtman, e.g. [8, 9, 14], wa s to study con-\ncentration of /ba∇dblX/ba∇dbldirectly using Lipschitz properties of /ba∇dbl·/ba∇dbl(where/ba∇dbl·/ba∇dblis the norm of the\nspace in question). One actually considered a random point on the sp here as opposed\nto a Gaussian random vector, the Gaussian approach being made po pular by Pisier [13],\nbut the point is that regardles of the randomness used, the funct ion under consideration\nwas the norm. A trick that will be useful in the context of /ba∇dbl·/ba∇dbl=|·|ω,pis to study con-\ncentration of |X|p\nω,p=/summationtextωiXp\n[i]instead of |X|ω,p(using Gaussian concentration and the\nprocedure from [1] just mentioned), and then to convert the res ult back to a bound on\n|X|ω,pby transforming the distribution under the action of s/ma√sto→s1/p. As functions on Rn,\n|·|ω,pand|·|p\nω,pare fundamentally different in terms of their local-Lipschitz propert ies: the\nfirst function achieves its Lipschitz constant on any neighbourhoo d of the origin, while\nfor the second function, the problematic points where the norm of the gradient is large\nhave been moved far away from the origin so that a convexity argum ent can be used for\npoints within a certain convex body containing the origin.\nSo, wewillneedtoboundthedistributionofagradient, whichcomesd owntobounding\ntheexpression/summationtexti−2rX2(p−1)\n[i]. Sincethetermsofthissumarenotindependent, onecannot\nuse the classical theory of sums of independent random variables. Forp≥3/2 one can\nwritesuchaquantity intermsofanormanduseGaussianconcentra tionappliedtonorms.\nFor 1≤p<3/2 one cannot write the gradient as a function of a norm, and one the refore\ncannot use the equation Lip( /ba∇dbl·/ba∇dbl) =b(/ba∇dbl·/ba∇dbl). This causes difficulties, but one can bound\nthe gradient above by a function of a norm, which is an interesting pr oblem in its own\nright, especially in the case p= 2−2r. We postpone this discussion until Section 7.\n5 Notation and once-off explanations\nThe symbols Candcdenote positive universal constants that may take on different\nvalues at each appearance. MandEdenote median and expected value. nandkwill\ntypically denote natural numbers and this will not always be stated e xplicitly but should\nbe clear from the context. Lip( f)∈[0,∞] denotes the Lipschitz constant of any function\nf:Rn→Rwith respect to the Euclidean norm./summationtextb\ni=af(i) denotes summation over\nalli∈Nsuch thata≤i≤b, regardless of whether a,b∈N. 1{·}denotes the indicator\nfunctionofasetorcondition. Whenprovingaprobabilityboundofth eformCexp(−ct2),\n6we may take Csufficiently large and assume in the proof that, say, t≥1, because for\nt<1 the resulting probability bound is greater than 1 and the result hold s trivially. After\nsuch a bound is proved we may replace Cwith 2 using the fact that there exists c′>0\nsuch that\nmin/braceleftbig\n1,Cexp/parenleftbig\n−ct2/parenrightbig/bracerightbig\n≤2exp/parenleftbig\n−c′t2/parenrightbig\nLastly, by making an all-round change of variables we may take c′to be, say, 1 or 1 /2.\nThe constants in the final probability bound will therefore (without further explanation)\nnot always match what appears to come from the proof.\n6 Lemmas\nWe start with basic estimates for the lower incomplete gamma functio n suited to our\npurposes.\nLemma 8 For allb,q∈[0,∞),\nc1+qmin{1+q,b}1+q≤/integraldisplayb\n0e−ωωqdω≤C1+qmin{1+q,b}1+q\ncebb1+q\n1+q+b≤/integraldisplayb\n0eωωqdω≤Cebb1+q\n1+q+b\nProof. The first integrand increases on [0 ,q] and decreases on [ q,∞), so forb≤q,\ncomparing the integral to the area of a large rectangle,\n/integraldisplayb\n0e−ωωqdω≤be−bbq≤b1+q\nwhile forb≥1+q,\n/integraldisplayb\n0e−ωωqdω≤Γ(1+q)≤C1+q(1+q)1+q≤C1+qb1+q\nIfq≥1 this also holds for q <b<1+q, since in that case (1+ q)1+q≤C1+qb1+q. So all\nthat remains for the upper bound is the case where 0 ≤q <1 andq <b<1+q, which\nimplies/integraldisplayb\n0e−ωωqdω≤/integraldisplayb\n0ωqdω≤b1+q\nWe now consider the lower bound. For b≤q, comparing the integral to the area of a\nsmaller rectangle,/integraldisplayb\n0e−ωωqdω≥b\n2e−b/2/parenleftbiggb\n2/parenrightbiggq\n≥c1+qb1+q\nForb≥qandq≥1, using what we have just proved,\n/integraldisplayb\n0e−ωωqdω≥/integraldisplayq\n0e−ωωqdω≥c1+qq1+q≥c1+q(1+q)1+q\n7Forb≥qand 0≤q<1, we consider two sub-cases: firstly b≤1+q, in which case\n/integraldisplayb\n0e−ωωqdω≥c/integraldisplayb\n0ωqdω≥c1+qb1+q\nand secondly b>1+q, in which case\n/integraldisplayb\n0e−ωωqdω≥c/integraldisplay1\n0ωqdω≥c≥c1+q(1+q)1+q\nThesecondintegralwith eωinsteadofe−ωcanbeestimatedbywriting eωωq= exp(ω+qlnω)\nand using ln ω≤(ω−b)/b+lnb, valid for all ω∈(0,b], and lnω≥2(ω−b)/b+lnb, valid\nfor allω∈[b/2,b]. Here we also use cez/(1+z)≤(ez−1)/z≤Cez/(1+z) valid for all\nz >0.\nWe will use the fact that for any non-increasing function f: [1,n]→R,\n1\n2/parenleftbigg\nf(1)+/integraldisplayn\n1f(x)dx/parenrightbigg\n≤n/summationdisplay\ni=1f(i)≤f(1)+/integraldisplayn\n1f(x)dx\nLemma 9 For alla,q∈[0,∞)and alln≥2the following is true: If a∈[0,1]then\nn/summationdisplay\ni=1i−a/parenleftBig\nlnn\ni/parenrightBigq\n≤C1+qn1−a(1+q)1+q(lnn)1+q\n((1−a)lnn+1+q)1+q\nand ifa∈[1,∞)the sum is bounded above by\nC(lnn)1+q\n(a−1)lnn+1+q+(lnn)q\nThe corresponding lower bounds hold by replacing Cwithc. Whenq= 0andi=nin the\nsum, we consider 00= 1.\nProof.We focus on the upper bounds; the lower bounds follow the same ste ps. Integrals\nare estimated using Lemma 8, and we use the fact that min {x,y}is the same order of\nmagnitude as xy/(x+y). First, let a∈(0,∞). Peeling off the first term, comparing the\nremaining sum to an integral using monotonicity, and setting es/a=n/x,\nn−an/summationdisplay\ni=1/parenleftBign\ni/parenrightBiga/parenleftBig\nlnn\ni/parenrightBigq\n≤(lnn)q+n1−a\na1+q/integraldisplayalnn\n0exp/parenleftbigg/parenleftbigg\n1−1\na/parenrightbigg\ns/parenrightbigg\nsqds\nIfa∈(1,∞) setw= (1−1/a)sto get\n(lnn)q+n1−a\n(a−1)1+q/integraldisplay(a−1)lnn\n0ewwqdw\nIfa= 1 we get (ln n)q+/integraltextlnn\n0sqdswhich can be absorbed into either the case a∈(0,1)\nor the case a∈(1,∞). Ifa∈(0,1) then set w=−(1−1/a)sto get\n(lnn)q+n1−a\n(1−a)1+q/integraldisplay(1−a)lnn\n0e−wwqdw\n8Ifa= 0, setting w= ln(n/x),\nn/summationdisplay\ni=1i−a/parenleftBig\nlnn\ni/parenrightBigq\n≤(lnn)q+/integraldisplayn\n1/parenleftBig\nlnn\nx/parenrightBigq\ndx≤(lnn)q+n/integraldisplaylnn\n0e−wwqdw\nFora∈[0,1] the factor (ln n)qgets absorbed into the remaining term since\nC1+qn1−a(1+q)1+q(lnn)1+q\n((1−a)lnn+1+q)1+q=C1+qn1−a(lnn)1+q\n/parenleftBig\n1+(1−a)lnn\n1+q/parenrightBig1+q\nThe following lemma interpolates between the case a= 1 anda/\\e}atio\\slash= 1.\nLemma 10 For all(a,T)∈R×[1,∞),\nc1+T1−a\n1+|1−a|lnTlnT≤/integraldisplayT\n1x−adx≤C1+T1−a\n1+|1−a|lnTlnT\nProof.First assume a/\\e}atio\\slash= 1 andT/\\e}atio\\slash= 1 and write\n/integraldisplayT\n1x−adx=exp((1−a)lnT)−1\n(1−a)lnTlnT\nThen interpret s−1(exp(s)−1) as the slope of a secant line andbound it above andbelow\nbyC(1+es)/(1+|s|) in the cases s≤ −1,s∈(−1,1)\\{0}and 1≤s. Then notice\nthat the estimate also holds when a= 1 and/or T= 1.\nDefineξ1: [0,1]→[0,1] byξ1(t) =et(1−t), from which it follows, see [4], that\nξ−1\n1(t)≤min/braceleftBig/radicalbig\n2(1−t),1−e−1t/bracerightBig\n: 0≤t≤1\nThe following lemma is taken from [4], which is based on basic estimates fo r the bino-\nmial distribution and the R´ enyi representation of order statistic s from the exponential\ndistribution. Recall that the order statistics of a vector x∈Rnare denoted/parenleftbig\nx(i)/parenrightbign\n1(the\nnon-decreasing rearrangement of its coordinates), and the non -increasing rearrangement\nof the absolute values of the coordinates of xare denoted/parenleftbig\nx[i]/parenrightbign\n1. So if all coordinates of\nxare non-negative, then x[i]=x(n−i+1).\nLemma 11 Let(γi)n\n1be an i.i.d. sample from (0,1)with corresponding order statistics/parenleftbig\nγ(i)/parenrightbign\n1and lett>0. With probability at least 1−3−1π2exp(−t2/2), the following event\noccurs: for all 1≤i≤n,\nγ(i)≤1−n−i+1\nn+1/parenleftbigg\n1−ξ−1\n1/parenleftbigg\nexp/parenleftbigg−t2−4ln(n−i+1)\n2(n−i+1)/parenrightbigg/parenrightbigg/parenrightbigg\n(2)\nand with probability at least 1−Cexp(−t2/2)the following event occurs: for all 1≤i≤n,\nγ(i)≤1−n−i\nnexp\n−cmax\n\n/parenleftBig\nt+√\nlni/parenrightBig√\ni\n/radicalbig\nn(n−i+1),t2+lni\nn−i+1\n\n\n (3)\n9In Lemma 11, (2) is preferable for i>n/2 while (3) is preferable for i≤n/2.\nLemma 12 Letn≥3,t≥0, and letXandYbe independent random vectors in\nRn, each with the standard normal distribution. Let T:Rn→Rnbe the function that\narranges the coordinates of a vector in non-decreasing orde r. Then with probability at\nleast1−Cexp(−t2), the following event occurs: for all 1≤i≤(n+1)/2,\nX[i]≤C/parenleftbigg\nlnn\ni+t2\ni/parenrightbigg1/2\n(4)\nand\n|TX−TY|∞≤Cmin/braceleftbiggt2\n√\nlnn,t/bracerightbigg\n(5)\nand with probability at least 0.51the following event occurs: for all 1≤i≤(n+1)/2,\nc/radicalbigg\nlnn\ni≤X[i]≤C/radicalbigg\nlnn\ni(6)\nProof.We will not repeat ‘with probability...’ as it is clear that there are probab ilities\nassociated to the events in question, and what those probabilities a re. To prove estimates\nfor a general distribution based on estimates for the uniformdistr ibution, transformunder\nthe action of the inverse cumulative distribution, which is increasing a nd preserves the\noperation of arranging in non-increasing order. So we may write X[i]= Φ−1/parenleftBig1+γ(n−i+1)\n2/parenrightBig\n,\nwhere (γi)n\n1is an i.i.d. sample from the uniform distribution on (0 ,1), and we use the\nbound\nc/radicalbigg\nln1\n1−x≤Φ−1/parenleftbigg1+x\n2/parenrightbigg\n≤C/radicalbigg\nln1\n1−x(7)\nvalid for all x∈(1/3,1), say. (4) now follows from (2) and (7), and includes the upper\nboundin(6)asaspecial case. Togetalower boundonanorderstat istic, weapplyLemma\n11 to the i.i.d. random variables (1 −γi)n\n1, which are also uniformly distributed in (0 ,1)\nand whose vector of order statistics is/parenleftbig\n1−γ(n−i+1)/parenrightbign\n1. An upper bound on 1 −γ(n−i+1)\ntranslates to a lower bound on γ(n−i+1). So, from (3) and (7),\nX[i]≥c/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtln\n1−n−i\nnexp\n−C/parenleftBig\n1+√\nlni/parenrightBig√\ni\nn\n\n−1\n≥c/radicalbigg\nlnn\ni\nwhich is seen to hold when n>n0(usinge−z≥1−z), and when n≤n0this is bounded\nbelow byc2≥c/radicalbig\nln(n/i).\nWe now consider (5). Its proof, which occupies the next two pages , may later be\nremoved and placed in another paper. It follows from the bounds re lating Φ and φ= Φ′\nthatd\ndxΦ−1(x) =1\nφ(Φ−1(x))≤C\nmin{x,1−x}/radicalBig\nlnmin{x,1−x}−1(8)\n10A difference between our current calculations for (5) and what has been done for (4) and\n(6) above, is that there are no absolute values involved in the definit ion ofX(i), and we\nwriteX(i)= Φ−1/parenleftbig\nγ(i)/parenrightbig\n, where (γi)n\n1is an i.i.d. uniform sample. Here we are re-using\nnotation, and this ( γi)n\n1is not the same as the previous ( γi)n\n1, which is inconsequential\nsince we are now doing a new calculation. So, by (8) and (2), for all n/2< i≤n,\nΦ−1/parenleftbig\nγ(i)/parenrightbig\n−Φ−1/parenleftbigi\nn+1/parenrightbig\nis bounded above by\nC/integraldisplay1−n−i+1\nn+1/parenleftbigg\n1−ξ−1\n1exp/parenleftbigg\n−t2−4ln(n−i+1)\n2(n−i+1)/parenrightbigg/parenrightbigg\ni/(n+1)(1−x)−1/parenleftbig\nln(1−x)−1/parenrightbig−1/2dx\n=C/integraldisplaylnn+1\nn−i+1−ln/parenleftbigg\n1−ξ−1\n1exp/parenleftbigg\n−t2−4ln(n−i+1)\n2(n−i+1)/parenrightbigg/parenrightbigg\nlnn+1\nn−i+1s−1/2ds\n≤C/parenleftbigg\nlnn+1\nn−i+1/parenrightbigg−1/2\nln/parenleftbigg\n1−ξ−1\n1exp/parenleftbigg−t2−4ln(n−i+1)\n2(n−i+1)/parenrightbigg/parenrightbigg−1\nThe function s/ma√sto→ −ln/parenleftbig\n1−ξ−1\n1exp(−s)/parenrightbig\nbehaves like√snear 0 and like s+1 whensis\nlarge. We then consider two cases, depending on whether\nt2+4ln(n−i+1)\n2(n−i+1)(9)\nlies in (0,1) or [1,∞), and in either case the estimate is bounded above by Ct2/√\nlnn.\nHere we have used the fact that for all a,b≥1,\nln(1+ab)≤Cln(1+a)ln(1+b)\nwhich is true since 1+ ab≤(1+a)(1+b) and for positive numbers uniformly bounded\naway from 0 a product dominates a sum up to a constant. Therefor e\nln/parenleftbigg\n1+n\nn−i+1/parenrightbigg\nln(1+n−i+1)≥clnn\nwhich can be modified be deleting the leftmost 1+ and changing the fac tor ln(1+n−i+1)\nto something larger such as n−i+1 or (n−i+1)/ln(n−i+1). A zero in the denominator\ndoesn’t hurt since we are in practice considering the reciprocals. An d assuming as we\nmay thatt≥1, 1+t≤Ct2. So this is where the upper bound Ct2/√\nlnncomes from.\nObviously this bound can be improved significantly for individual order statistics; we\nhaven’t bothered to write out such bounds since for our purposes we need a uniform\nestimate over all i. A similar calculation with a lower bound for γ(i)follows from (3)\napplied to/parenleftbig\n1−γ(i)/parenrightbign\n1: Φ−1/parenleftbigi−1\nn/parenrightbig\n−Φ−1/parenleftbig\nγ(i)/parenrightbig\nis bounded above by\nC/integraldisplay(i−1)/n\ni−1\nnexp/parenleftBigg\n−cmax/braceleftBigg(t+√\nln(n−i+1))√n−i+1\nn,t2\nn/bracerightBigg/parenrightBigg(1−x)−1/parenleftbig\nln(1−x)−1/parenrightbig−1/2dx\n=C/integraldisplaylnn\nn−i+1\n−ln/bracketleftBigg\n1−i−1\nnexp/parenleftBigg\n−cmax/braceleftBigg(t+√\nln(n−i+1))√n−i+1\nn,t2\nn/bracerightBigg/parenrightBigg/bracketrightBiggs−1/2ds\n11Using/integraltextb\nas−1/2ds= 2(b−a)/(√\nb+√a)≤C(b−a)/√\nbvalid for 0<a<b, applied to this\nlast quantity,\n√\nb=/parenleftbigg\nlnn\nn−i+1/parenrightbigg1/2\nandb−ais equal to\nCln\n1+i−1\nn−i+1\n1−exp\n−cmax\n\n/parenleftBig\nt+/radicalbig\nln(n−i+1)/parenrightBig√n−i+1\nn,t2\nn\n\n\n\n\n\n≤Ci−1\nn−i+1\n1−exp\n−cmax\n\n/parenleftBig\nt+/radicalbig\nln(n−i+1)/parenrightBig√n−i+1\nn,t2\nn\n\n\n\n\nThe expression 1 −e−sbehaves like sfor 0≤s≤1 and like 1 when s >1. We now\nconsider two cases depending on whether\ncmax\n\n/parenleftBig\nt+/radicalbig\nln(n−i+1)/parenrightBig√n−i+1\nn,t2\nn\n\n\nlies in [0,1] or (1,∞). In the first case, for the entire expression C(b−a)/√\nb, we get the\nsame bound Ct2/√\nlnnas before, using similar simplifications. In the second case we get\nCn\n(n−i+1)/radicalbiglnn\nn−i+1\nHowever for the expression defining this case to be >1,\nt≥cmin/braceleftBigg\n√n,n/radicalbig\nln(n−i+1)/bracerightBigg\nand regardless of which term defines this minimum we end up with the sa me bound\nCt2/√\nlnn. We must now handle the discrepancy between i/(n+1) and (i−1)/nin the\ncomputations above. From (8),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleΦ−1/parenleftbiggi\nn+1/parenrightbigg\n−Φ−1/parenleftbiggi−1\nn/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C\nn/parenleftbigg\n1−i\nn+1/parenrightbigg−1/parenleftBigg\nln/parenleftbigg\n1−i\nn+1/parenrightbigg−1/parenrightBigg−1/2\n≤C√\nlnn\nAll of this implies that /vextendsingle/vextendsingle/vextendsingle/vextendsingleX(i)−Φ−1/parenleftbiggi\nn+1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ct2\n√\nlnn\nSimilar bounds in the case 1 ≤i≤n/2 now follow by symmetry, and they also hold for\nYsinceYhas the same distribution as X, and a bound on |TX−TY|∞follows by the\ntriangle inequality. For large values of tthis can be improved as follows. Let x∗∈Rnbe\ndefined as\nx∗\ni= Φ−1/parenleftbiggi\nn+1/parenrightbigg\n12SinceTacts as an isometry on each of the n! overlapping regions of Rndetermined by\nthe order of the coordinates of a vector, Tis 1-Lipschitz on Rn. Sox/ma√sto→Tx/ma√sto→Tx−x∗/ma√sto→\n|Tx−x∗|∞is the composition of 1-Lipschitz functions and by Gaussian concent ration,\n|TX−x∗|∞≤M|TX−x∗|∞+Ct\nwhich also applies to TY, and the result follows again by the triangle inequality.\nLemma 13 Let0≤r <∞,1≤p <∞,n≥2, and letXbe a random vector in Rn\nwith the standard normal distribution. If r∈[0,1]then\nMn/summationdisplay\ni=1i−rXp\n[i]≤Cppp/2n1−r(lnn)1+p/2\n[p+(1−r)lnn]1+p/2\nand ifr∈[1,∞)then\nMn/summationdisplay\ni=1i−rXp\n[i]≤Cp(lnn)1+p/2\n1+(r−1)lnn+Cp(lnn)p/2\nwith the reverse inequalities holding with Creplaced by c.\nProof.The result follows from Eq. (6) of Lemma 12, together with Lemma 9.\nLemma 14 Let0≤r<∞and1≤p<∞. Then\nsup\n\n/parenleftBiggn/summationdisplay\ni=1i−rθp\n[i]/parenrightBigg1/p\n:θ∈Sn−1\n\n=/braceleftBigg/parenleftbig/summationtextn\n1i−2r/(2−p)/parenrightbig(2−p)/2p:p∈[1,2)\n1 : p∈[2,∞)\nForp∈[1,2)this can be bounded above by\n1+C/parenleftbigglnn\n1+|2−2r−p|lnn/parenrightbigg2−p\n2p/parenleftBig\n1+n2−2r−p\n2p/parenrightBig\nand below by the same quantity with Creplaced with c. Forr < r0(for any universal\nconstantr0>1) the leftmost ‘1+’ can be deleted.\nProof.Forp∈[1,2) an upper bound follows by H¨ older’s inequality for ℓn\n2/(2−p)−ℓn\n2/p\nduality, with equality when θi=i−r/(2−p)/parenleftBig/summationtextn\nj=1j−2r/(2−p)/parenrightBig−1/2\n. Now\n/parenleftBiggn/summationdisplay\n1i−2r/(2−p)/parenrightBigg(2−p)/2p\n≤/parenleftbigg\n1+/integraldisplayn\n1x−2r/(2−p)dx/parenrightbigg(2−p)/2p\nwhich is bounded using Lemma 10 and noting that 0 <(2−p)/(2p)≤1/2 and that\nc<(2−p)(2−p)<C. Forp∈[2,∞),/parenleftBig/summationtextn\ni=1i−rθp\n[i]/parenrightBig1/p\n≤/parenleftBig/summationtextn\ni=1θp\n[i]/parenrightBig1/p\n≤1 with equality\nwhenθ=e1.\n13We shall use the fact that for all b∈[1,n], not necessarily an integer,\nn/summationdisplay\ni=1i−rxp\n[i]≤2n\nbb/summationdisplay\ni=1i−rxp\n[i]\nwhere the sum on the right is over all i∈Nsuch that 1 ≤i≤b.\nLemma 15 Letψ:Rn/ma√sto→Rand letA⊂Rnbe convex set with nonempty interior. Let E\ndenote the collection of all x∈Rnsuch that the coordinates of xare distinct and non-zero.\nAssume that ψis continuous on Aand differentiable on A∩E. Then, as an element of\n[0,∞],\nLip(ψ|A) = sup\nx∈A∩E|∇ψ(x)|\nProof.(Sketch) We focus on proving that LHS≤RHSwhenRHS <∞; the rest comes\ndown to approximating a supremum. Note that Eis dense in Rn, and by considering sets\nof the form conv {z,B}wherez∈AandBis any ball of non-zero radius contained in A,\nwe see that int( A) is dense in A. So, suppose |∇ψ(z)| ≤Lfor allz∈A∩Eand some\nL<∞, and consider any x1,y1∈Awithx1/\\e}atio\\slash=y1, and anyε>0. We consider sequences\nx2,x3,x4andy2,y3,y4such that each xi+1is sufficiently close to its predecessor xiand by\ncontinuity each ψ(xi+1) is sufficiently close to ψ(xi). Similarly so with the y’s. This can\nbe done so that x2,y2∈int(A),x3,y3∈int(A) and that all coordinates of x3andy3are\ndistinct (2ndistinct coordinates in total), and that x4,y4∈A∩Esuch thatx4andy4also\nhave completely distinct coordinates between the two of them. We b ound|ψ(x1)−ψ(y1)|\nup to a term involving εby integrating over the line segment joining x4andy4, which\ncontains at most n+n(n−1)/2 points not in E. Details are left to the reader.\n7 Order statistics, norms and quasi-norms\nAs part of the proof of Theorem 17, towards estimating the distrib ution of a gradient\nin order to derive concentration inequalities, we need estimates for the distribution of\ncertain functionals of order statistics. However, in order to avoid the blowup of a quasi-\nnorm constant as p→1 we need more. We need:\n•A deterministic bound of the form\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤ϕ(/ba∇dblx/ba∇dbl)\nwhere/ba∇dbl·/ba∇dblis a norm,ϕ: [0,∞)→[0,∞), and the inequality is valid for all x∈Rn,\n•a bound on the distribution of ϕ(/ba∇dblX/ba∇dbl), whereXis a random vector in Rnwith the\nstandard normal distribution.\nSince these estimates will affect the bounds we end up with in Theorem 17, the problem\nis not purely existential; we want good estimates. Of greatest inter est is the case p∈\n14(1,3/2) where the quantity/summationtextn\ni=1i−2rx2(p−1)\n[i]is not already a function of a norm of x.\nGeometrically, the problem is related to finding a convex subset of a g iven non-convex\nset whose complement has comparable Gaussian measure to the com plement of the given\nnon-convex set.\nFigure 2\nTheorem 16 LetXbe a random vector in Rn,n≥3, with the standard normal distri-\nbution,0≤r<∞,1≤p<∞andt>0. In each of the following four cases, definitions\nare given for R,Sare|·|♯, and in each case,\nP/braceleftBig\n|X|♯≤S/bracerightBig\n≥1−Cexp/parenleftbig\n−t2/2/parenrightbig\nand |X|♯≤S⇒n/summationdisplay\ni=1i−2rX2(p−1)\n[i]≤R\nCase I: Ifp∈[3/2,∞)then for all x∈Rnset\n|x|♯=/parenleftBiggn/summationdisplay\ni=1i−2rx2(p−1)\n[i]/parenrightBigg1\n2(p−1)\nandR=S2(p−1)=A+Bt2(p−1)where\nA=Cpppn1−2r(lnn)p\n[p+(1−2r)lnn]p1{0≤r≤1/2}+/parenleftbiggCp(lnn)p\n1+(2r−1)lnn+Cp(lnn)p−1/parenrightbigg\n1{1/2<r<∞}\nand\nB=C/bracketleftBigg\n1+/parenleftbigglnn\n1+|2−2r−p|lnn/parenrightbigg2−p/parenleftbig\n1+n2−2r−p/parenrightbig/bracketrightBigg\n1{3/2≤p<2}+Cp1{2≤p<∞}\n15In this case |·|♯is a norm.\nCase II: Ifp∈[1,3/2)then for all x∈Rn,\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤C\nn/e/summationdisplay\ni=1i−2r/parenleftbigg\nlnn\ni+t2\ni/parenrightbiggp−1\n3−2p\n|x|2(p−1)\n♯\nwhere\n|x|♯=n/e/summationdisplay\ni=1i−2rx[i]\n/parenleftbig\nlnn\ni+t2\ni/parenrightbig3−2p\n2(10)\nand\nR=S=Cn/e/summationdisplay\ni=1i−2r/parenleftbigg\nlnn\ni+t2\ni/parenrightbiggp−1\n≤Cn1−2r(lnn)p\n[1+(1−2r)lnn]p1{0≤r≤1/2}+C/parenleftbigg(lnn)p\n1+(2r−1)lnn+(lnn)p−1/parenrightbigg\n1{1/2<r<∞}\n+C/parenleftbigg\n1+1+n2−2r−p\n1+|2−2r−p|lnnlnn/parenrightbigg\nt2(p−1)\nUnder the added assumption that p≥3/2−2r, and because the sums have been restricted\nto1≤i≤n/e,|·|♯is a norm.\nCase III: Ifp<3/2−2r(so necessarily 0≤r<1/4and1≤p<3/2), then for all\nx∈Rn,\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤Cn(1−2r)(3−2p)|x|2(p−1)\n♯\nwhere\n|x|♯=n/summationdisplay\ni=1i−2rx[i]\nand\nS=Cn1−2r+Cn1−4r\n2/parenleftbigglnn\n1+(1−4r)lnn/parenrightbigg1\n2\nt\nR=Cn(1−2r)(3−2p)S2(p−1)≤Cn1−2r+Cn2−2r−p/parenleftbigglnn\n1+(1−4r)lnn/parenrightbiggp−1\nt2(p−1)\nCase IV: Whenr∈(1/4,1/2]andp= 2(1−r), in which case p∈[1,3/2), the result\nin Case II can be improved as follows:\nCase IVa: If(1−2r)lnn≥e(which excludes the case p= 1), then for all x∈Rn,\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤C(lnn)3−2p|x|2(p−1)\n♯\n16where\n|x|♯=n/e/summationdisplay\ni=1β−(3−2p)\n2(p−1)\nii−r\np−1x[i],\nand\nβi=(1−2r)plnn\nn1−2ri−2r/parenleftBig\nlnn\ni/parenrightBigp−1\n+i−1\nThe coefficient β−(3−2p)\n2(p−1)\nii−r\np−1is non-increasing in ifor1≤i≤n/e(so that|·|♯is a norm).\nIn this case\nS=C1\np−1(1−2r)−p\n2(p−1)(lnn)−(3−2p)\n2(p−1)n1\n2+C1\np−1(lnn)1\n2t\nand\nR=C(lnn)3−2pS2(p−1)≤C(1−2r)−1n1−2r+C(lnn)2−pt2(p−1)\nCase IVb: If(1−2r)lnn<e, then for all x∈Rn,\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤C(lnn)|x|2(p−1)\n♯\nwhere|·|♯=|·|is the standard Euclidean norm, S=Cn1\n2+t, and\nR=C(lnn)S2(p−1)≤C(lnn)t2(p−1)\nProof. Case I: p∈[3/2,∞). In this case |·|2r,2(p−1)is a norm and it follows by classical\nGaussian concentration that with probability at least 1 −Cexp(−t2/2),\nn/summationdisplay\ni=1i−2rX2(p−1)\n[i]=|X|2(p−1)\n2r,2(p−1)≤/bracketleftBig\nM|X|2r,2(p−1)+tLip|·|2r,2(p−1)/bracketrightBig2(p−1)\nEstimates for M|X|2r,2(p−1)and Lip|·|2r,2(p−1)follow from Lemmas 13 and 14, and we\nleave the computation to the reader (with the reminder that we are applying these results\nwith 2rand 2(p−1) instead of randp). Certain numerical simplifications can be made\nbased on the values of pandrand the existance of the factor Cp.\nProof. Case II: We consider p∈(1,3/2) and reclaim the case p= 1 by taking a limit.\nFor allx∈Rn, by H¨ older’s inequality,/summationtextn\n1i−2rx2(p−1)\n[i]is bounded above by\nCn/e/summationdisplay\ni=1i−4r(p−1)x2(p−1)\n[i]\n(ln(n/i)+t2/i)(p−1)(3−2p)i−2r(3−2p)/parenleftbig\nln(n/i)+t2/i/parenrightbig(p−1)(3−2p)\n≤C\nn/e/summationdisplay\ni=1i−2rx[i]\n(ln(n/i)+t2/i)(3−2p)/2\n2(p−1)\nn/e/summationdisplay\ni=1i−2r/parenleftbigg\nlnn\ni+t2\ni/parenrightbiggp−1\n3−2p\n=C|x|2(p−1)\n♯\nn/e/summationdisplay\ni=1i−2r/parenleftbigg\nlnn\ni+t2\ni/parenrightbiggp−1\n3−2p\n17Eq. (4) from Lemma 12 then implies that\n|X|♯=n/e/summationdisplay\ni=1i−2rX[i]\n(ln(n/i)+t2/i)(3−2p)/2≤C′n/e/summationdisplay\ni=1i−2r/parenleftbigg\nlnn\ni+t2\ni/parenrightbiggp−1\n=S\nC′′\nAssuming this event occurs, the above calculation involving H¨ older’s inequality implies\nthat/summationtextn\n1i−2rX2(p−1)\n[i]is bounded above by S. Lastly,\nS≤Cn/e/summationdisplay\ni=1i−2r/parenleftBig\nlnn\ni/parenrightBigp−1\n+Ct2(p−1)n/e/summationdisplay\ni=1i−2r−p+1\nwhich is bounded above using Lemmas 9 and 10.\nProof. Case III: p<3/2−2r(so necessarily 0 ≤r<1/4 and 1≤p<3/2). Forp/\\e}atio\\slash= 1,\nby H¨ older’s inequality,\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤/parenleftBiggn/summationdisplay\ni=1i−2rx[i]/parenrightBigg2(p−1)/parenleftBiggn/summationdisplay\ni=1i−2r/parenrightBigg3−2p\n≤Cn(1−2r)(3−2p)|x|2(p−1)\n♯\nand forp= 1 the same bound is seen to hold. An upper bound on the quantiles of |X|♯\nfollows from Gaussian concentration (making use of Lemmas 10 and 1 3).\nProof. Case IV: p= 2(1−r) andr∈(1/4,1/2], in which case p∈[1,3/2). The\nsub-case (1 −2r)lnn<eis clear enough by H¨ older’s inequality for p/\\e}atio\\slash= 1,\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤/parenleftBiggn/summationdisplay\ni=1/parenleftbig\ni−2r/parenrightbig1\n2−p/parenrightBigg2−p/parenleftBiggn/summationdisplay\ni=1/parenleftBig\nx2(p−1)\n[i]/parenrightBig1\np−1/parenrightBiggp−1\nand then noting that the exponent −2r/(2−p) =−1 and applying classical Gaussian\nconcentration to |·|. Forp= 1 there is nothing to show. In this sub-case,\n(lnn)2−p\nlnn= exp(−(1−2r)lnlnn)∈/bracketleftbig\ne−1,1/bracketrightbig\nn1−2r= exp((1 −2r)lnn)∈[1,ee)\nwhich is how we simplify the exponents of ln n.\nThe rest of the proof deals with the other sub-case (1 −2r)lnn≥e. Throughout, we\nmake use of the relation p= 2(1−r) which is not always explicitly re-stated, and the\nreader should make a mental note of this. The case p= 1 is automatically excluded from\nthis sub-case. By taking the constant Cin the probability bound to be at least√ewe\nmay assume that t≥1. For any sequence ( αi)⌊n/e⌋\n1withαi>0, by H¨ older’s inequality,\nn/summationdisplay\ni=1i−2rx2(p−1)\n[i]≤C\nn/e/summationdisplay\ni=1α1\n3−2p\ni\n3−2p\nn/e/summationdisplay\ni=1i−r\np−1α−1\n2(p−1)\nix[i]\n2(p−1)\n=C\nn/e/summationdisplay\ni=1βi\n3−2p\nn/e/summationdisplay\ni=1i−r\np−1β−(3−2p)\n2(p−1)\nix[i]\n2(p−1)\n18whereβi=α1\n3−2p\ni. This vector β∈R⌊n/e⌋is considered a variable for now, and its value\nwill later be fixed to match the value quoted in the statement of the r esult. Summing\nonly up ton/ewill ensure that i2r\n3−2pβiis non-decreasing in i, which then implies that\n|x|♯=n/e/summationdisplay\ni=1i−r\np−1β−(3−2p)\n2(p−1)\nix[i] (11)\nis a norm. By classical Gaussian concentration applied to |·|♯, with probability at least\n1−Cexp(−t2/2),\nn/summationdisplay\ni=1i−2rX2(p−1)\n[i]≤\nn/e/summationdisplay\ni=1βi\n3−2p/bracketleftBig\nM|X|♯+tLip/parenleftBig\n|·|♯/parenrightBig/bracketrightBig2(p−1)\n(12)\nThe median can be estimated using (4) and the Lipschitz constant co mputed as the\nEuclidean norm of the gradient, which gives\nM|X|♯≤Cn/e/summationdisplay\ni=1i−r\np−1β−(3−2p)\n2(p−1)\ni/parenleftBig\nlnn\ni/parenrightBig1/2\nLip/parenleftBig\n|·|♯/parenrightBig\n=\nn/e/summationdisplay\ni=1i−2r\np−1β−(3−2p)\np−1\ni\n1/2\nWe temporarily assume that/summationtextβi= 1, which we may do by homogeneity, although this\ncondition will later be relaxed. We wish to minimize the function\nψ(β) =n/e/summationdisplay\ni=1i−r\np−1β−(3−2p)\n2(p−1)\ni/parenleftBig\nlnn\ni/parenrightBig1/2\n+t\nn/e/summationdisplay\ni=1i−2r\np−1β−(3−2p)\np−1\ni\n1/2\nover the collection of all β∈R⌊n/e⌋such thati2r\n3−2pβiis positive and non-decreasing in\niand such that/summationtextβi= 1. The method of Lagrange multipliers leads us to solve the\nequations\n∂ψ(β)\n∂βi=−λ\nwhich can be written as\nB1i−r\np−1/parenleftBig\nlnn\ni/parenrightBig1/2\nβ−1\n2(p−1)\ni+B2i−2r\np−1β−(2−p)\np−1\ni= 1\nwhereB1andB2are positive values that do not depend on i. This implies that\n1/2≤max/braceleftbigg\nB1i−r\np−1/parenleftBig\nlnn\ni/parenrightBig1/2\nβ−1\n2(p−1)\ni,B2i−2r\np−1β−(2−p)\np−1\ni/bracerightbigg\n≤1\n19and therefore\nβi≤max/braceleftbigg\n22(p−1)B2(p−1)\n1i−2r/parenleftBig\nlnn\ni/parenrightBigp−1\n,2p−1\n2−pBp−1\n2−p\n2i−2r\n2−p/bracerightbigg\nwith the reverse inequality holding when 22(p−1)and 2p−1\n2−pare deleted. At this point, and\nby homogeneity, we remove the condition/summationtextβi= 1 and are led to the definition\nβi=Ai−2r/parenleftBig\nlnn\ni/parenrightBigp−1\n+i−1\nfor someA>0.B1,B2and the powers of 2 dissapear since they do not depend on iand\nwe have re-scaled β, and we have used the equation p= 2(1−r) to simplify the exponent\n−2r/(2−p). We now minimize over A. It follows from Lemma 9 that\nn/e/summationdisplay\ni=1βi≤CA(1−2r)−pn1−2r+Clnn\nWith an eye on (12), it is clear that the bounds for M|X|♯and Lip/parenleftBig\n|·|♯/parenrightBig\nare decreasing\ninA. It therefore does not help to let Aslip below the point where\nCA(1−2r)−pn1−2r=Clnn\nbecause as Acontinues to decrease beyond this point/summationtextβistays the same order of mag-\nnitude while M|X|♯+tLip/parenleftBig\n|·|♯/parenrightBig\nincreases. We may therefore assume that\nA≥c(1−2r)plnn\nn1−2rn/e/summationdisplay\ni=1βi≤CA(1−2r)−pn1−2r\nIf we look back at (12) with our new bound for/summationtextβiand our definition of βi, and we take\nAout of the expression for/summationtextβiand move it into the powers of βiwith corresponding\nexponents−(3−2p)\n2(p−1)and−(3−2p)\np−1in the expression M/ba∇dblX/ba∇dbl+tLip(/ba∇dbl·/ba∇dbl), we see that these\npowers ofβibecome\n/parenleftbigg\ni−2r/parenleftBig\nlnn\ni/parenrightBigp−1\n+A−1i−1/parenrightbigg−(3−2p)\n2(p−1)/parenleftbigg\ni−2r/parenleftBig\nlnn\ni/parenrightBigp−1\n+A−1i−1/parenrightbigg−(3−2p)\np−1\nSo, in our current range for A, the expression to be minimized (or at least the bound that\nwe have for it) is increasing. This leads us to take\nA=(1−2r)plnn\nn1−2r(13)\nRecall thatfor |·|♯tobeanorm, see(11), it issufficient for i2r/(3−2p)βitobenon-decreasing\nini, equivalently for i−r/(p−1)β−(3−2p)/(2p−2)\ni to be non-increasing. Writing\nωi=i−r\np−1β−(3−2p)\n2(p−1)\ni=\nAn4r(p−1)\n3−2p/parenleftBigg\nn\ni/parenleftBig\nlnn\ni/parenrightBig−(3−2p)\n4r/parenrightBigg−4r(p−1)\n3−2p\n+ip−1\n3−2p\n−(3−2p)\n2(p−1)\n20and noting that z(lnz)−(3−2p)/(4r)is increasing for z≥exp((3−2p)/(4r)), we see that ωi\nis decreasing. We now bound M|X|♯and Lip/parenleftBig\n|·|♯/parenrightBig\n. From the definition of βi,\nβ−1\ni≤min/braceleftbigg\nA−1i2r/parenleftBig\nlnn\ni/parenrightBig−(p−1)\n,i/bracerightbigg\n(14)\nwhich leads us to solve,\nA−1i2r/parenleftBig\nlnn\ni/parenrightBig−(p−1)\n=i\nKeeping in mind that 1 −2r=p−1, the above equation holds precisely when\nn\ni/parenleftBig\nlnn\ni/parenrightBig−1\n=A1\np−1n (15)\nThe function z/ma√sto→z/lnzis increasing on [ e,∞) and we will show that provided n > n 0\n(for a universal constant n0>1),\n1\n2e2≤A1\np−1n≤2e\n3lnnn3\n2e (16)\nso that (15) has exactly one solution for i∈[n1−3\n2e,n/e2], denotedA0(not necessarily an\ninteger) which satisfies\nn1−3\n2e≤A0≤e−2n (17)\nA0lnn\nA0=A−1\np−1 (18)\nln/parenleftbig\nA1/(p−1)n/parenrightbig\n= lnn\nA0−lnlnn\nA0(19)\nln/parenleftbig\nA1/(p−1)n/parenrightbig\n≤ln/parenleftbiggn\nA0/parenrightbigg\n≤/parenleftbigg\n1−1\ne/parenrightbigg\nln/parenleftbig\nA1/(p−1)n/parenrightbig\n(20)\nThe assumption n > n 0does not limit our generality since the result is directly seen\nto hold when n≤n0in which case many of the coefficients involved are bounded by\nconstants. From (20) and the defining inequality of the current su b-case, it follows that\n1\n2ln((1−2r)lnn)≤(1−2r)ln/parenleftbiggn\nA0/parenrightbigg\n≤/parenleftbigg\n1−1\ne/parenrightbigg\nln((1−2r)lnn)\nFor the left inequality we used the fact that p−1∈(0,1/2) andzlnz≥ −1/efor\nz∈(0,1/2). The right inequality is more straightforward. We now verify (16) . Recalling\n(13) and the fact that 1 −2r=p−1, which we are using constantly, the lower bound in\n(16) holds provided\nlnn≥e(2−ln2)(p−1)\n(p−1)p\nFrom the definition of the current sub-case, ln n≥e/(p−1), so a sufficient condition for\nthe above inequality to hold is\n2−ln2≤1\np−1+ln(p−1)\n21which is true by considering 1 /z+lnzforz∈(0,1/2). The upper bound in (16) holds\nprovided\n(p−1)lnn≤exp/parenleftbiggp−1\npln/parenleftbigg2e\n3n3\n2e/parenrightbigg/parenrightbigg\nwhich holds by applying ez≥ez. This completes the task of verifying (16). From (14) it\nfollows that\nβ−1\ni≤/braceleftbiggCi :i≤A0\nCA−1i2r/parenleftbig\nlnn\ni/parenrightbig−(p−1):i>A0(21)\nIn an integral where the integrand grows or decays at a controlled rate, one can change\nan upper bound of A0+1 toA0at the expense of a constant. Using\n/integraldisplayb\nae−ωω1/2dω≤C(b−a)e−aa1/2\n1+b−a\nvalid as long as 1 ≤a≤b, and\n/integraldisplayb\n0e−ωωp−1dω≤C\nwhich gives the correct order of magnitude for (say) b≥1/2,M|X|♯is bounded above by\nC1\np−1A0/summationdisplay\ni=1i−1\n2/parenleftBig\nlnn\ni/parenrightBig1\n2+C1\np−1A−(3−2p)\n2(p−1)n/e/summationdisplay\ni=A0i−2r/parenleftBig\nlnn\ni/parenrightBigp−1\n≤C1\np−1n−1\n2/integraldisplayA0\n1/parenleftBign\nx/parenrightBig1/2/parenleftBig\nlnn\nx/parenrightBig1/2\ndx+C1\np−1A−(3−2p)\n2(p−1)n−2r/integraldisplayn/e\nA0/parenleftBign\nx/parenrightBig2r/parenleftBig\nlnn\nx/parenrightBigp−1\ndx\n≤C1\np−1n1\n2/integraldisplay1\n2lnn\n1\n2lnn\nA0e−ωω1/2dω+C1\np−1A−(3−2p)\n2(p−1)(1−2r)−pn1−2r/integraldisplay(1−2r)lnn\nA0\n1−2re−ωωp−1dω\n≤C1\np−1A1\n2\n0/parenleftbigg\nlnn\nA0/parenrightbigg1\n2\n+C1\np−1A−(3−2p)\n2(p−1)(1−2r)−pn1−2r\n≤C1\np−1A−(3−2p)\n2(p−1)(1−2r)−pn1−2r\nWe claim that\n/integraldisplayb\n0eωω−(3−2p)dω≤/braceleftbiggC(p−1)−1b2(p−1): 0≤b≤1\nC(p−1)−1+Cebb−(3−2p):b≥1\nFor 0≤b≤1 this is clear. For b≥3 this follows because on [2 ,∞) the local exponential\ngrowth rate of the integrand is\nd\ndω[ω−(3−2p)lnω] = 1−3−2p\nω∈[0.5,1]\n22and for 1< b<3 the bound follows by monotonicity in b. Using the claim just proved,\nLip/parenleftBig\n|·|♯/parenrightBig\nis bounded above by\n/bracketleftBigg\nC1\np−1lnA0+C1\np−1A−(3−2p)\np−1n−4r/integraldisplayn/e\nA0/parenleftBign\nx/parenrightBig4r/parenleftBig\nlnn\nx/parenrightBig−(3−2p)\ndx/bracketrightBigg1\n2\n≤C1\np−1(lnA0)1\n2+C1\np−1A−(3−2p)\n2(p−1)(4r−1)1−pn1−4r\n2/parenleftBigg/integraldisplay(4r−1)lnn\nA0\n4r−1eωω−(3−2p)dω/parenrightBigg1\n2\nIf (4r−1)ln(n/A0)<1 then this is bounded by\nC1\np−1(lnn)1\n2+C1\np−1A−(3−2p)\n2(p−1)n1−4r\n2/parenleftbigg\nlnn\nA0/parenrightbiggp−1\n≤C1\np−1(lnn)1\n2\nTo see why the last inequality is true, note that the inequality\n(lnn)1\n2≥A−(3−2p)\n2(p−1)n1−4r\n2/parenleftbigg\nlnn\nA0/parenrightbiggp−1\nreduces to\nA3−2p\n2\n0/parenleftbigg\nlnn\nA0/parenrightbigg1\n2\n≤n4r−1\n2(lnn)1\n2\nwhich in turn follows since 1 ≤A0≤nand 3−2p= 4r−1. If (4r−1)ln(n/A0)≥1\nthen using 4 r−1 = 3−2pandA0ln(n/A0) =A−1/(p−1), we get the same bound, i.e.\nLip/parenleftBig\n|·|♯/parenrightBig\n≤C1\np−1(lnn)1\n2\nGoing all the way back to (12), regardless of whether (4 r−1)ln(n/A0) lies in [0,1) or\n[1,∞),\nn/summationdisplay\ni=1i−2rX2(p−1)\n[i]≤C(1−2r)−pn1−2r+Ct2(p−1)(lnn)2−p\nThen note that\n(1−2r)−p\n(1−2r)−1= exp(−(p−1)ln(p−1))∈(c,1)\n8 Statement and proof of the main result\nThe following diagram indicates the various cases considered in Theor em 17; it will be\nuseful to refer back to it when reading the proof.\n23rp\n01\n41\n21 213\n22\nFigure 1iaib* ib**\niiaiib* iib**\niii\nivCase ia:3\n2≤p<∞, 0≤r≤1\n2\nCase ib*:3\n2≤p<∞,1\n2<r≤1\nCase ib**:3\n2≤p<∞, 1<r≤2\nCase iia: 1 ≤p<3\n2,3−2p\n4≤r≤1\n2,p/\\e}atio\\slash= 2−2r\nCase iib*: 1 ≤p<3\n2,1\n2<r≤1\nCase iib**: 1 ≤p<3\n2, 1<r≤2\nCase iii: 1 ≤p<3\n2,p<3\n2−2r\nCase iv: 1 ≤p<3\n2,p= 2−2r\n: boundary of a region : other relevant line\nTheorem 17 There exist universal constants C,c>0and a function (r,p)/ma√sto→cr,pfrom\n[0,2]×[1,∞)to(0,∞)such that the following is true. Let\n(n,k,ε,r,p )∈N×N×(0,1/2)×[0,2]×[1,∞)\nand letGbe a random n×kmatrix with i.i.d. standard normal random variables as en-\ntries. Cases i-iv will be defined as in Figure 1 above. In each c ase variables EandFwill be\ndefined, and as long as k≤min{E,F}, with probability at least 1−Cexp(−min{E,F})\nthe following event occurs: for all x∈Rk,\n(1−ε)Mr,p|x| ≤ |Gx|r,p≤(1+ε)Mr,p|x|\nwhereMr,pdenotes the median of |Ge1|r,p.EandFare defined as follows:\nIn Case ia :3\n2≤p<∞,0≤r≤1\n2and\nE=cpn(lnn)2[p+(1−2r)lnn]pε2\n(p+lnn)2+p≥/braceleftbiggcr,pnε2:r/\\e}atio\\slash= 1/2\ncr,pn(lnn)−pε2:r= 1/2\nF=cpn2(1−r)\np(lnn)1+2\npε2\np\n/parenleftBig\n1+n2−2r−p\np/parenrightBig\n(p+lnn)1+2\np/parenleftbigg1+|2−2r−p|lnn\nlnn/parenrightbiggmax{2−p\np,0}\n≥\n\ncr,pnε2\np :p<2−2r\ncr,pn(lnn)−(2\np−1)ε2\np:p= 2−2r\ncr,pn2(1−r)\npε2\np:p>2−2r\n24In Case ib* :3\n2≤p<∞,1\n2<r≤1and\nE=cpppn2(1−r)(lnn)2[1+(2r−1)lnn]ε2\n[p+(1−r)lnn]2+p≥/braceleftbigg\ncr,pn2(1−r)(lnn)−(p−1)ε2:r/\\e}atio\\slash= 1\ncr,pn2(1−r)(lnn)3ε2:r= 1\nF=cpn2(1−r)\np(lnn)1+2\npε2\np\n[p+(1−r)lnn]1+2\np≥/braceleftBigg\ncr,pn2(1−r)\npε2\np:r/\\e}atio\\slash= 1\ncr,pn2(1−r)\np(lnn)1+2\npε2\np:r= 1\nIn Case ib** :3\n2≤p<∞,1<r≤2and\nE=cp(lnn)3ε2\n[1+(r−1)lnn]2≥/braceleftbiggcr,p(lnn)ε2:r/\\e}atio\\slash= 1\ncr,p(lnn)3ε2:r= 1\nF=c(lnn)1+2\npε2\np\n[1+(r−1)lnn]2\np≥/braceleftBigg\ncr,p(lnn)ε2\np:r/\\e}atio\\slash= 1\ncr,p(lnn)1+2\npε2\np:r= 1\nIn Case iia :1≤p<3\n2,3−2p\n4≤r≤1\n2,p/\\e}atio\\slash= 2−2rand\nE=cn[1+(1−2r)lnn]pε2\n(lnn)p ≥/braceleftbiggcr,pnε2:r/\\e}atio\\slash= 1/2\ncr,pn(lnn)−pε2:r= 1/2\nF=cn2(1−r)\npε2\np\n1+n2−2r−p\np/parenleftbigg1+|2−2r−p|lnn\nlnn/parenrightbigg1/p\n≥\n\ncr,pnε2\np:p<2−2r\ncr,pn(lnn)−1\npε2\np:p= 2−2r\ncr,pn2(1−r)\npε2\np:p>2−2r\nIn Case iib* :1≤p<3\n2,1\n2<r≤1and\nE=cn2(1−r)(lnn)2[1+(2r−1)lnn]ε2\n[1+(1−r)lnn]2+p≥/braceleftbigg\ncr,pn2(1−r)(lnn)−(p−1)ε2:r/\\e}atio\\slash= 1\ncr,pn2(1−r)(lnn)3ε2:r= 1\nF=cn2(1−r)\np(lnn)1+1\np[1+|2−2r−p|lnn]1\npε2\np\n[1+(1−r)lnn]1+2\np≥/braceleftBigg\ncr,pn2(1−r)\npε2\np:r/\\e}atio\\slash= 1\ncr,pn2(1−r)\np(lnn)1+2\npε2\np:r= 1\nIn Case iib** :1≤p<3\n2,1<r≤2and\nE=c(lnn)3ε2\n[1+(r−1)lnn]2≥cr,p(lnn)ε2\nF=c(lnn)1+2\npε2\np\n[1+(r−1)lnn]2\np≥cr,p(lnn)ε2\np\nIn Case iii :1≤p<3\n2,p<3\n2−2rand\nE=cnε2\nF=cn[1+(1−4r)lnn]1−1\npε2\np\n(lnn)1−1\np≥cr,pnε2\np\n25In Case iv :1≤p<3\n2,p= 2−2rand\nE=cn[1+(1−2r)lnn]ε2\nlnn≥/braceleftbiggcr,pnε2:r/\\e}atio\\slash= 1/2\ncr,pn(lnn)−1ε2:r= 1/2\nF=cnε2\np\n(lnn)2−p\np\nThe parameters EandFin Theorem 17 can be bounded as follows:\nE≥\n\ncr,pnε2: 0≤r<1/2\ncr,pn(lnn)−pε2:r= 1/2\ncr,pn2(1−r)(lnn)−(p−1)ε2: 1/2<r<1\ncr,p(lnn)3ε2:r= 1\nand\nF≥\n\ncr,pnε2\np: 0≤r≤1/2,p<2−2r\ncr,pn(lnn)1−2\npε2\np: 0≤r≤1/2,p= 2−2r\ncr,pn2(1−r)\npε2\np: 0≤r≤1/2,p>2−2r\ncr,pn2(1−r)\npε2\np: 1/2<r<1\ncr,p(lnn)1+2\npε2\np: r= 1\nForn≥n0(r,p), the coefficient cr,pcan be written explicitly in terms of randp, as can\nn0(r,p).\nProof. LetGbe a random matrix with i.i.d. standard normal random variables as\nentries, and let θ∈Sn−1.Gθtherefore has the standard normal distribution in Rn.\nSettingψ(x) =/summationtexti−rxp\n[i],\n|∇ψ(x)|=p/parenleftBiggn/summationdisplay\ni=1i−2rx2(p−1)\n[i]/parenrightBigg1/2\n(22)\nwhich is valid for all xwith distinct non-zero coordinates. Fix any t >0. With (22) in\nmind, forj∈ {0,1}set\nAj=/braceleftBigg\nx∈Rn:|x|♯≤/parenleftbigg4\n3/parenrightbiggj\nS/bracerightBigg\nwhere|·|♯andS(andRbelow) are as in Theorem 16. The cases in that theorem overlap,\nwhich is not a problem as long as you pick a case that applies to the value s ofpandrin\nquestion, and stick with that case. We shall apply the cases as follow s:\n•If 3/2≤p<∞use Case I.\n•If 1≤p<3/2 andp≥3/2−2randp/\\e}atio\\slash= 2−2ruse Case II.\n•If 1≤p<3/2 andp<3/2−2ruse Case III.\n26•If 1≤p<3/2 andp= 2−2ruse Case IV (either IVa or IVb, whichever applies).\nSincer≤2, the bounds in Theorem 16 simplify slightly, and can be written as follo ws.\nIn each of the cases below, definitions are given for AandB, and in each case R≤\nA+Bt2(p−1), whereRis as defined in Theorem 16 (with the same value of tthat appears\nhere). Conditions defining the case (i.e. i, ii, iii, iv) come first and are eit her without\nbracketsorwithsquarebrackets[...], thesquarebracketsindicatin garedundantcondition.\nConditions defining the sub-case (i.e. a, b) come last and are in paren theses (...).\nCase ia: 3/2≤p<∞(and 0≤r≤1/2),\nA=Cpppn1−2r(lnn)p\n[p+(1−2r)lnn]p\nB=Cp/parenleftbigglnn\n1+|2−2r−p|lnn/parenrightbiggmax{2−p,0}/parenleftbig\n1+n2−2r−p/parenrightbig\nCase ib: 3/2≤p<∞(and 1/2<r≤2),\nA=Cp(lnn)p\n1+(2r−1)lnn\nB=Cp/parenleftbigglnn\n1+|2−2r−p|lnn/parenrightbiggmax{2−p,0}/parenleftbig\n1+n2−2r−p/parenrightbig\nCase iia: 1≤p<3/2,p≥3/2−2r,p/\\e}atio\\slash= 2−2r(and 0≤r≤1/2),\nA=Cn1−2r(lnn)p\n[1+(1−2r)lnn]p\nB=C1+n2−2r−p\n1+|2−2r−p|lnnlnn\nCase iib: 1≤p<3/2, [p≥3/2−2r], [p/\\e}atio\\slash= 2−2r] (and 1/2<r≤2),\nA=C(lnn)p\n1+(2r−1)lnn\nB=C1+n2−2r−p\n1+|2−2r−p|lnnlnn\nCase iii: 1≤p<3/2 andp<3/2−2r,\nA=Cn1−2r\nB=Cn2−2r−p/parenleftbigglnn\n1+(1−4r)lnn/parenrightbiggp−1\nCase iv: 1≤p<3/2 andp= 2−2r,\nA=Cmin/braceleftbig\n(1−2r)−1,lnn/bracerightbig\nn1−2r\nB=C(lnn)2−p\n27ByTheorem16, γn(A1)≥γn(A0)≥1−Cexp(−t2/2). SinceA1isconvex, aboundon\nthe gradient transfers directly to a bound on the Lipschitz consta nt (using Lemma 15 to\nignore points of non-differentiability), and Lip(ψ|A1)≤CpR1/2. We may now extend the\nrestrictionψ|A1to a Lipschitz function ψ∗:Rn→Rsuch thatLip(ψ∗) =Lip(ψ|A1). By\na result of Schechtman (see his comments near the end of the pape r), as long as k≤ct2,\nwith probability at least 1 −Cexp(−ct2), the following event occurs: for all θ∈Sk−1,\n|ψ∗(Gθ)−Eψ∗(Gθ)| ≤tLip(ψ∗) (23)\nHere it is essential to have a result that applies to Lipschitz function s besides just norms\n(the most typical application of Schechtman’s result is to norms). W e now show that for\nallθ∈Sk−1,Gθ∈A1and therefore ψ∗(Gθ) =ψ(Gθ). To do this, consider a 1 /4-net\nN ⊂Sk−1with cardinality |N| ≤12k. By the union bound, with probability at least\n1−12kCexp(−ct2)≥1−Cexp(−c2t2), for allω∈ N,Gω∈A0. Now for any θ∈Sk−1\nwriteθ=/summationtext∞\ni=0εiωi, where 0 ≤εi≤4−iandωi∈ N. Then\n|Gθ|♯≤∞/summationdisplay\ni=04−i|Gωi|♯≤4\n3S\nwhich shows that Gθ∈A1. (23) can then be written as\n|ψ(Gθ)−Eψ∗(Gθ)| ≤tCpR1/2\nvalid for all θ∈Sk−1. It is an elementary calculation that concentration about any point\nimplies concentration about the median, with slightly modified constan t, so\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ(Gθ)\nMψ(Gθ)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CptR1/2\nMψ(Gθ)(24)\nChoosetso that\n4p−1CptR1/2\nMψ(Gθ)=ε (25)\nand assume that ε∈(0,2/p) so thatpε/4<1/2 (in order to satisfy |1−u| ≤1/2 below).\nThe bounds for ε∈[2/p,1/2) will follow from the bounds for ε∈(0,2/p) by monotonicity\nand by changing the value of Cthat appears in Cp. Using/vextendsingle/vextendsingle1−u1/p/vextendsingle/vextendsingle≤p−121−1/p|1−u|,\nwhich holds when |1−u| ≤1/2, withu=ψ(Gθ)/Mψ(Gθ), (24) implies\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ(Gθ)1/p\nMψ(Gθ)1/p−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2p−1CptR1/2\nMψ(Gθ)<ε\nIt then follows from homogeneity that\n(1−ε)|x|M|Ge1|r,p≤ |Gx|r,p≤(1+ε)|x|M|Ge1|r,p\nfor allx∈Rk. (25) can be used to bound tin terms of ε, sinceRhas been expressed\nin terms of tin Theorem 16, and in Lemma 13 Mψ(Gθ) is bounded below in t. The\nsufficient condition k≤ct2and the probability bound 1 −Cexp(−ct2) can then be\n28written in terms of ε. The inversion can be simplified by converting a sum to a max and\nusing the fact that if f(x) = max {g(x),h(x)}withg,hcontinuous and increasing, then\nf−1(x) = min{g−1(x),h−1(x)}. The result is that,\nt2≥min\n\ncpA−1/parenleftBigg\nMn/summationdisplay\ni=1i−rXp\n[i]/parenrightBigg2\nε2,cB−1/p/parenleftBigg\nMn/summationdisplay\ni=1i−rXp\n[i]/parenrightBigg2/p\nε2/p\n\n\nwhereAandBare as defined in cases i-iv above, and X=Ge1follows the standard\nnormal distribution in Rn. Cases ib and iib split into cases ib*, ib**, iib* and iib**,\ndepending on whether 1 /2< r≤1 or 1<r≤2. The final bounds can then be written\nas in the statement of the theorem.\n9 The general case\nHere we study the norm\n|x|ω,p=/parenleftBiggn/summationdisplay\ni=1ωixp\n[i]/parenrightBigg1/p\nwhere (ωi)n\n1is any non-increasing sequence in [0 ,1] withω1= 1. Our main interest is in\nthe case 1 ≤p<∞, however our proof will force us to consider also 0 <p<1. The proof\nof Lemma 14 generalizes easily and we see that\nsup\n\n/parenleftBiggn/summationdisplay\ni=1ωiθp\n[i]/parenrightBigg1/p\n:θ∈Sn−1\n\n=\n\n/parenleftbigg/summationtextn\n1ω2\n2−p\ni/parenrightbigg2−p\n2p\n:p∈[1,2)\n1 : p∈[2,∞)\nand generalizing Lemma 13, for 0 <p<∞,\nM|X|ω,p≤C/parenleftBiggn/summationdisplay\ni=1ωi/parenleftBig\nlnn\ni/parenrightBigp/2/parenrightBigg1/p\nwith the reverse inequality with Creplaced by c. For 1≤p<∞,|·|ω,pis a norm, while\nfor 0<p<1 the triangle inequality is replaced with\n|x+y|ω,p≤21/p/parenleftBig\n|x|ω,p+|y|ω,p/parenrightBig\n(proof just as in the classical case of ℓn\np), and|·|ω,pis no longer a norm but a quasi-norm.\nFor an infinite sum, using induction one can show that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\ni=1xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω,p≤∞/summationdisplay\ni=12i/p|xi|ω,p\nAssometimes happens, thegeneralresult iseasier tostateandpr ovethanisthespecial\ncase, since the complexity is hidden. The proof of Theorem 18 below is a variation of the\nproof of Theorem 17, using the quasi-norm property instead of th e triangle inequality.\n29Theorem 18 There exists a universal constant c>0such that the following is true. Let\n(n,k,ε,p)∈N×N×(0,1/2)×[1,∞)\nand let(ωi)n\n1be any non-increasing sequence in [0,1]withω1= 1. Let|·|ω,pdenote the\ncorresponding Lorentz norm, and let Gbe a random n×kmatrix with i.i.d. standard\nnormal random variables as entries. If p/\\e}atio\\slash= 1set\nd=/parenleftbigg\n1+1\np−1/parenrightbigg−1\nmin\n\ncp/parenleftBig/summationtextn\ni=1ωi/parenleftbig\nlnn\ni/parenrightbigp/2/parenrightBig2\nε2\n/summationtextn\ni=1ω2\ni/parenleftbig\nlnn\ni/parenrightbigp−1,cB−1/p/parenleftBiggn/summationdisplay\ni=1ωi/parenleftBig\nlnn\ni/parenrightBigp/2/parenrightBigg2/p\nε2/p\n\n\nwhere\nB=\n\n/summationtextn\ni=1ω2\nii−(p−1): 1<p<3/2/parenleftbigg/summationtextn\ni=1ω2\n2−p\ni/parenrightbigg2−p\n: 3/2≤p<2\n1 : 2 ≤p<∞\nand ifp= 1set\nd=c/parenleftBig/summationtextn\ni=1ωi/parenleftbig\nlnn\ni/parenrightbig1/2/parenrightBig2\nε2\n/summationtextn\ni=1ω2\ni\nAssume that k≤d. With probability at least 1−2e−dthe following event occurs: for all\nx∈Rk,\n(1−ε)|x|M|Ge1|ω,p≤ |Gx|ω,p≤(1+ε)|x|M|Ge1|ω,p\nProof.First, assume that p/\\e}atio\\slash= 1 (we leave the case p= 1 to the reader). For x∈Rn, set\nψ(x) =n/summationdisplay\ni=1ωixp\n[i]|x|♯=/parenleftBiggn/summationdisplay\ni=1ω2\nix2(p−1)\n[i]/parenrightBigg1\n2(p−1)\nand so\n|∇ψ(x)|=p/parenleftBiggn/summationdisplay\ni=1ω2\nix2(p−1)\n[i]/parenrightBigg1/2\nwhichisvalidforall xwithdistinctnon-zerocoordinates. Thepointsofnon-differentiab ility\nare not a problem, by Lemma 15. Consider any θ∈Sn−1. SetR=A+CpBt2(p−1), where\nA=Cp/summationtextn\ni=1ω2\ni/parenleftbig\nlnn\ni/parenrightbigp−1andBisdefinedinthestatementofthetheorem. For j∈ {0,1},\nset\nAj=/braceleftBigg\nx∈Rn:|x|♯≤/parenleftbigg4q\n3/parenrightbiggj\nR1\n2(p−1)/bracerightBigg\nwhere\nq=/braceleftbigg\n2−1/(p−1): 1<p<3/2\n1 : 3/2≤p<∞\nFor 1< p <3/2 we now use (4) from Lemma 12 on order statistics from the standa rd\nnormal distribution, as used in Case II of Theorem 16, and for 3 /2≤p <∞we use\n30Gaussian concentration applied to |·|♯. The result of this is that γn(A1)≥γn(A0)≥\n1−Cexp(−ct2). It follows by definition of A1and by its convexity, and the expression\nfor|∇ψ|that Lip(ψ|A1)≤CpR1/2. By the extension property of real valued Lipschitz\nfunctions on metric spaces, ψ|A1can be extended to a function ψ∗:Rn/ma√sto→Rsuch that\nLip(ψ∗) = Lip(ψ|A1). By Schechtman’s result (Theorem 7 here), as long as k≤ct2, the\nfollowing event occurs with probability at least 1 −Cexp(−ct2): for allθ∈Sk−1,\n|ψ∗(Gθ)−Mψ∗(Gθ)| ≤CtLip(ψ∗)\nWe now show that for all θ∈Sk−1,Gθ∈A1and therefore ψ(Gθ) =ψ∗(Gθ). Let\nN ⊂Sk−1be aq/4-net inSk−1. The standard volumetric bound shows that Ncan\nbe chosen so that |N| ≤(12/q)k. By the bound on γn(A0) and the union bound, with\nprobabilityatleast1 −C(12/q)kexp(−ct2)≥1−Cexp(−c′t2), thefollowingevent occurs:\nfor allθ∈ N,Gθ∈A0, i.e.\n|Gθ|♯≤R1\n2(p−1)\nNow for any θ∈Sk−1, writeθ=/summationtext∞\n1εiθi, where 0 ≤εi≤(q/4)iandθi∈ N, to conclude\n(using the quasi-norm property),\n|Gθ|♯≤R1\n2(p−1)∞/summationdisplay\ni=0(q)−(i−1)(q/4)i≤4q\n3R1\n2(p−1)\nSoGθ∈A1as desired, and ψ∗(Gθ) =ψ(Gθ). So, conditioning on the events dealt with\nabove, for all θ∈Sk−1,\n|ψ(Gθ)−Mψ∗(Gθ)| ≤CtLip(ψ)\nThe rest of the proof is identical to the proof of Theorem 17, and w e may change the\ncoefficient of e−dfromCto 2 by changing the value of c.\nThe bounds in Theorem 18 are non-optimal when:\n•the coefficient sequence ( ωi)n\n1approximates ( i−r)n\n1, forp= 2−2r(1/4<r<1/2), and\nin this case we refer the reader to Theorem 17 and Corollary 2,\n•whenr>1 in which case we refer the reader to Theorem 3, and when\n•p→1 (p/\\e}atio\\slash= 1), which is due to the presence of the factor 1 /(1 + 1/(p−1)), but this\nis not an issue in the asymptotic case n→ ∞whileωandpremain fixed. We refer the\nreader to Theorem 17 and Corollary 4 for estimates that do not inclu de this extra factor.\nReferences\n[1] Bobkov, S. G., Nayar, P., Tetali, P.: Concentration properties of restricted measures\nwith applications to non-Lipschitz functions. Geometric aspects of functional analysis,\n25-53, Lecture Notes in Math. 2169, Springer, Cham, 2017.\n[2] Fresen, D. J.: Explicit Euclidean embeddings in permutation invarian t normed spaces.\nAdv. Math. 266, 1–16 (2014).\n31[3] Fresen, D.J.: Variationsandextensions ofthe Gaussianconcen tration inequality, Part\nI. To appear in Quaest. Math. (pending revision). An older version o f Part I and Part\nII combined available at arXiv:1812.10938.\n[4] Fresen, D.J.: Variationsandextensions ofthe Gaussianconcen tration inequality, Part\nII. To appear in Electron. J. Probab. (pending revision). An older v ersion of Part I\nand Part II combined available at arXiv:1812.10938.\n[5] Gordon, Y.: Some inequalities for Gaussian processes and applicat ions. Israel J. Math.\n50, 265-289, 1985.\n[6] Huang, H., Wei, F.: UpperboundfortheDvoretzkydimension inMilma n-Schechtman\ntheorem. 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Anal. 265 (9), 2074–2088 (2013)\n32"    },    {        "title": "2201.07842v1.Active_tuning_of_plasmon_damping_via_light_induced_magnetism.pdf",        "content": "Title: Active tuning of plasmon damping via light induced magnetism .  \n \nAuthor list :  \nOscar Hsu -Cheng Cheng1*, Boqin Zhao1*, Zachary Brawley3, Dong Hee Son1,2, Matthew \nSheldon1,3 \n1Department of Chemistry, Texas A&M University, College Station, TX, USA.  \n2 Center for Nanomedicine, Institute for Basic Science and Graduate Program of Nano \nBiomedical Engineering, Advanced Science Institute, Yonsei University, Seoul, Republic of Korea.  \n3Department of Material Science and Engineering, Texas A&M University, College Station, \nTX, USA.  \n*These authors contributed equally to this work.  \ne-mail: dhson@chem.tamu.edu; sheldonm@tamu.edu  \n Keywords: plasmonic, inverse Faraday effect, Raman thermometry, magnetoplasmon,  \n Abstract  \nCircularly  polarized optical excitation of  plasmonic  nanostructures causes \ncoherent circulating motion of their electrons, which in turn, gives rise to strong optically \ninduced magnetization — a phenomenon known as  the inverse Faraday  effect  (IFE). In \nthis study we report  how the IFE also significantly decreases  plasmon damping. By \nmodulating the optical  polarization state incident on achiral plasmonic nanostructures \nfrom linear to circular , we observe reversible increases of reflectance by 78% as well as \nsimultaneous  increases of optical field concentration by 35.7%  under 10\n9 W/m2 \ncontinuous wave (CW) optical excitation. These signatures of decreased plasmon \ndamping  were also monitored in the presence of an externally  applied magnetic field (0.2 \nT). The combined interactions  allow  an estimate of the light -induced magnetization , which \ncorresponds to an effective magnetic field  of ~1.3 T during circularly  polarized CW \nexcitation (109 W/m2). We rationalize the observed decreases  in plasmon damping  in \nterms of the Lorentz forces acting on the circulating electron trajectories . Our results \noutline strategies  for actively modulating  intrinsic  losses in the metal , and thereby , the \noptical mode quality  and field concentration via opto- magnetic  effects  encoded in the \npolarization state of incident light.  \n Introduction \nThe r eversible modulation of  plasmonic  resonances in metal nanostructures using \nexternal stimul i – i.e., “ active plasmonics” —  is currently of great interest for potential \napplications in sensing,  optoelectronic  devices, and light-based information processing\n1. \nCommonly explored strategies modulate  plasmon resonance frequenc ies by altering the \nsurrounding dielectric environment  of nanostructures, for  example, with \nthermoresponsive materials2-4. Modification of plasmonic  modes based on distance -\ndependent optical coupling  between nanostructures  in compliant  media under stress and \nstrain has also been demonstrated5-7. Alternatively, t he optical properties  of the metal \ncomprising a nanostructure can be reversibly modulated.  Within the D rude model, the \ncomplex dielectric  function of a  metal at angular frequency, 𝜔𝜔, is well described in terms \nof the electrical carrier density,  𝑛𝑛, and the damping constant,  𝛾𝛾, for the carrier oscillations : 𝜀𝜀(𝜔𝜔) = 1−𝜔𝜔𝑝𝑝2\n𝜔𝜔2+𝑖𝑖𝜔𝜔𝑖𝑖 (1) \nwhere 𝜔𝜔𝑝𝑝=�𝑛𝑛𝑒𝑒2\n𝜀𝜀𝑜𝑜𝑚𝑚𝑒𝑒 is the bulk plasma frequency, also depending on the electron charge, \n𝑒𝑒, effective mass,  𝑚𝑚𝑒𝑒, and vacuum permittivity, 𝜀𝜀𝑜𝑜8. Researchers have shown that  \ncapacitive surface charging  of metals  when they are integrated into electrochemical cells \nresult s in reversible shifts of their plasmon resonance frequency through the dependence \non 𝑛𝑛9-11. In the time domain, pulsed laser excitation can similarly perturb electronic carrier \npopulations giving rise to transient modulation of plasmonic behavior12-17. \n In comparison, the possibility of actively tuning plasmon damping  in the steady \nstate, and the opportunities  for manipulating  plasmonic behavior, has been studied much \nless18-21. In equation (1) , 𝛾𝛾 reflects several different  microscopic processes connected \nwith the conductivity and mean  free path of electrical carriers in the metal such as  \nelectron- electron scattering , electron- phonon scattering , surface scattering22, and \nchemical interface damping with surface adsorbates23, in addition to other loss pathways \nsuch as  radiation  damping24 and Landau damping25,26. Usually, 𝛾𝛾 is considered to be an \nintrinsic property that is determined by  the chemical identity of the metal27,28, surface \nchemistry  and morphology, such as the crystal grain size29-32, or the modal  behavior at a \nparticular  frequency19,33, e.g. near field localization versus far field out -coupling . However , \nchanges  in plasmon damping have a profound impact  on the ability of a nanostructure to  \nconcentrate optical power  in a particular  mode , known as  the quality factor or “ Q” factor. \nDecreasing 𝛾𝛾 lowers the imaginary part of the permittivity , decreasing ohmic losses from \ncarrier motion at the optical frequency  and increasing overall optical scattering or \nreflectance. L ower damping also provides  greater field enhancement  at sub-wavelength \n“hot spots” , improving efficiency for localized  sensing, photochemistry, or heating  via \nphotothermalization.  \nIn a series of recent studies,  the Vuong laboratory  reported anomalously large \nmagneto- optical (MO) responses from colloidal Au nanoparticles under small magnetic \nfields (~1 mT) and low intensity circularly polarized (CP) excitation (<1 W/ cm2)34,35. The \nlarge MO response was  attributed to the interaction between ext ernal magnetic fields and \ncirculating drift  current s in the metal  that were  resonantly excited during  CP excitation  \n(Fig. 1a). The generation of drift current s from coherent charge density waves that \ncirculate  in metals  during CP optical cycle s has been studied extensively theoretically36-\n38, and is understood to contribute to optically induced static  (DC) magnetism . This \nbehavior is also known as the Inverse Faraday Effect (IFE)36-41. Our recent  experimental \nstudy  of 100 nm Au nanoparticle colloids showed ultrafast modulation of effective  \nmagnetic fields up to 0.038 T under a peak intensity of 9.1×1013 W/m2, confirming MO \nactivity  and optically -induced magnetism  many orders of magnitude great er than for bulk \nAu42. Notably,  Vuong  et al.  also reported an apparent increase of the volume averaged \nelectrical conductivity  when an external magnetic field was aligned with the light -induced \nmagnetic field. Considering  several recent theoretical  and experiment al magnetic circular \ndichroism  (MCD) studies43-45, these results can be interpreted as a decrease in plasmonic  \ndamping when the microscopic electron motion in a plasmon resonance contributes  to \nDC magnetization.  \nIn this study we show that plasmon damping is indeed strongly modified by \nmagnetic interactions , whether magnetism  is induced externally using an applied magnetic field or created internally with light via the IFE. We observe that the normal \nincidence backscattering , i.e. the reflectance, of array s of non-chiral Au nanostructures is \nincreased by 78%  when controlling  the ellipticity  of incident light  from linearly polarized \n(LP) to CP  during 109 W/m2 continuous wave (CW) excitation. Further, we query the \noptical field concentration at hot spots in the metal  by taking advantage of recently refined \nelectronic Raman  thermometry techniques46-53. Localized photothermal  heating induces \ntemperatures ~2 3 K greater during CP excitation than during LP excitation under  an \nexcitation intensity of ~109 W/m2, suggesting active modulation of optical field \nenhancement  by 35.7% at hot spots. The local heating is further modulated by the \npresence of an externally applied magnetic  field (~0.2 T) , supporting  the underlying \nmagnetic origin of these  phenomena and allowing estimation of the light -induced effective \nmagnetic fields  at hot spots  (~1.3 T). Taken together, our results indicate reversible \nmodulation of the plasmon damping that can exceed ~ 50%, solely  by controlling the \nellipticity of the incident  radiation.  \n \n \n \nFig. 1  Sample overview . (a) Relationship between incident light helicity, induced \nelectronic motion, induced magnetization from the IFE  (MIFE), and induced magnetization  \n(Mind) from an external static magnetic field (Bapp) in a plasmonic nanostructure. When \nincident light has left -handed circular polarization (LHCP), M IFE and Mind are parallel. (b) \nSchematic of the Au nanodisk array sample (not to scale). ( c) Top view and ( d) side view \nof the local field enhancement during 532 nm excitation. Width and height not to scale in (d). (e) Optical image of the array on an Au film (f ) SEM image. Scale bar s: (c) 100 nm \n(e) 40 µm and ( f) 1 µm.  \n    \nResult and Discussion  \nSamples consisting of 100 µm x 100 µm arrays of 400 nm diameter by 100 nm \nheight disk-shaped gold nanostructures in a square lattice pattern (700 nm pitch) were \ndeposited on 38 nm thick Al 2O3 layer on top of 100  nm thick gold  films using electron-\nbeam lithography ( Fig. 1b, e, f , see Methods section). The nanodisk shape supports \ncirculating electronic currents during CP excitation (Fig. 1a). Periodic arrays provide high \nabsorpt ivity across the visi ble spectrum (Fig.  1e, Fig.  2a), aiding photothermal heating for \nthe Raman thermometry studies detailed below. The overall sample geometry is achiral, \nhighly symmetric, and exhibits no polarization or ellipticity dependence for absorption or scattering (neglecting non- linear effects), as confirmed by full wave optical simulations \n(FDTD method, see S upporting Information).  \n  \n  \nFig. 2 Spectroscop y of gold nanodisk array s. (a) Absorption spectrum. The dashed \nline indicates 532 nm. (b) Electronic Raman spectrum during 532 nm CW excitation. Different spectral regions provide information analyzed in this study. Orange box: anti -\nStokes signal used for temperature fitting. Purple box: 532 nm backscattering (Rayleigh \nline, filtered here) used to quantify ellipticity -dependent reflection. Green box: the broad \nenergy distribution of the Stokes region is  fit for the plasmon damping, 𝛾𝛾. \n \nFig. 3 Confocal spectral mapping under different helicit ies. (a, d) Confocal Raman \nand (b, e) backscattering intensity map of the  gold nanodisk  array under 532 nm CW \nexcitation with (a, b) CP or (d, e) LP. (c, f) Line scans  of the backscattering efficiency \nalong the region between (blue) or over (red) nanodisk s with LP (dashed trace) or CP \n(solid trace) excitation. Inset: optical image of the sample  array. The green box indicates \nthe region of the confocal map.  Scale bar: 4 µm.   \n \n \nSamples were mounted onto a piezo- driven microscope stage, and confocal maps \nof the electronic Raman (eR) spectrum (Fig . 3a, d ) or the backscattering intensity at 532 \nnm (Fig. 3b, e) were collected as a function of position over the edge of an array during linearly polarized (LP) or circularly polarized (CP) excitation. A representative eR spectrum is shown in Fig . 2b.  In comparison with typical Raman signals that result from \ninelas tic scattering with vibrational modes in a sample, the eR signal is due to inelastic \ninteractions with the electron gas at the metal surface. The broad eR signal therefore \nprovides information about the energetic distribution of electrons, such as their \ntemperature. As clearly seen by comparison of F ig. 3a, d , and Fig. 1 c, d, the eR signal is \nstrongest at “hot spots” at the edge of nanodisks where field enhancement is greatest . In \ncontrast, the sample backscattering, or reflectance,  (Fig. 3b, e) is lower ov er the array  \ndue to pronounced plasmonic absorption, ultimately giving rise to localized photothermal \nheating.  \nThe backscattering efficiency of the nanodisks is strongly modulated based on the \npolarization state (ellipticity) of the incident light. As displayed in F ig. 3c, f, we compared \nthe backscattered light intensity from different locations over the nanodisk array and the \nadjacent Au film during LP (dashed trace) or CP (solid trace) excitation. The signal \nintensity was converted to backscattering effic iency by referencing the Au film region to \na smooth Ag mirror, in order to rule out any polarization- dependent instrument response. \n(see Methods section for details). In all locations over the array the backscattering \nefficiency is generally larger for CP versus LP, with a maximum relative increase of \nbackscattering up to 78 % at the edge of individual  nano disks during CP excitation.  \nWe rule out the possibility that this trend is due to inherent differences in the \nellipticity -dependent scattering efficiency based on sample geometry, because the \nsample is not chiral. Neglecting optically induced magnetization or other non- linear effects, \nthe total absorption and scattering of the nanostructure array is expected to show no dependence on beam ellipticity. Indeed, we have performed linear, full wave optical simulations (FDTD method, see Supporting information) tha t confirm no difference in the \nabsorption or scattering efficiency based on LP or CP excitation. We hypothesize that the large difference in backscatteri ng efficiency observed experimentally results from \nellipticity -dependent modulation of the plasmon damping. This interpretation is \nqualitatively supported by additional optical simulations ( see Supporting information) that \nshow a comparable increase of bac kscattering , depending on sample location,  when \ndamping is decreased by  50% or greater . \nDecreased plasmon damping also results in more concentrated optical fields at hot \nspots, which can lead to more pronounced local photothermal heating. In terms of \nequat ion (1) , as the damping constant 𝛾𝛾 decreases the imaginary part of the dielectric \nfunction 𝐼𝐼𝑚𝑚(𝜀𝜀(𝜔𝜔)) also decreases, giving rise to a larger Q -factor and greater local field \nenhancement\n54. The local power density for heating,  𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧), depends on field \nenhancement as55,56: \n𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧) = 1\n2𝐼𝐼𝑚𝑚[𝜀𝜀(𝑥𝑥,𝑦𝑦,𝑧𝑧)]𝜀𝜀0|𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧)|2 (2) \nwhere 𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧) is the local electric field.  Thus, lower plasmon damping provides a net \nincrease of heating power and correspondingly larger temperatures at locations with \nstrong field enhancement. Intuitively, lower damping increases the cross section that \nfunnels light energy into a plasmonic hot  spot.  See simulations  of this effect for the \nnanodisk array  geometry in  Supporting information.  \nRaman signal intensity also depends on local field enhancement, scaling as |𝐸𝐸|4 \n57. Therefore Raman- based thermometry techniques primarily probe the nanostructure \ntemperature at hot  spots. We measured the sample temperature at hot spots by adapting \nan anti -Stokes (aS) Raman thermometry method developed by Xie et al46. Experimentally, \nit has been shown that the spectral intensity of the aS eR signal, 𝑆𝑆(∆𝜔𝜔), is thermally \nactivated according to a Bose- Einstein distribution. The aS spectrum collected from a \nsample at an unknown temper ature, 𝑇𝑇𝑙𝑙, can be normalized by a spectrum collected at a \nknown temperature, 𝑇𝑇0, according to  \n𝑆𝑆(∆𝜔𝜔)𝑇𝑇𝑙𝑙\n𝑆𝑆(∆𝜔𝜔)𝑇𝑇0 = 𝑒𝑒𝑒𝑒𝑝𝑝 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘𝐵𝐵𝑇𝑇0� −1\n𝑒𝑒𝑒𝑒𝑝𝑝 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘𝐵𝐵𝑇𝑇𝑙𝑙� −1 (3) \nwhere ℎ is Plank constant, 𝑐𝑐 is the speed of light, ∆𝜔𝜔 is the wavenumber (negat ive for \nanti-Stokes), and 𝑘𝑘𝐵𝐵 is Boltzmann constant. Spectral features that do not change with \nthermal activation, such as the frequency -dependent signal enhancement factor, cancel \nout, so that 𝑇𝑇𝑙𝑙 is the only unknown fitting parameter.  \nWith fixed linear polarization, we measured the aS spectra of samples as a function \nof excitation power to induce variable amounts of photothermal heating , and we fit for 𝑇𝑇 𝑙𝑙. \nThe spectra were normalized by a spectrum collected at the lowest possible power that \npreserved good signal -to-noise ( I = 7.2 × 106 W/m2), with the goal of inducing minimal heating above room temperature, i.e. 𝑇𝑇0 ≈ 298 K. We observe that  𝑇𝑇𝑙𝑙 increases linearly \nwith LP excitation intensity (Fi g. 4, blue line), in good agreement wit h many other reports \nof gold and copper nanostructures46,50,58. The linear fit to the temperature trend shows a \ny-intercept near room temperature, at the limit of zero incident power, further confirming \nthe accuracy of the thermometry technique. The fitted slope of the trend ( 4.6×10-8 K·m2/W) \ndescribes the “heating efficiency” of the sample under LP excitation.  \nWe determined changes in the sample temperature, and hence modification of the \nplasmon damping, by measuring the eR response while varying the ellipticity of the \nexcitation beam. However, given the complex spectral dependence of the eR signal, \nsimilar procedures as those described for the backscattering study could not be used to correct for the ellipti city-dependent instrument response. Instead, we devised a dual \nbeam configuration (see M ethods section for details). In summary, two separate CW 532 \nnm laser beams were coincident on the sample. A low power “Beam 1” was maintained with linear polarization.  A second, higher power “Beam 2” was used to induce variable \nmagnetization and damping in the sample by controlling excitation ellipticity. The eR \nspectrum resultant from Beam 1 was isolated and fit for 𝑇𝑇\n𝑙𝑙 by collecting a spectrum with \nboth Beam 1 and Beam 2 incident, and then subtracting the spectrum collected with only \nBeam 2 incident. This procedure allowed us to probe the sample temperature using a \nbeam that had non- changing incident power and linear polarization (Beam 1) while the \nsample was excited with variable ellipticity and power. The accuracy of this dual -beam \nmethod is confirmed in Fig . 4 (red trace).  The fitted 𝑇𝑇𝑙𝑙 are reported as a function of the \ntotal incident power of Beam 1 and Beam 2, with the power of Beam 1 held constant and both bea ms maintained in linear polarization. A similar y -intercept and heating efficiency \nis observed using either thermometry technique.  \n \n \nFig. 4 Photothermal heating with a single - or dual-beam geometr y. (a) Fitted \nnanostructure temperature, 𝑇𝑇\n𝑙𝑙 as function of excitation intensity, 𝐼𝐼, when only Beam 1 \n(blue) or when both Beam  1 and Beam 2 (red) were  incident . Blue linear  fit: 𝑇𝑇𝑙𝑙 (𝐾𝐾) =\n 4.6×10−8𝐼𝐼 + 299. For the dual -beam  study the  intensity of Beam 1 was kept at 4.8×108 \n(W/m2), and the intensi ty of Beam 2 varied between 6.6 –  9.5×108 (W/m2). Red linear  fit: \n𝑇𝑇𝑙𝑙 (𝐾𝐾) = 4.6×10−8𝐼𝐼+299.  \n \nFig. 5. Ellipticity dependent photothermal heating. Fitted nanostructure temperature \nas a function of Beam 2  ellipticity . The intensity of Beam 1 was kept at 4.8×108 W/m2. The \nintensity of Beam 2 was kept at 9.5×108 W/m2. The dashed line is a guide for the eye. \nSee Fig . S8 for a detailed explanation of ellipticity  and ellipticity angle.  \n \n \nWe then examined the dependence on the ellipticity of Beam 2 while the power of \nboth beams was held constant (Fig . 5). For both right -handed or left -handed circular \npolarization (RHCP or LHCP), 𝑇𝑇𝑙𝑙 increases with increasing ellipticity. For the same \nmagnitude ellipticity but opposite helicity, the increase of  𝑇𝑇𝑙𝑙 is simila r. This trend indicates \nenhanced field concentration at hot spots when the ellipticity -dependent magnetization in \nthe sample is increased. The maximum increase of temperature observed for CP compared with LP is ~23 K. Based on the heating efficiency determ ined under LP \nexcitation (Fig . 4a, red line), this same temperature increase would be expected if the \nsample received 5 × 10\n8 W/m2 more incident power. Since the total excitation power was \nkept constant at 1.4 × 109 W/m2, we conclude that the switch from LP to CP excitation \nequivalently increases the heating power at hot spots by 35.7%. We emphasize that field enhancement is not expected to depend on excitation ellipticity, neglecting non- linear \neffects, because the sampl e is not chiral.  \nWe also studied sample behavior under an externally applied magnetic field. For \nthis experiment the ellipticity of Beam 2 was kept at either 0 (LP), +0.67 (LHCP), or - 0.67 \n(RHCP). Note that the ellipticity for RHCP was limited to this ran ge based on our \nexperimental geometry (see Methods section). An external magnetic field B\napp = 0.2 T \nwas applied parallel to the direction of light propagation (Fig. 1a), and 𝑇𝑇𝑙𝑙 was measured \nas a function of total incident power. As summarized in Fig. 6,  the fitted 𝑇𝑇𝑙𝑙 are larger under \nLHCP excitation compared to RHCP, while both polarizations cause greater heating than LP. This effect can be rationalized in terms of the direction between  the optically induced \nmagnetization , M\nIFE, and the applied magneti c field , Bapp. When MIFE and B app are anti -\nparallel, as for RHCP, the optically induced circular electron motion is opposite the direction favored by the Lorentz forces from the external magnetic field, resulting in an increase of damping and lower optical  field enhancement.  In support of this picture, Gu. \net al.  theoretically analyzed the behavior of a free electron gas in a nanoparticle under \nCP excitation and predicted that the optically induced magnetic moment is enhanced \n(suppressed) when an external m agnetic field is aligned (anti -aligned), due to the Lorentz \nforces on individual electrons that perturb their circular movement59. Theoretical studies \nof magnetoplasmons also predict decreases in damping when rotating surface charge \ndensity waves provide magnetization parallel with externally applied magnetic fields43. \nThe difference in heating efficiency during LHCP and RHCP excitation allows an \nestimate of the strength of the optically induced magnetization, MIFE, at hot spots  in terms \nof the magnetization, Mind, that results  from Bapp (see full calculations in Supporting \nInformation) . Assuming that the temperature increase compared to LP excitation is \nlinearly proportional to the net magnetization Mind + MIFE we determine an  “effective” \nmagnetic field, B eff, at hot spots to be 1.3T  for the highest incident power of 1.45× 109 \nW/m2 and 0.67 ellipticity . Note that B eff is not the  magnetic field produced by optically \nexciting the nanostructure, but rather , corresponds to the field strength of a hypothetical  \nexternal magnet that would produce the  same magnetization in the dark  as observed \nduring  CP optical excitation with no B app. This estimate also assumes that M ind and MIFE \nare either aligned or anti -aligned, though their orientation may be more complex \nmicroscopically40. When n ormalized for optical power density, the observed  \nmagnetization is in good agreement with our previous time- resolved studies of ensembles \nof Au colloids42 (Table S1) . \nFinally, we comment that the plasmon damping can als o be estimated directly by \nfitting to the Stokes side eR spectrum (green box, Fig. 2b). As discussed in detail in a \nrecent report from our laboratory53, the eR spectrum reflects the approximately Lorentzian \ndistribution of non- thermal electron -hole pairs that have been generated during the \nplasmon damping process, i.e., the natural linewidth of the excited plasmon. The fitted \ndamping observed under LP ( 1.42×109 W/m2) was  34.1 meV and the lowest damping \nobserved under CP ( 1.42× 109 W/m2, ellipticity of 0.94) was  31.6 meV. These values can \nequivalently be reported as a plasmon dephasing time of 19.3 fs  (LP) or 20.8 fs (CP) and \nare comparable to values commonly reported in ultrafast transient absorption studies of Au nanostructures\n21. While this fitted estimate of the ellipticity -dependent change in \ndamping is somewhat smaller c ompared to the estimate based on computational \nmodeling of the sample backscattering study discussed above, both measures consistently indicate a significant decrease of plasmon damping during CP excitation.    \nFig. 6. Photothermal heating under an external magnetic field and different \nhelicit ies. Fitted  nanostructure temperature (𝑇𝑇𝑙𝑙) as a function of total intensity (𝐼𝐼) with \nBapp = 0.2 T  and variable excitation helicity . The magnitude of ellipticity is 0.67 for b oth \nLHCP and RHCP. The linear fits to each temperature trend are LHCP: 𝑇𝑇𝑙𝑙 (𝐾𝐾) =\n 8.1×10−8𝐼𝐼+293; RHCP: 𝑇𝑇 𝑙𝑙 (𝐾𝐾) = 7.3×10−8𝐼𝐼+292; LP: 𝑇𝑇𝑙𝑙 (𝐾𝐾) = 4.8×10−8𝐼𝐼+305.  \n \n \nConclusion  \nWe have demonstrated the ability to modulate plasmon damping in achiral \nplasmonic gold nanodisk arrays by controlling incident light ellipticity. Confocal mapping \nrevealed that CP excitation leads to enhanced efficiency for backscattering, consistent \nwith an overall decrease of damping. A dual -beam Raman thermometry technique \nquantified localized heating in samples. We observe more efficient photothermal heating \nwhen the ellipticity of incident light increases, regardless of helicity  (RHCP or LHCP ), \nindicat ing greater field enhancement at hot  spots. The simultaneous increase of \nscattering and absorption is a telltale signature of decreased damping in plasmonic \nabsorbers. In comparison, under an external magnetic field, RHCP and LHCP excitation \nprovide differ ent amounts of heating. This behavior suggests that the microscopic origin \nof decreased damping is the interaction between the optically driven coherent electron \nmotion and Lorentz forces from DC magnetic fields, whether magnetic fields are optically \ninduc ed or externally applied. Our results provide further insight into electron dynamics \ninside plasmonic nanostructures during CP excitation and suggest multiple new strategies \nfor controllably modulating heating, magnetization, reflectance, damping, and rela ted \nphotophysical effects.  \n \n \n \n \n \n \nMethods  \n \nNanostructure Fabrication    \nPrior to fabrication, a silicon substrate was cleaned using a combination of base \npiranha and UV -ozone. A 100 nm gold mirror was then thermally evaporated (Lesker  PVD \nelectron- beam evaporator) onto the silicon substrate. A  38 nm thick Al 2O3 dielectric layer \nwas then deposited on the gold mirror by RF sputtering (Lesker PVD RF sputterer) and \nthe thickness was determined using  ellipsometry. Next, 950 PMMA A4 was spin- coated \nonto the Al 2O3 as the electron beam resist layer. Electron- beam lithography (TESCAN \nMIRA3 EBL) was performed to pattern the 100 µm × 100 µm nanodisk array into the e-\nbeam resist. After development, a 5 nm chrome adhesion layer was thermally deposit ed \non the surface of the exposed PMMA, followed by a 100 nm layer of gold. Finally, liftoff was performed in acetone using a combination of pipet pumping and sonication, leaving only the nanodisk array on the surface of the substrate. The morphology of gol d nanodisks \narray is confirmed by SEM (Fig . 1f). \n Raman Spectroscopy  \nRaman spectra were taken using a confocal microscope system (Witec  RA300) \nand spectrometer (UHTS300, grating = 300 g/mm). For Raman spectral mapping, the excitation source was 532 nm CW Nd:YAG laser and  the Raman spectra were collected by a 100x objective (Zeiss EC Epiplan Neofluar, NA = 0.9, WD = 0.31mm). A pair  \nconsisting  of a holographic 532 nm notch filter (RayShield Coupler, Witec) and a 532 nm \nnotch filter (NF533- 17, Thorlab) were added to prevent saturation of the spectrometer. \nThe obtained Raman spectra were corrected by the transmission spectra of the notch filter. The resolution of the Raman map was 50 nm in lateral dimensions (both x and y \ndirection) and 100 nm in the z direction. The optimal z height was determined by \nmaximizing the Raman Stokes signal. The 532 nm backscattering efficiency under \ndifferent elli pticity was obtained by referencing to a silver mirror.  \nFor the dual-beam Raman spectroscopy experiment, two 532 nm CW laser were \nused. Beam 2 was coupled through free space, and was a 532 CW diode laser with a spot size of 2 um\n2 on the sample. The ellipt icity of Beam 2 was controlled by a half \nwaveplate and a quarter waveplate optically in series . The function of Beam 2 was to \ngenerate circular current s in the gold nanostructures and actively tune the damping \nconstant. The difference in the highest achiev able ellipticity for LHCP and RHCP was a \nresult of the limitation of the beam splitter. Beam 1 was coupled through a fiber coupler \n(Rayshield coupler) and was sourced by a  532 nm CW Nd:YAG laser , with a spot size of \n0.55 µm2 when focused.  Beam 1 always had lower intensity than Beam 2 and was linearly \npolarized. Both beams were focused by a 100x objective (Zeiss EC Epiplan Neofluar, NA = 0.9, WD = 0.31mm) on the gold nanostructures. Below the sample, there was a slot for inserting a magnet, which has magneti c field parallel to the incident light (point ing \ndownward) with magnetic field strength = 0.2 T near the surface of gold nanostructure.  \n     Acknowledgement s \nWe thank  Prof. Luat Vuong  for helpful discussions . This work was funded in part by the \nNational Science Foundation (Grant DMR -2004810). M.S. also acknowledges support \nfrom the Welch Foundation (A -1886).  \n \n \nAuthor contributions  \nO.H.- C.C. and B.Z. carried out the measurements  and analyzed the data.  O.H.-C.C. \ndrafted the manuscript. 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Plasmon enhanced direct and IFE in nonmagnet ic nanocomposites. J. Opt. Soc. Am. B  \n27 (2010).  \n Supporting information for \nActive tuning of plasmon damping via light induced magnetism  \n \nAuthor list:  \nOscar Hsu -Cheng Cheng1*, Boqin Zhao1*, Zachary Brawley3, Dong Hee Son1,2, \nMatthew Sheldon1,3 \n1Department of Chemistry, Texas A&M University, College Station, TX, USA.  \n2 Center for Nanomedicine, Institute for Basic Science and Graduate Program \nof Nano Biomedical Engineering, Advanced Science Institute, Yonsei University, Seoul, Republic  of Korea  \n3Department of Material Science and Engineering, Texas A&M University, \nCollege Station, TX, USA.  \n*These authors contributed equally to this work.  \ne-mail: dhson@chem.tamu.edu; sheldonm@tamu.edu  \n \n \n \n   \n \n  \n \n   \n \n  \n \n   \n    \n \n \n \n   1. Full wave electromagnetic simulations  \n1.1 Simulations Setup  \nThree -dimensional full wave electromagnetic simulations were performed using \nfinite difference time domain (FDTD) methods (Lumerical, Ansys Inc.). A 100 \nnm height and 2 18 nm radius  (determined by the SEM image)  Au nanodisk on \na 40 nm thick Al 2O3 layer  (with a 5 nm Cr adhension layer in between)  on top \nof Au film  defined  the simulation geometry . Periodic boundary conditions were \napplied on all sides perpendicular to the substrate with a simulation region of \n700 nm x 700 nm, representing the periodicity of the nanostructure.  A plane \nwave  source was injected from above,  normal to the substrate.  The \nbackscattered radiation  was collected by a 2- D monitor placed above the plane \nwave source.  \n \nThe refractive index  of Al 2O3 was calculated as a linear combination of the \nexperimental index values from Palik1 for Al 2O3 and Al, so that the imaginary \npart at 532 nm is close to 0. 2, to account for excess Al in the layer during \nnanostructure fabrication as determined in control studies . The permittivity \nvalues for Au was modeled with the analytical function based on Drude- Lorentz \nmodel2: \n𝜀𝜀(𝜔𝜔)=1−𝑓𝑓0𝜔𝜔p,02\n𝜔𝜔2+𝑖𝑖𝛤𝛤0𝜔𝜔+∑𝑓𝑓j𝜔𝜔p,j2\n𝜔𝜔j2−𝜔𝜔2−𝑖𝑖𝛤𝛤j𝜔𝜔𝑗𝑗max\n𝑗𝑗=1 (S1) \nwhere 𝜀𝜀(𝜔𝜔) is the relative permittivity, 𝛤𝛤0 is the intraband damping constant, \n𝛺𝛺p=�𝑓𝑓0𝜔𝜔p,0  is the plasma frequency associated with intraband transition. \n𝑗𝑗max  is the number of Lorentz oscillators (in our case, 5) with strength 𝑓𝑓j , \nfrequency 𝜔𝜔j and damping constant 𝛤𝛤j for each individual oscillator.  \n In order to systematically study the effect of modulation of damping on optical \nproperties observed in the experiment, we studies the dependence on the \nintraband damping term  𝛤𝛤\n0, and all the interband damping terms 𝛤𝛤j in equation \n(S1).  Bulk Au values  were used for simulations  represent ing LP excitation in the \nexperiment.  These damping terms  were reduced up to 50% of their original \nvalues to generate new permittivity values to represent the CP excitation in the \nexperiment, applied to the Au film and the entire Au nanodisk.  \n 1.2 Backscattering simulations  \nThe confocal backscattering map in the experiment ( Fig. 3) show s the spatial \ndependence of backscattering  on the nanodisk array. In order to reproduce this \nspatial dependence behavior in the simulation, we mapped out the distribution \nof Poynting vector along z direction (Pz) of the scattered field on a x -y plane \nslightly  above the top surface of the nanodisk . We then averaged around each point  with a Gaussian -like decay which approximates the effect of Gauss ian \nbeam excitation  in the experiment ( Fig. S1a ). The backscattering intensity is \nthe largest near the center of  the nanodisk  and lowest between the gaps.  When \ncomparing the backscattering spatial distribution at different damping ( Fig. S1b ), \nit is clear that decreasing damping increases the backscattering intensity along \nthe black dash line in Fig. S1a , which is consistent with experimental results \n(Fig. 3c, f ). \n However, with 50% damping modulation, the magnitude of backscattering \nchange pr edicted by the simulation was not as large as that observed in the \nexperiment  at all locations . We hypothesize that the damping modulation is \nhighly spatially non- uniform  in the experimental  system  with maxima near hot \nspots,  unlike the assumption of  uniform damping modulation across the \nnanostructure in the modeling.  A spatial dependent change in damping in \nexperiments is likely a key factor that is not accounted for in these simulations, limiting the quantitative accuracy of the result. Nonetheless,  the modeling \ncorroborates that the observed experimental trend is consistent with decreased damping in the metal.  \n \n \n \nFig. S1  a) Simulated backscattering map on the x -y plane above the nanodisk \ntop surface with Gaussian- like weighting.  The r ed line circles region of the \nnanodisk.  b) Backscattering intensity plotted along the black dashed line in a), \nat bulk damping and 50% damping, as well as the backscattering from  an Au \nfilm for reference.  \n \n \nIn addition to the backscattering map, the overal l backscattering intensity from \nthe nanostructure, essentially the average of the spatial dependent backscattering intensity, was also simulated ( Fig. S2 ). The overall value also \nincreases by around 0.5% with damping values decreased by half , further \nhighlighting the increased backscattering efficiency with decreasing damping.  \n \nIf we label the overall backscattered  percentage of power as R, the absor bed \npercentage of power  would be 1- R, due to the opacity of the nanostructure.  \nTherefore, the absorption spectrum of the nanostructure can be computed \naccordingly . Fig. S3 shows the computed absorption spectrum of the \nnanostructure . Although i t does not exactly match the experimental absorption \nspectrum in Fig . 2a, but the general shape is very similar.  \n \nWe emphasize that simulation does not  model  any nonlinear effects . If we only \nchange the ellipticity of the excitation source in the simulations , without \nmodulating Au  permittivity  values , the absorption and the backscattering \nintensity  values  of the nanostructure remain constant . This suggests that the \ndifference in absorption and scattering seen in the simulation must come from some nonlinear effects , such as modulation of damping . \n  \n \nFig. S2  Overall backscattering efficiency of gold nanodisk array s with damping \nof 100% and 50% of bulk damping.  \n \nFig. S3  Simulated absorption spectrum of gold nanodisk array s with the bulk \ndamping constant.  \n \n \n1.3 Plasmonic heating simulations  \nWe mentioned in the main article that the local power density for heating,  \n𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧), depends on field enhancement as : \n𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧) = 1\n2𝐼𝐼𝐼𝐼[𝜀𝜀(𝑥𝑥,𝑦𝑦,𝑧𝑧)]𝜀𝜀0|𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧)|2 (S3) \nwhere 𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧) is the local electric field. We simulated the local electric field \ndistribution 𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧) on the nanodisk with damping modulation to verify the \nexperimentally observed increase in photothermal heating with decreasing \ndamping.  Here, we mapped the electric field intensity (|E|2) distribution on th e \ntop surface of the nanodisk , multiplied by the imaginary permittivity, which is \nproportional to the local heating power.  Fig. S 4 shows the calculated results for \nboth bulk damping and 50% damping.  At 50% damping, the local heating  at hot \nspots  is significantly  higher than the bulk damping case.  \n \n \nFig. S4  Heating map  on the top surface of the nanodisk a) with bulk damping, \nb) with 50% damping and c) the difference between 50% damping and bulk damping.  \n2. Raman fitting methodology  \nThe method used in the manuscript to extract  the temperature of the plasmonic \nnanostructure was adopted from Xie et al3 and from many of our previous \nreports4-7. Basically, the anti -Stokes electronic Raman scattering signal from the \nplasmonic nanostructure is associated with the thermalized electron distribution \nthat is approximately thermally equilibrated with the metal lattice. Therefore, the \nspectral intensity of the anti -stokes spectrum  𝑆𝑆(∆𝜔𝜔) can be characterized with \na Bose- Einstein distribution:  \n𝑆𝑆(∆𝜔𝜔)∝𝐶𝐶×𝐷𝐷(∆𝜔𝜔)×1\nexp�−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇�−1 (S3) \nwhere ℎ  is Plank constant, 𝑐𝑐  is the speed of light, ∆𝜔𝜔  is the wavenumber \n(negative for anti -Stokes), 𝑘𝑘B  is Boltzmann constant , and 𝑇𝑇  is the \ntemperature of the sample. 𝐶𝐶 is a scaling factor. 𝐷𝐷(∆𝜔𝜔) is a correction factor \nproportional to the photonic density of states (PDOS) of the nanostructure. In \npractice, the PDOS is approximated with reflection4-6, absorption,  or dark -field \nscattering8 spectrum. However,  there is not a direct experimental measurement \nof the PDOS . Therefore, we follow the procedure of Xie et al3 and have \nnormalized the Raman spectrum  at an unknown temperature 𝑇𝑇𝑙𝑙 to a reference  \nRaman spectrum at  a known temperature 𝑇𝑇0, to remove the dependence of \n𝐷𝐷(∆𝜔𝜔): \n𝑆𝑆(∆𝜔𝜔)𝑇𝑇𝑙𝑙\n𝑆𝑆(∆𝜔𝜔)𝑇𝑇0 = 𝐶𝐶1×𝐷𝐷(∆𝜔𝜔)×(𝑒𝑒𝑒𝑒𝑒𝑒  �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇0� −1)\n𝐶𝐶2×𝐷𝐷(∆𝜔𝜔)×(𝑒𝑒𝑒𝑒𝑒𝑒  �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇l� −1)=𝐶𝐶×𝑒𝑒𝑒𝑒𝑒𝑒  �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇0� −1\n𝑒𝑒𝑒𝑒𝑒𝑒  �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇l� −1 (S4) \nThe reference spectrum is  usually chosen to be the spectrum at lowest \nillumination intensity, assuming no laser heating, and therefore the sample \ntemperature is assumed to be at room temperature.  \n By correctly normalizing the spectral counts to the illumination power and integration time, 𝐶𝐶 is theoretically a value of  1. Therefore, the only parameter \nto solve for is the  unknown temperature 𝑇𝑇\nl . In practice, fitting accuracy is \nimproved if 𝐶𝐶 is a free fit parameter  (usually very close to 1) to account  for \nsmall changes in the collection efficiency during the course of the measurement.  \n            3. The accuracy of Raman -fitted  temperature s \n \nFig. S5 Fitted temperature of the nanostructure compared to the thermal \nstage temperature.  Blue asterisk with error bar: average fitted temperature \nand standard deviation. Red dotted line: reference line with T stage = T fit. \n \n We have shown that the sample temperature extracted from the Raman spectrum with equation ( 3) is physically accurate. The same nanostructure \nsample used in the manuscript  was heated to a pre- set temperature on a \nthermal stage ( TS1500, Linkam). Raman spectra were taken at different \nthermal stage temperatures with only beam 1 present. The intensity of beam 1 \nwas set to be very low for negligible  laser heating.  The Raman spectrum \ncollected at 298 K thermal stage temperature was chosen for the reference spectrum and 𝑇𝑇\n0 was set to be 298 K in the fitting.  \n The fitted temperatures were compared to the corresponding thermal stage \ntemperatures, as shown in Fig. S 5. Standard deviation (the blue bar around the \ndata points)  was calculated based on the fitted temperatures for s everal spectra \nat the same thermal stage temperature,  which is around 1 K, showing the high  \nprecision of the measurements. The data points lie close to the red reference line with T\nstage = T fit, confirming our ability to accurately report the nanostructure \ntemperatures  with anti-Stokes Raman thermometry.  \n  4. Determination  of optically -induced magnetism \n𝑀𝑀IFE at hot spots  \nAlthough bulk Au has been reported to display diamagnetism, there have been \nnumerous  reports pointing to observed paramagnetic, or even ferromagnetic \nbehavior in various nanoscale Au systems9-12. According to the theory of the \nIFE13, 14 and by studying the direction of magnetization in this report, we \nconcluded that the Au nanodisk arrays behave paramagnetically  (under \ncircularly polarized optical pumping) . However, the (volume) susceptibility is not \nknown. Here, we may label it symbolically with 𝜒𝜒V  and carry it through our \nsubsequent calculations . In addition, our previous experimental study15 also \nindicates a paramagnetic behavior in nanoscale Au. Therefore, we model the \nmagnetization of Au under an external magnetic field (in the dark) as follows : \n𝐵𝐵app=𝜇𝜇0�1\n𝜒𝜒V�𝑀𝑀ind (S5) \nwhere 𝜇𝜇0 is the vacuum permeability, 𝐵𝐵app is the external applied magnetic \nfield and 𝑀𝑀ind is the induced magnetization.  \n \nThe induced magnetization in response to the external magnetic field 𝑀𝑀ind is \na separate contribution to the sample magnetism in addition to  the light -induced \nmagnetism 𝑀𝑀IFE (Fig. 1a) , and both effects are expected to contribute to the \ndamping experienced by coherently driven, circulating electrons during optical pumping . Therefore, the temperature increase measured in the experiment (Fig. \n6, Fig. S6 ) is assumed  to scale linearly with the total induced magnetism 𝑀𝑀\nind+\n𝑀𝑀IFE. \n First, we calculate the strength of 𝑀𝑀\nind under a 0.2 T applied external magnetic \nfield. By applying equation (S5), with  𝜇𝜇0=1.257 ×10−6 H/m, we can derive \nthat 𝑀𝑀ind=1.6×105𝜒𝜒V A/m. We determine 𝑀𝑀IFE in the experiment based the \nfitted temperature data at the highest incident intensity in Fig. 6  (also  Fig. S6 ). \nNote  that the temperature without an external magnetic field , when 𝑀𝑀ind=0, \nwould have been 405 K  for both LHCP and RHCP.  This means the samples \nexperience a temperature increase of ~ 33 K during CP compared to LP  when \nonly the 𝑀𝑀IFE alters the damping in the sample. The additional 5 K increase or \ndecrease of temperature (see Fig. S6) during  LHCP  or RHCP , respectively,  is \ndue to the interaction with  𝑀𝑀ind . Therefore, 𝑀𝑀IFE  is calculated to be  \napproximately 6.6x larger than 𝑀𝑀ind, or 1.05×106𝜒𝜒V A/m, which corresponds \nto induced magnetic flux density of 1.3𝜒𝜒V T. According to equation (S5), t his is \nequivalent to 𝑀𝑀ind  that would be produced under  a 1.3 T external applied \nmagnetic field, which we label as B eff. \n Lastly, we can calculate the magnetic moment per Au atom m\nAu based on the \nrelationship  \n𝑀𝑀IFE=𝑁𝑁Au\n𝑉𝑉𝐼𝐼Au (S6) \nIn an Au crystal lattice, t he number density of Au atoms 𝑁𝑁Au\n𝑉𝑉  is 58.9 nm-3. \nTherefore, 𝐼𝐼Au=1.9𝜒𝜒V𝜇𝜇B. \n We compare our  results with  our previous  experimental study on Au \nnanocolloids,  as well as two experiment measurements  on Au film,  summarized in Table. S1 , assuming that the IFE magnetism is linearly proportional to the \nincident optical power (as suggested by theoretical studies on the IFE13, 14, 16). \nOur analysis  in this study only reveal s the magnetic behavior at hot spots, \nbecause most of the Raman signal comes from hot spots on the nanodisk. In \nRef. [1 5], the magnetic moment per Au atom was averaged over the entire \nnanoparticle, which could be the reason for a slightly  smaller value for induced \nmagnetic moment per atom  compared to this study.  Both nanoscale Au system \npossess excitation intensity normalized magnetic moment per atom ( 𝐼𝐼Au/𝐼𝐼) of \nalmost 4 orders of magnitude larger than Au film,  indicating an enhancement of \nthe IFE  phenomenon in nanoscale compared to bulk . \n \nFor reference, i f 𝜒𝜒V is on the order of  10−5 (a typical value for bulk Au) , the \nIFE induced  magnetic field at hot spots  during in the study  is estimated to be \non the order of 10−5 T. \n \n Table. S1 Comparison between induced magnetic moment per atom due to the IFE in different reports.  \n \n System  Excitation \nIntensity  \n𝐼𝐼 (W/m2) 𝑀𝑀IFE (A/m)  𝐼𝐼Au (𝜇𝜇B) 𝐼𝐼Au/𝐼𝐼 \n(10−10𝜇𝜇B/(W\nm2)) \nThis work  Au nanodisk \narray  ~109 (CW)  1.05×106𝜒𝜒V 1.9𝜒𝜒V b19𝜒𝜒V \nRef. [15]15 Au \nnanopartic le \ncolloid  ~9×1013 \n(pulsed)  3.0×104𝜒𝜒V 2.8×104𝜒𝜒V 3.1𝜒𝜒V \nRef. [1 7]17  Au film  ~13×1013 \n(pulsed)  a4.5×106𝜒𝜒V 8.2𝜒𝜒V 6.3×10−4𝜒𝜒V \nRef. [1 8]18  Au film  ~13×1013 \n(pulsed)  a1.1×107𝜒𝜒V 20𝜒𝜒V 1.5×10−3𝜒𝜒V \na Several assumptions were made to calculate this value. The path length in Au film is estimated by the \nskin depth (1/absorption coefficient). The verdet  constant of Au film is extracted from Ref. [17]19. The \ninduced magnetization was also normalized to the excitation frequency. 𝜒𝜒𝑉𝑉 of Au f ilm was used as bulk \nAu value.  \nb At hot spots.   \nFig. S6 Diagram to aid induced magnetic field calculation.  \n \n \n   \n \n \n   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n5. Optical setup  \n \nFig. S7. Optical setup for dual -beam Raman spectroscopy.  CW: continuous \nwave; LP: linear polarizer; HWP: half waveplate; QWP: quarter waveplate; BS: \nbeam splitter.  \n  6. Definition of ellipticity  \n \nFig. S8. Ellipticity explanation. This figure shows a left -handed elliptically \npolarized light. φ is the ellipticity angle. Ellipticity is defined as the intensity \nratio between the short and long axis. Ellipticity angle = 0, - 45\no, +45o or \nEllipticity = 0, 1, - 1 represent LP, LHCP, RHCP , respectively.  \n \nReferences  \n1. Palik, E. D., Handbook of optical constants of solids. Academic Press Handbook Series  1985.  \n2. Rakic, A. D.; Djurisic, A. B.; Elazar, J. M.; Majewski, M. L., Optical properties of metallic films \nfor vertical -cavity optoelectronic devices. Appl. Opt.  1998 . 37, (22), 5271- 5283.  \n3. Xie, X.; Cahill, D. G., Thermometry of plasmonic nanostructures by anti -Stokes electronic Raman \nscattering. Appl. Phys. Lett.  2016. 109, (18),  \n4. Hogan, N.; Wu, S.; Sheldon, M., Photothermalization and Hot Electron Dynamics in the  Steady \nState. J. Phys. Chem. C  2019. 124, (9), 4931- 4945.  \n5. Wu, S. X.; Hogan, N.; Sheldon, M., Hot Electron Emission in Plasmonic Thermionic Converters. \nACS Energy Lett.  2019. 4, (10), 2508- 2513.  \n6. Hogan, N.; Sheldon, M., Comparing steady state photothe rmalization dynamics in copper and gold \nnanostructures. J. Chem. Phys.  2020. 152, (6), 061101.  \n7. Wu, S. X.; Cheng, O. H. C.; Zhao, B.; Hogan, N. Q.; Lee, A.; Son, D. H.; Sheldon, M., The \nconnection between plasmon decay dynamics and the surface enhanced R aman spectroscopy background: \nInelastic scattering from non -thermal and hot carriers. J. Appl. Phys.  2021. 129, (17),  \n8. Cai, Y . Y .; Sung, E.; Zhang, R.; Tauzin, L. J.; Liu, J. G.; Ostovar, B.; Zhang, Y .; Chang, W. S.; \nNordlander, P.; Link, S., Anti -Stoke s Emission from Hot Carriers in Gold Nanorods. Nano Lett.  2019. \n19, (2), 1067- 1073.  \n9. Ulloa, J. A.; Lorusso, G.; Evangelisti, M.; Camon, A.; Barbera, J.; Serrano, J. L., Magnetism of \nDendrimer -Coated Gold Nanoparticles: A Size and Functionalization Study.  J. Phys. Chem. C  2021. 125, \n(37), 20482- 20487.  \n10. Nealon, G. L.; Donnio, B.; Greget, R.; Kappler, J. P.; Terazzi, E.; Gallani, J. L., Magnetism in gold \nnanoparticles. Nanoscale  2012. 4, (17), 5244- 5258.  \n11. Tuboltsev, V .; Savin, A.; Pirojenko, A.; Raisan en, J., Magnetism in nanocrystalline gold. ACS Nano  \n2013. 7, (8), 6691- 6699.  \n12. Trudel, S., Unexpected magnetism in gold nanostructures: making gold even more attractive. Gold \nBull.  2011 . 44, (1), 3 -13. \n13. Hertel, R., Theory of the inverse Faraday effect  in metals. J. Magn. Magn. Mater.  2006. 303, (1), \nL1-L4. \n14. Hertel, R.; Fahnle, M., Macroscopic drift current in the inverse Faraday effect. Phys. Rev. B  2015. \n91, (2),  \n15. Cheng, O. H. C.; Son, D. H.; Sheldon, M., Light -induced magnetism in plasmonic gold \nnanoparticles. Nat. Photonics  2020. 14, (6), 365- +. \n16. Sinha -Roy, R.; Hurst, J.; Manfredi, G.; Hervieux, P. A., Driving Orbital Magnetism in Metallic \nNanoparticles through  Circularly Polarized Light: A Real -Time TDDFT Study. ACS Photonics  2020 . 7, \n(9), 2429- 2439.  \n17. Kruglyak, V . V .; Hicken, R. J.; Ali, M.; Hickey, B. J.; Pym, A. T. G.; Tanner, B. K., Measurement \nof hot electron momentum relaxation times in metals by femtos econd ellipsometry. Phys. Rev. B  2005 . \n71, (23),  \n18. Kruglyak, V . V .; Hicken, R. J.; Ali, M.; Hickey, B. J.; Pym, A. T. G.; Tanner, B. K., Ultrafast third -\norder optical nonlinearity of noble and transition metal thin films. Journal of Optics a- Pure and Ap plied \nOptics  2005. 7, (2), S235 -S240.  \n19. Mcgroddy, J. C.; Mcaliste.Aj; Stern, E. A., Polar Reflection Faraday Effect in Silver and Gold. \nPhysical Review  1965. 139, (6a), 1844- &.  "    },    {        "title": "1708.03685v1.On_the_Small_Mass_Limit_of_Quantum_Brownian_Motion_with_Inhomogeneous_Damping_and_Diffusion.pdf",        "content": "arXiv:1708.03685v1  [cond-mat.stat-mech]  11 Aug 2017On the Small Mass Limit of Quantum Brownian Motion with\nInhomogeneous Damping and Diffusion\nSoon Hoe Lim,1,2,∗Jan Wehr,1,2Aniello Lampo,3\nMiguel´Angel Garc´ ıa-March,3and Maciej Lewenstein3,4\n1Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA\n2Program in Applied Mathematics, University of Arizona, Tuc son, AZ 85721, USA\n3ICFO - Institut de Ciencies Fotoniques,\nThe Barcelona Institute of Science and Technology,\n08860 Castelldefels (Barcelona), Spain\n4ICREA - Instituci´ o Catalana de Recerca i Estudis Avan¸ cats ,\nLluis Companys 23, E-08010 Barcelona, Spain\n(Dated: September 29, 2018)\nWe study the small mass limit (or: the Smoluchowski-Kramers limit) of a class of quantum Brow-\nnian motions with inhomogeneous damping and diffusion. For O hmic bath spectral density with\na Lorentz-Drude cutoff, we derive the Heisenberg-Langevin e quations for the particle’s observables\nusing a quantum stochastic calculus approach. We set the mas s of the particle to equal m=m0ǫ,\nthe reduced Planck constant to equal /planckover2pi1=ǫand the cutoff frequency to equal Λ = EΛ/ǫ, wherem0\nandEΛare positive constants, so that the particle’s de Broglie wa velength and the largest energy\nscale of the bath are fixed as ǫ→0. We study the limit as ǫ→0 of the rescaled model and derive\na limiting equation for the (slow) particle’s position vari able. We find that the limiting equation\ncontains several drift correction terms, the quantum noise-induced drifts , including terms of purely\nquantum nature, with no classical counterparts.\nI. INTRODUCTION\nMultiscale analysis of both classical and quantum systems has been a subject of active\ninvestigation in recent years. The underlying idea is that due to the p resence of widely sep-\narated characteristic time scales in the system, one can obtain a sim plified, reduced model\nthat often captures the essential dynamics on a coarse-grained time scale [ 1–5]. Depending\non the nature ofthe systems, different approachescan be under taken to implement this idea.\nFor instance, Markovian limits such as weak coupling limit and repeated interaction limit\nwere studied in [ 6–12] to justify the use of effective equations such as quantum Langev in\nequations in modeling quantum systems arising in quantum optics and q uantum electro-\ndynamics [ 13–15]. Adiabatic elimination type problems for open quantum systems were\nstudied in [ 16–24] and perturbative methods were considered in [ 25–28].\nOf particular interest is the small mass limit (or the Smoluchoswki-Kra merslimit [ 29,30])\nof noisy dynamical systems. It has been extensively studied and is w ell understood for\nclassical systems; see for instance, [ 31–37]. On the other hand, analogous questions for\nquantum systems [ 38,39] are more intricate, and there were few attempts to study the\nsmall mass limit, or the related strong friction limit for quantum syste ms. Such study was\ninitiatedandrefinedintheseriesofworks[ 40–44]forthe Caldeira-Leggettmodelofquantum\nBrownian motion (QBM) [ 45–48]. In these works, a quantum Smoluchowski equation, an\nequation for the coordinate-diagonal elements of the density ope rator (i.e. the position\nprobability distribution), was derived in the overdamped regime.\n∗shoelim@math.arizona.edu2\nThe results of these works (see for instance, [ 40]) say that the strong friction limit of\nquantummechanicsisessentiallyclassicalmechanicsasthe quantum effectsareburiedin the\nfast momentum variable, which is adiabatically eliminated due to separa tion of associated\ntime scales in the limit. Such limit is the opposite of the weak coupling limit [ 49,50], and\nits result can be viewed as a consequence of decoherence due to th e strong coupling. In\n[51–54], more careful analysis and related applications were presented, w hereas in [ 55] a\nHeisenberg approach was used. All these attempts rely on asympt otic expansions to study\na restricted class of QBM, where the coupling of the system to the e nvironment is linear\nin the system’s position. One important message from these asympt otic expansions is that\nthe leading correction term to the Smoluchowski equation is a quant um correction that\ndominates the classical ones in the low temperature regime, revealin g the important role of\nzero-point quantum fluctuations.\nSimilar studies for QBM in inhomogeneous environments are even more interesting [ 56].\nSuch study was conducted in [ 57,58], where a semi-classical Langevin approach was em-\nployed. In [ 59], using the Fokker-Planck equation, the authors derived a limiting c-number\nLangevin equation for the position variable in the overdamped limit. Wh ile the limiting\nequation obtained contains interesting quantum correction terms , these studies are not sat-\nisfactory for two main reasons. First, an ad-hoc Markovian appro ximation is made before\nthe overdamped limit was studied. Second, a semi-classical approac h is used and assumed\nthat the quantum fluctuations around the mean value of the syste m’s observables are small.\nTherefore, a more systematic study that takes into account the full quantum nature of the\nmodel, including the noise, is necessary. Motivated by this and our go al to generalize the\nstudy of small mass limit to quantum dissipative systems, this paper p resents a quantum\nstochastic calculus approach to study a related limit for a class of non-Markovian QBM\nwith inhomogeneous damping and diffusion. In particular, we will model the noise using\nthe fundamental noise processes of the theory of quantum stoc hastic calculus, introduced by\nHudson and Parthasarathy in the seminal paper [ 60]. Rigorous justification of the results\nin this paper will be presented elsewhere.\nThe paper is organizedas follows. In Section II, we introduce a QBM field model to model\ninteraction of a quantum particle with an equilibrium quantum heat bat h at a positive tem-\nperature. In Section III, we present the exact Heisenberg equations of motion for the par -\nticle’s observables. In Section IV, we review some basic results from Hudson-Parthasarathy\n(H-P) quantum stochastic calculus and fix our notations. Modeling t he stochastic force, ap-\npearing in the Heisenberg equations, using Hudson-Parthasarath yquantum noise processes,\nwe derive a quantum stochastic differential equation (QSDE) versio n of the QBM model\nin Section V. We identify the characteristic time scales of the model and study it s rescaled\nversion in Section VI. Our main result is the derivation of the effective equation (see eqn.\n(112)) for the (slow) position variable in the limit as all the characteristic t ime scales of the\nmodel tend to zero at the same rate. The derivations, as well as dis cussions of the results,\nare presented in Section VII. We end the paper by stating the conclusions and making some\nremarks in Section VIII.\nII. QBM MODEL\nIn this section, we introduce a one-dimensional Hamiltonian model to study the dynamics\nof a quantum Brownian particle coupled to an equilibrium heat bath. Th e particle interacts\nwith the heat bath via a coupling, which is a function of the position var iables. This\nfunction can be nonlinear in the system’s position, in which case the pa rticle is subject to\ninhomogeneousdampinganddiffusion[ 38,47,61]. Themodelcanbeviewedasafieldversion3\nofthe onestudied in [ 47,61], a generalizationof the Pauli-Fierzmodel [ 62–64], ora quantum\nanalog of the one studied in Appendix A of [ 36]. It is a fundamental model which not only\nallowssimpleanalytictreatmentsandprovidesphysicalinsights,but alsorealisticallymodels\nmany open qantum systems — for instance, an atom in an electromag netic field.\nThe full dynamics of the model is described by the Hamiltonian:\nH=HS⊗I+I⊗HB+HI+Hren⊗I, (1)\nwhereHSandHBare Hamiltonians for the particle and the heat bath respectively, giv en\nby\nHS=P2\n2m+U(X), HB=/integraldisplay\nR+/planckover2pi1ωb†(ω)b(ω)dω, (2)\nHIis the interaction Hamiltonian given by\nHI=−f(X)⊗/integraldisplay\nR+[c(ω)b†(ω)+c(ω)b(ω)]dω, (3)\nandHrenis the renormalization Hamiltonian given by\nHren=/parenleftbigg/integraldisplay\nR+|c(ω)|2\n/planckover2pi1ωdω/parenrightbigg\nf(X)2. (4)\nHereXandPare the particle’s position and momentum operators, mis the mass of the\nparticle,U(X) is a smooth confining potential, b(ω) andb†(ω) are the bosonic annihila-\ntion and creation operator of the boson of frequency ωrespectively and they satisfy the\nusual canonical commutation relations (CCR): [ b(ω),b†(ω′)] =δ(ω−ω′),[b(ω),b(ω′)] =\n[b†(ω),b†(ω′)] = 0.We assume that the operator-valued function f(X) is positive and can\nbe expanded in a power series, and c(ω) is a complex-valued coupling function (form factor)\nthat specifies the strength of the interaction with each frequenc y of the bath. It determines\nthe spectral density of the bath and therefore the model for da mping and diffusion of the\nparticle. The heat bath is initially in the Gibbs thermal state, ρβ=e−βHB/Tr(e−βHB), at\nan inverse temperature β= 1/(kBT).\nIn this paper, we consider the coupling function:\nc(ω) =/radicalbigg\n/planckover2pi1ω\nπΛ2\nω2+Λ2, (5)\nwhere Λ is a positive constant. The bath spectral density is given by :\nJ(ω) :=|c(ω)|2\n/planckover2pi1=ω\nπΛ2\nω2+Λ2, (6)\nwhich is the well-known Ohmic spectral density with a Lorentz-Drude cutoff [49]. The\nrenormalization potential Hrenis needed to ensure that the bare potential acting on the\nparticle isU(X) and that the Hamiltonian can be written in a positively defined form:\nH=HS⊗I+HB−I, whereHB−Iis given by\nHB−I=/integraldisplay\nR+/planckover2pi1ω/parenleftbigg\nb(ω)−c(ω)\n/planckover2pi1ωf(X)/parenrightbigg†/parenleftbigg\nb(ω)−c(ω)\n/planckover2pi1ωf(X)/parenrightbigg\ndω. (7)4\nIII. HEISENBERG-LANGEVIN EQUATIONS\nIn this section, we present the Heisenberg equations of motion and study the stochastic\nforce term appearing in the equation. This will pave the way to model the action of the\nheat bath on the particle by appropriate quantum colored noises int roduced in the next sec-\ntions. Our final goal is the construction of dissipative non-Markov ian Heisenberg-Langevin\nequations driven by appropriatethermal noises, which are built fro m H-P fundamental noise\nprocesses. From now on, Idenotes identity operatoron an understood spaceand 1 Adenotes\nindicator function of the set A.\nDefine the particle’s velocity, V(t) =P(t)\nmand note that f′(X) =−i[f(X),P]//planckover2pi1. Starting\nfrom the Heisenberg equations of motion and eliminating the bath var iables, one obtains the\nfollowing equations for the particle’s observables (see Appendix Afor details of derivations):\n˙X(t) =V(t), (8)\nm˙V(t) =−U′(X(t))−f′(X(t))/integraldisplayt\n0κ(t−s){f′(X(s)),V(s)}\n2ds\n+f′(X(t))·(ζ(t)−f(X)κ(t)), (9)\nwhere\nκ(t) =/integraldisplay\nR+dω2|c(ω)|2\n/planckover2pi1ωcos(ωt) =/integraldisplay\nR+dω2J(ω)\nωcos(ωt) (10)\nis thememory kernel ,\nζ(t) =/integraldisplay\nR+dωc(ω)(b†(ω)eiωt+b(ω)e−iωt) (11)\nis astochastic force whose correlation function depends on the coupling function, c(ω), and\nthe distribution of the initial bath variables, b(ω) andb†(ω) – let us remind that we initially\nconsider a thermal Gibbs state. The term f′(X(t))f(X)κ(t) is the initial slip term [ 65]. The\ninitial position and velocity are given by XandVrespectively.\nThe above equations are exact, non-Markovian, operator-value d and describe completely\npositive dynamics. Note that in the damping term which is nonlocal in tim e, we have an\nanti-commutator, which does not appear in the corresponding clas sical equation or in the\nequationforthelinearQBMmodel(where f(X) =X). Thepresenceoftheanti-commutator\nis thus a quantum feature of the inhomogeneous damping.\nThe initial preparation of the total system, which fixes the statist ical properties of the\nbath operators and of the system’s degrees of freeedom, turns the forceζ(t) into a random\none [66]. We specify a preparation procedure to fix the properties of the s tochastic force.\nTo this end, we absorb the initial slip term into the stochastic force, defining:\nξ(t) :=ζ(t)−f(X)κ(t). (12)\nWith this, in the nonlinear coupling case, the equation for the particle ’s velocity is driven by\nthe multiplicative noise f′(X(t))ξ(t). From now on, we refer to ξ(t) as thequantum noise .\nThe statistics of ξ(t) depends on the distributions of the initial bath variables ( b(ω),b†(ω))\nand the initial system variable f(X).\nDenoting by Eβthe expectation with respect to the thermal Gibbs state ρβat the tem-5\nperatureT, we have\nEβ[(b†(ω)eiωt+b(ω)e−iωt)(b†(ω′)eiω′s+b(ω′)e−iω′s)]\n=/bracketleftBig\n(1+νβ(ω))e−iω(t−s)+νβ(ω)eiω(t−s)/bracketrightBig\nδ(ω−ω′), (13)\nwhereνβ(ω) is given by the Planck’s law\nνβ(ω) =1\nexp(β/planckover2pi1ω)−1. (14)\nSince we absorbed the initial slip term into the stochastic force, ξ(t) no longer has a\nstationary correlation when averaged with respect to ρβ[67]. However, ξ(t) is stationary\nand Gaussian when conditionally averagedwith respect to ρ′\nβ=e−βHB−I/Tr(HB−I),where\nHB−IisthequadraticHamiltoniandefinedin( 7)andtheaverageisconditionedontheinitial\nposition variable X.\nWe will work in a Fock vacuum representation using the H-P quantum stochastic calculus\napproach (see Section IV). In particular, our goal is to describe the quantum noise as a\nquantum stochastic process satisfying certain QSDE such that th e symmetric correlation\nfunction of the stochastic process with respect to the vacuum st ate on an enlarged Fock\nspace coincides with that of ξ(t) with respect to ρ′\nβ. As a preparation to achieve this goal,\nwe studyE′\nβ[ξ(t)ξ(s)] in the following. We write:\nE′\nβ[ξ(t)ξ(s)] =/integraldisplay\nR+dω/planckover2pi1J(ω)/parenleftbigg\ncoth/parenleftbigg/planckover2pi1ω\n2kBT/parenrightbigg\ncos(ω(t−s))−isin(ω(t−s))/parenrightbigg\n(15)\n=:D1(t−s)−iD(t−s), (16)\nwhereD1is the noise kernel given by\nD1(t−s) :=E′\nβ[{ξ(t),ξ(s)}/2], (17)\ni.e. the symmetric correlation function of ξ(t) with respect to ρ′\nβ, andDis the dissipation\nkernel given by\nD(t−s) :=iE′\nβ[[ξ(t),ξ(s)]/2], (18)\nwhich is related to linear susceptibility. Expanding, one gets for small /planckover2pi1(or similarly, for\nlargeT),E′\nβ[ξ(t)ξ(s)] =kBTκ(t)+O(/planckover2pi1),which is the classical Einstein’s relation.\nFor our choice of c(ω) (see (5)), the memory kernel, κ(t), is exponentially decaying with\ndecay rate Λ, i.e. κ(t) = Λe−Λt.Moreover, one can compute, for t>0:\nD1(t) =/planckover2pi1Λ\n2cot/parenleftbigg/planckover2pi1Λ\n2kBT/parenrightbigg\nκ(t)+∞/summationdisplay\nn=12kBTΛ2νn\nν2n−Λ2e−νnt, (19)\nwhereνn=2πnkBT\n/planckover2pi1are the bath Matsubara frequencies [ 68]. Also, the dissipation kernel is\nD(t) =/planckover2pi1Λ3\n2e−Λt. (20)\nIn this paper, we consider the case kBT >/planckover2pi1Λ/π, so that cot( /planckover2pi1Λ/2kBT) and all the terms6\nin the series in ( 19) are positive.\nWe end this section with the following remark. For T→0 we have instead:\nE′\nβ[{ξ(t),ξ(s)}/2]→−/planckover2pi1Λ2\n2π(e−Λ(t−s)Ei(Λ(t−s))+eΛ(t−s)Ei(−Λ(t−s))),(21)\nwhereEiis the exponential integral function defined as follows:\n−Ei(−x) = ˆγ(0,x) =/integraldisplay∞\nxe−t/tdt. (22)\nHereEi(x) =1\n2(Ei+(x)+Ei−(x)),Ei+(x) =Ei(x+i0),Ei−(x) =Ei(x−i0). The symmet-\nric correlation function obtained above can be interpreted as follow s. As the temperature\nTdecreases, the Matsubara frequencies νnget closer to each other, so at zero temperature\nall of them contribute and the sum in ( 19) may be replaced by an integral, which turns out\nto have expression in terms of the Eifunctions [ 69]. In fact, in this case the symmetric\ncorrelation function decays polynomially for large times [ 70]. In other words, systems at\nzero temperature are strongly non-Markovian and the technique s in this paper cannot be\napplied to study them.\nIV. PRELIMINARIES ON QUANTUM STOCHASTIC CALCULUS\nTo ensure that our exposition is self-contained, as well as to fix the notations and ter-\nminologies, we review some basic ideas and results from H-P quantum s tochastic calculus,\nwhich is a boson Fock space stochastic calculus based on the creatio n, conservation and\nannihilation operators of free field theory. For details of the calculu s, we refer to the mono-\ngraphs [71] and [72]. For recent developments, perspectives and applications of the c alculus\nto the study of open quantum systems, we refer to [ 73–84]. In the following, we use Dirac’s\nbra-ket notation.\nThe natural space to support the quantum noise, which models the effective action of the\nenvironment on the system, is a bosonic Fock space. It describes s tates of a quantum field\n(heat bath in our case) consisting of an indefinite number of identica l particles. The bosonic\nFock space, over the one-particle space H, is defined as\nΓ(H) =C⊕H⊕H◦2⊕···⊕H◦n⊕..., (23)\nwhere C, denoting the one-dimensional space of complex numbers, is called t hevacuum\nsubspace andH◦n, denoting the symmetric product of ncopies of H, is called the n-particle\nsubspace. For anynelements |u1/an}bracketri}ht,|u2/an}bracketri}ht,...,|un/an}bracketri}htinH, the vector ⊗n\nj=1|uj/an}bracketri}htis known as the\nFock vector. Since the particles constituting the noise space (and in each of the n-particle\nspace) are bosons, in order to describe the n-particle state (i.e. to belong to the n-particle\nspace,H◦n), a Fock vector has to be symmetrized:\n|u1/an}bracketri}ht◦|u2/an}bracketri}ht◦···◦|un/an}bracketri}ht=1\nn!/summationdisplay\nσ∈Pn|uσ(1)/an}bracketri}ht⊗|uσ(2)/an}bracketri}ht⊗···⊗|uσ(n)/an}bracketri}ht, (24)\nwherePnis the set of all permutations σ, of the set {1,2,...,n}.7\nImportant elements of the bosonic Fock space, Γ( H), are the exponential vectors :\n|e(u)/an}bracketri}ht= 1⊕|u/an}bracketri}ht⊕|u/an}bracketri}ht⊗2\n√\n2!⊕···⊕|u/an}bracketri}ht⊗n\n√\nn!⊕..., (25)\nwhere|u/an}bracketri}ht ∈ Hand|u/an}bracketri}ht⊗ndenotes the tensor product of ncopies of |u/an}bracketri}ht. The exponential\nvectors satisfy the following scalar product formula:\n/an}bracketle{te(u)|e(v)/an}bracketri}ht=e/an}bracketle{tu|v/an}bracketri}ht(26)\nfor every |u/an}bracketri}ht,|v/an}bracketri}ht ∈ H,with the same notation for scalar products in appropriate spaces.\nThe linear span, E, of all exponential vectors forms a dense subspace of Γ( H). We refer\ntoEas theexponential domain . Any bounded linear operator on Γ( H) can be determined\nby its action on the exponential vectors. Note that |ψ(u)/an}bracketri}ht=e−/an}bracketle{tu|u/an}bracketri}ht/2|e(u)/an}bracketri}htis a unit vector.\nThe pure state with the density operator |ψ(u)/an}bracketri}ht/an}bracketle{tψ(u)|is called the coherent state associated\nwith|u/an}bracketri}ht. We call |Ω/an}bracketri}ht:=|e(0)/an}bracketri}httheFock vacuum vector , which corresponds to the state with\nno particles. In the special case when H=C, the coherent states on Γ( H) =C⊕C⊕···\nare sequences of the form:\n|ψ(α)/an}bracketri}ht=e−|α|2/2/parenleftbigg\n1,α,α2\n√\n2!,···,αn\n√\nn!.../parenrightbigg\n. (27)\nConsider now the case when the one-particle space is H=L2(R+), leading to the bosonic\nFockspaceΓ( L2(R+)). Wewillbeformulatingadifferential(intime)descriptionofproces ses\non the Fock space, and R+represents the time semi-axis. We emphasize that this has\nto be distinguished from the frequency representation, as adopt ed when writing the bath\nHamiltonian HBin Section II. One can think of Γ( L2(R+)) as the space describing a single\nfield channel coupled to the system.\nIn general, one can consider the bosonic Fock space over the one- particle space, L2(R+)⊗\nZ=L2(R+;Z), where Zis a complex separable Hilbert space, equipped with a complete\northonormal basis ( |zk/an}bracketri}ht)k≥1. The space Zis called the multiplicity space of the noise. An\nelement of L2(R+)⊗ Zis a square integrable function from R+intoZ. Physically, the\ndimension of Zis the number of field channels coupled to the system. When Z=C(one-\ndimensional), the corresponding bosonic Fock space describes a sin gle field channel [ 79].\nWhenZ=Cdand the |zi/an}bracketri}ht= (0,...,0,1,0,...,0) with 1 in the i-th slot,i= 1,2,...,d,\nis fixed as a canonical orthonormal basis in Cd, the corresponding Fock space describes d\nfield channels coupled to the system. Since the dimension of Zcan be infinite, it allows\nconsidering infinitely many field channels coupled to the system. To ta ke advantage of\nthis generality, we take the quantum noise space to be the bosonic F ock space Γ( H) over\nH=L2(R+)⊗Zin the following.\nThe canonical observables on the bosonic Fock space are the creation andannihilation\noperators associated to a vector |u/an}bracketri}ht ∈ H, denoteda†(u) anda(u) respectively. They satisfy\nthe commutation relations: [ a(u),a(v)] = 0, [a†(u),a†(v)] = 0 and [ a(u),a†(v)] =/an}bracketle{tu|v/an}bracketri}ht=/integraltext\nR+u(s)v(s)ds, for|u/an}bracketri}ht,|v/an}bracketri}ht ∈ H. Their action on the exponential vectors is defined by:\na(u)|e(v)/an}bracketri}ht=/an}bracketle{tu|v/an}bracketri}ht|e(v)/an}bracketri}ht, (28)\na†(u)|e(v)/an}bracketri}ht=d\ndǫ|e(v+ǫu)/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nǫ=0, (29)8\nfor all|u/an}bracketri}ht,|v/an}bracketri}ht ∈ H. Note that in the special case u=v, we havea(u)|ψ(u)/an}bracketri}ht=/an}bracketle{tu|u/an}bracketri}ht|ψ(u)/an}bracketri}ht,\nwhich is an eigenvalue relation similar to the one that defines the coher ent state as eigen-\nvector of annihilation operator in quantum optics [ 85]. Moreover, we have\na(u)|e(0)/an}bracketri}ht= 0, (30)\na(u)|v/an}bracketri}ht⊗n=√n/an}bracketle{tu|v/an}bracketri}ht|v/an}bracketri}ht⊗(n−1), (31)\na†(u)|v/an}bracketri}ht⊗n=1√n+1n/summationdisplay\nr=0|v/an}bracketri}ht⊗r⊗|u/an}bracketri}ht⊗|v/an}bracketri}ht⊗(n−r). (32)\nSince vectors of the form |v/an}bracketri}ht⊗nlinearly span the n-particle space, this shows that a(u)\nmaps then-particle subspace into the ( n−1)-particle subspace while a†(u) maps the n-\nparticle subspace into the ( n+1)-particle subspace, justifying their names as annihilation\nand creation operators respectively.\nThe basic idea of H-P quantum stochastic calculus comes from the continuous tensor\nproduct factorization property of bosonic Fock space. The tensor product factorizat ion\nproperty says that when the one-particle space is given by a direct sum,H=H1⊕H2, we\nhave the factorization property for the corresponding Fock spa ce: Γ(H) = Γ(H1⊕ H2) =\nΓ(H1)⊗Γ(H2). In our setup,\nL2(R+;Z) =L2([0,t];Z)⊕L2([t,∞);Z) (33)\nfor everyt >0. We exploit this property to describe the total space on which the system\nand the noise evolve jointly. Denote, for all 0 <s<t:\nFt]=HS⊗Γ(L2([0,t];Z)),F[s,t]= Γ(L2([s,t];Z)),F[t= Γ(L2([t,∞);Z)),(34)\nwithF0]=HS(the initial space describing the system) and F=HS⊗Γ(H) (the total space\nfor the time evolution of the system in the presence of quantum nois e). We then have the\nnatural identification F=Ft]⊗F[tbased on the factorization of the exponential vectors:\n|ψ/an}bracketri}ht⊗|e(u)/an}bracketri}ht=|ψ/an}bracketri}ht⊗|e(u[0,t])/an}bracketri}ht⊗|e(u[t,∞))/an}bracketri}ht, (35)\nwhere|ψ/an}bracketri}ht ∈ HS,u[0,t](τ) = 1[0,t](τ)u(τ),u[t,∞)(τ) = 1[t,∞)(τ)u(τ). Note that Ft]andF[t\nembed naturally into Fas subspaces by tensoring with the vacuum vector.\nAny vector |u/an}bracketri}ht ∈ Hmay be regarded as a Z-valued function. For a fixed basis of Z(e.g.\nin the case when Zis the space Cdwith the canonical basis |zk/an}bracketri}ht), we setuk(t) =/an}bracketle{tzk|u(t)/an}bracketri}htZ\nfork≥1, where /an}bracketle{t·|·/an}bracketri}htZdenotes scalar product on Z. We define the creation and annihilation\nprocesses associated with the orthonormal basis {|zk/an}bracketri}ht}k≥1as follows:\nA†\nk(t) =a†(1[0,t]⊗zk), A k(t) =a(1[0,t]⊗zk), (36)\nfork= 1,2,..., where 1 [0,t]denotes indicator function of [0 ,t] as an element of L2(R+).\nEachAk(respectively, A†\nk) is defined on a distinct copy of the Fock space Γ( L2(R+)) and\ntherefore, the Ak’s (respectively, A†\nk) are commuting. Physically, each of them represents a\nsingle channel of quantum noise input coupled to the system. Note t hat in the special case\nZ=C, the above construction only gives a single pair of creation and annih ilation process\nand in the case Z=Cd, we havedpairs of creation and annihilation processes associated\nwithddistinct noise inputs. The actions of the Ak(t) on the exponential vectors are given9\nby the eigenvalue relations:\nAk(t)|e(u)/an}bracketri}ht=/parenleftbigg/integraldisplayt\n0uk(s)ds/parenrightbigg\n|e(u)/an}bracketri}ht, (37)\nand theA†\nk(t) are the corresponding adjoint processes:\n/an}bracketle{te(v)|A†\nk(t)|e(u)/an}bracketri}ht=/parenleftbigg/integraldisplayt\n0vk(s)ds/parenrightbigg\n/an}bracketle{te(v)|e(u)/an}bracketri}ht. (38)\nThe above processes, which are time integrated versions of instan taneous creation and\nannihilation operators are two of the three kinds of fundamental noise processes intro-\nduced by Hudson and Parthasarathy. They satisfy an integrated version of the CCR:\n[Ak(t),A†\nl(s)] =δklmin(t,s), [Ak(t),Al(s)] = [A†\nk(t),A†\nl(s)] = 0.\nFor eachk, their ‘future pointing’ infinitesimal time increments, dA#\nk(t) :=A#\nk(t+dt)−\nA#\nk(t), where # denotes either creation or annihilation processes, with r espect to the time\ninterval [t,t+dt], areindependent processes. The independence is due to the fact that time\nincrements with respect to non-overlapping time intervals are comm uting since they are\nadaptedwith respect to F, i.e. they act non-trivially on the factor F[t,t+dt]of the space\nF=Ft]⊗F[t,t+dt]⊗F[t+dtand trivially, as identity operator on the remaining two factors.\nIn other words, for a fixed k,\ndA#\nk(t)|e(u)/an}bracketri}ht= (A#\nk(t+dt)−A#\nk(t))|e(u)/an}bracketri}ht (39)\n=e(u[0,t])⊗a#(1[t,t+dt]⊗zk)e(u[t,t+dt])⊗e(u[t+dt,∞)),(40)\nwhere the operators a#are defined in ( 36). Therefore, any Hermitian noise processes M(t)\nthat areappropriatecombinations ofthe A#\nk(t) (for instance, the quantum Wiener processes\nintroduced later in ( 44)) have independent time increments, i.e. if we define the character is-\ntic function of Mwith respect to the coherent state, |ψ(u)/an}bracketri}ht, asϕM(λ) :=/an}bracketle{tψ(u)|eiλM|ψ(u)/an}bracketri}ht,\nthen for any two times s≤t, we see that their joint characteristic function with respect to\nthe coherent states is the product of individual characteristic fu nctions:\nϕM(s),M(t)−M(s)(λs,λt) :=/an}bracketle{tψ(u)|eiλsM(s)+iλt(M(t)−M(s))|ψ(u)/an}bracketri}ht\n=ϕM(s)(λs)ϕM(t)−M(s)(λt). (41)\nThis property is a quantum analog of the notion of processes with ind ependent increments\nin classical probability.\nRemark IV.1. In quantum field theory, the operators A†\nk(t) andAk(t) are called the\nsmeared field operators and are usually written formally as:\nAk(t) =/integraldisplayt\n0bk(s)ds, A†\nk(t) =/integraldisplayt\n0b†\nk(s)ds, (42)\nwhere thebk(t) =1√\n2π/integraltext\nRˆbk(ω)e−iωtdωandb†\nk(t) =1√\n2π/integraltext\nRˆb†\nk(ω)eiωtdωare the idealized\nBose field processes satisfying the singular CCR: [ bk(t),b†\nl(s)] =δklδ(t−s) [15]. Physically,\nsinceb†\nk(s) creates a particle at time sthrough the kth noise channel, A†\nk(t) creates a\nparticle that survives up to time t. These formal expressions for the annihilation and\ncreation processes are simpler to work with than the more regular in tegrated processes10\ndefined in ( 37)-(38). As remarked on page 39 of [ 79], the more fundamental processes from\nthe underlying physics point of view are the quantum field processes , not the rigorously\ndefined, more regular integrated processes.\nExploiting the structure of bosonic Fock space and the properties of the fundamental\nnoise processes outlined above, Hudson and Parthasarathy deve loped and studied quantum\nstochastic integrals withrespecttothesefundamentalprocessesforasuitableclass ofadapted\nintegrand processes, in analogy with the constructions of the clas sical Itˆ o theory. The most\nimportantresultofthe calculusisthe quantum Itˆ o formula , whichdescribeshowthe classical\nLeibnitz formula for the time-differential of a product of two funct ions gets corrected when\nthese functions depend explicitly on the fundamental processes. In the vacuum state, the\nquantum Itˆ o formula can be summarized by:\ndAk(t)dA†\nl(t) =δkldt (43)\nand all other products of differentials that involve dAk(t),dA†\nk(t) anddtvanish. This can\nbe viewed as a chain rule with Wick ordering [ 86] and as a quantum analogueof the classical\nItˆ o formula.\nIn particular, with respect to the initial vacuum state, the field qua draturesW0\nk(t) =\nAk(t) +A†\nk(t) (k= 1,2,...) are mean zero Hermitian Gaussian processes with variance t.\nTherefore, they can be viewed as quantum analogue of classical Wie ner processes and their\nformal time derivatives, dW0\nk(t)/dt=bk(t) +b†\nk(t), are quantum analogues of the classical\nwhite noises. If one takes Z=Cd, then (W0\n1,W0\n2,...,W0\nd) is a collection of commuting\nprocesses and thus form a quantum analogue of d-dimensional classical Wiener process in\nthe vacuum state. Moreover, one has dW0\ni(t)dW0\nj(t) =δijdt, which is the classical Itˆ o\ncorrection formula for Wiener process. These results hold for a mo re general class of field\nobservables:\nWθ\nk(t) =e−iθkAk(t)+eiθkA†\nk(t), (44)\nwhereθk∈Ris a phase angle. We refer to the Wθ\nk(t) asquantum Wiener processes , as they\nare operator-valued processes on a Fock space, analogous to cla ssical Wiener processes.\nIn contrast to closed quantum system’s unitary evolution, interac tion with an environ-\nment leads to randomness in the unitary evolution of an open quantu m system. Using the\nquantum Itˆ o formula, Hudson and Parthasarathy deduced the g eneral form of a unitary,\nreversible, Markovian evolution for a system interacting with an env ironment described by\nthe fundamental noise processes. The unitary evolution operato r,V(t), of the whole system,\nin the interaction picture with respect to the free field dynamics, is f ound to satisfy an Itˆ o\nSDE of the following form:\ndV(t) =/bracketleftBigg/parenleftBigg\n−i\n/planckover2pi1He−1\n2/summationdisplay\nkL†\nkLk/parenrightBigg\ndt+/summationdisplay\nk/parenleftBig\nL†\nkdAk(t)−LkdA†\nk(t)/parenrightBig/bracketrightBigg\nV(t),(45)\nV(0) =I, (46)\nassociated to the system operators ( He,{Lk}), whereHe=H†\neis an effective Hamiltonian\nand theLkare Lindblad coupling operators. It can be viewed as a noisy Schrodin ger\nequation. The evolution of a noisy system observable, X, initially defined on HS, can\nalso be obtained. By applying the quantum Itˆ o formula, one can ded uce that its evolution,11\njt(X) =V(t)†(X⊗I)V(t), onFisdescribedby thefollowingHeisenberg-Langevinequation:\ndjt(X) =jt(L(X))dt+/summationdisplay\nk/parenleftbigg\njt([X,Lk])dA†\nk(t)+jt([L†\nk,X])dAk(t)/parenrightbigg\n,(47)\nj0(X) =X⊗I, (48)\nwhereLis the well-known Lindblad generator [ 87]:\nL(X) =i\n/planckover2pi1[He,X]+1\n2/summationdisplay\nk([L†\nk,X]Lk+L†\nk[X,Lk]). (49)\nOne can also obtain the evolution of field observables in this way and st udy the relation\nbetween input and output field processes [ 15]. We call such equation for an observable a\nquantum stochastic differential equation (QSDE) and its solution is a quantum stochastic\nprocess, which is a noncommutative analogue of classical stochastic proces s. To obtain the\nLindblad master equation for reduced system density operator, ρS(t), we first take the Fock\nvacuum conditional expectation of jt(X) to obtain the evolution of the reduced system\nobservable, Tt(X), defined via\n/an}bracketle{tψ|Tt(X)|φ/an}bracketri}ht=/an}bracketle{tψ⊗e(u)|jt(X)|φ⊗e(v)/an}bracketri}ht, (50)\nso thatdTt(X) =Tt(L(X))dt, then the Lindblad master equation:\ndρS(t) =L∗(ρS(t))dt (51)\nis obtained by duality.\nRemark IV.2. Following Remark IV.1, one can introduce the notion of quantum colored\nnoise[88,89]. Fork= 1,2,..., define\nb†\ng,k(t) :=1√\n2π/integraldisplay\nRˆb†\nk(ω)eiωtˆg(ω)dω, b g,k(t) :=1√\n2π/integraldisplay\nRˆbk(ω)e−iωtˆg(ω)dω,(52)\nwhere ˆg(ω) andˆbk(ω) denote the Fourier transform of g(t) andbk(t) respectively. Note that\nb†\ng,k(t) is the inverse Fourier transform of ˆb†\nk(ω)ˆg(ω) and so by the convolution theorem we\nhave:\nb†\ng,k(t) =1√\n2π/integraldisplay\nRg(t−s)b†\nk(s)ds=1√\n2π/integraldisplay\nRg(t−s)dA†\nk(s), (53)\nwhere theA†\nk(s) are creation processes. This is reminiscent of the formula for clas sical\ncolored noise defined via filtering of white noise [ 90]:\n/integraldisplay\nRγ(t−s)1{s≤t}(s)dBs=/integraldisplayt\n−∞γ(t−s)dBs, (54)\nwhereγ(t) describes the filter and B= (Bs) is a classical Wiener process. In the limit g→1\n(flat spectrum limit), the b†\ng,k(t) converge to the fundamental noise process b†\nk(t) =dA†\nk/dt.\nSimilarremarksapply to bg,k(t) and toappropriatelinearcombinationsof bg,k(t) andb†\ng,k(t),\nwhich therefore deserve to be called quantum colored noise proces ses.12\nV. QSDE’S FOR QUANTUM NOISE\nGuided by formula of the symmetric correlation function in ( 19) and the plan outlined in\nSectionIII, we model the quantum noise by:\n∞/summationdisplay\nk=0ηk(t), (55)\nwhere theηk(t) are independent quantum Ornstein-Uhlenbeck processes [91], satisfying the\nSDEs:\ndηk(t) =−αkηk(t)dt+/radicalbig\nλkdWθ\nk(t), ηk(0) =ηk. (56)\nHere theWθ\nkare independent quantum Wiener processes defined in Section IVand theηk\nare initial variables on a copy of Fock space. For a fixed θ, independence and commutation\nfor these processes can be achieved by realizing the ηk(t) on distinct copies of Fock space,\ni.e.\n∞/summationdisplay\nk=0ηk(t) =η0(t)⊗I⊗I⊗···+I⊗η1(t)⊗I⊗···+... (57)\non/circlemultiplytext∞\nk=0Γ(L2(R+)) = Γ(L2(R+)⊗ K) where the multiplicity space Kis a sequence space\nwhose elements are of the form ( x0,x1,x2,...), with each xi∈C. From now on, each ηkis\nunderstood to be\nI⊗···⊗I/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nkcopies⊗ηk⊗I⊗··· (58)\nand similarly for each Wθ\nk.\nThe formal solution to the SDE ( 56) is given by:\nηk(t) =ηke−αkt+/radicalbig\nλk/integraldisplayt\n0e−αk(t−s)dWθ\nk(s). (59)\nSince there is a unique stationary solution of the SDEs ( 56), for allkands∈[0,t]:\nE′′\n∞[η2\nk] =λk\n2αk, E′′\n∞[ηkWθ\nk(s)] =E′′\n∞[Wθ\nk(s)ηk] = 0, (60)\nwhereE′′\n∞denotes expectation with respect to the vacuum state associate d with Ω ⊗Ω⊗···\non the enlarged Fock space Γ( L2(R+)⊗K). Then, with the parameters αnandλndefined\nby\nαn=νn1{n≥1}+Λ1{n=0}>0 (61)\nand\nλn=4ν2\nnΛ2kBT\nν2n−Λ21{n≥1}+/planckover2pi1Λ3cot/parenleftbigg/planckover2pi1Λ\n2kBT/parenrightbigg\n1{n=0}>0, (62)13\nit can be verified that\nE′′\n∞/bracketleftbigg{/summationtext\nkηk(t),/summationtext\nlηl(s)}\n2/bracketrightbigg\n=D1(t−s), (63)\nwhereD1is given in ( 19).\nEquations ( 61) and (62) establish a link between the quantum noise as introduced in\neqn. (55) and the physical model of Section II. We remark that there is freedom in the above\nconstruction of quantum noise, as the driving noise process, ( Wθ\nk), is a family of quantum\nWiener processes parametrized by θ. On the one hand, the choice of the parameter should\nbe fixed by physical considerations, i.e. by the nature of the field th at the system couples\nto in the microscopic model. On the other hand, one would like to show t hat the quantum\nnoises describe a Markovian system, so one should write the SDEs ( 56) in a H-P QSDE\nform.\nTo this end, let ξk(t) andηk(t) be canonical conjugate bath observables that obey the\ncommutation relation [ ξj(t),ηk(t)] =i/planckover2pi1δjkIfor allt≥0. Suppose that the evolution of each\npair (ξk(t),ηk(t)) is Markovian and can be described by the H-P QSDEs associated wit h\n(Hk,Lk), where\nHk=η2\nk\n2+αk\n4{ξk,ηk}, Lk=√λk\n/planckover2pi1ξk+iαk\n2√λkηk, (64)\nwhereαkandλkare given as before. Therefore, they solve the H-P QSDEs:\ndξk(t) =ηk(t)dt+/planckover2pi1αk\n2√λkdWπ\nk(t), (65)\ndηk(t) =−αkηk(t)dt+/radicalbig\nλkdW−π/2\nk(t), (66)\nwhere\nWπ\nk(t) =−(Ak(t)+A†\nk(t)), W−π/2\nk(t) =i(Ak(t)−A†\nk(t)) (67)\nare noncommuting, conjugate quantum Wiener processes satisfy ing [Wπ\nk(t),W−π/2\nk(s)] =\n2iδ(t−s)I. Modulo the negative factor, one can view the formal time derivativ es of the\nWπ\nk(t) andW−π/2\nk(t) asthe noisesarisingfromthe positionand momentum field observab les\nrespectively. We fix the freedom in our construction by taking the M arkovian system ( 65)-\n(66) as the model for noise. Therefore, we take/summationtext\nkηk(t) to be the quantum colored noise\nthat models the action of the heat bath on the evolution of the syst em’s observables.\nPhysically, one can think ofour quantum noisemodel asequivalent to a model ofinfinitely\nmanynon-interactingancillasthatconvertthewhitenoisetocolore dnoisethroughachannel\nat each Matsubara frequency [ 92]. That one needs infinitely many ancillas is due to the fact\nthat there are infinitely many transition (Bohr) energies, each of w hich equals the energy\nof a boson with a particular Matsubara frequency in the bath. Acco rding to our noise\nmodel, when a boson with the Matsubara frequency νkis created or annihilated, the energy\ntransition does not occur instantaneously but happens on the time scale of 1/αkvia a\nchannel associated with νk.14\nVI. THE RESCALED MODEL AND SDE’S\nWe setm=m0ǫ, Λ =EΛ/ǫand/planckover2pi1=ǫ, wherem0andEΛare fixed positive constants\nandǫ >0 is a small parameter, so that /planckover2pi1Λ =EΛ(the maximum energy of bosons in the\nbath) andm//planckover2pi1=m0(proportional to the de Broglie wavelength of the Brownian particle )\nare fixed in the Hamiltonian.\nThese scalings of the model parameters are motivated as follows. A n atom’s mass is often\nsmall and so are the characteristic time scales of the quantum bath . On the other hand,\nfor small Planck constant the equipartition theorem is valid, so the m ean kinetic energy of\nthe system is O(1) (i.e. of order 1) as ǫ→0. This has to be compared to the classical case,\nwhere the fact that the kinetic energy is O(1) leads to the presenc e of noise-induced drift\nin the small mlimit when the original system is subject to state-dependent dampin g and\ndiffusion [ 4]. Hence, this suggests that the scalings give meaningful effective d ynamics in\nthe limitǫ→0.\nNext, we elucidate our scalings in the context of separation of time s cales. Taking ǫ→0\nis equivalent to taking the joint limit of small mass ( m→0), the memoryless limit (Λ → ∞)\n(which also implies the small noise correlation time limit, due to the quant um fluctuation-\ndissipation relation) and the classical limit ( /planckover2pi1→0). Note that in the limit the spectral\ndensityJ(ω) becomes strictly Ohmic, since the cutoff is removed as ǫ→0. In other words,\nthe inertial time scale, the memory time scale, the noise correlation t ime scale and the\nquantum time scale vanish simultaneously at the same rate as we take ǫ→0 in the rescaled\nmodel. This limit is a quantum version of the one studied in [ 36].\nUpon applying the above rescalings, the parameters in the QSDEs fo r theξn(t) andηn(t)\nin Section Vbecome\nαn=2πnkBT\nǫ1{n≥1}+EΛ\nǫ1{n=0}=:1\nǫan, (68)\nand\nλn=4kBT(2πnkBT)2\n(2πnkBT)2−E2\nΛE2\nΛ\nǫ21{n≥1}+EΛcot/parenleftbiggEΛ\n2kBT/parenrightbiggE2\nΛ\nǫ21{n=0}=:1\nǫ2Σ2\nn.(69)\nNotice that in the following the relevant parameters are anand Σ nas defined in eqns. ( 68)\nand (69). The rescaled version of the resulting Heisenberg-Langevin equa tions (8)-(9) can\nbe cast as the following system of SDEs on the total space F=HS⊗Γ(L2(R+)⊗K):\ndX(t) =V(t)dt, (70)\nm0ǫdV(t) =−U′(X(t))dt+f′(X(t))∞/summationdisplay\nn=0ηn(t)dt−f′(X(t))Y(t)dt, (71)\ndZ(t) =Y(t)dt, (72)\nǫdY(t) =−EΛY(t)dt+EΛf′(X(t))V(t)dt−iEΛ\n2m0f′′(X(t))dt, (73)\ndξn(t) =ηn(t)dt+ǫan\n2ΣndWπ\nn(t), n= 0,1,2,..., (74)\nǫdηn(t) =−anηn(t)dt+ΣndW−π/2\nn(t), n= 0,1,2,..., (75)15\nwhere we have defined the auxiliary quantum stochastic process\nY(t) =EΛ\nǫ/integraldisplayt\n0e−EΛ\nǫ(t−s){f′(X(s)),V(s)}\n2ds (76)\nand used the commutation relation[P,f′(X)]\n2m=−if′′(X)\n2m0to rearrange the order, so that\nV(t) appears last in ( 73). Note that it is crucial that we have the scaling/planckover2pi1\nm=1\nm0so\nthat the last term on the right hand side of ( 73) isO(1). It should be clear from the\ncontext which factor of the total space the operators act non- trivially on. For instance,\nX=X(t= 0) =X(t= 0)⊗I0⊗I1⊗...;ξn(t) =I⊗I0⊗···⊗In−1⊗ξn(t)⊗In+1⊗...,\nforn= 0,1,...; etc, where Iis identity operator on HSandInis identity operator on the\nnth copy of Fock space.\nFrom now on, vectors and matrices whose elements are operators will be denoted by\nbold letters. For an operator matrix A= (Aij)i,j=0,1,2,..., its transpose, denoted byT,\nis defined as ( Aij)T\ni,j=0,1,2,...:= (Aji)i,j=0,1,2,....The action of Aon an operator vector\nx= (xj)j=0,1,2,..., written as Ax, results in another operator vector (/summationtext\njAijxj)i=0,1,2,.... If\nAis diagonal, we write it as diag( Ak)k=0,1,2,....\nIntroducing the operator vectors\nXt= [X(t)Z(t)ξ0(t)···ξN(t)···]T,Vt= [V(t)Y(t)η0(t)···ηN(t)···]T,(77)\nwe rewrite the above system in a more compact way:\ndXt=Vtdt+ǫµdWπ\nt, (78)\nǫdVt=−ˆγ(X(t))Vtdt+F(X(t))dt+σdW−π/2\nt, (79)\nwith the initial conditions Xt=XandVt=V.\nIn the above, ˆγ(X(t)) (the superoperator that acts on Vt) denotes the block operator\nmatrix, whose entries depend on X(t), given by\nˆγ(X(t)) =/bracketleftbigg\nA(X(t))B(X(t))\n0 D/bracketrightbigg\n, (80)\nwith\nA(X(t)) =/bracketleftbigg\n0f′(X(t))\nm0\n−EΛf′(X(t))EΛ/bracketrightbigg\n, (81)\nB(X(t)) =/bracketleftbigg\n−f′(X(t))\nm0−f′(X(t))\nm0···\n0 0 ···/bracketrightbigg\n,0=\n0 0\n0 0\n......\n, (82)\nD= diag(an)n=0,1,...is the diagonal operator matrix,\nF(X(t)) =/bracketleftbigg\n−U′(X(t))\nm0−iEΛ\n2m0f′′(X(t)) 0 0 ···/bracketrightbiggT\n, (83)16\nµis the block operator matrix given by\nµ=/bracketleftbigg\n0T\nµ1/bracketrightbigg\n,withµ1= diag/parenleftbiggan\n2Σn/parenrightbigg\nn=0,1,..., (84)\nσis the block operator matrix given by\nσ=/bracketleftbigg\n0T\nΣ/bracketrightbigg\n,withΣ= diag(Σ n)n=0,1,..., (85)\nand\nWπ\nt= [Wπ\n0(t)Wπ\n1(t)···]TandW−π/2\nt= [W−π/2\n0(t)W−π/2\n1(t)···]T.(86)\nIn the above, the scalar-looking entries are really scalar multiples of appropriate identity\noperators. In particular, 0 denotes the zero operator on appro priate space.\nVII. FORMAL DERIVATION OF LIMITING EQUATION\nWe are interested in the limit as ǫ→0 of (78)-(79). These equations are similar to the\nones studied in [ 32] and we adapt the techniques employed there and use the main resu lts\nfrom quantum Itˆ o calculus outlined in Section IVto study the limit problem.\nIn the limit ǫ→0, we expect that Xtis a slow variable compared to Vt. In the following,\nwe formally derive the limiting equation for the first component of Xt, i.e. the particle’s\npositionX(t), in the limit ǫ→0. To give meanings to our derivations, one considers the\naction of an operator, say Z(t), on a vector of the form ψ⊗e(u), i.e.Z(t)(ψ⊗e(u)), where\nψ∈ HSande(u) is the exponential vector associated with u∈L2(R+)⊗K. We suppress\nthis interpretation of operators in the following and work directly wit h the operators.\nA rewriting of ( 79) leads to:\nVtdt=−ǫˆγ−1(X(t))dVt+ˆγ−1(X(t))F(X(t))dt+ˆγ−1(X(t))σdW−π/2\nt,(87)\nwhereˆγ−1satisfiesˆγ−1ˆγ=ˆγˆγ−1=Iand can be verified to be given by the following block\noperator matrix:\nˆγ−1(X(t)) =/bracketleftbigg\nA−1(X(t))−A−1(X(t))B(X(t))D−1\n0 D−1/bracketrightbigg\n, (88)\nwhere\nA−1(X(t)) =/bracketleftbigg\nm0[f′(X(t))]−2−[a0f′(X(t))]−1\nm0[f′(X(t))]−10/bracketrightbigg\n, (89)\n−A−1(X(t))B(X(t))D−1=/bracketleftbigg[a0f′(X(t))]−1[a1f′(X(t))]−1···\na−1\n0 a−1\n1.../bracketrightbigg\n,(90)\nand\nD−1(X(t)) = diag(a−1\nn)n=0,1,.... (91)17\nAsdXt=Vtdt+ǫµdWπ\nt, it follows that we can write Xtin the integral form:\nXt=X−/integraldisplayt\n0ǫˆγ−1(X(s))dVs+/integraldisplayt\n0ˆγ−1(X(s))F(X(s))ds+/integraldisplayt\n0ˆγ−1(X(s))σdW−π/2\ns\n+ǫµ(Wπ\nt−Wπ). (92)\nThe only terms that depend explicitly on ǫon the right hand side above are the second\nterm and the last term. Therefore, we study the asymptotic beha vior of these terms as\nǫ→0. The last term will tend to zero as ǫ→0. For the second term, we consider the\ncomponents of the operator process/integraltextt\n0ǫˆγ−1(X(s))dVs= [D1(t)D2(t)···]T, where\nD1(t) =/integraldisplayt\n0[f′(X(s))]−2m0ǫdV(s)−/integraldisplayt\n0[a0f′(X(s))]−1ǫdY(s)\n+∞/summationdisplay\nn=0/integraldisplayt\n0[anf′(X(s))]−1ǫdηn(s), (93)\nD2(t) =/integraldisplayt\n0[f′(X(s))]−1m0ǫdV(s)+∞/summationdisplay\nn=0/integraldisplayt\n0ǫ\nandηn(s), (94)\nDn+3(t) =ǫ\nan(ηn(t)−ηn), n= 0,1,2,.... (95)\nIn particular, the first component of Xtis given by:\nX(t) =X−D1(t)−/integraldisplayt\n0[f′(X(s))]−2U′(X(s))ds+i\n2m0/integraldisplayt\n0[f′(X(s))]−1f′′(X(s))ds\n+/integraldisplayt\n0∞/summationdisplay\nn=0Σn\nan[f′(X(s))]−1dW−π/2\nn(s). (96)\nIntegrating by parts, we can write the first integral in ( 93) as:\n/integraldisplayt\n0[f′(X(s))]−2m0ǫdV(s) = [f′(X(t))]−2m0ǫV(t)−[f′(X)]−2m0ǫV\n−/integraldisplayt\n0d\nds/parenleftbig\n[f′(X(s))]−2/parenrightbig\nm0ǫV(s)ds. (97)\nNext, we make a remark on taking derivatives of operator-valued f unctions. Let h(X(s))\nbe a function, depending on the position process X(s), which can be expanded in a power\nseries. The formula for the derivative of the operator inverse rea ds\nd\nds([h(X(s))]−1) =−[h(X(s))]−1/parenleftbiggd\nds[h(X(s))]/parenrightbigg\n[h(X(s))]−1. (98)\nForh(X(s)) =X(s)p, wherep= 2,3,..., rearranging the order to move V(s) to the right,\none obtains:\nd\ndsX(s)p=pX(s)p−1V(s)−i/planckover2pi1\nmX(s)p−2c(p), (99)18\nwherec(p) is a constant depending on p. From this, one deduces:\nd\nds[h(X(s))] =/parenleftbigg∂\n∂X(s)h(X(s))/parenrightbigg\nV(s)−i/planckover2pi1\nmg(X(s)), (100)\nfor some function g, where∂\n∂X(s)denotes formal derivative with respect to X(s). Using\nthis, it can be shown that, for some function k,\nd\nds([h(X(s))]−1) =−[h(X(s))]−1/parenleftbigg∂\n∂X(s)h(X(s))/parenrightbigg\n[h(X(s))]−1V(s)+i/planckover2pi1\nmk(X(s))\n=∂\n∂X(s)([h(X(s))]−1)V(s)+i/planckover2pi1\nmk(X(s)). (101)\nNote that /planckover2pi1/m= 1/m0is independent of ǫand the above remark allows us to apply the\nfollowing chain rule for operators:\nd\nds/parenleftbig\n[f′(X(s))]−2/parenrightbig\n=∂\n∂X(s)([f′(X(s))]−2)V(s)+i\nm0l(X(s)), (102)\nfor some function l, so that\n/integraldisplayt\n0[f′(X(s))]−2m0ǫdV(s) = [f′(X(t))]−2m0ǫV(t)−[f′(X)]−2m0ǫV\n−/integraldisplayt\n0/parenleftbigg∂\n∂X(s)[f′(X(s))]−2/parenrightbigg\nm0ǫV(s)2ds−i/integraldisplayt\n0l(X(s))ǫV(s)ds\n(103)\nGuided by the estimates in the classical case [ 36], we expect that the terms in the above\nexpression, which contain the momentum process, ǫV(s),s∈[0,t], tend to zero as ǫ→0\nand the terms containing the “kinetic energy”, ǫV(s)2, areO(1) asǫ→0. Physically, these\nstatements can be justified by arguing that the momentum proces s is a fast variable that\nequilibrates rapidly and the equipartition theorem becomes valid in the considered limit,\nrespectively. It is these contributions from ǫV(s)2that invalidate the naive procedure to\nobtain the limiting equation by simply setting ǫto zero in the pre-limit equations; one\nexpects to obtain correction drift terms in the limiting equation for p article’s position.\nSimilarly, we can repeat the abovecalculations and arguments for th e other integralterms\nin (93). We are thus left with the problem of deriving the limiting expressions forǫV(s)2,\nǫV(s)Y(s),ǫV(s)ηn(s),n= 0,1,...asǫ→0.\nTo derive them, we apply quantum Itˆ o formula to\nǫ2VsVT\ns=ǫ2\nV(s)2V(s)Y(s)... V(s)ηN(s)···\nY(s)V(s)Y(s)2... Y(s)ηN(s)···\n............\nηN(s)V(s)ηN(s)Y(s)... η2\nN(s)...\n............\n, (104)\nwhere the entries should be interpreted as tensor products of op erators.19\nThis gives:\nd[(ǫVs)(ǫVT\ns)] =d[ǫVs]ǫVT\ns+ǫVsd[ǫVT\ns]+d[ǫVs]d[ǫVT\ns]\n= [−ˆγ(X(s))Vs+F(X(s))+σdW−π/2\ns]ǫVT\nsds\n+ǫVs[−ˆγ(X(s))Vs+F(X(s))+σdW−π/2\ns]Tds+σσTds,(105)\nwhere we have used the quantum Itˆ o formula ( 43) to compute:\ndW−π/2\nsd(W−π/2\ns)T=Ids. (106)\nRearranging, we obtain the following operator Lyapunov equation [93,94]:\nˆT(J) :=ΓJ+JΓT=B1+B2+B3, (107)\nwhere\nΓ=ˆγ(X(s)),J=ǫVsVT\nsds,B1=−d[ǫ2VsVT\ns], (108)\nB2=ǫ[(σdW−π/2\ns+F(X(s)))VT\ns+Vs(d(W−π/2\ns)TσT+F(X(s))T)]ds,B3=σσTds.\n(109)\nThe formal solution to this equation can be written as\nJ=ˆT−1(B1)+ˆT−1(B2)+ˆT−1(B3). (110)\nBy previous arguments on the asymptotic behavior of the momentu m process, we expect\nthatˆT−1(B1) andˆT−1(B2) tend to zero as ǫ→0.\nTherefore, in the limit ǫ→0,ǫVsVT\nsconverges to the solution, ¯J, of the operator\nLyapunov equation:\nˆγ(¯X(s))¯J+¯Jˆγ(¯X(s))T=σσT. (111)\nSolving the above equation for ¯Jallows us to extract the limits of ǫV(s)2,ǫV(s)Y(s) and\nǫV(s)ηn(s),n= 0,1,..., which we denote by J1,1,J1,2,J1,n+3forn= 0,1,..., respectively.\nWe refer to Appendix Bfor the solution of this equation. This is what we need to determine\nthe asymptotic behavior of X(t) in (96) asǫ→0. The following limiting equation for ¯X(t)\nis the main result of this paper. It is derived for a large class of non-M arkovian QBM with\ninhomogeneous damping and diffusion and is valid for positive temperat ure.\nThe Main Result. In the limit ǫ→0, the particle’s position, X(t), converges to the\nsolution, ¯X(t), of the following equation:\nd¯X(t) =−[f′(¯X(t))]−2U′(¯X(t))dt+i\n2m0[f′(¯X(t))]−1f′′(¯X(t))dt+S(¯X(t))dt\n+[f′(¯X(t))]−1/parenleftBigg/radicalBigg\na0cot/parenleftbigga0\n2kBT/parenrightbigg\ndW−π/2\n0(t)+∞/summationdisplay\nn=1/radicalBigg\n4a2\n0kBT\na2n−a2\n0dW−π/2\nn(t)/parenrightBigg\n,\n(112)20\nwhereS(¯X(t)) is the quantum noise-induced drift , arising in the limit of simultaneously\nvanishing inertial, memory, noise correlation and quantum time scales , given by:\nS(¯X) =/parenleftbigg∂\n∂¯X[f′]−2/parenrightbigga0\n2cot/parenleftbigga0\n2kBT/parenrightbigg\n+/parenleftbigg∂\n∂¯X[f′]−2/parenrightbigg∞/summationdisplay\nn=1/braceleftBigg\n2kBTa2\n0\na2n−a2\n0/parenleftbigg\nI+an\nm0a0(an+a0)(f′)2/parenrightbigg/parenleftbigg\nI+a0\nm0an(an+a0)(f′)2/parenrightbigg−1/bracerightBigg\n−/parenleftbigg∂\n∂¯X[f′]−1/parenrightbigg∞/summationdisplay\nn=1/braceleftBigg\n2kBTa0\nm0an(an+a0)/parenleftbigg\nI+a0\nm0an(an+a0)(f′)2/parenrightbigg−1/bracerightBigg\nf′,(113)\nwherea0=EΛ,an= 2πnkBTand theW−π/2\nn(t) =i(An(t)−A†\nn(t)),n= 0,1,..., are\nindependent quantum Wiener processes introduced in ( 44) of Section IV.\nWe make a few remarks about the limiting equation ( 112).\nFirst, as in the classical case, it contains drift correction terms ind uced by vanishing of\nall the characteristic time scales. The presence of such noise-indu ced drift is a consequence\nof nonlinear coupling in the QBM model. Expanding S(¯X) about large T, we see that:\nS(¯X) =∂\n∂¯X/parenleftbig\n[f′(¯X)]−2/parenrightbig\nkBT+O(1/T), (114)\nand so the form of the zeroth order contribution coincides with the classical noise-induced\ndrift obtained in the case of equilibrium bath where the Einstein’s relat ion is satisfied (c.f.\n(101) in [ 32] withD= [f′]−2kBT). Moreover, the zeroth order contribution after expanding\nin largeTfor the noise terms reads\n[f′]−1/radicalbig\n2kBTdW−π/2\n0(t), (115)\nwhose form (modulo the quantum nature of the noise W−π/2\n0) is identical to that in the\nclassicalresult. Therefore, wesee that ourquantum noise-induc eddrift consistsofa classical\ncounterpart (the first term in the formula for S(¯X) above), which reduces to the classical\nnoise-induced drift in the high temperature regime, and also severa l drift terms that are\npurely quantum in origin. The latter drifts depend explicitly on both ba th parameters an\nandEΛ. Note that, in contrast to the classical results and to the linear co upling case, we\nhave an additional term (the contribution involving the imaginary num ber) in the limiting\nequation due to inhomogeneous nature of the bath.\nSecond, in the linear coupling ( f(¯X) =¯X) case, the limiting equation reduces to:\nd¯X(t) =−U′(¯X(t))dt+/radicalBigg\nEΛcot/parenleftbiggEΛ\n2kBT/parenrightbigg\ndW−π/2\n0(t)+∞/summationdisplay\nn=1/radicalBigg\n4E2\nΛkBT\na2n−E2\nΛdW−π/2\nn(t).(116)\nIn contrast to the results obtained in the literature (see for insta nce, eqn. (14) in [ 42]), the\nlimiting equation is not a classical SDE, but a QSDE driven by quantum th ermal noises. At\nlowtemperature, allthequantumnoisetermsintheaboveexpress ioncontributesignificantly\nto the limiting dynamics. While this is in agreement with the finding in [ 42] that quantum\nfluctuations play an important role at low temperatures, the detaile d expression of this\nrole obtained here is different. On the other note, in this special cas eS(¯X) = 0, so drift\ncorrection terms are absent in the limiting equation and the formal p rocedure of setting ǫ\nto zero in ( 70)-(75) yields a correct limiting equation.21\nThird, to demonstrate the relations between the noise coefficients and the parameters in\nthe noise-induced drifts, one can rewrite:\nd¯X(t) =−[f′(¯X(t))]−2U′(¯X(t))dt+i\n2m0[f′(¯X(t))]−1f′′(¯X(t))dt+S(¯X(t))dt\n+/radicalbig\n4kBT[f′(¯X(t))]−1∞/summationdisplay\nn=0βndW−π/2\nn(t), (117)\nwhere\nS(¯X) = 2kBT/bracketleftbigg∂\n∂¯X/parenleftbig\n[f′]−2/parenrightbig\nβ2\n0+∂\n∂¯X/parenleftbig\n[f′]−2/parenrightbig∞/summationdisplay\nn=1/braceleftBigg\nβ2\nn/parenleftbigg\nI+β2\nn\nm0cn(f′)2/parenrightbigg/parenleftbigg\nI+β2\nn\nm0dn(f′)2/parenrightbigg−1/bracerightBigg\n−∂\n∂¯X/parenleftbig\n[f′]−1/parenrightbig∞/summationdisplay\nn=1/braceleftBigg\nβ2\nn\nm0dn/parenleftbigg\nI+β2\nn\nm0dn(f′)2/parenrightbigg−1/bracerightBigg\nf′/bracketrightbigg\n, (118)\nwith\nβ0=/radicalBigg\na0\n4kBTcot/parenleftbigga0\n2kBT/parenrightbigg\n, βn=/radicalBigg\na2\n0\na2n−a2\n0, dn=ana0\nan−a0, cn=a2\n0\nana0\nan−a0.(119)\nThe formulae for the noise parameters βn(n= 1,2,...) resembles those derived in [ 95]. In\nparticular, β2\nncan be written as 1 /(eǫ(n)/kBT−1), withǫ(n) = 2kBTln(an/a0)>0 and\nsimilarly for dn/an=ancn/a2\n0. Therefore, information about expected number of bosons in\nan energy state of energy ǫ(n) is encoded in the noise coefficients of the limiting equation\nand, more importantly, since the quantum noise-induced drifts dep end explicitly on β2\nn, they\nalso encode such information about the heat bath.\nFourth, for the nonlinear coupling case the limiting equation does not describe a Marko-\nvian dynamics and so cannot be cast as a QSDE in a H-P form, due to th e presence of\nnonzero contribution containing the imaginary expression in the equ ation. On the other\nhand, for the linear coupling case, the limiting equation can be cast int o a H-P QSDE.\nHowever, the H-P form is not unique since the associated effective H amiltonian and Lind-\nblad operators can only be deduced from the form of Heisenberg-L angevin equation for a\nsingle observable and the Lindblad form is invariant under certain tra nsformations of the\nHamiltonian and Lindblad operators. One way to choose a Lindblad for m is to argue as\nfollows. In the usual weak coupling limit the effective dynamics would be a Markovian non-\ndissipative dynamics with the Lindblad operator given by L=Xand the effective system\nHamiltonian HeffequalHS(plus possibly a correction term proportional to X2). Since\nhere we are taking small mass limit together with a white noise limit, one w ould expect to\nobtain Markovian dynamics associated with a modified Land a modified He. In this way,\nonepostulates the Lindblad operators to be:\nLn=¯X−/radicalbig\n4kBTβnP//planckover2pi1, n= 0,1,2,... (120)\nand the effective system Hamiltonian to be:\nHe={−U′(¯X),P}/2. (121)22\nVIII. CONCLUSIONS AND FINAL REMARKS\nIn this paper, we study the small mass limit of QBM model using a quant um stochastic\ncalculus approach, extending analogous studies for classical mode ls. More precisely, in\nthe limit considered, the particle’s momentum is a fast variable that ca n be adiabatically\neliminated to obtain a reduced dynamics described by evolution of pos ition alone, while at\nthesametimememoryeffectsreducetoadditionaldrifts. Ourmainr esultisderivationofthe\nlimiting equation ( 112) for the particle’s position. This equation exhibits strong quantum\neffects. It is driven by thermal noises which are linear combinations o f H-P fundamental\nprocesses. Its most important feature is the presence of quantum noise-induced drift given\nin eqn. ( 113), which corrects the equation one would obtain by naively setting ǫto zero in\nthe pre-limit equations. This correction consists of terms which hav e classical counterparts,\nas well as drifts that are purely quantum in origin.\nWe expect that such quantum noise-induced drifts will lead to intere sting effects in exper-\nimental studies of small mass quantum system at low temperatures , such as an impurity in\na ultracold Bose gas. In [ 47,96–98], it has been shown that the Hamiltonian of this system\nmay be cast in the form of a QBM model. Here, the impurity plays the ro le of the Brownian\nparticle, while the environment is represented by the Bogoliubov exc itations of the gas. In\ngeneral, the coupling between the impurity and the bath shows a non linear dependence on\nthe position of the former. Accordingly, this system is a good candid ate to detect a quantum\nnoise-induced drift. However, the spectral density of an impurity in a Bose gas cannot be\nreduced to an Ohmic one. For instance, in [ 98] it has been shown that, in the case in which\nthe gas is homogeneous, i.e. its density is space-independent, the s pectral density shows the\nfollowing behavior:\nJ(ω)∼ωd+2, (122)\nwheredis the dimension of the system. Therefore, it would be interesting to extend the\npresent study to a QBM model where the bath spectral density is d ifferent from the one\nconsidered here, in particular the non-Ohmic ones [ 99,100]. We will leave these further\nexplorations to future work.\nACKNOWLEDGEMENTS\nS. Lim and J. Wehr were partially supported by NSF grant DMS 161504 5. This work has\nbeen funded by a scholarship from the Programa M´ asters d’Excel- l´ encia of the Fundaci´ o\nCatalunya-La Pedrera, ERC Advanced Grant OSYRIS, EU IP SIQS, EU PRO QUIC, EU\nSTREP EQuaM (FP7/2007-2013, No. 323714), Fundaci´ o Cellex, th e Spanish MINECO\n(SEVERO OCHOA GRANT SEV-2015-0522, FOQUS FIS2013-46768, F ISICATEAMO\nFIS2016-79508-P), and the Generalitat de Catalunya (SGR 874 an d CERCA/Program).\nIX. BIBLIOGRAPHY\n[1] S. Bo and A. Celani, Multiple-scale stochastic processes: decimation, averag ing and beyond ,\nPhysics Reports (2016).\n[2] G. Pavliotis and A. Stuart, Multiscale methods , Texts in Applied Mathematics, Vol. 53\n(Springer, New York, 2008).23\n[3] D. Givon, R. Kupferman, and A. Stuart, Extracting macroscopic dynamics: model problems\nand algorithms , Nonlinearity 17, R55 (2004).\n[4] G. Volpe and J. 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Petersen, Quantum filter for a\nclass of non-Markovian quantum systems , inDecision and Control (CDC), 2015 IEEE 54th\nAnnual Conference on (IEEE, 2015) pp. 7096–7100.\n[93] R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX- XB= Y , Bulletin\nof the London Mathematical Society 29, 1 (1997).\n[94] M. Rosenblum et al. ,On the operator equation BX−XA=Q, Duke Mathematical Journal\n23, 263 (1956).\n[95] S. Attal and A. Joye, The Langevin equation for a quantum heat bath , Journal of Functional\nAnalysis 247, 253 (2007).\n[96] J. Bonart and L. F. Cugliandolo, From nonequilibrium quantum Brownian motion to impurity\ndynamics in one-dimensional quantum liquids , Phys. Rev. A 86, 023636 (2012).\n[97] J. Bonart and L. F. Cugliandolo, Effective potential and polaronic mass shift in a trapped\ndynamical impurityLuttinger liquid system , EPL (Europhysics Letters) 101, 16003 (2013).\n[98] A. Lampo, S. H. Lim, M. ´Angel Garc´ ıa-March, and M. Lewenstein, Bose\npolaron as an instance of quantum Brownian motion , arXiv:1704.07623 (2017),\narXiv:1704.07623 [cond-mat.quant-gas] .\n[99] G. Ford and R. OConnell, Anomalous diffusion in quantum Brownian motion with colored\nnoise , Physical Review A 73, 032103 (2006).\n[100] D. K. Efimkin, J. Hofmann, and V. Galitski, Non-Markovian quantum friction of bright\nsolitons in superfluids , Physical review letters 116, 225301 (2016).\nAPPENDICES\nAppendix A: Derivation of Heisenberg Equations for Particl e’s Observables\nIn this appendix we derive equations ( 8)-(9). Let\nb(ω) =/radicalbiggω\n2/planckover2pi1/parenleftbigg\nx(ω)+i\nωp(ω)/parenrightbigg\n, b†(ω) =/radicalbiggω\n2/planckover2pi1/parenleftbigg\nx(ω)−i\nωp(ω)/parenrightbigg\n,(A1)\n[x(ω),p(ω′)] =i/planckover2pi1δ(ω−ω′)I, (A2)\nwhere we have normalized the masses of all bath oscillators.27\nThe Heisenberg equation of motion gives\n˙X(t) =i\n/planckover2pi1[H,X(t)] =P(t)\nm, (A3)\n˙P(t) =i\n/planckover2pi1[H,P(t)]\n=−U′(X(t))+f′(X(t))/integraldisplay\nR+dωc(ω)/radicalbigg\n2ω\n/planckover2pi1xt(ω)−2f(X(t))f′(X(t))/integraldisplay\nR+r(ω)dω,\n(A4)\n˙xt(ω) =i\n/planckover2pi1[H,xt(ω)] =pt(ω), ω∈R+, (A5)\n˙pt(ω) =i\n/planckover2pi1[H,pt(ω)] =−ω2xt(ω)+/radicalbigg\n2ω\n/planckover2pi1c(ω)f(X(t)), ω∈R+, (A6)\nwherer(ω) =|c(ω)|2/(/planckover2pi1ω) andf′(X) = [f(X),P]/(i/planckover2pi1).\nNext we eliminate the bath degrees of freedom from the equations f orX(t) andP(t).\nSolving for xt(ω),ω∈R+, gives:\nxt(ω) =x0(ω)cos(ωt)+p0(ω)sin(ωt)\nω/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nx0\nt(ω)+/integraldisplayt\n0sin(ω(t−s))\nω/radicalbigg\n2ω\n/planckover2pi1c(ω)f(X(s))ds.(A7)\nSubstituting this into the equation for P(t) results in:\n˙P(t) =−U′(X(t))+f′(X(t))/integraldisplay\nR+dωc(ω)/radicalbigg\n2ω\n/planckover2pi1x0\nt(ω)\n+2\n/planckover2pi1f′(X(t))/integraldisplay\nR+dω|c(ω)|2/integraldisplayt\n0dssin(ω(t−s))f(X(s))−2f(X(t))f′(X(t))/integraldisplay\nR+dωr(ω).\n(A8)\nUsing integration by parts, we obtain\n/integraldisplayt\n0dssin(ω(t−s))f(X(s)) =f(X(t))\nω−f(X)cos(ωt)\nω−/integraldisplayt\n0cos(ω(t−s))\nωd\nds(f(X(s)))ds\n(A9)\nand therefore,\n˙P(t) =−U′(X(t))+f′(X(t))/integraldisplay\nR+dωc(ω)(b†\nt(ω)+bt(ω))\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nζ(t)\n−f′(X(t))/integraldisplayt\n0ds/integraldisplay\nR+dω2r(ω)cos(ω(t−s))\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nκ(t−s)d\nds(f(X(s)))\n−f′(X(t))f(X)/integraldisplay\nR+dω2r(ω)cos(ωt)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nκ(t), (A10)28\nwhere\nd\nds(f(X(s))) =i\n/planckover2pi1[H,f(X(s))] ={f′(X(s)),P(s)}\n2m, (A11)\nbt(ω) =b(ω)e−iωt,b†\nt(ω) =b†(ω)eiωtand{·,·}denotes anti-commutator.\nAppendix B: Solving the Operator Lyapunov Equation\nWe outline the derivation of the solution, ¯J, to the operator Lyapunov equation:\nˆγ(¯X(s))¯J+¯Jˆγ(¯X(s))T=σσT, (B1)\nwhereˆγandσareblockoperatormatrices,definedinSection VI.First,weobservethatupon\ntaking transpose on both sides of the equation, we have ˆγ(¯X(s))¯JT+¯JTˆγ(¯X(s))T=σσT,\nso uniqueness of the solution implies ¯J=¯JT, i.e.Jk,l=Jl,kfor allk,l.\nWe write ¯Jin the block-structure form:\n¯J=/bracketleftbiggJ1J2\nJT\n2J4/bracketrightbigg\n, (B2)\nwhere\nJ1=/bracketleftbigg\nJ1,1J1,2\nJ1,2J2,2/bracketrightbigg\n,J2=/bracketleftbigg\nJ1,3J1,4···\nJ2,3J2,4···/bracketrightbigg\nandJ4=\nJ3,3J3,4···\nJ4,3J4,4···\n.........\n.(B3)\nWorking out the matrix multiplications of the block operator matrices in the equation\ngives\nJ4=1\n2D−1Σ2, (B4)\nwhich is a diagonal block operator matrix, and the following Sylvester -type equations:\nAJ2+J2D=−1\n2BD−1Σ2, (B5)\nAJ1+J1AT=−BJT\n2−J2BT. (B6)\nEqn. (B5) gives a system of linear equations for J1,n+3andJ2,n+3, forn= 0,1,...:\nf′\nm0J2,n+3+anJ1,n+3=f′\n2m0Σ2\nn\nan, (B7)\n−a0f′J1,n+3+(a0+an)J2,n+3= 0, (B8)29\nwhich has the solution:\nJ2,n+3=Σ2\nn\n2m0a0\na2n(a0+an)/bracketleftbigg\nI+a0\nm0an(a0+an)(f′)2/bracketrightbigg−1\n(f′)2, (B9)\nJ1,n+3=Σ2\nn\n2m0a2n/bracketleftbigg\nI+a0\nm0an(a0+an)(f′)2/bracketrightbigg−1\nf′, (B10)\nwhere we have used the fact that h(X)g(X) =g(X)h(X) for any functions g,h. Similarly,\neqn. (B6) gives:\nJ1,2=∞/summationdisplay\nn=0J1,n+3, J2,2=1\n2{f′,∞/summationdisplay\nn=0J1,n+3}, (B11)\nJ1,1=/parenleftbigg\n(f′)−1+1\nm0a0f′/parenrightbigg∞/summationdisplay\nn=0J1,n+3−1\nm0a0∞/summationdisplay\nn=0J2,n+3. (B12)\nSubstituting the expressions for J2,n+3andJ1,n+3from (B9)-(B10) into the above equation\ngives the formula for J1,2,J2,2andJ1,1. In particular,\nJ1,1=∞/summationdisplay\nn=0/braceleftBigg\nΣ2\nn\n2m0a2n/bracketleftbigg\nI+an\nm0a0(a0+an)(f′)2/bracketrightbigg/bracketleftbigg\nI+a0\nm0an(a0+an)(f′)2/bracketrightbigg−1/bracerightBigg\n.(B13)\nWe remark that upon taking the limit, the contributions involvingthe J1,2andJ1,3cancel\neach other, as in the classical situation, and so the contributions c oming from J1,n(n≥4)\nare indeed correction drift terms induced by purely quantum noises ."    },    {        "title": "1206.1565v1.From_resolvent_estimates_to_damped_waves.pdf",        "content": "arXiv:1206.1565v1  [math.AP]  7 Jun 2012FROM RESOLVENT ESTIMATES TO DAMPED WAVES\nHANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\nAbstract. In this paper we show how to obtain decay estimates for the\ndamped wave equation on a compact manifold without geometri c control via\nknowledge of the dynamics near the un-damped set. We show tha t if replacing\nthe damping term with a higher-order complex absorbing potential gives an\noperator enjoying polynomial resolvent bounds on the real a xis, then the “re-\nsolvent” associated to our damped problem enjoys bounds of t he same order.\nIt is known that the necessary estimates with complex absorb ing potential\ncan also be obtained via gluing from estimates for correspon ding non-compact\nmodels.\n1.Introduction\nOn a compact, connected Riemannian manifold without boundary ( X,g), we\nconsider the non-selfadjoint Schr¨ odinger operator\n(1.1) P(h) =h2∆g+iha\nwherea∈C∞(X) is a non-negative function, and ∆ g=d∗dis thenon-negative\nLaplacian associated to the metric g.This paper mainly addresses the question of\nthe semiclassical analysis of the resolvent of P(h),\nRz(h) := (P(h)−z)−1\nforzin a complex h−dependent neighborhood of 1. For non-selfadjoint operators,\nit is well known that the norm of the resolvent /ba∇dblRz(h)/ba∇dblL(L2,L2)may be large, even\nfarfromthe spectrum [18], and abetter understandingofthe res olventpropertiesof\nnon-selfadjoint operators remains a challenging problem [29]. In this paper we are\nparticularly interested in (polynomial) upper bounds in hfor the resolvent. These\nbounds are especially useful when studying the stabilization problem , which deals\nwith the rate of the energy decay of the solution of the damped wave equation on\nX:\n(1.2)/braceleftbigg/parenleftbig\n∂2\nt+∆g+a(x)∂t/parenrightbig\nu(x,t) = 0,(x,t)∈X×(0,∞)\nu(x,0) =u0∈H1(X), ∂tu(x,0) =u1∈H0(X).\nIt has been shown (see [23]) that if a >0 somewhere, then the energy of the waves,\nE(u,t) =1\n2/integraldisplay\nX|∂tu|2+|∇u|2dx\nsatisfiesE(u,t)t→∞− −− →0 for any initial data ( u0,u1)∈H1×H0. If some monotone\ndecreasing function f(t) can be found such that\nE(u,t)/lessorequalslantf(t)E(u,0),\nDate: August 29, 2021.\nThe authors are grateful for partial support from NSF grants DMS-0900524 (HC), DMS-\n1068742 (AV), DMS-1001463 (JW).\n12 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\nso-called strong stabilization occurs. It is not hard to show that th is is equivalent\nto a uniform exponential decay : ∃C,β >0 such that for any usolution of (1.2),\nE(u,t)/lessorequalslantCe−βtE(u,0).\nIn pioneering works of Rauch, Taylor, Bardos and Lebeau [2,23,26 ], it has been\nshown in various settings that strong stabilization is equivalent to th e geometric\ncontrol condition (GCC) : there exists T0>0 such that from every point in Σ =\n{|ξ|2\ng= 1} ⊂T∗X, the bicharacteristic of P(h) reaches {a >0}in time/lessorequalslantT0. By\ncontrast, when the manifold Xis no longer controlled by a, decay rate estimates\nusually involve additional regularity of the initial data. They take the form\nE(u,t)/lessorequalslantfs(t)/ba∇dblu/ba∇dbl2\nHs\nfors >0 and\n/ba∇dblu/ba∇dblHs=/ba∇dblu(0)/ba∇dbl2\nH1+s+/ba∇dbl∂tu(0)/ba∇dbl2\nHs.\nThe question of exponential energy decay reduces to the study o f high-frequency\nphenomena, in particular the issue of the spectral properties in th e semiclassical\nlimith→0 of certain non-selfadjoint operators approximately of the form∗(1.1),\non a fixed energy layer. For instance, when geometric control hold s, there exist\nh0>0,C,c >0 such that for z∈[1−δ,1+δ]+i[−ch,ch] we have\n(1.3) /ba∇dblRz(h)/ba∇dblL(L2,L2)/lessorequalslantC/h, h < h 0.\nStandard arguments then show that this resolvent estimate implies the uniform ex-\nponential decay for the energy. Similar arguments will apply in the ca se considered\nhere, of resolvent estimates with loss.\n1.1.Motivation. While our main motivation for studying resolvent estimates for\nP(h) come from the stabilization problem, our approachin this paper is or iented by\ngeometric considerations, as we explain now. As discussed above, in the presence\nof geometric control, the resolvent ( P(h)−z)−1enjoys a polynomial bound in a\nneighbourhoodofsize charoundthe realaxis, and this propertyimplies exponential\ndamping. When geometric control no longer holds, it is then a natura l question\nto ask what type of estimate can be satisfied by /ba∇dbl(P(h)−z)−1/ba∇dbl, and, crucially, in\nwhat type of complex neighbourhood of the real axis a resolvent es timate can be\nobtained. The properties of the undamped set\n(1.4) N={ρ∈S∗X:∀t∈R,a◦etHp(ρ) = 0}\nare of central importance for this question. Here, Hpdenotes the Hamiltonian\nvector field generated by the principal symbol p=σh(P(h)) =|ξ|2\ngof the operator\nP(h),andS∗X=p−1(1) denotes the unit cosphere bundle. We remind the reader\nthat the flow generated by HpinS∗Xis simply the geodesic flow.\nWe now review some known results in the case N /\\e}atio\\slash=∅. In [7], the case when\nNis a single hyperbolic orbit is analyzed, and a polynomial resolvent estim ate\nfor/ba∇dbl(P−z)−1/ba∇dblis shown in a h/|logh|-size neighbourhood of the real axis. As a\nconsequence, the energy decay is sub-exponential : with the abo ve notation, one\ncan getfs(t) = e−βs√\nt. It is known from recent work [3] that this decay is sharp. If\n∗Strictly speaking, in order to apply resolvent bounds to the damped wave equation, we also\nneed the imaginary part of the Schr¨ odinger operator to be mi ldlyzdependent, with\nRz(h) = (h2∆g+iha√z−z)−1;\nthis will be handled by perturbation (see Corollary 4.3, Sec tion 6, as well as references [9,21,23]).FROM RESOLVENT ESTIMATES TO DAMPED WAVES 3\nthe curvature of Xis assumed to be negative, and if the relative size ofthe damping\nfunction ais sufficiently large, then the resolvent obeys a polynomial bound in a\nsizehneighbourhood of the real axis, and as a result, exponential deca y for regular\ninitial data occurs [27,28]. Indeed, the hypotheses in [28] is much mo re general,\nrequiring only undamped sets of small pressure; the hyperbolic geo desic is a special\ncase. We note also that for constant negative curvature, the need for an arbitrarily\nlarge damping function ahas been recently removed by Nonnenmacher, by using\ndifferent methods [24].\nA natural question raised by the above remarks, is the following: to what extent\ndoes the geometry of the trapped set alonedetermine a type of decay? In other\nwords, given a trapped geometry, what type of resolvent estimat e do we expect,\nand in what complex neighbourhood of the real axis? This amounts in m any cases\nto a potentially rather crude decay rate for the energy, as it only d epends on the\nstructure of Nand not on the global dynamics of geodesics passing through the\ndamping; in certain cases, however, our results can be seen to be s harp.†\nMotivated by the “black box” approach of Burq-Zworski [4] (cf. e arlier work\nof Sj¨ ostrand-Zworski [31]) as well as recent work on the gluing o f resolvent esti-\nmates by Datchev-Vasy [15], we give a recipe for taking information f romresolvent\nestimates obtained for a noncompact problem in which the set Nconsists of all\ntrapped geodesics—those not escaping to infinity—and investigate what these esti-\nmates imply for the compact problem with damping. In practice, as re cent results\nof Datchev-Vasy [15] have shown the resolvent estimates on man ifolds with, say,\nasymptotically Euclidean ends to be equivalent to estimates on a comp act manifold\nwith acomplex absorbing potential substituting for the noncompactends, we choose\nfor the sake of brevity and elegance to take this latter model as ou r “noncompact”\nsetting. As will be discussed below, these complex absorbing potent ials have the\neffect of annihilating semiclassical wavefront set along geodesics pa ssing through\nthem; this is why they are roughly interchangeable with noncompact ends, along\nwhich energy can flow off to infinity never to return.\nWe thus formulate our main question as follows. Assuming that ( X,g) andaare\ngiven, we consider a model operator of the form\nP1(h) =h2∆g+iW1\nin which the damping is replaced by a complex absorbing potential. We assume that\nthis model operator enjoys a given “resolvent” estimate‡on the real axis:\n(1.5) /ba∇dbl(P1(h)−z)−1/ba∇dbl/lessorequalslantCα(h)\nh, z∈[1−δ,1+δ].\nVariousexamples of such estimates alreadyappear in the literature , see for instance\n[5,7,8,11,12,25,34,36] and the referencestherein. Given (1.5 ) we then aim to obtain\nanalogous estimates for the inverse of the operator with damping, i.e., on\n(P(h)−z)−1= (h2∆g+iha−z)−1,\nwhenthecomplexabsorbingpotentialhascruciallybeenreplacedby anO(h) damp-\ning term. In this paper, we address this question using a control th eory argument\n†A very natural further question would then be : given a trappe d geometry, what kind of global\nassumptions on the manifold can improve the crude decay rate obtained when only Nis known?\n‡We refer to this as a resolvent estimate owing to its close rel ationship with the estimate for\nthe resolvent in scattering problems.4 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\nmotivated by [4], together with a recently improved estimate on reso lvents trun-\ncated away from the trapped set on one side [17], and show that we o btain the\nsame order of estimate as for the model operator. In the next su bsection we state\nthe precise results.\n1.2.Results. As above, we take Hpto be the Hamilton vector field of p=|ξ|2\ng−1\nand etHpits bicharacteristic flow inside p−1(1) =S∗X(i.e., geodesic flow). We\ncontinue to take\n(1.6) N={ρ∈S∗X:∀t∈R,a◦etHp(ρ) = 0},\nand will add the further assumption that\nπ(N)∩suppa=∅,\nwhereπis projection T∗X→X.Thus there exists a non-empty open set O1such\nthat supp( a)⊂X\\O1, andπ(N)⋐O1. The following Theorem is our main “black\nbox” spectral estimate.\nTheorem 1.1. Assume that for some δ∈(0,1)fixed and K∈Z, there is a function\n1/lessorequalslantα(h) =O(h−K)such that\n/ba∇dbl(h2∆g+ia−z)−1/ba∇dblL2→L2/lessorequalslantα(h)\nh,\nforz∈[1−δ,1+δ]. Then there exists C,c0>0such that\n/ba∇dbl(h2∆g+iha−z)−1/ba∇dblL2→L2/lessorequalslantCα(h)\nh,\nforz∈[1−δ,1+δ]+i[−c0,c0]h/α(h).\nWhenN=∅, one has α(h) =O(1), while for N /\\e}atio\\slash=∅, one has α(h)→ ∞as\nh→0. As a general heuristic, the “larger” the trapped set is, the larg er isα(h)\nwhenh→0, and the weaker the above global estimate is—see section 5 below f or\nexamples.\nRemark 1.2.As discussed above, instead of the assumption on the model opera tor\nh2∆g+iW1with complex absorption, we could just as well, by the results of [15],\nhave made an assumption on a model operator in which the set O1is “glued” to\nnon-compact ends of various forms. In particular, it would suffice t o know the\ncut-off resolvent estimate on the real axis for the limiting resolvent\n/vextenddouble/vextenddoubleχ(h2∆′−z+i0)−1χ/vextenddouble/vextenddouble\nL(L2,L2)/lessorequalslantα(h)\nh\nfor a localizer χequal to 1 on O1and for ∆′the Laplacian on a manifold with\nEuclidean ends whose trapped set is contained in a set O′\n1isometric to O1.\nRemark 1.3.The hypotheses of Theorem 1.1 can be weakened to phase space hy-\npotheses, with aa pseudodifferential operator (as in [30]). We have chosen to keep\nthe damping as a function on the base in accordancewith tradition an d for the sake\nof brevity.\nRemark 1.4.The assumption that α(h) =O(h−K) is of a technical nature. It does\nnot appear to be too restrictive, however, since every knownestimate for weakly\nunstable trapping satisfies this assumption (see Section 5 below). I ndeed, if the\nundamped set Nis at least weakly semi-unstable, the results of [11] suggest that\nin factα(h) is always Oǫ(h−1−ǫ) for any ǫ >0.FROM RESOLVENT ESTIMATES TO DAMPED WAVES 5\nRemark 1.5.Ifα(h) is not of polynomial nature, the proof of Theorem 1.1 has to\nbe slightly modified (see below). As a result, the final estimate we can obtain is\nweaker : there exists C,c0>0 such that\n/ba∇dbl(h2∆g+iha−z)−1/ba∇dblL2→L2/lessorequalslantCα2(h)\nh,\nforz∈[1−δ,1+δ]+i[−c0,c0]h/α2(h).\nIn section 5 we describe three different settings in which our gluing re sults apply,\nin which the dynamics in a neighborhood of the trapped set are respe ctively\n(1) Normally hyperbolic\n(2) Degenerate hyperbolic\n(3) Hyperbolic with a condition on topological pressure.\nIn addition to proving resolvent estimates in these settings, we disc uss applications\nto decay rates for solutions to the damped wave equation.\n2.Operators with complex absorbing potentials\nIn this section, we collect some standard results about operators with complex\nabsorbing potentials. Such a potential has a much stronger effect than damping,\nnamely (in the microlocal absence of forcing), that of annihilating se miclassical\nwavefront set completely along bicharacteristics passing through it in the forward\ndirection.\nWe collect basic results about the resolvent of an operator with com plex absorb-\ning potential. This includes both existence of the family and the basic p ropagation\nestimates, which tell us that the complex absorbing potential kills off wavefront set\nunder forward propagation. We begin with the definition of the “res olvent:”\nLemma 2.1. LetW∈C∞(X),W/greaterorequalslant0andWnot identically zero. Suppose that\nP1=h2∆g+iW\nonX. Then(P1−z)−1is a meromorphic family of bounded operators on L2for\nallz∈C,analytic in the closed lower half-plane.\nProof.Wesimplyremarkthat h2∆+1isinvertible,andapplytheanalyticFredholm\ntheorem to conclude that there exists a meromorphic resolvent fa mily. By the\nFredholm alternative, any pole has to correspond to nullspace of P1−z.Since\nIm/a\\}b∇acketle{tP1u,u/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tWu,u/a\\}b∇acket∇i}ht−(Imz)/ba∇dblu/ba∇dbl2,\nforuto be in the nullspace would imply that Im z/greaterorequalslant0; equality would further\nrequire that /a\\}b∇acketle{tWu,u/a\\}b∇acket∇i}ht= 0 which is forbidden by unique continuation. /square\nFinally, we recall the microlocal bound of propagation through trap ping by\nDatchev-Vasy [16], as well as basic backward propagation of singula rities in the\npresence of complex absorption (see also Lemma A.2 of [25] for the la tter). In this\ncontext we will say that a bicharacteristic γ(by bicharacteristic we always mean an\nintegral curve of HRep1in the characteristicset of Re p1−Rezwherep1is the semi-\nclassical principal symbol of P1), or a point on γ, isnon-trapped ifW(γ(T))>0\nfor some T∈R, and is trappedotherwise. We say it is forward non-trapped if\nW(γ(T))>0 for some T >0.In the terminology of [16] we say that§the resolvent\n§We have opposite signs for imaginary parts of p1relative to [16], so incoming and outgoing\nare interchanged.6 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\n(P1−z)−1issemiclassically incoming with a loss of h−1provided that whenever\nq∈T∗Xis on a forward non-trapped bicharacteristic γofP1andf=O(1) on\nγ|[0,T]then (P1−z)−1fisO(h−1) atq.\nLemma 2.2. (See [16].) Suppose that P1=h2∆g+iWonX,W/greaterorequalslant0, andz∈C\nsuch that Imz=O(h∞),Rez∈[1−δ,1+δ]for0< δ <1fixed. Assume that the\nresolvent is polynomially bounded, i.e.,\n/vextenddouble/vextenddouble(P1−z)−1/vextenddouble/vextenddouble\nL(L2,L2)/lessorequalslantCh−Kfor some K.\nThen(P1−z)−1is semiclassically incoming with a loss of h−1. In particular, if\nW(γ(T))>0for some T >0andWF/planckover2pi1(f)is disjoint from γ|[0,T], then(P1−\nz)−1f=O(h∞)atq.\nFurther, if χ∈C∞(X)andT∗\nsuppχXcontains no trapped points, then χ(P1−\nz)−1χisO(h−1).\nA further result that is of crucial importance in avoiding losses in our estimates\nis the following result of [17]:\nLemma 2.3. (See [17].) With the notation of Lemma 2.2, if\n/vextenddouble/vextenddouble(P1−z)−1/vextenddouble/vextenddouble\nL(L2,L2)/lessorequalslantα(h)\nh\nand ifχ∈C∞(X)andT∗\nsuppχXcontains no trapped points, then for some C >0,\n/vextenddouble/vextenddouble(P1−z)−1χ/vextenddouble/vextenddouble\nL(L2,L2)/lessorequalslantC/radicalbig\nα(h)\nh.\n3.Propagation and damping estimates\nWe now switch from complex absorbing potentials back to damping: se t\nP=P(h) =h2∆g+iha\nand consider the equation\n(P−z)u=f , z∈[1−δ,1+δ].\nWe also set Σ = p−1(0)⊂T∗Xwherep=σh(P). Let us start by recalling\na classical result about propagation estimates (see, e.g. Theorem 12.5 of [37] for\na proof by conjugation to normal form; an alternative is the usual commutator\nargumentasdescribedin [22]in thehomogeneoussettingand[8]inth e semiclassical\nsetting):\nLemma 3.1. Suppose q∈Σand for some T >0the forward bicharacteristic\nexp([0,T]Hp)(q)is disjoint from a compact set K. Then there are Q,Q′which\nare elliptic at q, resp.exp(THp)(q), withWF′\n/planckover2pi1(Q′)∩K=∅such that /ba∇dblQu/ba∇dbl/lessorequalslant\n/ba∇dblQ′u/ba∇dbl+Ch−1/ba∇dblf/ba∇dbl,u= (P−z)−1f.\n4.Proof of Theorem 1.1\nFor the moment, we consider only the case where z∈[1−δ,1+δ]. We have\nP=h2∆g+iha\nand we additionally write\nP1=h2∆g+iaFROM RESOLVENT ESTIMATES TO DAMPED WAVES 7\nfor the operator with damping replaced by absorption. Choose an o pen setV1such\nthatN⋐V1⋐O1. LetB1,ϕ∈C∞\n0(X) be smooth functions with B1|V1= 1,\nsuppB1⊂O1,ϕ= 1 on supp ∇B1and supp ϕ∩ N=∅. We observe that N\nsatisfies the assumptions of Lemma 2.3, so that\n/ba∇dbl(P1−z)−1ϕu/ba∇dbl/lessorequalslantCh−1/radicalbig\nα(h)/ba∇dblϕu/ba∇dbl.\nThen, noticing that aandB1have disjoint supports, we have\n(4.1)/ba∇dblB1u/ba∇dbl=/ba∇dbl(P1−z)−1(P1−z)B1u/ba∇dbl\n=/ba∇dbl(P1−z)−1(P−z)B1u/ba∇dbl\n=/ba∇dbl(P1−z)−1(B1(P−z)+[P,B1])u/ba∇dbl\n/lessorequalslant/ba∇dbl(P1−z)−1B1(P−z)u/ba∇dbl+/ba∇dbl(P1−z)−1ϕ[P,B1]ϕu/ba∇dbl\n/lessorequalslantα(h)\nh/ba∇dblB1(P−z)u/ba∇dbl+C/radicalbig\nα(h)/ba∇dblϕu/ba∇dbl\nSince for ρ /∈ Nthe curve etHp(ρ) passes through {a >0},each such bicharac-\nteristic curve must certainly enter the compact set X\\O1.Thus by compactness,\nthere exists ǫ0>0 such that every such curve passes through {a/greaterorequalslantǫ0}.We now\ntake a cutoff function χ/greaterorequalslant0 with supp χ⊂ {a > ǫ0/2},andχ= 1 whenever a/greaterorequalslantǫ0;\nhence every controlled geodesic passes through {χ= 1}.\nWe next recall a classical lemma concerning the propagation of singu larities in\nthe presence of geometric control. (See [4], which builds on a semiclas sical version\nof [22], proved in [32].) This is a slight variation on Lemma 3.1, and can of co urse\nalso be proved using the original positive commutator argument (se e [8]).\nLemma 4.1. (See [4]) Let Ube an open neighbourhood of N,χ∈C∞(X)as\nabove. If B0= Ψ0,0\nh(X)is such that WF′\nh(B0)⊂T∗X\\U, then for zreal near 1,\n/ba∇dblB0u/ba∇dbl/lessorequalslantC\nh/ba∇dbl(P−z)u/ba∇dbl+/ba∇dblχu/ba∇dbl+O(h∞)/ba∇dblu/ba∇dbl\nSince supp ϕand supp(1 −B1) lie inside the controlled region, we can write :\n/ba∇dbl(I−B1)u/ba∇dbl/lessorequalslantCh−1/ba∇dbl(P−z)u/ba∇dbl+C/ba∇dblχu/ba∇dbl+O(h∞)/ba∇dblu/ba∇dbl\nand\n/ba∇dblϕu/ba∇dbl/lessorequalslantCh−1/ba∇dbl(P−z)u/ba∇dbl+C/ba∇dblχu/ba∇dbl+O(h∞)/ba∇dblu/ba∇dbl.\nBut\n/ba∇dblχu/ba∇dbl2/lessorequalslantC/a\\}b∇acketle{tau,u/a\\}b∇acket∇i}ht\n=Ch−1Im/a\\}b∇acketle{t(P−z)u,u/a\\}b∇acket∇i}ht\n/lessorequalslantCh−1/ba∇dbl(P−z)u/ba∇dbl/ba∇dblu/ba∇dbl.\nStarting with (4.1), we deduce from the above inequalities that\n/ba∇dblB1u/ba∇dbl2/lessorequalslantC/parenleftBigα2(h)\nh2/ba∇dbl(P−z)u/ba∇dbl2+Cα(h)/ba∇dblϕu/ba∇dbl2/parenrightBig\n/lessorequalslantC/parenleftBigα2(h)\nh2/ba∇dbl(P−z)u/ba∇dbl2+Cα(h)/ba∇dblχu/ba∇dbl2+O(h∞)/ba∇dblu/ba∇dbl2/parenrightBig\n.8 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\nHence, we have\n/ba∇dblu/ba∇dbl2/lessorequalslantC(/ba∇dblB1u/ba∇dbl2+/ba∇dbl(I−B1)u/ba∇dbl2)\n/lessorequalslantC/parenleftBigα2(h)\nh2/ba∇dbl(P−z)u/ba∇dbl2+α(h)/ba∇dblχu/ba∇dbl2+O(h∞)/ba∇dblu/ba∇dbl2/parenrightBig\n/lessorequalslantC/parenleftBigα2(h)\nh2/ba∇dbl(P−z)u/ba∇dbl2+α(h)\nh/ba∇dbl(P−z)u/ba∇dbl/ba∇dblu/ba∇dbl\n+O(h∞)/ba∇dblu/ba∇dbl2/parenrightBig\n/lessorequalslantC/parenleftBigα2(h)\nh2/ba∇dbl(P−z)u/ba∇dbl2+4ǫ−1α2(h)\nh2/ba∇dbl(P−z)u/ba∇dbl2+ǫ/ba∇dblu/ba∇dbl2\n+O(h∞)/ba∇dblu/ba∇dbl2/parenrightBig\n.\nIfǫis small, we can absorb the last two terms in the above inequality on the left\nhand side, and we obtain\n/ba∇dblu/ba∇dbl/lessorequalslantCα(h)\nh/ba∇dbl(P−z)u/ba∇dbl.\nNow simply observe that by the triangle inequality this bound is still valid if we\nadd tozan imaginary part that satisfies\n|Imz|/lessorequalslanthα−1(h)C′\nforC′such that C′C <1, and this concludes the proof of the theorem.\nRemark 4.2.Ifα(h) is not of polynomial nature, then Lemma 2.3 cannot be used.\nAs a result, the square root in Equation (4.1) must be removed. The rest of the\nargument is the same, and we end up with the estimate given in Remark 1.5. Note\nalso that the energy decay rates for the damped wave equation ar e of course weaker\nthan in the case where Lemma 2.3 can be applied.\nIn order to apply Theorem 1.1 to the situation of the stationary dam ped wave\noperator, we now state a corollary where the Schr¨ odinger opera tor depends mildly\nonz.\nCorollary 4.3. LetXbe a compact manifold without boundary, and let /tildewideP(h,z)be\nthe modified operator\n/tildewideP(h,z) =h2∆g+ih√za−z.\nAssume that for some δ∈(0,1)fixed there is a function 1/lessorequalslantα(h) =O(h−K),\nfor some K∈Z, such that\n/ba∇dbl(h2∆g+ia−z)−1/ba∇dblL2→L2/lessorequalslantα(h)\nh,\nforz∈[1−δ,1+δ]. Then there exists C,c0>0such that\n/ba∇dbl/tildewideP(h,z)−1/ba∇dblL2→L2/lessorequalslantCα(h)\nh,\nfor\nz∈[1−c0/α(h),1+c0/α(h)]+i[−c0,c0]h/α(h).FROM RESOLVENT ESTIMATES TO DAMPED WAVES 9\nProof.The real part of the perturbation is manifestly bounded by a small m ultiple\nofh/α(h) and can thus be perturbed awayby Neumann series. It thus only r emains\nto check that the size of the imaginary part of the perturbation:\nRe(ha−√zha)−Imz\ncan also be made much smaller than h/α(h).\nTake√z= 1+r+iβforr,βtobedetermined. Then z= (1+r)2−β2+2(1+r)iβ.\nThen\nRe(ha−√zha)−Imz=ha(1−(1+r))−2(1+r)β=ǫh/α(h)\nforǫ >0 small if, say, |β|/lessorequalslantǫh/2α(h) and|r|/lessorequalslantα−1(h). Squaring, we obtain\nz∈[1−c0/α(h),1+c0/α(h)]+i[−c0,c0]h/α(h).\n/square\n5.Examples\nIn this section, we briefly outline some known microlocal resolvent es timates,\nstate the two different stationary damped wave operator estimat es, and then draw\nconclusions about solutions to the damped wave equation (1.2).\n5.1.A normally hyperbolic trapped set. In this section, we treat the case in\nwhich the trapped set is a smooth manifold in S∗Xaround which the dynamics\nisnormally hyperbolic . In this case, estimates for the resolvent with a complex\nabsorbing potential have been obtained by Wunsch-Zworski [36]. A particular case\nof interest is the “photon sphere” for the Kerr black hole geometr y, where the phase\nspace is 6-dimensional Nis a symplectic submanifold, diffeomorphic to T∗S2—see\nsection 2 of [36] for details on this application, and [33] for placing it in a ctual\nKerr-de Sitter space. Another special case is of course that of a single hyperbolic\nclosed geodesic (discussed further in the following section).\nThe precise formulationofnormalhyperbolicityused here is asfollow s: wedefine\nthe backward-forward trapped sets by\nΓ±={ρ∈T∗X:∀t≷0,a◦etHp(ρ) = 0}\nThen of course\nN= Γ+∩Γ−,\nwhere we have now ceased to restrict to a single energy surface (s oN ⊂T∗Xis\na homogeneous subset in view of the homogeneity of p) in order to employ the\nterminology of symplectic geometry more easily.\nWe make the following assumptions on this intersection:\n(1) Γ±are codimension-one smooth manifolds intersecting transversely a tN.\n(It is not difficult to verify that Γ ±must then be coisotropic and Nsym-\nplectic.)\n(2) The flow is hyperbolic in the normal directions to Kwithin the energy\nsurfaceS∗X: there exist subbundles E±ofTN(Γ±) such that at p∈S∗X\nTNΓ±=TN ⊕E±,\nwhere\nd(exp(Hp) :E±→E±10 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\nN{a >0}\nFigure 1. The “peanut of rotation”.\nand there exists θ >0 such that\n(5.1) /ba∇dbld(exp(Hp)(v)/ba∇dbl/lessorequalslantCe−θ|t|/ba∇dblv/ba∇dblfor allv∈E∓,±t/greaterorequalslant0.\nAs discussed in [36], these hypotheses as stated are not structu rally stable, but they\ndo follow (at least up to loss of derivatives) from the more stringent hypothesis that\nthe dynamics be r-normally-hyperbolic for every rin the sense of [20, Definition\n4]. This implication, and the structural stability of the hypotheses, follows from a\ndeep theorem of Hirsch-Pugh-Shub [20] and Fenichel [19].\nAs a consequence of the estimates of [36] for resolvents, we the n obtain the\nfollowing estimate for the damped operator:\nTheorem 5.1. Let(X,g)satisfy the dynamical conditions enumerated above. Then\nwe have\n/ba∇dbl(h2∆g+iha−z)−1/ba∇dblL2→L2/lessorequalslantC|logh|\nh,\nforz∈[1−δ,1+δ]+i[−c0,c0]h/|logh|.\nThis estimate, or more precisely its refinement in Corollary 4.3, provid e a corre-\nsponding energy decay estimate for solutions to the damped wave e quation (1.2).\nIn order to avoid irritating issues of projecting away from constan t subspaces, etc.,\nwe assume that u0≡u(x,0) = 0.\nCorollary 5.2. Assume the hypotheses of Theorem 5.1 hold, and let ube a solution\nto(1.2)withu0= 0, andu1∈Hsfor some s∈(0,2]. Then there exists a constant\nC=Cs>0such that\nE(u,t)/lessorequalslantCe−st1/2/C/ba∇dblu1/ba∇dbl2\nHs.\nA simple situation in which the hypotheses of Theorem 1.1 are satisfied is that\nof a connected compact manifold of the form X=X0∪X1withX1open and X0\nisometric to a warped product Ru×Sn−1\nθwith metric\ng=du2+cosh2udθ2.\nWe takea∈C∞(X) to be identically 1 on X1as well as equal to 1 for |x|>1 inX0.\nThis class of manifolds thus includes the “peanut of rotation” shown in Figure 1 as\nwell as its higher dimensional generalizations.\nThen the trapped set is easily seen to be N={u= 0,ξ= 0}whereξis the\ncotangant variable dual to u,and the function x= 2−u2satisfies the convexity\nhypotheses. The flow on Nis normally hyperbolic, with the stable and unstableFROM RESOLVENT ESTIMATES TO DAMPED WAVES 11\nmanifolds being the two components of the set\n/parenleftbig\nξ2+|η|2\nSn−1\ncosh2u/parenrightbig\n=|η|2\nSn−1\ni.e., by the intersection of the condition that energy and angular mom entum match\ntheir values on N.This is an example of a normally hyperbolic trapped set, and\nhence both parts of Theorem 5.1 apply.\n5.2.A trapped set with degenerate hyperbolicity. In this section, we study\na variant of the normally hyperbolic case, in which the intersection of stable and\nunstable manifolds is no longer transverse, hence the results of [36 ] no longer apply.\nThis is the case of a surface of rotation with a degenerate hyperbolic closed orbit.\nOur example manifold is a topological torus X= [−1,1]x×S1\nθ, equipped with\nthe metric\n(5.2) ds2=dx2+A2(x)dθ2\nwhere near x= 0,\nA(x) = (1+ |x|2m)1\n2m\nandmis an integer /greaterorequalslant1. This manifold has a “fatter” part and a “thinner” part.\nAt the thickest, there is an elliptic geodesic, and at the thinnest par t, where x= 0,\nthere is an unstable geodesic, which we denote by γ. Ifm/greaterorequalslant2, the Gaussian\ncurvature is chosen to vanish to a finite order at the unstable geod esic, hence the\ngeodesic is degenerately hyperbolic . If the Gaussian curvature is strictly negative\n(m= 1) in a neighbourhood of the thinnest part, the geodesic is non-de generate;\nthe geometry of a single closed hyperbolic geodesic has been extens ively studied\nin [3,4,6,7,10,13,14] and others. As is seen in the previous sectio n, in this non-\ndegeneratehyperboliccasethe energydecayssub-exponentially with derivativeloss;\nin [3], it is shown that the sub-exponential decay rate is sharp. Base d on the sharp\npolynomial loss in local smoothing and resolvent estimates in [12], The orem 1.1\nshows that for the degenerate hyperbolic periodic geodesic, we ha ve the following\nestimates.\nTheorem 5.3. LetXbe as above, and suppose a(x)controls Xgeometrically\noutside a sufficiently small neighbourhood U⊃γ, so that N={γ}. Then\n/ba∇dbl(h2∆g+iha−z)−1u/ba∇dbl/lessorequalslantCh−2m/(m+1)/ba∇dblu/ba∇dbl\nforz∈[1−δ,1+δ]+i[−c0,c0]h2m/(m+1).\nAs in the previous subsection, we deduce from this resolvent estima te an energy\ndecay estimate for solutions to the damped wave equation.\nCorollary 5.4. Assume the hypotheses of Theorem 5.3 hold, and let ube a solution\nto(1.2)withu0= 0, andu1∈Hsfor some s∈(0,2]. Then there exists a constant\nC=Cs>0such that\nE(u,t)/lessorequalslantC\ntm+1\nm−1\n(log(2+ t))3(m+1)2\n2(m−1)2\n−s\n/ba∇dblu1/ba∇dbl2\nHs.12 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\n5.3.Hyperbolic trapped set with small topological pressure. In this sec-\ntion, we assume that the trapped set Nhas a hyperbolic structure, and that the\ntopological pressure of half the unstable Jacobian on the trapped set is negative.\nRoughly, this means that the set Nis rather thin, or filamentary: in dimension\n2, this is for instance equivalent to require that Nhas Hausdorff dimension <2.\nThe simplest case to have in mind is a single, closed hyperbolic orbit. We t hen\nbuild on [25] to get resolvent estimates near the trapped set, which we extend to\nthe global manifold with our different methods.\nWe briefly recall here the above dynamical notions. By definition, th e hyperbol-\nicity ofN ⊂S∗Xmeans that for any ρ∈ N, the tangent space TρNsplits into\nflow,stableandunstable subspaces\nTρN=RHp⊕Es\nρ⊕Eu\nρ.\nIfXis of dimension d, the spaces Es\nρandEu\nρared−1 dimensional, and are\npreserved under the flow map:\n∀t∈R, detHp(Es\nρ) =Es\netHp(ρ), detHp(Eu\nρ) =Eu\netHp(ρ).\nMoreover, there exist C,λ >0 such that\ni)/ba∇dbldetHp(v)/ba∇dbl/lessorequalslantCe−λt/ba∇dblv/ba∇dbl,for allv∈Es\nρ, t/greaterorequalslant0\nii)/ba∇dblde−tHp(v)/ba∇dbl/lessorequalslantCe−λt/ba∇dblv/ba∇dbl,for allv∈Eu\nρ, t/greaterorequalslant0. (5.3)\nOne can show that there exist a metric on T∗Xcall theadapted metric , for which\none can take C= 1 in the preceding equations.\nThe above properties allow us to define the unstable Jacobian. The a dapted\nmetricon T∗XinducesavolumeformΩ ρonanyddimensionalsubspaceof T(T∗\nρX).\nUsing Ω ρ, we can define the unstable Jacobian at ρfor time t. Let us define the\nweak-stable and weak-unstable subspaces at ρby\nEs,0\nρ=Es\nρ⊕RHp, Eu,0\nρ=Eu\nρ⊕RHp.\nWe set\nJu\nt(ρ) = detde−tHp|Eu,0\netHp(ρ)=Ωρ(de−tHpv1∧···∧de−tHpvd)\nΩetHp(ρ)(v1∧···∧vd), Ju(ρ) :=Ju\n1(ρ),\nwhere (v1,...,v d) can be any basis of Eu,0\netHp(ρ). While we do not necessarily have\nJu(ρ)<1, it is true that Ju\nt(ρ) decays exponentially as t→+∞.\nWe denote by Pr Nthe topological pressure functional on the closed, invariant\nsetN. We briefly recall a definition, see [35], [25] for more material. If fis a\ncontinuous function on N,nan integer and ǫ >0, define\nZn,ǫ(f) = sup\nS\n\n/summationdisplay\nρ∈Sexpn−1/summationdisplay\nk=0f◦ekHp(ρ)\n\n\nwherethesupremumistakenoverallthe( ǫ,n)separatedsubsets S. Thetopological\npressure of fonNis then\nPrN(f) := lim\nǫ→0limsup\nn→∞1\nnlogZn,ǫ(f).FROM RESOLVENT ESTIMATES TO DAMPED WAVES 13\nOur main assumption here is that the topological pressure of1\n2logJuonNis\nnegative, namely:\nPrN(1\n2logJu)<0.\nFor some δ >0 small enough, this imply the following resolvent estimate:\n(5.4) ∀z∈[1−δ,1+δ],/ba∇dbl(h2∆g+ia−z)−1/ba∇dbl/lessorequalslantC|logh|\nh\nThis estimate is already contained in [25], modulo two minor simplifications\nin our case: the manifold is compact, and infinity is replaced with the ab sorbing\npotentiali a, whichcontroleverythingoutsidethe trappedset –[15]showsex plicitly\nthat the estimate, with the more complicated geometry at infinity, o f [25] implies\nthe slightly simpler complex absorption result. Using Theorem 1.1, we im mediately\ndeduce the following result:\nTheorem 5.5. LetXbe a compact manifold, and suppose that a/greaterorequalslant0controlsX\nexcept on N, which is assumed to be hyperbolic with the property that\nPrN(1\n2logJu)<0.\nForδ >0small enough, there is h0andc0>0such that for h/lessorequalslanth0andz∈\n[1−δ,1+δ]+i[−c0,c0]h\n|logh|we have\n/ba∇dbl(h2∆g+iha−z)−1/ba∇dbl/lessorequalslantC|logh|\nh.\nIn particular, there is no spectrum near the real axis in a region of s izeh/|logh|.\nAs the resolvent estimate is the same order as that in Theorem 5.1, w e deduce the\nsame energy decay estimates as in Corollary 5.2.\n6.From resolvent estimates to the damped wave equation and\nenergy decay\nIn this section, we show how to move from a high energy resolvent es timate to\nan energy decay estimate for the damped wave equation, proving C orollaries 5.2\nand 5.4. To estimate the energy decay for the damped wave equatio n, as usual we\nrewrite it as a first-order evolution problem : if u= (u,∂tu) one can write (1.2) as\n(6.1) ∂tu= iBu,B=/parenleftbigg\n0−iId\ni∆gia/parenrightbigg\n.\nThe evolution group eitBmaps initial data ( u0,u1)∈H:=H1(X)×H0(X) to a\nsolution ( u,∂tu) of (6.1) where usolves (1.2). For s >0, define\n/ba∇dblu/ba∇dbls:=/ba∇dblu0/ba∇dblH1+s(X)+/ba∇dblu1/ba∇dblHs(X)\nIt is not hard to see that if we can prove\n(6.2)/vextenddouble/vextenddoubleeitB(1−iB)−s/vextenddouble/vextenddouble2\nL2(X)→L2(X)/lessorequalslantf(t)\nthen we can deduce a decay rate for the energy :\nE(u,t)/lessorequalslantf(t)/ba∇dblu/ba∇dbl2\ns\nIt turns out that we can obtain bounds such as (6.2) if we have estim ates on the\nhigh-frequency resolvent ( λ−B)−1,|λ| → ∞.14 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\nTo see this, we recall the following setup from [9]. Now suppose ( λ− B)−1\ncontinues holomorphically to a neighbourhood of the region\nΩ =/braceleftbigg\nλ∈C:|Imλ|/lessorequalslant/braceleftbiggC1, |Reλ|/lessorequalslantC2\nP(|Reλ|),|Reλ|/greaterorequalslantC2,/bracerightbigg\n,\nwhereP(|Reλ|)>0 and is monotone decreasing (or constant) as |Reλ| → ∞,\nP(C2) =C1, and assume for simplicity that ∂Ω is smooth. Assume\n/ba∇dbl(λ−B)−1/ba∇dblH→H/lessorequalslantG(|Reλ|) (6.3)\nforλ∈Ω, where G(|Reλ|) =O(|Reλ|N) for some N/greaterorequalslant0.\nTheorem 6.1. Suppose Bsatisfies all the assumptions above, and let k∈N,k >\nN+1. Then for any F(t)>0, monotone increasing, satisfying\nF(t)(k+1)/2/lessorequalslantexp(tP(F(t))), (6.4)\nthere is a constant C >0such that/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeitB\n(1−iB)k/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH→H/lessorequalslantCF(t)−k/2. (6.5)\nIn all cases considered in this paper, we have semiclassical resolven t estimates\n/ba∇dbl(h2∆g+i√zha−z)−1/ba∇dblL2→L2/lessorequalslantα(h)\nh, z∼1+ih/α(h),\nIf we rescale\nτ2=z\nh2,\nthen our resolvent estimates become\n/ba∇dbl(∆g+iτa−τ2)−1/ba∇dblL2→L2/lessorequalslantα(|τ|−1)\n|τ|.\nfor Imτ∼(α(|Reτ|−1))−1. By interpolation, this implies for 0 /lessorequalslantj/lessorequalslant2,\n/ba∇dbl(∆g+iτa−τ2)−1/ba∇dblHs→Hs+j/lessorequalslant|τ|j−1α(|τ|−1).\nHence, with Bas above and H=H1×H0, a simple calculation yields\n/ba∇dbl(λ−B)−1/ba∇dblH→H/lessorequalslantα(|λ|−1).\nFor Corollary 5.2, we take α(|λ|−1) = log(2+ |λ|). Then k= 2 suffices, P(r) =\nlog−1(r), and hence we may take\nF(t) = et1/2/C.\nThis recovers the endpoint estimate s= 2. To get the intermediate estimates for\ns∈(0,2) we interpolate with the trivial estimate\nE(u,t)/lessorequalslantE(u,0).\nFor Corollary 5.4, we have α(|λ|−1) =|λ|(m−1)/(m+1),N= (m−1)/(m+1)<1,\nso thatk= 2, and P(r) =r(1−m)/(m+1). We try\nF(t) =ts\nlogq(t),\nand insist\nt3s/2log−3q/2(t)/lessorequalslantexp(tts(1−m)/(m+1)logq(m−1)/(m+1)(t)),FROM RESOLVENT ESTIMATES TO DAMPED WAVES 15\nwhich is satisfied if\ns=m+1\n(m−1)\nand\nq=3(m+1)2\n2(m−1)2.\nAgain interpolating with the trivial estimate proves the Corollary.\nReferences\n[1] N. Anantharaman. Spectral deviations for the damped wav e equation. G.A.F.A. , 20:593–\n626, 2010.\n[2] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient condit ions for the observation, con-\ntrol and stabilization of waves from the boundary. SIAM J. Control and Optimization ,\n30(5):1024–1065, 1992.\n[3] N. Burq and H. Christianson. Imperfect geometric contro l and overdamping for the damped\nwave equation. In preparation.\n[4] N. Burq and M. Zworski, Geometric control in the presence of a black box, Journal of the\nA.M.S.17(2):443–471, 2004.\n[5] F. Cardoso, G. Popov, and G. Vodev. 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Spectra and pseudospectra, the behaviour of non-normal\nmatrices and operators . Princeton University Press, 2005.\n[19] N. Fenichel. Persistence and smoothness of invariant m anifolds for flow. Indiana Univ. Math.\nJ.21 No. 3 (1972), 193–226.\n[20] M.W. Hirsch, C.C. Pugh and M. Shub, Invariant manifolds Lecture Notes in Mathematics,\nVol. 583. Springer-Verlag, Berlin-New York, 1977.\n[21] M.Hitrik, Eigenfrequencies and expansions fordamped wave equations. Methods Appl. Anal.\n10 (2003), no. 4, 543–564.\n[22] L. H¨ ormander. On the existence and the regularity of so lutions of linear pseudo-differential\nequations. Enseignement Math. (2)17(1971), 99–163.16 HANS CHRISTIANSON, EMMANUEL SCHENCK, ANDR ´AS VASY, AND JARED WUNSCH\n[23] G. Lebeau, ´Equation des ondes amorties , volume 19 of Algebraic and geometric methods in\nmathematical physics . Kluwer Acad. Publ., Dordrecht, 1993.\n[24] S. Nonnenmacher. Spectral theory of damped quantum sys tems.Preprint, arXiv:1109.0930 ,\n2011.\n[25] S. Nonnenmacher and M. Zworski, Quantum decay rates in c haotic scattering. Acta Math. ,\n203(2):149–233, 2009.\n[26] J. Rauch and M. Taylor. Decay of solutions to nondissipa tive hyperbolic systems on compact\nmanifolds. Comm. in Pure and Appl. Anal. , 28:501–523, 1975.\n[27] E. Schenck. Energy decay for the damped wave equation un der a pressure condition. Comm.\nMath. Phys. , 300(2):375–410, 2010.\n[28] E. Schenck. Exponential stabilization without geomet ric control. Math. Res. Lett. ,\n18(2):379–388, 2011.\n[29] J.Sj¨ ostrand.Some resultsonnonselfadjointoperato rs: asurvey. Further progress in analysis ,\n45–74, World Sci. Publ., Hackensack, NJ, 2009.\n[30] J. Sj¨ ostrand. Asymptotic distribution of eigenfrequ encies for damped wave equations. Publ.\nRes. Inst. Math. Sci. , 36(5):573–611, 2000.\n[31] J. Sj¨ ostrand, Johannes and M. Zworski, Complex scaling and the distribution of scattering\npoles, J. Amer. Math. Soc. 4 (1991), no. 4, 729–769.\n[32] J. Sj¨ ostrand and M. Zworski. Quantum monodromy and sem i-classical trace formulæ. J.\nMath. Pures Appl. (9) 81, 1–33, 2002.\n[33] A. Vasy. Microlocal analysis of asymptotically hyperb olic and Kerr-de Sitter spaces.\nPreprint, arXiv:1012.4391 , 2010. With an appendix by S. Dyatlov.\n[34] A. Vasy and M. Zworski. Semiclassical estimates in asym ptotically Euclidean scattering.\nComm. Math. Phys. 212 (2000), no. 1, 205–217.\n[35] P. Walters. An introduction to ergodic theory , Springer, 1982.\n[36] J. Wunsch and M. Zworski, Resolvent estimates for normally hyperbolic trapped sets , Ann.\nHenri Poincar´ e, 12(2011), 1349–1385.\n[37] M. Zworski, Semiclassical analysis , AMS Graduate Studies in Mathematics, 2012.\nDepartment of Mathematics, University of North Carolina\nD´epartement de Math ´ematiques, Universit ´e Paris 13\nDepartment of Mathematics, Stanford University\nDepartment of Mathematics, Northwestern University\nE-mail address :hans@unc.edu\nE-mail address :schenck@math.univ-paris13.fr\nE-mail address :andras@math.stanford.edu\nE-mail address :jwunsch@math.northwestern.edu"    },    {        "title": "1703.00172v1.Behaviors_of_the_energy_of_solutions_of_two_coupled_wave_equations_with_nonlinear_damping_on_a_compact_manifold_with_boundary.pdf",        "content": "arXiv:1703.00172v1  [math.AP]  1 Mar 2017BEHAVIORS OF THE ENERGY OF SOLUTIONS OF TWO COUPLED\nWAVE EQUATIONS WITH NONLINEAR DAMPING ON A COMPACT\nMANIFOLD WITH BOUNDARY.\nM. DAOULATLI\nAbstract. In this paper we study the behaviors of the the energy of solut ions of coupled\nwave equations on a compact manifold with boundary in the cas e of indirect nonlinear\ndamping. Onlyone ofthetwoequationsis directlydampedbya localized nonlineardamping\nterm. Under geometric conditions on both the coupling and th e damping regions we prove\nthat the rate of decay of the energy of smooth solutions of the system is determined from\na first order differential equation .\n1.Introduction and Statement of the results\nLet(Ω,g0)beaC∞compactconnectedn-dimensionalRiemannianmanifoldwith boundary\nΓ.Wedenoteby∆theLaplace-Beltrami operatoronΩforthemetr icg0.Weconsiderasystem\nof coupled wave equations with nonlinear damping\n\n\n∂2\ntu−∆u+b(x)v+a(x)g(∂tu) = 0 in R∗\n+×Ω\n∂2\ntv−∆v+b(x)u= 0 inR∗\n+×Ω\nu=v= 0 onR∗\n+×Γ\n(u(0,x),∂tu(0,x)) = (u0,u1) and (v(0,x),∂tv(0,x)) = (v0,v1) in Ω,(1.1)\nwhereg:R−→Ris a continuous, monotone increasing function, g(0) = 0. In addition we\nassume that\ng(y)y≤M0y2,|y|<1\nmy2≤g(y)y≤My2,|y| ≥1\n/ba∇dblg′/ba∇dblL∞≤M1,\nfor some positive real numbers M0, m, M andM1. Inthis paper, we deal with real solutions,\nthe general case can be treated in the same way. With the syste m above we associate the\nenergy functional given by\nEu,v(t) =1\n2/integraldisplay\nΩ|∇u(t,x)|2+|∇v(t,x)|2+|∂tu(t,x)|2+|∂tv(t,x)|2dx\n+/integraldisplay\nΩb(x)u(t,x)v(t,x)dx.(1.2)\nWe assume that aandbare two nonnegative smooth functions such that\n/ba∇dblb/ba∇dbl∞≤1−δ\nλ2, (1.3)\nDate: January 20, 2018.\n2000Mathematics Subject Classification. Primary: 35L05, 35B35; Secondary: 35B40, 93B07 .\nKey words and phrases. Coupled wave, Energy decay, Stabilization, nonlinear damp ing.\n12 M. DAOULATLI\nfor someδ>0,whereλis the Poincar´ e’s constant on Ω .Under these assumptions we have\nEu,v(0) =1\n2/integraldisplay\nΩ|∇u0(x)|2+|∇v0(x)|2+|u1(x)|2+|v1(x)|2dx+/integraldisplay\nΩb(x)u0(x)v0(x)dx\n≥δ\n2/integraldisplay\nΩ|∇u0(x)|2+|∇v0(x)|2+|u1(x)|2+|v1(x)|2dx,\n(1.4)\nfor all (u0,v0,u1,v1)∈ H=H1\n0(Ω)×H1\n0(Ω)×L2(Ω)×L2(Ω).\nThe nonlinear evolution equation (1 .1) can be rewritten under the form\n/braceleftiggd\ndtU+AU+BU= 0\nU(0) =U0∈ H(1.5)\nwhere\nU=\nu\nv\n∂tu\n∂tv\n,U0=\nu0\nv0\nu1\nv1\n,\nand the unbounded operator AonHis defined by\nA=\n0 0 −Id0\n0 0 0 −Id\n−∆b0 0\nb−∆ 0 0\n\nwith domain\nD(A) ={U∈ H;AU∈ H}\n=/parenleftbig\nH1\n0(Ω)∩H2(Ω)/parenrightbig\n×/parenleftbig\nH1\n0(Ω)∩H2(Ω)/parenrightbig\n×H1\n0(Ω)×H1\n0(Ω),\nand\nBU=\n0\n0\na(x)g(∂tu)\n0\n\nUnder our assumptions and from the nonlinear semi-group the ory (see for example [5]), we\ncan infer that for U0∈ H,the problem (1 .5) admits a unique solution U∈C0(R+,H).\nMoreover we have the following energy estimate\nEu,v(t)−Eu,v(0) =−/integraldisplayt\n0/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds (1.6)\nfor allt≥0.In addition, since g′∈L∞(R),then ifU0∈D(A),we haveU∈C(R+,D(A))\nand\nE∂tu,∂tv(t)−E∂tu,∂tv(0) =−/integraldisplayt\n0/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds. (1.7)\nThe systems like (1 .1) appear in many physical situations. Indirect damping of r eversible\nsystems occurs in several applications in engineering and m echanics. In general it is impos-\nsible or too expansive to damp all the components of the state , so it is important to study\nstabilization properties of coupled systems with a reduced number of feedbacks.\nThe case of a linear damping and constant coupling b in (1 .1) has already been treated\nin [3]. They showed that the System (1 .1) cannot be exponentially stable and that theSTABILIZATION OF COUPLED WAVE 3\nenergy decays polynomially. In [2] Alabau et al generalized these results to cases for which\nthe coupling b=b(x) and the damping term a=a(x) satisfy the Piecewise Multipliers\nGeometric Condition (PMGC) [2]. This geometric assumption is a generalization of the\nusual multiplier geometric condition (or Γ-condition) of [ ?, 16] and is much more restrictive\nthan the sharp Geometric Control Condition (GCC). In [1] Ala bau et al generalized this\nresult and they proved that the system (1 .1) is polynomially stable when the regions {a>0}\nand{b>0}both satisfy the Geometric Control Condition and the coupli ng term satisfies a\nsmallness assumption. This result has generalized by Aloui et al [7], by assuming a more\nnatural smallness condition on the infinity norm of the coupl ing termb.Finally we quote the\nresult of Fu [13] in which he shows the logarithmic decay prop erty without any geometric\nconditions on the effective damping domain.\nThe problem of the indirect nonlinear damping has been studi ed by Alabau et Al [5] when\nthe system is coupled by the velocity. In this case they show t hat the energy of these kinds\nof system decays as fast as that of the corresponding scalar n onlinearly damped equation.\nHence, the coupling through velocities allows a full transm ission of the damping effects. To\nour knowledge no results seems to be known in the case of indir ect nonlinear damping for a\ncoupled system coupled in displacements.\nThegoal ofthispaperistodeterminetherateofdecay ofthee nergyofcoupledwavesystem\nwith indirect nonlinear damping and coupled in displacemen ts. More precisely, we prove,\nunder some geometric conditions on the localized damping do main and the localized coupling\ndomain, that the rate of decay of the energy is determined fro m a first order differential\nequation . In addition, we obtain that if the behavior of the d amping is close to the linear\ncase, then the linear and the nonlinear case has the same rate of decay. In the other case\nwe find that the rate of decay of the coupled system is close to t he one obtained for a single\ndamped wave equation.\nThe optimality of our results is a open questions. Lower ener gy estimates have been\nestablished in [4, 5] and [6] for scalar one-dimensional wav e equations, scalar Petrowsky\nequations in two-dimensions and one-dimensional wave syst ems coupled by velocities. These\nresults can beextended to the case of one-dimensional wave s ystems coupled by displacement.\nInourcase we obtain aquasi-optimal energy decay formulawh enthebehavior of thedamping\nis not close to the linear one.\nA natural necessary and sufficient condition to obtain contro llability for wave equations is\nto assume that the control set satisfies the Geometric Contro l Condition (GCC) defined in\n[8, 18]. For a subset ωof Ω andT >0, we shall say that ( ω,T) satisfies GCC if every geodesic\ntraveling atspeedoneinΩmeets ωinatimet<T. Wesaythat ωsatisfiesGCCifthereexists\nT >0 such that ( ω,T) satisfies GCC. We also set Tω=inf{T >0;(ω,T) satisfiesGCC}.\nWe denote by ω={a(x)>0}the control set and by O={b(x)>0}the coupling set.\nAssumption (A1): :Unique continuation property:\nThere exists T0>0,such that the only solution of the system\n\n\n∂2\ntu1−∆u1+b(x)u2= 0 in (0 ,T0)×Ω\n∂2\ntu2−∆u2+b(x)u1= 0 in (0 ,T0)×Ω\nu1=u2= 0 on (0,T0)×Γ\na(x)u1= 0 on (0,T0)×Ω\nu1∈H1((0,T0)×Ω) andu2∈L2((0,T0)×Ω),(1.8)\nis the null one u1=u2= 0.4 M. DAOULATLI\nNotethattheuniquecontinuationassumptionaboveisvalid ifweassumethat ω∩Osatisfies\nthe GCC (see [7]). Also according to Alabau et al. [1, Proposi tion 4.7] we have the following\nresult: We assume that ωandOsatisfy the GCC, then if /ba∇dblb/ba∇dbl∞≤min/parenleftbigg\n1\n5λ2,1\n50λ√\nCTω/parenrightbigg\n,there\nexistsT∗≥max(Tω,TO) such that if T0> T∗then the only solution of the system (1 .8) is\nthe null one.\nIn order to characterize decay rates for the energy, we need t o introduce several special\nfunctions, whichinturnwilldependonthegrowthof gneartheorigin. Accordingto[14]there\nexists a concave continuous, strictly increasing function h0, linear at infinity with h0(0) = 0\nsuch that\nh0(g(s)s)≥ǫ0/parenleftig\n|s|2+|g(s)|2/parenrightig\n,|s| ≤1, (1.9)\nwhereǫ0is a positive constant .We set\nh(s) =ma(Ω)h0/parenleftig\ns\nma(Ω)/parenrightig\n,wherema=a(x)dx. (1.10)\nAssumption (A2):We assume that there exists 0 < r0≤1 such that the function\nh−1∈C3((0,r0]) and strictly convex. In addition we suppose that\nlim\ns→0h−1(s) = lim\ns→0/parenleftbig\nh−1/parenrightbig′(s) = lim\ns→0s/parenleftbig\nh−1/parenrightbig′′(s) = lim\ns→0s2/parenleftbig\nh−1/parenrightbig′′′(s) = 0, (1.11)\nand there exist β >1 andα0>0,such that\nlim\ns→∞s/parenleftbig\nh−1/parenrightbig′/parenleftbig\n1/sβ/parenrightbig\n=α0,\n/parenleftbig\nh−1/parenrightbig′(s)≤βs/parenleftbig\nh−1/parenrightbig′′(s),for alls∈[0,r0],/parenleftbig\nβ2−β/parenrightbig\ns/parenleftbig\nh−1/parenrightbig′′(s)+β2s2/parenleftbig\nh−1/parenrightbig′′′(s)≥0,for alls∈[0,r0].(1.12)\nMoreover, we assume that if βs/parenleftbig\nh−1/parenrightbig′′(s)−/parenleftbig\nh−1/parenrightbig′(s)>0,for alls∈(0,r0],then\nthere exists α1>0,such that\n(h−1)′(s)/parenleftBig\n(β2−β)s(h−1)′′(s)+β2s2(h−1)′′′(s)/parenrightBig\nβs(h−1)′′(s)−(h−1)′(s)≤α1,for alls∈[0,r0]. (1.13)\nWe know that in the case of linear damping we have\nEu,v(t)≤c\nt/summationtext1\ni=0E∂i\ntu,∂i\ntv(0), t>0,\nU0= (u0,v0,u1,v1)∈/parenleftbig\nH1\n0(Ω)∩H2(Ω)/parenrightbig2×/parenleftbig\nH1\n0(Ω)/parenrightbig2,\nso we cannot expect to obtain a better rate of decay in the case of nonlinear damping. More\nprecisely we have the following result.\nTheorem 1. We suppose that aandbare two smooth non-negative functions and the con-\nditions (1.3) and the assumption A2 hold. In addition, we assu me thatωandOsatisfy the\nGCC and the assumption A1 holds .The solution U(t) = (u(t),v(t),∂tu(t),∂tv(t))of the\nsystem(1.1)then satisfies\nEu,v(t)≤C/parenleftig\n1+/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig\n(ϕ(t))−1, t>0,\nU0= (u0,v0,u1,v1)∈/parenleftbig\nH1\n0(Ω)∩H2(Ω)/parenrightbig2×/parenleftbig\nH1\n0(Ω)/parenrightbig2,(1.14)STABILIZATION OF COUPLED WAVE 5\nwhereCis positive constant and ϕis a solution of the following ODE\ndϕ\ndt−ǫ0\n2C1ϕ/parenleftbig\nh−1/parenrightbig′/parenleftbig\n1/ϕβ/parenrightbig\n= 0,0<ϕ−β(0)≤r0such that\n/parenleftbig\nh−1/parenrightbig′/parenleftbig\nϕ−β(0)/parenrightbig\n<inf/parenleftig\n2C1δ\nǫ0/parenleftbig\n8CT+2λ2+1/parenrightbig−1,1\nǫ0/parenleftbig1\nm+M2/parenrightbig−1/parenrightig\n,(1.15)\nwhereC1=C(T,/ba∇dblb/ba∇dbl∞,/ba∇dbla/ba∇dbl∞). Moreover,\nforU0∈ H,lim\nt→∞Eu,v(t) = 0.(1.16)\nRemark 1. The smallness condition on the infinity norm of bis required to ensure that the\nonly solution of the system\n\n−∆u+b(x)v= 0\n−∆v+b(x)u= 0\nu=v= 0onΓ(1.17)\nis the null one.\nRemark 2. We note that θ(t) =1\nϕ(t)is a solution of the following ODE\ndθ\ndt+ǫ0\n2C1θ/parenleftbig\nh−1/parenrightbig′/parenleftbig\nθβ/parenrightbig\n= 0. (1.18)\nTo prove our result it is sufficient to show the integrability o fϕ′Eu,von (0,∞). For this\npurpose we show an estimate on a functional X(t) which is equivalent to the weighted energy\nfunctional (See [11] for similar idea). Also we prove a weigh ted observability estimate for the\nwave equation with a potential. In addition, we use the uniqu e continuation hypotheses (A1)\nto prove a weak observability estimate of the weighted L2-norm of the solution.\n1.1.Some examples of decay rates and lower energy estimates. We give some\nexamples of feedback growths together with the resulting en ergy decay rate when apply-\ning our results. For clarity of exposition we will deal with t he damping which satisfies\nstrict bounds, i.e. by saying g(s)s≃f(s) we will mean there are constants m,Mso that\nmf(s)≤g(s)s≤Mf(s). In the sequel ˜Cdenotes a generic positive constant which is inde-\npendents of the energy of the initial data and setting E0= 1+/summationtext1\ni=0E∂i\ntu,∂i\ntv(0). Below we\nassume that ϕ(0) verifies the condition (1 .15).\nFirst we give an explicit upper bound of solutions of the ordi nary differential equation\n(1.18).\nLemma 1. We assume that there exists 0<r0≤1such that the function h−1∈C2((0,r0]),\nmonotone increasing and strictly convex. In addition we supp ose that there exists α>0,such\nthat/parenleftbig\nh−1/parenrightbig′(s)≤αs/parenleftbig\nh−1/parenrightbig′′(s),for alls∈(0,r0]. (1.19)\nLetθbe a solution of the following ODE\ndθ\ndt+Cθ/parenleftbig\nh−1/parenrightbig′/parenleftig\nθβ/parenrightig\n= 0,such that 0<θ(0)≤r0,\nwhereCis a positive constant and β≥1. We have\nθ(t)≤/parenleftbigg/parenleftig/parenleftbig\nh−1/parenrightbig′/parenrightig−1/parenleftig\nα/βC\nt+k0/parenrightig/parenrightbigg1/β\n,for allt≥0,6 M. DAOULATLI\nwhere\nα/βC\n(h−1)′(r0)≤k0≤α/βC\n(h−1)′/parenleftbig\nθβ(0)/parenrightbig. (1.20)\nProof.Let\nψ(t) =/parenleftbigg/parenleftig/parenleftbig\nh−1/parenrightbig′/parenrightig−1/parenleftbiggα/βC\nt+k0/parenrightbigg/parenrightbigg1/β\n,for allt≥0.\nDirect computations and (1 .12), give\nψ′(t)≥ −α\nβψ(t)\nt+k0,for allt≥0.\nOn the other hand, it is easy to see that\nψ(t)/parenleftbig\nh−1/parenrightbig′/parenleftig\nψβ(t)/parenrightig\n=α\nβCψ(t)\nt+k0,for allt≥0.\nTherefore, using (1 .20), we conclude that\ndψ\ndt+Cψ/parenleftbig\nh−1/parenrightbig′/parenleftig\nψβ/parenrightig\n≥0, ψ(0)≥θ(0).\nThe desired result follows from [10, Lemma 1]. /square\nNow we give some examples.\nExample 1 (Linearly bounded case) .Supposeg(s)s≃s2. According to (1.9), auxiliary\nfunctionh0which may be defined as h0(y) = (cy)γwith1/2< γ < 1and for suitable\nconstantc>0. We use the ODE\ndϕ\ndt−ǫ0\n2C1ϕ/parenleftbig\nh−1\n0/parenrightbig′/parenleftig\n1\nma(Ω)ϕ−γ\n1−γ/parenrightig\n= 0.\nConsequently,\nEu,v(t)≤˜CE0\nt+1, t≥0.\nExample 2. Supposeg(s)s≃s2(ln(1/s))−p,0<|s|<1/2for somep >0. According to\n(1.9), auxiliary function h0which may be defined as h0(y) = (cy)γwith1/2<γ <1and for\nsuitable constant c>0. We use the ODE\ndϕ\ndt−ǫ0\n2C1ϕ/parenleftbig\nh−1\n0/parenrightbig′/parenleftig\n1\nma(Ω)ϕ−γ\n1−γ/parenrightig\n= 0.\nConsequently,\nEu,v(t)≤˜CE0\nt+1, t≥0.\nExample 3 (The Polynomial Case) .Supposeg(s)s≃ |s|p+1,0<|s|<1for somep >1.\nAccording to (1.9), auxiliary function h0which may be defined as h0(y) = (cy)2/(p+1)for\nsuitable constant c>0(determined by the coefficients in the polynomial bound on the damping\ng(s)). We use the ODE\ndϕ\ndt−ǫ0\n2C1ϕ/parenleftbig\nh−1\n0/parenrightbig′/parenleftig\n1\nma(Ω)ϕ−β/parenrightig\n= 0.\nConsequently,\nEu,v(t)≤˜CE0\n(t+1)2/β(p−1), t≥0,STABILIZATION OF COUPLED WAVE 7\nwhere\nβ >1ifp≥3\nβ=2\nP−1ifp<3.\nExample 4 (Exponential damping at the origin) .Assume:g(s) =s3e−1/s2,0<|s|<1.\nFirst we need to determine h0according to (1.9). Setting h0(g(y)y) =cy2, we see that\nh−1\n0(y) =/radicalbig\ny/c·g/parenleftig/radicalbig\ny/c/parenrightig\n=c−2y2exp(−c/y)\nWe use the ODE\ndϕ\ndt−ǫ0\n2C1ϕ/parenleftbig\nh−1\n0/parenrightbig′/parenleftig\n1\nma(Ω)ϕ−β/parenrightig\n= 0,\nto obtain\nEu,v(t)≤˜CE0\n(ln(t+2))1/β, t≥0,\nfor allβ >1.\nExample 5 (Exponential damping at the origin) .Assume:g(s) =s3e−e1/s2\n,0<|s|<1.\nFirst we need to determine h0according to (1.9). Setting h0(g(y)y) =cy2, we see that\nh−1\n0(y) =/radicalbig\ny/c·g/parenleftig/radicalbig\ny/c/parenrightig\n=c−2y2exp(−exp(c/y))\nWe use the ODE\ndϕ\ndt−ǫ0\n2C1ϕ/parenleftbig\nh−1\n0/parenrightbig′/parenleftig\n1\nma(Ω)ϕ−β/parenrightig\n= 0,\nto obtain\nEu,v(t)≤˜CE0\n(lnln(t+e2))1/β, t≥0,\nfor allβ >1.\nWe finish this part by giving a result on the lower estimate of t he energy of the one-\ndimensional coupled wave system.\nProposition 1. We suppose that Ω = (0,1)andgis a odd function. We set\nh−1(s) =g(√s)√s,fors≥0.\nWe assume that aandbare two smooth non-negative functions and the conditions (1 .3) and\nthe assumption A2 hold. In addition, we suppose that ωandOsatisfy the GCC and the\nassumption A1 holds .LetU(t) = (u(t),v(t),∂tu(t),∂tv(t))be the solution of the system\n(1.1),then there exists T0>0such that\nEu,v(t)≥/parenleftbigg\nψ(t)\n4√\nE∂tu,∂tv(0)/parenrightbigg2\n, t≥T0,\nU0= (u0,v0,u1,v1)∈/parenleftbig\nH1\n0(Ω)∩H2(Ω)/parenrightbig2×/parenleftbig\nH1\n0(Ω)/parenrightbig2,\nwhereψis a solution of the following ODE\ndψ\ndt+/ba∇dbla/ba∇dblL∞ψ/parenleftbig\nh−1/parenrightbig′(ψ) = 0,0<ψ(0)≤4/radicalig\nE∂tu,∂tv(0)Eu,v(T0).(1.21)8 M. DAOULATLI\nProof.We proceed as in [5] and using the fact that\nh−1(s)≤s/parenleftbig\nh−1/parenrightbig′(s),\nwe see that there exists T0>0 such that\nd/radicalbigEu,v\ndt+/ba∇dbla/ba∇dblL∞/radicalbig\nEu,v/parenleftbig\nh−1/parenrightbig′/parenleftbigg\n4/radicalig\nE∂tu,∂tv(0)/radicalbig\nEu,v/parenrightbigg\n≥0, t≥T0.\nSinceψis a solution of (1 .21),then using [10, Lemma 1] we conclude that\n/radicalig\nEu,v(t)≥ψ(t)\n4/radicalbig\nE∂tu,∂tv(0), t≥T0.\n/square\n2.Proof of Theorem 1\nFirst we give the following weighted observability estimat e for the wave equation with\npotential.\nProposition 2. Letγ,δ >0andχ∈L∞(Ω)satisfying\n•χ≥0or else,\n• /ba∇dblχ/ba∇dbl∞≤γ2−δ\nλ2.\nLetφbe a positive function in C2(R+)such that\nfor everyI⊂⊂R+,there existm,M >0\nm≤φ(s)≤M,for alls∈I.(2.1)\nIn addition, if φ′is not the null function, we assume that there exists a positi ve constant K\nsuch that\nsup\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′(t)\nφ′(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤K. (2.2)\nMoreover we suppose that the function\nt/ma√sto−→/vextendsingle/vextendsingle/vextendsingleφ′(t)\nφ(t)/vextendsingle/vextendsingle/vextendsingleis decreasing and lim\nt→+∞/vextendsingle/vextendsingle/vextendsingleφ′(t)\nφ(t)/vextendsingle/vextendsingle/vextendsingle= 0. (2.3)\nWe consider also ψnonnegative smooth function on Ωsuch that the set (V:={ψ(x)>0},T)\nsatisfies the GCC. Then there exists CT>0,such that for all (u0,u1)∈H1\n0(Ω)×L2(Ω),\nf∈L2\nloc/parenleftbig\nR+,L2(Ω)/parenrightbig\nand allt>0,the solution of\n\n\n∂2\ntu−γ2∆u+χ(x)u=finR∗\n+×Ω\nu= 0 onR∗\n+×Γ\n(u(0,x),∂tu(0,x)) = (u0,u1)inΩ(2.4)\nsatisfies with\nEu(t) =1\n2/integraldisplay\nΩγ2|∇u(t,x)|2+|∂tu(t,x)|2+χ(x)|u(t,x)|2dx,\nthe inequality\n/integraldisplayt+T\ntφ(s)Eu(s)ds≤CT/parenleftbigg/integraldisplayt+T\ntφ(s)/integraldisplay\nΩ/parenleftig\nψ(x)|∂tu(s,x)|2+|f(s,x)|2/parenrightig\ndxds/parenrightbigg\n.(2.5)STABILIZATION OF COUPLED WAVE 9\nProof.First we remark that\nEu(t)≃1\n2/integraldisplay\nΩ|∇u(t,x)|2+|∂tu(t,x)|2dx.\nTo prove the estimate (2 .5),we argue by contradiction. We assume that there exist a posit ive\nsequence (tn),a sequence of functions fn∈L2\nloc/parenleftbig\nR+,L2(Ω)/parenrightbig\nand a sequence ( un(t)) of\nsolutions of the system (2 .4) with initial data ( u0,n,u1,n)∈H1\n0(Ω)×L2(Ω),such that\n/integraltexttn+T\ntnφ(s)Eun(s)ds\n≥n/parenleftbigg/integraldisplaytn+T\ntnφ(s)/integraltext\nΩψ(x)|∂tun(s,x)|2+|fn(s,x)|2dxds/parenrightbigg\n.(2.6)\n1stcase:tn−→\nn→+∞∞.Setting\nλ2\nn=/integraltexttn+T\ntnφ(s)Eun(s)dsandvn(t) =1\nλn(φ(s))1/2un(t+tn),\nThereforevnis a solution of the following system\n\n\n∂2\ntvn−γ2∆vn+χ(x)vn=1\nλn(φ(tn+t))1/2fn(tn+t,x)+f1\nn(t,x) inR∗\n+×Ω\nvn= 0 onR∗\n+×Γ\n(vn(0,x),∂tvn(0,x)) = (vn,0,vn,1) in Ω\nwhere\nf1\nn(t,x) =1\n2φ′′(t+tn)(φ(t+tn))−1vn−1\n4/parenleftbig\nφ′(t+tn)/parenrightbig2φ(t+tn)−2vn\n+1\nλnφ′(t+tn)(φ(t+tn))−1/2∂tun.\nThanks to (2 .6),we get\n/integraldisplayT\n0/integraldisplay\nΩ|∇vn(s,x)|2+φ(s+tn)\nλ2\nn|∂tun(s+tn,x)|2dxds= 1,\n1\nλ2\nn/integraldisplayT\n0/integraldisplay\nΩφ(s+tn)/parenleftig\nψ(x)|∂tun(tn+s,x)|2+|fn(tn+s,x)|2/parenrightig\ndxds≤1\nn.(2.7)\nUsing Poincare’s inequality we deduce that\n/integraldisplayT\n0/integraldisplay\nΩ|vn(s,x)|2dxds≤λ2.\nUtilizing the first part of (2 .7) and the estimate above, we deduce that there exists\nα1>0,such that\n/integraldisplayT\n0Evn(s)ds≤α1.\nA combination of the first part of (2 .7),(2.2) and (2.3),gives\nlim\nn→∞/integraldisplayT\n0/integraldisplay\nΩ/vextendsingle/vextendsinglef1\nn(tn+s,x)/vextendsingle/vextendsingle2dxds= 0.10 M. DAOULATLI\nIt is easy to see that\n/integraldisplayT\n0/integraldisplay\nΩψ(x)|∂tvn(s,x)|2dxds\n≤/integraldisplayT\n0/integraldisplay\nΩ1\nλ2\nnφ(tn+s)ψ(x)|∂tun(tn+s,x)|2+/vextendsingle/vextendsingle/vextendsingleφ′(tn)\nφ(tn)/vextendsingle/vextendsingle/vextendsingle2\nψ(x)|vn(s,x)|2dxds\n≤/integraldisplayT\n0/integraldisplay\nΩ1\nλ2\nnφ(tn+s)ψ(x)|∂tun(tn+s,x)|2dxds+α2/vextendsingle/vextendsingle/vextendsingleφ′(tn)\nφ(tn)/vextendsingle/vextendsingle/vextendsingle2\n−→\nn→+∞0,\nnoting that in the last result we have used the second part of ( 2.7) and (2.3).\nOn the other hand, According to [7], we know that\n/integraldisplayT\n0/integraldisplay\nΩ|∇vn(s,x)|2+|∂tvn(s,x)|2dxds\n≤CT/integraldisplayT\n0/integraldisplay\nΩψ(x)|∂tvn(s,x)|2+1\nλ2\nnφ(tn+s)|fn(tn+s,x)|2+/vextendsingle/vextendsinglef1\nn(s,x)/vextendsingle/vextendsingle2dxds\nand the contradiction follows from the fact that the RHS of th e estimate above goes\nto zero as n goes to infinity and\n1 =/integraldisplayT\n0/integraldisplay\nΩ|∇vn(s,x)|2+φ(s+tn)\nλ2\nn|∂tun(s+tn,x)|2dxds\n≤2/integraldisplayT\n0/integraldisplay\nΩ|∇vn(s,x)|2+|∂tvn(s,x)|2dxds+α2/vextendsingle/vextendsingle/vextendsingleφ′(tn)\nφ(tn)/vextendsingle/vextendsingle/vextendsingle2\n.\n.\n2ndcase:The sequence ( tn) is bounded. Setting\nλ2\nn=/integraldisplaytn+T\ntnEun(s)dsandvn(t) =1\nλnun(t+tn).\nUsing the fact that the sequence ( tn) is bounded and (2 .1),we infer that there exist\nα0, α1>0,such that\n0<α0≤/integraldisplayT\n0Evn(s)ds≤α1,\nand/integraldisplayT\n0/integraldisplay\nΩψ(x)|∂tvn(s,x)|2+1\nλ2\nn|fn(s+tn,x)|2dxds−→\nn→+∞0.\nTo finish the proof we need to proceed as in the first case.\n/square\nThe result below is a week weighted observability inequalit y and we need it to control the\nL2norm of the solution.\nProposition 3. We assume that ωandOsatisfy the GCC and the assumption A1 holds.\nLetT >max(Tω,TO,T0)andα>0.Letφbe a positive function in C2(R+)such that\nfor everyI⊂⊂R+,there existm,M >0\nm≤φ(s)≤M,for alls∈I.(2.8)\nIn addition, if φ′is not the null function, we assume that there exists a positi ve constant K\nsuch that\nsup\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′(t)\nφ′(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤K. (2.9)STABILIZATION OF COUPLED WAVE 11\nMoreover we suppose that\nthe function t/ma√sto−→/vextendsingle/vextendsingle/vextendsingleφ′(t)\nφ(t)/vextendsingle/vextendsingle/vextendsingleis decreasing and lim\nt→+∞/vextendsingle/vextendsingle/vextendsingleφ′(t)\nφ(t)/vextendsingle/vextendsingle/vextendsingle= 0. (2.10)\nThen there exists CT,α>0,such that for all (u0,v0,u1,v1)∈ H,and allt>0,the solution\nof the system\n\n∂2\ntu−∆u+b(x)v+a(x)g(∂tu) = 0 inR∗\n+×Ω\n∂2\ntv−∆v+b(x)u= 0 inR∗\n+×Ω\nu=v= 0 onR∗\n+×Γ\n(u(0,x),∂tu(0,x)) = (u0,u1)and(v(0,x),∂tv(0,x)) = (v0,v1)inΩ(2.11)\nsatisfies the inequality\n/integraldisplayt+T\ntφ(s)/integraldisplay\nΩ/parenleftig\n|v(s,x)|2+|u(s,x)|2/parenrightig\ndxds\n≤CT,α/integraldisplayt+T\ntφ(s)/integraldisplay\nΩa(x)/parenleftig\n|g(∂tu(s,x))|2+|∂tu(s,x)|2/parenrightig\ndxds\n+α/integraldisplayt+T\ntφ(s)Eu,v(s)ds.(2.12)\nProof.To prove the estimate (2 .12),we argue by contradiction. We assume that there exist a\npositive sequence ( tn) and a sequence ( Un(t) = (un(t),vn(t),∂tun(t),∂tvn(t))) of solutions\nof the system (2 .11) with initial data ( u0,n,v0,n,u1,n,v1,n)∈ H,such that\n/integraltexttn+T\ntnφ(s)/integraldisplay\nΩ|vn(s,x)|2+|un(s,x)|2dxds\n≥n/integraldisplaytn+T\ntnφ(s)/integraldisplay\nΩa(x)/parenleftig\n|g(∂tun(s,x))|2+|∂tun(s,x)|2/parenrightig\ndxds+α/integraldisplaytn+T\ntnφ(s)Eun,vn(s)ds.\n(2.13)\n1stcase:The sequence ( tn) is bounded. Setting\nλ2\nn=/integraldisplaytn+T\ntn/integraldisplay\nΩ/parenleftig\n|vn(s,x)|2+|un(s,x)|2/parenrightig\ndxdsand\nVn(t) =/parenleftbig\nwn(t), yn(t), ∂twn(t), ∂tyn(t)/parenrightbig\n=1\nλnUn(t+tn),\nUsing the fact that the sequence ( tn) is bounded, (2 .8) and (2.13),we infer that there\nexistα0, α1, α2>0,such that\n/integraldisplayT\n0/integraldisplay\nΩ|wn(s,x)|2+|yn(s,x)|2dxds≥α0>0,\n/integraldisplayT\n0/integraldisplay\nΩa(x)/parenleftig\n|∂twn(s,x)|2+1\nλ2\nn|g(∂tun(tn+s,x))|2/parenrightig\ndxds≤α2\nn\nand/integraldisplayT\n0Ewn,yn(s)ds≤α1.\nTo finish the proof we use the unique continuation hypotheses (A1) and we proceed\nas in [7, Proof of lemma 7].\n2ndcase:tn−→\nn→+∞∞.Setting\nλ2\nn=/integraltexttn+T\ntnφ(s)/integraldisplay\nΩ/parenleftig\n|vn(s,x)|2+|un(s,x)|2/parenrightig\ndxdsand\nVn(t) =/parenleftbig\nwn(t), yn(t), ∂twn(t), ∂tyn(t)/parenrightbig\n=1\nλn(φ(t+tn))1/2Un(t+tn),12 M. DAOULATLI\nTherefore\n\n∂2\ntwn−∆wn+b(x)yn+1\nλna(x)(φ(tn+t))1/2g(∂tun) =fn(t,x) in R∗\n+×Ω\n∂2\ntyn−∆yn+b(x)wn=f1\nn(t,x) inR∗\n+×Ω\nwn=yn= 0 onR∗\n+×Γ\n(wn(0,x),∂twn(0,x)) = (wn,0,wn,1) and (yn(0,x),∂tyn(0,x)) = (yn,0,yn,1) in Ω\nwhere\nfn=1\n2φ′′(t+tn)(φ(t+tn))−1wn−1\n4/parenleftbig\nφ′(t+tn)/parenrightbig2φ(t+tn)−2wn\n+1\nλnφ′(t+tn)(φ(t+tn))−1/2∂tun,\nf1\nn=1\n2φ′′(t+tn)(φ(t+tn))−1yn−1\n4/parenleftbig\nφ′(t+tn)/parenrightbig2φ(t+tn)−2yn\n+1\nλnφ′(t+tn)(φ(t+tn))−1/2∂tvn.\nThanks to (2 .13) we get\n/integraldisplayT\n0/integraldisplay\nΩ|wn(s,x)|2+|yn(s,x)|2dxds= 1,\n1\nλ2\nn/integraldisplayT\n0/integraldisplay\nΩa(x)φ(tn+s)/parenleftig\n|∂tun(tn+s,x)|2+|g(∂tun(s+tn,x))|2/parenrightig\ndxds≤1\nn\nand/integraldisplayT\n0φ(s+tn)\nλ2\nnEun,vn(s)ds≤1/α.(2.14)\nNow using the estimates above, we deduce that there exist α1>0,such that\n/integraldisplayT\n0Ewn,yn(s)ds≤α1.\nUtilizing (2 .14),(2.9) and (2.10),we can show that\nlim\nn→∞/integraldisplayT\n0/integraldisplay\nΩ|fn(s,x)|2dxds= lim\nn→∞/integraldisplayT\n0/integraldisplay\nΩ/vextendsingle/vextendsinglef1\nn(s,x)/vextendsingle/vextendsingle2dxds= 0.\nOn the other hand, it is easy to see that\n/integraldisplayT\n0/integraldisplay\nΩψ(x)|∂tvn(s,x)|2dxds\n≤/integraldisplayT\n0/integraldisplay\nΩ1\nλ2\nnφ(tn+s)ψ(x)|∂tun(tn+s,x)|2+/vextendsingle/vextendsingle/vextendsingleφ′(tn)\nφ(tn)/vextendsingle/vextendsingle/vextendsingle2\nψ(x)|wn(s,x)|2dxds\n≤/integraldisplayT\n0/integraldisplay\nΩ1\nλ2\nnφ(tn+s)ψ(x)|∂tun(s,x)|2dxds+α2/vextendsingle/vextendsingle/vextendsingleφ′(tn)\nφ(tn)/vextendsingle/vextendsingle/vextendsingle2\n−→\nn→+∞0,\nnoting that in the last result we have used the second part of ( 2.14) and (2.10). To\nfinish the proof we use the unique continuation hypotheses (A 1) and we proceed as\nin [7, Proof of lemma 7].\n/square\nLemma 2. Letϕ∈C2(R+)and non-decreasing. Let ube a solution of the system (1.1)with\ninitial data U0= (u0,v0,u1,v1)∈D(A).We set\nX(t) =ϕ′(t)/integraldisplay\nΩu(t,x)∂tu(t,x)+v(t,x)∂tv(t,x)dx\n+k′\n1ϕ(t)/integraldisplay\nΩ∂2\ntu(t,x)∂tv(t,x)−∂2\ntv(t,x)∂tu(t,x)dx\n+ϕ(t)Eu,v(t)+kϕ′(t)E∂tu,∂tv(t),(2.15)STABILIZATION OF COUPLED WAVE 13\nwherek, Tandk1are positive constants. Then\nX(t+T)−X(t)+/integraldisplayt+T\ntϕ′(s)Eu,v(s)ds+/integraldisplayt+T\nt/integraldisplay\nΩa(x)ϕ(s)g(∂tu(s,x))∂tu(s,x)dxds\n+k/integraldisplayt+T\nt/integraldisplay\nΩa(x)ϕ′(s)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n≤2/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|∂tu(s,x)|2+|∂tv(s,x)|2dxds\n+/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)/parenleftig\n|g(∂tu(s,x))|2+|u(s,x)|2/parenrightig\ndxds\n+(kE∂tu,∂tv(0)+k0Eu,v(0))/integraldisplayt+T\nt|ϕ′′(s)|ds\n+k1/bracketleftbigg\nϕ′(t)/integraldisplay\nΩ∂2\ntu(s,x)∂tv(s,x)−∂2\ntv(s,x)∂tu(s,x)dx/bracketrightbiggs=t+T\ns=t.\n(2.16)\nProof.We differentiate X(t) with respect to t, we obtain\nX′(t) =ϕ′(t)/integraldisplay\nΩ|∂tu(t,x)|2+|∂tv(t,x)|2dx\n+ϕ′(t)/integraldisplay\nΩu(t,x)∂2\ntu(t,x)+v(t,x)∂2\ntv(t,x)dx\n+ϕ′′(t)/integraldisplay\nΩu(t,x)∂tu(t,x)+v(t,x)∂tv(t,x)dx\n+k1d\ndt/bracketleftbigg\nϕ′(t)/integraldisplay\nΩ∂2\ntu(t,x)∂tv(t,x)−∂2\ntv(t,x)∂tu(t,x)dx/bracketrightbigg\n+ϕ′(t)Eu,v(t)\n−ϕ(t)/integraldisplay\nΩa(x)g(∂tu(t,x))∂tu(t,x)dx−kϕ′(t)/integraldisplay\nΩa(x)g′(∂tu(t,x))/vextendsingle/vextendsingle∂2\ntu(t,x)/vextendsingle/vextendsingle2dx\n+kϕ′′(t)E∂tu,∂tv(t).\n(2.17)\nUsing the first and the second equations of (1 .1),we infer that\nϕ′(t)/integraldisplay\nΩ/parenleftbig\nu(t,x)∂2\ntu(t,x)+v(t,x)∂2\ntv(t,x)/parenrightbig\ndx\n=−ϕ′(t)/integraldisplay\nΩ|∇u(t,x)|2+|∇v(t,x)|2+a(x)g(∂tu(t,x))u(t,x)+2b(x)u(t,x)v(t,x)dx\n=−2ϕ′(t)Eu,v(t)+ϕ′(t)/integraldisplay\nΩ|∂tu(t,x)|2+|∂tv(t,x)|2dx−ϕ′(t)/integraldisplay\nΩa(x)g(∂tu(t,x))u(t,x)dx.\nThanks to Young’s inequality we get\nϕ′(t)/integraldisplay\nΩa(x)g(∂tu(t,x))u(t,x)dx≤ϕ′(t)\n2/integraldisplay\nΩa(x)/parenleftig\n|g(∂tu(t,x))|2+|u(t,x)|2/parenrightig\ndx.\nTo estimate the third term of the RHS of (2 .17),we use Poincare’s inequality and the fact\nthat the energy is decreasing\nϕ′′(t)/integraldisplay\nΩu(t,x)∂tu(t,x)+v(t,x)∂tv(t,x)dx≤k0|ϕ′′(s)|Eu,v(0)\nFor the last term of the RHS of (2 .17),we use the fact that\nE∂tu,∂tv(t)≤E∂tu,∂tv(0)14 M. DAOULATLI\nCombiningtheestimates above, makingsomearrangement and integrating theresultbetween\ntandt+T, we obtain (2 .16). /square\nLet\nH(x) =/braceleftbigg\nh−1(x) onR+\n∞ onR∗\n−.(2.18)\nNoting that according to [4], if h−1is a strictly convex C1function from [0 ,r0] toRsuch\nthath−1(0) =/parenleftbig\nh−1/parenrightbig′(0) = 0. Then the convex conjugate function of His defined by\nH∗(x) =x/parenleftig/parenleftbig\nh−1/parenrightbig′/parenrightig−1\n(x)−h−1/parenleftbigg/parenleftig/parenleftbig\nh−1/parenrightbig′/parenrightig−1\n(x)/parenrightbigg\n,on/bracketleftig\n0,/parenleftbig\nh−1/parenrightbig′(r0)/bracketrightig\n.(2.19)\nLemma 3. We assume that the assumption A2 holds. Let ϕbe a solution of the following\nODE\ndϕ\ndt−ǫ0\n2C1ϕ/parenleftbig\nh−1/parenrightbig′/parenleftbig\n1/ϕβ/parenrightbig\n= 0,0<ϕ−β(0)≤r0. (2.20)\nwhereǫ0andC1are positive constant. Then we have ϕis a concave strictly increasing\nfunction in C3(R+). In addition we have\n(1) lim\nt→∞ϕ(t) =∞.\n(2) lim\nt→∞ϕ′(t) =ǫ0α0\n2C1andlim\nt→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕ′′(t)\nϕ′(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\n(3)The function t/ma√sto−→/vextendsingle/vextendsingle/vextendsingleϕ′′(t)\nϕ′(t)/vextendsingle/vextendsingle/vextendsingleis decreasing.\n(4)Ifϕ′′is not the null function, then there exists K >0such that sup\nR+/vextendsingle/vextendsingle/vextendsingleϕ′′′(t)\nϕ′′(t)/vextendsingle/vextendsingle/vextendsingle≤K.\n(5)/integraldisplay∞\n0|ϕ′′(s)|ds=ϕ′(0)−ǫ0α0\n2C1.\n(6)ǫ0\n2C1/integraldisplay∞\n0ϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds≤1\nβ−1ϕ1−β(0).\nProof.Using the second part of (1 .12) and the fact that\n0<ϕ−β(t)≤ϕ−β(0)≤r0,for allt>0, (2.21)\nwe obtain\nϕ′′(t) =ǫ0\n2C1ϕ′(t)/parenleftig/parenleftbig\nh−1/parenrightbig′/parenleftbig\nϕ−β(t)/parenrightbig\n−βϕ−β(t)/parenleftbig\nh−1/parenrightbig′′/parenleftbig\nϕ−β(t)/parenrightbig/parenrightig\n≤0, (2.22)\nfor allt∈R+,which means that the function ϕis a concave on R+.\nIt is easy to see that the function ϕ∈C3(R+).\n(1) First we note that\nϕ−1(t) =2C1\nǫ0/integraldisplayt\nϕ(0)ds\ns(h−1)′(1/sβ).\nTherefore, using the fact that\ns/parenleftbig\nh−1/parenrightbig′/parenleftbig\n1/sβ/parenrightbig\n≤s/parenleftbig\nh−1/parenrightbig′/parenleftbig\n1/ϕβ(0)/parenrightbig\n,for alls∈[0,∞),\nwe deduce that\nlim\nt→∞ϕ−1(t) =∞,\nthus\nlim\nt→∞ϕ(t) =∞. (2.23)STABILIZATION OF COUPLED WAVE 15\n(2) Thanks to (2 .23) and (1.12) we get\nlim\nt→∞ϕ′(t) =ǫ0α0\n2C1. (2.24)\nDirect computations and using (1 .11),yield\nlim\nt→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕ′′(t)\nϕ′(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\n(3) From (2 .22),we deduce that\n/vextendsingle/vextendsingle/vextendsingleϕ′′(t)\nϕ′(t)/vextendsingle/vextendsingle/vextendsingle=ǫ0\n2C1/parenleftig\n−/parenleftbig\nh−1/parenrightbig′/parenleftbig\nϕ−β(t)/parenrightbig\n+βϕ−β(t)/parenleftbig\nh−1/parenrightbig′′/parenleftbig\nϕ−β(t)/parenrightbig/parenrightig\n.\nWe differentiate the estimate above and making some arrangeme nts, we obtain\nd\ndt/vextendsingle/vextendsingle/vextendsingleϕ′′(t)\nϕ′(t)/vextendsingle/vextendsingle/vextendsingle=−ǫ0ϕ′(t)\n2C1ϕβ+1(t)/parenleftig/parenleftbig\nβ2−β/parenrightbig/parenleftbig\nh−1/parenrightbig′′/parenleftbig\nϕ−β(t)/parenrightbig\n+β2ϕ−β(t)/parenleftbig\nh−1/parenrightbig′′′/parenleftbig\nϕ−β(t)/parenrightbig/parenrightig\n.\nFrom the estimate above, (2 .21) and (1.12),we see that the function t/ma√sto−→/vextendsingle/vextendsingle/vextendsingleϕ′′(t)\nϕ′(t)/vextendsingle/vextendsingle/vextendsingle\nis decreasing.\n(4) We differentiate the identity (2 .22) and making some arrangements, we obtain\nϕ′′′(t)\nϕ′′(t)=ǫ0\n2C1/parenleftig/parenleftbig\nh−1/parenrightbig′/parenleftbig\nϕ−β(t)/parenrightbig\n−βϕ−β(t)/parenleftbig\nh−1/parenrightbig′′/parenleftbig\nϕ−β(t)/parenrightbig/parenrightig\n+ǫ0(ϕ′(t))2\n2C1ϕ′′(t)ϕ(t)/parenleftig/parenleftbig\nβ2−β/parenrightbig\nϕ−β(t)/parenleftbig\nh−1/parenrightbig′′/parenleftbig\nϕ−β(t)/parenrightbig\n+β2ϕ−2β(t)/parenleftbig\nh−1/parenrightbig′′′/parenleftbig\nϕ−β(t)/parenrightbig/parenrightig\n.\nOn the other hand, from (2 .20) and (2.22),we infer that\n/vextendsingle/vextendsingle/vextendsingle(ϕ′(t))2\nϕ′′(t)ϕ(t)/vextendsingle/vextendsingle/vextendsingle=(h−1)′(ϕ−β(t))\nβϕ−β(t)(h−1)′′(ϕ−β(t))−(h−1)′(ϕ−β(t)).\nCombining the two estimates above, we see that/vextendsingle/vextendsingle/vextendsingleϕ′′′(t)\nϕ′′(t)/vextendsingle/vextendsingle/vextendsingle≤ǫ0\n2C1/parenleftig\nβϕ−β(t)/parenleftbig\nh−1/parenrightbig′′/parenleftbig\nϕ−β(t)/parenrightbig\n−/parenleftbig\nh−1/parenrightbig′/parenleftbig\nϕ−β(t)/parenrightbig/parenrightig\n+ǫ0(h−1)′(ϕ−β(t))\n2C1/parenleftbigg\n(β2−β)ϕ−β(t)(h−1)′′(ϕ−β(t))+β2ϕ−2β(t)(h−1)′′′(ϕ−β(t))\nβϕ−β(t)(h−1)′′(ϕ−β(t))−(h−1)′(ϕ−β(t))/parenrightbigg\n.\nSo from Assumption A2, we conclude that there exists K >0 such that\nsup\nR+/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕ′′′(t)\nϕ′′(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤K.\n(5) Using (2 .22) and (2.24),we obtain\n/integraldisplay∞\n0|ϕ′′(s)|ds=−/integraldisplay∞\n0ϕ′′(s)ds=ϕ′(0)−ǫ0α0\n2C1.\n(6) Thanks to (2 .19) and (2.20),we see that\nϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\n≤2C1\nǫ0ϕ′(s)/parenleftig/parenleftbig\nh−1/parenrightbig′/parenrightig−1/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\n=2C1\nǫ0ϕ′(s)\nϕβ(s),\ntherefore integrating the estimate above between zero and i nfinity and using (2 .23)\nand the fact that β >1 we obtain\nǫ0\n2C1/integraldisplay∞\n0ϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds≤1\nβ−1ϕ1−β(0).\n/square16 M. DAOULATLI\n2.1.Proof of Theorem 1. We assume that ωandOsatisfy the GCC and the assumption\nA1 holds.Let (u,v) be a solution of the system (1 .1) with initial data U0= (u0,v0,u1,v1)∈\nD(A).LetT >max(Tω,TO,T0).\nWe haveuis a solution of the nonhomogeneous wave equation with a loca lized nonlinear\ndamping and ( ω,T) satisfies the GCC. In addition, taking into account of lemma (3),we see\nthatϕ′satisfiestherequiredassumptionsofproposition(2) .Therefore, usingtheobservability\nestimate (2 .5) and (2.16),we deduce that\nX(t+T)−X(t)+/integraldisplayt+T\ntϕ′(s)Eu,v(s)ds+/integraldisplayt+T\nt/integraldisplay\nΩa(x)ϕ(s)g(∂tu(s,x))∂tu(s,x)dxds\n+k/integraldisplayt+T\nt/integraldisplay\nΩa(x)ϕ′(s)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n≤2/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|∂tv(s,x)|2dxds\n+(4CT+1)/parenleftbigg/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ/parenleftig\na(x)/parenleftig\n|g(∂tu(s,x))|2+|∂tu(s,x)|2/parenrightig\n+|b(x)v(s,x)|2/parenrightig\ndxds/parenrightbigg\n+3/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)|u(s,x)|2dxds+(kE∂tu,∂tv(0)+k0Eu,v(0))/integraldisplayt+T\nt|ϕ′′(s)|ds\n+k1/bracketleftbigg\nϕ′(t)/integraldisplay\nΩ∂2\ntu(s,x)∂tv(s,x)−∂2\ntv(s,x)∂tu(s,x)dx/bracketrightbiggs=t+T\ns=t.\n(2.25)\nTo estimate/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|∂tv(s,x)|2dxds,we first use the fact that vis a solution of the\nnonhomogeneous wave equation and ( O,T) satisfies the GCC, then from the observability\nestimate (2 .5),we infer that\n2/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|∂tv(s,x)|2dxds≤4CT/parenleftbigg/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩb(x)|∂tv(s,x)|2+|b(x)u(s,x)|2dxds/parenrightbigg\n.\n(2.26)\nNow we estimate/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩb(x)|∂tv(s,x)|2dxds.We have\n\n\n∂2\nt(∂tu)−∆(∂tu)+b(x)(∂tv)+a(x)g′(∂tu)∂2\ntu= 0 inR∗\n+×Ω\n∂2\nt(∂tv)−∆(∂tv)+b(x)(∂tu) = 0 in R∗\n+×Ω\n∂tu=∂tv= 0 onR∗\n+×Γ.(2.27)\nWe multiply the first equation of (2 .27) by (ϕ′(t)∂tv) and the second equation by ( ϕ′(t)∂tu)\nand integrating the difference of these results over Ω, we obta in\nϕ′(t)/integraldisplay\nΩb(x)|∂tv(t,x)|2dx=−d\ndt/parenleftbigg\nϕ′(t)/integraldisplay\nΩ/parenleftbig\n∂tv(t,x)∂2\ntu(t,x)−∂tu(t,x)∂2\ntv(t,x)/parenrightbig\ndx/parenrightbigg\n−ϕ′′(t)/integraldisplay\nΩ/parenleftbig\n∂tv(t,x)∂2\ntu(t,x)−∂tu(t,x)∂2\ntv(t,x)/parenrightbig\ndx\n+ϕ′(t)/integraldisplay\nΩb(x)|∂tu(t,x)|2dx−ϕ′(t)/integraldisplay\nΩa(x)g′(∂tu(t,x))∂2\ntu(t,x)∂tv(t,x)dx.STABILIZATION OF COUPLED WAVE 17\nUsing Young’s inequality, we infer that\nϕ′(t)/integraldisplay\nΩb(x)|∂tv(s,x)|2dx≤ǫϕ′(t)/integraldisplay\nΩa(x)|∂tv(s,x)|2dx\n−d\ndt/parenleftbigg\nϕ′(t)/integraldisplay\nΩ/parenleftbig\n∂tu(t,x)∂2\ntv(t,x)−∂tv(t,x)∂2\ntu(t,x)/parenrightbig\ndx/parenrightbigg\n+ϕ′′(t)/integraldisplay\nΩ/parenleftbig\n∂tu(t,x)∂2\ntv(t,x)−∂tv(t,x)∂2\ntu(t,x)/parenrightbig\ndx\n+ϕ′(t)/integraldisplay\nΩb(x)|∂tu(s,x)|2dx+1\nǫϕ′(t)/integraldisplay\nΩa(x)/vextendsingle/vextendsingleg′(∂tu(t,x))∂2\ntu(s,x)/vextendsingle/vextendsingle2dx.\nIntegrating the estimate above between tandt+T,we obtain\n/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩb(x)|∂tv(s,x)|2dxds\n≤ǫ/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)|∂tv(s,x)|2dxds+/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩb(x)|∂tu(s,x)|2dxds\n+1\nǫ/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)/vextendsingle/vextendsingleg′(∂tu(s,x))∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n+/integraldisplayt+T\ntϕ′′(s)/integraldisplay\nΩ/parenleftbig\n∂tu(t,x)∂2\ntv(t,x)−∂tv(t,x)∂2\ntu(t,x)/parenrightbig\ndxds\n−/bracketleftbigg\nϕ′(s)/integraldisplay\nΩ/parenleftbig\n∂2\ntu(s,x)∂tv(s,x)−∂2\ntv(s,x)∂tu(s,x)/parenrightbig\ndx/bracketrightbiggs=t+T\ns=t.\nNow using the observability estimate (2 .26) and taking ǫ=1\n4/bardbla/bardbl∞CT,we get\n/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩb(x)|∂tv(s,x)|2dxds\n≤2/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩb(x)|∂tu(s,x)|2+|b(x)u(s,x)|2dxds\n+16/ba∇dbla/ba∇dbl∞CT/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)/vextendsingle/vextendsingleg′(∂tu(s,x))∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n+2/integraldisplayt+T\ntϕ′′(s)/integraldisplay\nΩ/parenleftbig\n∂tu(t,x)∂2\ntv(t,x)−∂tv(t,x)∂2\ntu(t,x)/parenrightbig\ndxds\n−2/bracketleftbigg\nϕ′(s)/integraldisplay\nΩ/parenleftbig\n∂2\ntu(s,x)∂tv(s,x)−∂2\ntv(s,x)∂tu(s,x)/parenrightbig\ndx/bracketrightbiggs=t+T\ns=t.\nCombining the estimate above and (2 .26),we find that\n2/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|∂tv(s,x)|2dxds\n≤8CT/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩb(x)|∂tu(s,x)|2+|b(x)u(s,x)|2dxds\n+64/ba∇dbla/ba∇dbl∞(CT)2/ba∇dblg′/ba∇dblL∞/integraltextt+T\ntϕ′(s)/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n+8CT/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplayt+T\nt|ϕ”(s)|ds\n−8CT/bracketleftbigg\nϕ′(s)/integraldisplay\nΩ/parenleftbig\n∂2\ntu(s,x)∂tv(s,x)−∂2\ntv(s,x)∂tu(s,x)/parenrightbig\ndx/bracketrightbiggs=t+T\ns=t.(2.28)18 M. DAOULATLI\nNow using the observability estimate (2 .5),we infer that\n2/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|∂tv(s,x)|2dxds\n≤32C2\nT/ba∇dblb/ba∇dbl∞/parenleftbigg/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ/parenleftig\na(x)/parenleftig\n|g(∂tu(s,x))|2+|∂tu(s,x)|2/parenrightig/parenrightig\ndxds/parenrightbigg\n+/parenleftig\n8CT/ba∇dblb/ba∇dbl∞+32C2\nT/ba∇dblb/ba∇dbl2\n∞/parenrightig/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|u(s,x)|2+|v(s,x)|2dxds\n+64/ba∇dbla/ba∇dbl∞(CT)2/ba∇dblg′/ba∇dblL∞/integraltextt+T\ntϕ′(s)/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n+8CT/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplayt+T\nt|ϕ”(s)|ds\n−8CT/bracketleftbigg\nϕ′(s)/integraldisplay\nΩ/parenleftbig\n∂2\ntu(s,x)∂tv(s,x)−∂2\ntv(s,x)∂tu(s,x)/parenrightbig\ndx/bracketrightbiggs=t+T\ns=t.(2.29)\nUtilizing (2 .25),(2.29) and making some arrangements, we obtain\nX(t+T)−X(t)+/integraldisplayt+T\ntϕ′(s)Eu,v(s)ds+/integraldisplayt+T\ntϕ(s)/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds\n+k/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n≤/parenleftbig\n4CT+32C2\nT/ba∇dblb/ba∇dbl∞+1/parenrightbig/parenleftbigg/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ/parenleftig\na(x)/parenleftig\n|g(∂tu(s,x))|2+|∂tu(s,x)|2/parenrightig/parenrightig\ndxds/parenrightbigg\n+/parenleftig\n/ba∇dbla/ba∇dbl∞+12CT/ba∇dblb/ba∇dbl∞+32C2\nT/ba∇dblb/ba∇dbl2\n∞/parenrightig/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|v(s,x)|2+|u(s,x)|2dxds\n+64/ba∇dbla/ba∇dbl∞(CT)2/ba∇dblg′/ba∇dblL∞/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n+(k1−8CT)/bracketleftbigg\nϕ′(s)/integraldisplay\nΩ/parenleftbig\n∂2\ntu(s,x)∂tv(s,x)−∂2\ntv(s,x)∂tu(s,x)/parenrightbig\ndx/bracketrightbiggs=t+T\ns=t\n+(8CT+k)/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplayt+T\nt|ϕ′′(s)|ds.\nWe takek1= 8CT,we conclude that\nX(t+T)−X(t)+/integraldisplayt+T\ntϕ′(s)Eu,v(s)ds+/integraldisplayt+T\ntϕ(s)/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds\n+/parenleftig\nk−64/ba∇dbla/ba∇dbl∞(CT)2/ba∇dblg′/ba∇dblL∞/parenrightig/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n≤/parenleftig\n/ba∇dbla/ba∇dbl∞+12CT/ba∇dblb/ba∇dbl∞+32C2\nT/ba∇dblb/ba∇dbl2\n∞/parenrightig/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩ|v(s,x)|2+|u(s,x)|2dxds\n+/parenleftig\n32(CT)2/ba∇dblb/ba∇dbl∞+4CT+1/parenrightig/parenleftbigg/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)/parenleftig\n|∂tu(s,x)|2+|g(∂tu(s,x))|2/parenrightig\ndxds/parenrightbigg\n+(8CT+k+k0)/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplayt+T\nt|ϕ′′(s)|ds.\n(2.30)STABILIZATION OF COUPLED WAVE 19\nNow using (2 .12) withα=1\n2(/bardbla/bardbl∞+12CT/bardblb/bardbl∞+32C2\nT/bardblb/bardbl2\n∞), we get\nX(t+T)+1\n2/integraldisplayt+T\ntϕ′(s)Eu,v(s)ds+/integraldisplayt+T\ntϕ(s)/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds\n+/parenleftig\nk−64/ba∇dbla/ba∇dbl∞(CT)2/ba∇dblg′/ba∇dblL∞/parenrightig/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n≤C1/parenleftbigg/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)/parenleftig\n|∂tu(s,x)|2+|g(∂tu(s,x))|2/parenrightig\ndxds/parenrightbigg\n+X(t)+(8CT+k+k0)/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplayt+T\nt|ϕ′′(s)|ds.\n(2.31)\nwhere\nC1=C(T,/ba∇dblb/ba∇dbl∞,/ba∇dbla/ba∇dbl∞). (2.32)\nNow we have to estimate the first term of RHS of the estimate abo ve by the third term of\nthe LHS. We set for all fixed s≥0,Ωs={x,|∂tu(s,x)|<1}.Thanks to (1 .9),we have\n/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩsa(x)/parenleftig\n|∂tu(s,x)|2+|g(∂tu(s,x))|2/parenrightig\ndxds\n≤1\nǫ0/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩsh0(g(∂tu(s,x))∂tu(s,x))a(x)dxds\n≤1\nǫ0/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩh0(g(∂tu(s,x))∂tu(s,x))a(x)dxds\nSinceh0is concave, we can use (the reverse) Jensen’s inequality and we obtain\n/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩsa(x)/parenleftig\n|∂tu(s,x)|2+|g(∂tu(s,x))|2/parenrightig\ndxds\n≤1\nǫ0/integraldisplayt+T\ntϕ′(s)h/parenleftbigg/integraldisplay\nΩ(g(∂tu(s,x))∂tu(s,x))a(x)dx/parenrightbigg\nds.\nUnder our assumptions the function Hdefined by (2 .18) is convex and proper. Hence, we\ncan apply Young’s inequality [19]\n/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩsa(x)/parenleftig\n|∂tu(s,x)|2+|g(∂tu(s,x))|2/parenrightig\ndxds\n≤/parenleftbigg\nǫ0\n2C1/integraldisplayt+T\ntϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds+ǫ0\n2C1/integraldisplayt+T\ntϕ(s)/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds/parenrightbigg\n,\nwhereH∗is the convex conjugate of the function H.\nOn the other hand, using the fact that the function gis linearly bounded near infinity, we\ninfer that\n/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)/parenleftig\n|∂tu(s,x)|2+|g(∂tu(s,x))|2/parenrightig\ndxds\n≤1\n2C1/integraldisplayt+T\ntϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds+1\n2C1/integraldisplayt+T\ntϕ(s)/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds\n+/parenleftbig1\nm+M2/parenrightbig/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds.20 M. DAOULATLI\nThe estimate above combined with (2 .31), give\nX(t+T)+1\n2/integraldisplayt+T\ntϕ′(s)Eu,v(s)ds\n+/integraldisplayt+T\nt/parenleftig\n1\n2−/parenleftbig1\nm+M2/parenrightbig\nC1ϕ′(s)\nϕ(s)/parenrightig\nϕ(s)/integraldisplay\nΩa(x)g(∂tu(s,x))∂tu(s,x)dxds\n+/parenleftig\nk−64/ba∇dbla/ba∇dbl∞(CT)2/ba∇dblg′/ba∇dblL∞/parenrightig/integraldisplayt+T\ntϕ′(s)/integraldisplay\nΩa(x)g′(∂tu(s,x))/vextendsingle/vextendsingle∂2\ntu(s,x)/vextendsingle/vextendsingle2dxds\n≤X(t)+(8CT+k+k0)/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplayt+T\nt|ϕ′′(s)|ds+1\n2/integraldisplayt+T\ntϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds.\n(2.33)\nWe remind that\nX(t) =ϕ′(t)/integraldisplay\nΩu(t,x)∂tu(t,x)+v(t,x)∂tv(t,x)dx\n+k1ϕ′(t)/integraldisplay\nΩ∂2\ntu(t,x)∂tv(t,x)−∂2\ntv(t,x)∂tu(t,x)dx+ϕ(t)Eu,v(t)+kϕ′(t)E∂tu,∂tv(t),\nwithk1= 8CT.Using Young’s inequality and Poincar´ e inequality, it is ea sy to see that\nX(t)≤/parenleftig\nϕ(t)+ϕ′(t)\nδ/parenleftbig\nk1+2λ2+1/parenrightbig/parenrightig\nEu,v(t)+ϕ′(t)/parenleftig\nk+k1\nδ/parenrightig\nE∂tu,∂tv(t)\nX(t)≥/parenleftig\n1−ϕ′(t)\nδϕ(t)/parenleftbig\nk1+2λ2+1/parenrightbig/parenrightig\nϕ(t)Eu,v(t)+ϕ′(t)/parenleftig\nk−k1\nδ/parenrightig\nE∂tu,∂tv(t)(2.34)\nSo, takingksuch that\nk≥max/parenleftbigg8CT\nδ,64/ba∇dbla/ba∇dbl∞(CT)2/vextenddouble/vextenddoubleg′/vextenddouble/vextenddouble\nL∞/parenrightbigg\n,\nusing (2.33) and the fact that\n/parenleftig\n1\n2−C1/parenleftbig1\nm+M2/parenrightbigϕ′(t)\nϕ(t)/parenrightig\n≥0 and 1−ϕ′(t)\nδϕ(t)/parenleftbig\n8CT+2λ2+1/parenrightbig\n≥0,for allt≥0,\nwe deduce that X(t)≥0 and\nX(t+T)+1\n2/integraldisplayt+T\ntϕ′(s)Eu,v(s)ds\n≤X(t)+8CT/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplayt+T\nt|ϕ′′(s)|ds+1\n2/integraldisplayt+T\ntϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds.\n(2.35)\nfor allt≥0. Thus\nn−1/summationtext\ni=0/parenleftigg\nX((i+1)T)−X(iT)+1\n2/integraldisplay(i+1)T\niTϕ′(s)Eu,v(s)ds/parenrightigg\n≤(8CT+k+k0)/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplaynT\n0|ϕ′′(s)|ds+1\n2/integraldisplaynT\n0ϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds,STABILIZATION OF COUPLED WAVE 21\nand this gives\nX(nT)+1\n2/integraldisplaynT\n0ϕ′(s)Eu,v(s)ds\n≤X(0)+(8CT+k+k0)/parenleftig/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig/integraldisplaynT\n0|ϕ”(s)|ds\n+1\n2/integraldisplaynT\n0ϕ(s)H∗/parenleftig\n2C1ϕ′(s)\nǫ0ϕ(s)/parenrightig\nds,for alln∈N.\nUtilizing lemma 3, we conclude that there exists a positive c onstantCsuch that\n/integraldisplay+∞\n0ϕ′(s)Eu,v(s)ds≤C/parenleftig\n1+/summationtext1\ni=0E∂i\ntu,∂i\ntv(0)/parenrightig\n. (2.36)\nSince\nϕ(t)Eu,v(t)≤ϕ(0)Eu,v(0)+/integraldisplay+∞\n0ϕ′(s)Eu,v(s)ds, t≥0,\nthen (2.36),gives (1.14).\nFinally, using the density of D(A) inH, we obtain (1 .16).\nReferences\n[1] Alabau-Boussouira, Fatiha; L´ eautaud, Matthieu, Indi rect controllability of locally coupled wave-type\nsystems and applications. J. Math. Pures Appl. (9) 99 (2013) , no. 5, 544–576.\n[2] F. Alabau-Boussouira, M. L´ eautaud, Indirect stabiliz ation of locally coupled wave-type systems, ESAIM\nControl Optim. Calc. Var. 18 (2012) 548–582.\n[3] Alabau, Fatiha; Cannarsa, Piermarco; Komornik, Vilmos , Indirectinternal stabilization ofweakly coupled\nevolution equations. J. Evol. Equ. 2 (2002), no. 2, 127–150.\n[4] Alabau-Boussouira, Fatiha, A unified approach via conve xity for optimal energy decay rates of finite and\ninfinite dimensional vibrating damped systems with applica tions to semi-discretized vibrating damped\nsystems. J. Differ. Equations 248, No. 6, 1473-1517 (2010).\n[5] Alabau-Boussouira, Fatiha, Wang Zhiqiang and Yu Lixin, A one-step optimal energy decay formula for\nindirectly nonlinearly damped hyperbolic systems coupled by velocities. ESAIM: COCV 23 (2017) 721-\n749.\n[6] F. Alabau-Boussouira, New trends towards lower energy e stimates and optimality for nonlinearly damped\nvibrating systems. J. Differential Equations, 249: 1145-11 78, 2010.\n[7] Aloui, L.; Daoulatli, M., Stabilization of two coupled w ave equations on a compact manifold with bound-\nary. J. Math. Anal. Appl. 436, No. 2, 944-969 (2016).\n[8] Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient condit ions for the observation, control, and stabiliza-\ntion of waves from the boundary. SIAM J. Control Optim. 30, 10 24–1065 (1992).\n[9] L. Bociu and I. Lasiecka, Uniquenessof weak solutions fo r the semilinear wave equations with supercritical\nboundary/interior sources and damping, Discrete Contin. D yn. Syst., 22 (2008), 835-860.\n[10] Daoulatli, M. Behaviors of the energy of solutions of th e wave equation with damping and external force.\nJ. Math. Anal. Appl. 389 (2012), no. 1, 205–225.\n[11] Daoulatli, M. Energy decay rates for solutions of the wa ve equation with linear damping in exterior\ndomain. Evol. Equ. Control Theory 5 (2016), no. 1, 37–59.\n[12] Dehman, Belhassen; Le Rousseau, J´ erˆ ome; L´ eautaud, Matthieu, Controllability of two coupled wave\nequations on a compact manifold. Arch. Ration. Mech. Anal. 2 11 (2014), no. 1, 113–187.\n[13] X. Fu, Sharp decay rates for the weakly coupled hyperbol ic system with one internal damping, SIAM J.\nCONTROL OPTIM., 50 (2012), 1643-1660\n[14] I. Lasiecka and D. Tataru, Uniform boundary stabilizat ion of semi-linear wave equation with nonlinear\nboundary dissipation, Differential Integral Equations 6(1993), 507–533.\n[15] L´ eautaud, M. Spectral inequalities for non-selfadjo int elliptic operators and application to the null-\ncontrollability of parabolic systems. J. Funct. Anal. 258 ( 2010), no. 8, 2739–2778.22 M. DAOULATLI\n[16] J.-L. Lions. Controlabilite exacte, perturbations et stabilisation de systemes distribues. Tome 1, volume\n8 of Recherches en Mathematiques Appliquees. Masson, Paris , 1988.\n[17] M. Nakao, Energy decay for the linear and semilinear wav e equations in exterior domains with some\nlocalized dissipations, Math. Z. 238 (2001) 781–797.\n[18] Rauch, J., Taylor, M.: Exponential decay of solutions t o hyperbolic equations in bounded domains.\nIndiana Univ. Math. J. 24, 79–86 (1974).\n[19] R. T. Rockafellar, Convex Analysis Princeton Universi ty Press, Princeton, NJ, 1970.\nUniversity of Dammam, King Saudi Arabia & University of Cart hage, Tunisia\nE-mail address , M. Daoulatli: moez.daoulatli@infcom.rnu.tn"    },    {        "title": "1607.07932v1.Strongly_Enhanced_Effects_of_Lorentz_Symmetry_Violation_in_Yb_____and_Highly_Charged_Ions.pdf",        "content": "arXiv:1607.07932v1  [hep-ph]  27 Jul 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nStrongly Enhanced Effects of Lorentz-Symmetry Violation\nin Yb+and Highly Charged Ions\nM.S. Safronova\nDepartment of Physics and Astronomy, University of Delawar e\nNewark, DE 19716, USA\nJoint Quantum Institute, NIST and the University of Marylan d\nCollege Park, MD 20742, USA\nA Lorentz-symmetry test with Ca+ions demonstrated the potential of using\nquantum information inspired technology for tests of funda mental physics. A\nsystematic study of atomic-system sensitivities to Lorent z violation identified\nthe ytterbium ion as an ideal system with high sensitivity as well as excellent\nexperimental controllability. A test of Lorentz-violatin g physics in the electron-\nphoton sector with Yb+ions has the potential to reach levels of 10−23, five\norders of magnitude more sensitive than the current best bou nds. Similar\nsensitivities may be also reached with highly charged ions.\n1. Introduction\nLocal Lorentz invariance (LLI) is an important foundation of mode rn\nphysicsand hasbeen a subjectofmanystringentexperimental an dobserva-\ntional tests.1However, a number of theories unifying gravitywith the Stan-\ndard Model of particle physics suggest possible violation of Lorentz sym-\nmetry. While the suggested LLI-violation energy scale is much larger than\nthe energy currently attainable by particle accelerators, it might b e acces-\nsible with precision measurements at low energy. Therefore, high-p recision\ntests of LLI with photons and particles (protons, neutrons, elec trons) may\nprovide insight into possible new physics and set limits on various theor ies.\nExperimental breakthroughs in atomic, molecular and optical (AMO )\nphysics, including laser cooling and trapping of atoms, attainment of Bose-\nEinstein condensation, optical frequency combs, and quantum co ntrol led\nto extraordinaryadvancesinthe controlofmatter andlight. The seachieve-\nments, coupled with dramatic improvements in precision time and fre-\nquency metrology, measurement techniques such as atomic interf erometry\nand magnetometry, and advances in first-principles atomic and mole cularProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\ntheoryenabledaplethoraofnewapplicationsofAMO,includingnovel ways\nto test the fundamental physics laws. The availability of trapped ult racold\natoms and ions, subject to precise interrogation and control, pro vided for\nnew opportunities for tests of Lorentz symmetry. A diverse set o f AMO\nLorentz-symmetry tests involves experiments with atomic clocks,2other\nprecision spectroscopy measurements,3magnetometers,4,5electromagnetic\ncavities,6,7and quantum information trapped ion technologies.8A cold Cs\natom clock test of Lorentz invariance in the matter sector was car ried out\nin Ref. 2, setting the best limits on the tensor Lorentz-violating coe fficients\nfor the proton.\nIn 2015, an experiment with a pair of trapped calcium ions improved\nbounds on LLI-violatingStandard-Model Extension(SME) coefficie ntscJK\nfor electrons by a factor of 100 demonstrating the potential of q uantum in-\nformationtechniquesinthesearchforphysicsbeyondthe Standa rdModel.8\nThe same experiment can be interpreted as testing anisotropy in th e speed\nof light, improving a previous such bound6by a factor of 5, with the sen-\nsitivity similar to more recent work reported in Ref. 7.\n2. Ca+experiment\nLorentz-violation tests are analyzed in the context of the phenom enologi-\ncal framework known as the SME, which is an effective field theory th at\naugments the Standard-Model lagrangian with every possible comb ination\nof the Standard-Model fields that is not term-by-term Lorentz in variant,\nwhile maintaining gauge invariance, energy-momentum conservation , and\nobserver Lorentz invariance of the total action.9Violations of Lorentz in-\nvariance and the Einstein equivalence principle in bound electronic sta tes\nresult in a small shift of the hamiltonian that can be described by3,10\nδH=−/parenleftbigg\nC(0)\n0−2U\n3c2c00/parenrightbiggp2\n2−1\n6C(2)\n0T(2)\n0, (1)\nwhere we use atomic units, pis the momentum of a bound electron, and c\nis the speed of light. The parameters C(0)\n0,c00, andC(2)\n0are elements in the\ncµνtensorwhichcharacteriseshypotheticalLorentzviolationinthee lectron\nsector within the SME.1,10The nonrelativistic form of the T(2)\n0operator is\nT(2)\n0=p2−3p2\nz. Predicting the energy shift due to LLI violation involves\nthe calculation of the expectation value of the above hamiltonian for the\natomic states of interest. Therefore, the shift of the Ca+3d5/2energy\nlevel due to the cµνtensor depends on the values of /angbracketleft3d5/2|p2|3d5/2/angbracketrightand\n/angbracketleft3d5/2|T(2)\n0|3d5/2/angbracketrightmatrix elements.Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\nThe frequency difference (in Hz) between the shifts of the m= 5/2 and\nm= 1/2 substates of the 3 d5/2manifold was calculated in Ref. 8:\n1\nh/parenleftbig\nEm=5/2−Em=1/2/parenrightbig\n=/parenleftbig\n−4.45(9)×1015Hz/parenrightbig\n×C(2)\n0.(2)\nThe basic idea of the Ca+experiment is to monitor this energy difference\nbetween the magnetic mJ=|1/2|andmJ=|5/2|substatesofthe3 d5/2mJ\nmanifold over time to set the limit on potential violation of LLI. Theref ore,\nthe shift of the Ca+3d5/2energy level due to the cµνtensor depended only\non the value of /angbracketleft3d5/2|T(2)\n0|3d5/2/angbracketrightmatrix element, as the contribution of the\nscalar term canceled for the states of the same mJmanifold. Superposition\nof two ions prepared in a decoherence-free subspace\n|Ψ/angbracketright=1√\n2(|1/2,−1/2/angbracketright+|5/2,−5/2/angbracketright), (3)\nwhere|m1,m2/angbracketrightrepresents the 3 d5/2state withmJ=m1andm2for the\nfirst and second ion, respectively, allowed elimination of the largest p oten-\ntial systematic uncertainty cased by the fluctuation of the magne tic field.\nDetails of the experiment are given in Ref. 8.\n3. Proposal for LLI test with Yb+ions\nFurther improvement of LLI violation limits calls for a system with a long -\nlived (or ground) state that has a large /angbracketleftj|T(2)\n0|j/angbracketrightmatrix element. We\nhave carried out a systematic study of this quantity for various sy stems\nand identified general rules for the enhancement of the reduced m atrix\nelements of the T(2)operator.11Our calculations for ndstates in Ba+and\nYb+, which areheavieranaloguesofCa+, found only a smallincreaseofthe\nT(2)matrix elements in comparison with the Ca+case. However, Yb+has\nanother metastable level, 4 f136s2 2F7/2with theT(2)\n0matrix element that\nis over an order of magnitude larger than for the ndstates. We find that\ndeeper localizationofthe probeelectronleadsto enhancedsensitiv ityto the\ntensor Lorentz violation. Our study has shown that /angbracketleftψ|r|ψ/angbracketrightof∼0.8 a.u or\nbelow for the corresponding electron is a good indicator of the large value\nof the reduced T(2)matrix element. This condition is satisfied for the 4 f\nhole states, such as Yb+state considered here, or for highly charged ions\nwithnfvalence electrons and degree of ionization ∼15. The reduced T(2)\nmatrix elements in Yb+and Sm15+are 135 a.u. and 149 a.u., respectively,\nin comparison of 9.3 a.u for the 3 d5/2state in Ca+.Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n4\nThe Yb+4f136s2 2F7/2state also has an exceptionally long lifetime\non the order of several years.12Therefore, the Ramsey duration of the pro-\nposed experiment with Yb+is not limited by spontaneous decay during the\nmeasurement as in Ca+case. The electric-octupole E3 transition between\nthe 4f136s2 2F7/2excited state and the ground state is used as the basis for\nthe optical atomic clock with the single trapped Yb+ion, which presently\nhas the lowest uncertainty among the all of the optical ion clocks.13Yb+\nions are also used in quantum information research.14As a result, exper-\nimental techniques for precision control and manipulation of Yb+atomic\nstates are particulary well developed making it an excellent candidat e for\nsearches of Lorentz-violation signature.\nWe estimated that experiments with the metastable 4 f136s2 2F7/2state\nof Yb+can reach sensitivities of 1 .5×10−23for thecJKcoefficients,11\nover 105times more stringent than current best limits.8Moreover, the\nprojectedsensitivitytothe cTJcoefficientswillbe atthelevelof1 .5×10−19,\nbelowtheratiobetweentheelectroweakandPlanckenergyscales.11Similar\nsensitivities may potentially be reached for LV tests with highly charg ed\nions,givenfuturedevelopmentofexperimentaltechniquesforth esesystems.\nReferences\n1.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2016 edition, arXiv:0801.0287v9.\n2. P. Wolf, F. Chapelet, S. Bize, and A. Clairon, Phys. Rev. Le tt.96, 060801\n(2006).\n3. M.A. Hohensee, N. Leefer, D. Budker, C. Harabati, V.A. Dzu ba, and V.V.\nFlambaum, Phys. Rev. Lett. 111, 050401 (2013).\n4. M. Smiciklas, J.M. Brown, L.W. Cheuk, S.J. Smullin, and M. V. Romalis,\nPhys. Rev. Lett. 107, 171604 (2011).\n5. F. Allmendinger et al., Phys. Rev. Lett. 112, 110801 (2014).\n6. C. Eisele, A.Y. Nevsky, and S. Schiller, Phys. Rev. Lett. 103, 090401 (2009).\n7. M. Nagel et al., Nature Commun. 6, 8174 (2015).\n8. T.Pruttivarasin, M. Ramm, S.G.Porsev, I.I.Tupitsyn, M. S.Safronova, M.A.\nHohensee, and H. H¨ affner, Nature 517, 592 (2015).\n9. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 58, 116002 (1998).\n10. V.A. Kosteleck´ y and C.D. Lane, Phys. Rev. D 60, 116010 (1999).\n11. V.A. Dzuba, V.V. Flambaum, M.S. Safronova, S.G. Porsev, T. Pruttivarasin,\nM.A. Hohensee, and H. H¨ affner, Nature Physics 12, 465 (2016).\n12. N. Huntemann et al., Phys. Rev. Lett. 108, 090801 (2012).\n13. N. Huntemann, C. Sanner, B. Lipphardt, Chr. Tamm, and E. P eik, Phys.\nRev. Lett. 116, 063001 (2016).\n14. R. Islam et al., Science 340, 583 (2013)."    },    {        "title": "1309.3761v1.Comments_on_Lorentz_and_CPT_Violation.pdf",        "content": "arXiv:1309.3761v1  [hep-ph]  15 Sep 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n1\nIUHET 581, August 2013\nCOMMENTS ON LORENTZ AND CPT VIOLATION\nV. ALAN KOSTELECK ´Y\nPhysics Department, Indiana University\nBloomington, IN 47405, USA\nThis contribution to the CPT’13 meeting briefly introduces L orentz and CPT\nviolation and outlines two recent developments in the field.\n1. Introduction\nThe idea that small observable violations of Lorentz symmetry could pro-\nvide experimental access to Planck-scale effects continues to dra w atten-\ntion across several subfields of physics. In the three years since the previous\nmeeting in this series, considerable progress has been made on both experi-\nmental and theoretical fronts. This contribution to the CPT’13 pr oceedings\ncontains a brief introduction, followed by comments on two topics of recent\ninterest: nonminimal fermion couplings and Riemann-Finsler geometr y.\n2. Basics\nA satisfactory theoretical description of Lorentz violation must in corpo-\nrate coordinate independence, realism, and generality. A powerfu l approach\nuses effective field theory, starting with General Relativity coupled to the\nStandard Model and adding to the Lagrange density all observer- invariant\nterms that contain Lorentz-violating operators combined with con trolling\ncoefficients. This yields the comprehensive realistic effective field the ory\nfor Lorentz violation called the Standard-Model Extension (SME).1,2The\nSME also describes general CPT violation, which in the context of rea listic\neffective field theory is accompanied by Lorentz violation.3The full SME\ncontains operators of arbitrary mass dimension d, while the minimal SME\nrestricts attention to operators of renormalizable dimension d≤4.\nObservable signals of Lorentz violation are governed by the SME coe f-\nficients. Experiments typically search for particle interactions with back-\nground coefficient values, which can produce effects dependent on the par-Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n2\nticle velocity, spin, flavor, and couplings. Many investigations of this type\nhave been performed,4achieving impressive sensitivities that in some cases\nexceed expectations for suppressed Planck-scale effects. If th e SME coef-\nficients are produced by spontaneous Lorentz violation,5as is necessary\nwhen gravity is based on Riemann geometry,2then they are dynamical\nquantitiesthatmust incorporatemasslessNambu-Goldstonemode s.6These\nmodes have numerous interpretations, including serving as an alter native\norigin for the photon in Einstein-Maxwell theory6and the graviton in Gen-\neral Relativity,7or representing new spin-dependent8or spin-independent9\nforces, among other possibilities. Massive modes can also appear.10\n3. Nonminimal fermion sector\nIn the nonminimal sector of the SME, the number of Lorentz-violat ing\noperators grows rapidly with the mass dimension d. Systematically cata-\nloguing and characterizing the possibilities is therefore indispensible in the\nsearch for Lorentz violation. In the CPT’10 proceedings, I outlined some\nfeatures appearing in the treatment of quadratic operators of a rbitrary d\nin the photon Lagrange density.11In the intervening three-year period, in-\nvestigations of the quadratic fermion sector for arbitrary dhave also been\nperformed. The Lagrange density for propagation and mixing of an y num-\nber of fermions has been developed and applied to describe general Lorentz\nviolation in the neutrino sector.12More recently, quadratic operators of\narbitrary dhave been studied for a massive Dirac fermion.13\nMany nonminimal operators generate effects that are in principle ob -\nservable, and each such operator generates a distinct experimen tal signal.\nForquadraticoperators,whichcharacterizeparticlepropagatio nandphase-\nspace features of particle decays, the Lorentz-violating effects can include\ndirection dependence (anisotropy), wave-packet deformation ( dispersion),\nandmodesplitting(birefringence).Intheneutrinosector,forex ample,some\noperatorscontrolflavor-dependenteffects in neutrinoandant ineutrinomix-\ning, producing novel energy and direction dependences that involv e both\nDirac- and Majorana-type couplings. Others govern species-inde pendent\neffects, which can differ for neutrinos and antineutrinos and can pr oduce\npropagation times varying with energy and direction, in some cases e xceed-\ning that of light. A few operators produce ‘countershaded’ effect s14that\ncannot be detected via oscillations or propagation but change inter action\nproperties in processes such as beta decay.15\nAnalogous effects appear in the description of a massive Dirac fermio n\nin the presence of Lorentz violation, for which the exact dispersion relationProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n3\nfor arbitrary dis known in closed and compact form.13For example, the\nfermion group velocity is anisotropic and dispersive, while the fermion spin\nexhibits a Larmor-like precession caused by birefringent operator s. Using\nfield redefinitions to investigate observability reveals that many ope rators\nof dimension dproduce no effects or are physically indistinguishable from\nothers of dimensions dord±1. Nonetheless, the number of observable\ncoefficients grows as the cube of d. To date, almost all the nonminimal\ncoefficient space for fermions is experimentally untouched. This offe rs an\nopen arena for further exploration with a significant potential for discovery.\n4. Riemann-Finsler geometry\nThe surprising ‘no-go’ result that the conventional Riemann geome try of\nGeneral Relativity and its extension to Riemann-Cartan geometry a re both\nincompatible with explicit Lorentz violation raises the questions of whe ther\nan alternative geometry is involved and, if so, whether a correspon ding\ngravitational theory exists. The obstruction to explicit Lorentz v iolation,\nwhich disappears for the spontaneous case, can be traced to the generic\nincompatibility of the Bianchi identities with the external prescriptio n of\ncoefficients for Lorentz violation. It is therefore reasonable to co njecture\nthatanaturalgeometricalsettingwouldincludemetricdistancesd epending\nlocally on the coefficients in addition to the Riemann metric.2\nSupport for this conjecture has recently emerged with the discov ery\nthat a fermion experiencing explicit Lorentz violation tracks a geode sic in\na pseudo-Riemann-Finsler geometry rather than a conventional g eodesic\nin pseudo-Riemann spacetime.16Riemann-Finsler geometry is a well-\nestablished mathematical field with numerous physical applications ( see,\ne.g., Ref. 17), such as the famous Zermelo navigation problem of obt aining\nthe minimum-time path for a ship in the presence ofocean currents. A large\nclass of Riemann-Finsler geometries is determined by the geodesic mo tion\nof particles in the SME.16,18Among these are the canonical Randers ge-\nometry, which is related to the 1-form SME coefficient aµ, and numerous\nnovel geometries of simplicity comparable to the Randers case. One exam-\nple of the latter is bspace, a calculable Riemann-Finsler geometry that also\nis based on a 1-form and has Finsler structure complementary to th at of\nRanders space. Physically, this geometry underlies the geodesic mo tion of\na fermion in curved spacetime in the presence of chiral CPT-odd Lor entz\nviolation.All the SME-inspired geometriesexhibit mathematically inter est-\ning features connected to physical properties. For instance, wh en the SME\ncoefficients are covariantly constant, the trajectories are conv entional Rie-Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n4\nmann geodesics and special Riemann-Finsler geometries known as Be rwald\nspaces result. Many open challenges remain in this area, ranging fro m more\ntechnical questions such as resolving singularities or classifying geo metries\nto physical issues such as generalizingZermelo navigationor uncove ringim-\nplications for the SME. The prospects appear promising for furthe r insights\nto emerge from this geometrical approach to Lorentz violation.\nAcknowledgments\nThisworkwassupportedinpartbyU.S. D.o.E.grantDE-FG02-13ER4 2002\nand by the Indiana University Center for Spacetime Symmetries.\nReferences\n1. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n2. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n3. O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002).\n4.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2013 edition, arXiv:0801.0287v6.\n5. V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 39, 683 (1989); Phys. Rev.\nLett.63, 224 (1989); Phys. Rev. D 40, 1886 (1989); V.A. Kosteleck´ y and R.\nPotting, Nucl. Phys. B 359, 545 (1991); Phys. Rev. D 51, 3923 (1995); V.A.\nKosteleck´ y and R. Lehnert, Phys. Rev. D 63, 065008 (2001).\n6. R. Bluhm and V.A. Kosteleck´ y, Phys. Rev. D 71, 065008 (2005).\n7. S.M. Carroll et al., Phys. Rev. D 80, 025020 (2009); V.A. Kosteleck´ y and R.\nPotting, Phys. Rev. D 79, 065018 (2009); Gen. Rel. Grav. 37, 1675 (2005).\n8. N. Arkani-Hamed et al., JHEP0507, 029 (2005).\n9. J. Alfaro and L.F. Urrutia, Phys. Rev. D 81, 025007 (2010); B. Altschul et\nal., Phys. Rev. D 81, 065028 (2010); V.A. Kosteleck´ y and J.D. Tasson, Phys.\nRev. D83, 016013 (2011).\n10. R. Bluhm et al., Phys. Rev. D 77, 065020 (2008).\n11. V.A. Kosteleck´ y and M. Mewes, Ap. J. Lett. 689, L1 (2008); Phys. Rev. D\n80, 015020 (2009); Phys. Rev. Lett. 110, 201601 (2013).\n12. J.S. D´ ıaz et al., arXiv:1308.6344; V.A. Kosteleck´ y and M. Mewes, Phys. Rev .\nD85, 096005 (2012).\n13. V.A. Kosteleck´ y and M. Mewes, arXiv:1308.4973.\n14. V.A. Kosteleck´ y and J.D. Tasson, Phys. Rev. Lett. 102, 010402 (2009).\n15. J.S. D´ ıaz et al., arXiv:1305.4636.\n16. V.A. Kosteleck´ y, Phys. Lett. B 701, 137 (2011).\n17. S.-S. Chern and Z. Shen, Riemann-Finsler Geometry , World Scientific, Sin-\ngapore, 2005.\n18. D. Colladay and P. McDonald, Phys. Rev. D 85, 044042 (2012); V.A. Kost-\nelecky, N. Russell, and R. Tso, Phys. Lett. B 716, 470 (2012); V.A. Kost-\neleck´ y and N. Russell, Phys. Lett. B 693, 2010 (2010)."    },    {        "title": "1712.10006v1.Ternary_generalization_of_Pauli_s_principle_and_the_Z6_graded_algebras.pdf",        "content": "Ternary generalization of Pauli's principle\nand the Z6-graded algebras\nRichard Kerner\nLaboratoire de Physique Th\u0013 eorique de la Mati\u0012 ere Condens\u0013 ee (LPTMC),\nUnivesit\u0013 e Pierre et Marie Curie - CNRS UMR 7600\nTour 23-13, 5-\u0012 eme \u0013 etage, Bo^ \u0010te Courrier 121, 4 Place Jussieu, 75005 Paris, FRANCE\nAbstract\nWe show how the discrete symmetries Z2andZ3combined with the superposition principle result\nin theSL(2;C)-symmetry of quantum states. The role of Pauli's exclusion principle in the derivation\nof theSL(2;C) symmetry is put forward as the source of the macroscopically observed Lorentz\nsymmetry; then it is generalized for the case of the Z3grading replacing the usual Z2grading, leading\nto ternary commutation relations. We discuss the cubic and ternary generalizations of Grassmann\nalgebra. Invariant cubic forms are introduced, and their symmetry group is shown to be the SL(2;C)\ngroup The wave equation generalizing the Dirac operator to the Z3-graded case is constructed.\nIts diagonalization leads to a sixth-order equation. The solutions cannot propagate because their\nexponents always contain non-oscillating real damping factor. We show how certain cubic products\ncan propagate nevertheless. The model suggests the origin of the color SU(3) symmetry.\n1 Introduction\nIn modern physics, which was created by scienti\fc giants like Galileo, Kepler, Newton and Huygens, the\ndescription of the world surrounding us is based on three essential realms, which are Material bodies ,\nForces acting between them and Space and Time. Newton's third law:\na=1\nmF: (1)\nshows the relation between three di\u000berent realms which are dominant in our description of physical\nworld: massive bodies ( m), force \felds responsible for interactions between the bodies (\" F\") and space-\ntime relations de\fning the acceleration (\" a\"). Similar ingredients are found in physics of fundamental\ninteractions: we speak of elementary particles and \felds evolving in space and time.\nIn the formula (1) we deliberately have put the acceleration on the left-hand side, and the inverse of\nmass anf the force on the right-hand side in order to separate the directly observable entity a) from the\nproduct of two entities whose de\fnition is much less direct and clear.\nAlso, by putting the acceleration alone on the left-hand side, we underline the causal relationship\nbetween the phenomena: the force is the cause of acceleration, and not vice versa. In modern language,\nthe notion of force is generally replaced by that of a \feld. The fact that the three ingredients are related\nby the equation (1) may suggest that perhaps only two of them are fundamentally independent, the\nthird one being the consequence of the remaining two.\nThe three aspects of theories of fundamental interactions can be symbolized by three orthogonal\naxes, as shown in following \fgure, which displays also three choices of pairs of independent properties\nfrom which we are supposed to be able to derive the third one.\n1arXiv:1712.10006v1  [physics.gen-ph]  16 Dec 2017Figure 1: The three realms of physical world\nThe attempts to understand physics with only two realms out of three represented in (1) have a very\nlong history. They may be divided in three categories, labeled I,IIandIIIin the Figure above.\nIn the category Ione can easily recognize Newtonian physics, presenting physical world as collection\nof material bodies (particles) evolving in absolute space and time, interacting at a distance. Newton\nconsidered light being made of tiny particles, too; the notion of \felds was totally absent. Any change in\npositions and velocities of any massive material object was immediately felt by all other masses in the\nentire Universe.\nTheories belonging to the category IIassume that physical world can be described uniquely as a\ncollection of \felds evolving in space-time manifold. This approach was advocated by Kelvin, Einstein,\nand later on by Wheeler. As a follower of Maxwell and Faraday, Einstein believed in the primary role\nof \felds and tried to derive the equations of motion as characteristic behavior of singularities of \felds,\nor the singularities of the space-time curvature.\nThe category IIIrepresents an alternative point of view supposing that the existence of matter is\nprimary with respect to that of the space-time, which becomes an \\emergent\" realm - an euphemism for\n\\illusion\". Such an approach was advocated recently by N. Seiberg and E. Verlinde [2]. It is true that\nspace-time coordinates cannot be treated on the same footing as conserved quantities such as energy\nand momentum; we often forget that they exist rather as bookkeeping devices, and treating them as real\nobjects is a \\bad habit\", as pointed out by D. Mermin [1].\nSeen under this angle, the idea to derive the geometric properties of space-time, and perhaps its very\nexistence, from fundamental symmetries and interactions proper to matter's most fundamental building\nblocks seems quite natural.\nMany of those properties do not require any mention of space and time on the quantum mechanical\nlevel, as was demonstrated by Born and Heisenberg in their version of matrix mechanics, or by von\nNeumann's formulation of quantum theory in terms of the C\u0003algebras [3], [4]. The non-commutative\ngeometry is another example of formulation of space-time relationships in purely algebraic terms [5].\nIn what follows, we shall choose the latter point of view, according to which the space-time relations\nare a consequence of fundamental discrete symmetries which characterize the behavior of matter on the\nquantum level. In other words, the Lorentz symmetry observed on the macroscopic level, acting on what\nwe perceive as space-time variables, is an averaged version of the symmetry group acting in the Hilbert\nspace of quantum states of fundamental particle systems.\n2 Space-time as emerging realm\nIn standard textbooks introducing the Lorentz and Poincar\u0013 e groups the accent is put on the transfor-\nmation properties of space and time coordinates, and the invariance of the Minkowskian metric tensor\ng\u0016\u0017. But neither its components, nor the space-time coordinates of an observed event can be given an\nintrinsic physical meaning; they are not related to any conserved or directly observable quantities.\n2Under a closer scrutiny, it turns out that only time - the proper time of the observer - can be\nmeasured directly. The notion of space variables results from the convenient description of experiments\nand observations concerning the propagation of photons, and the existence of the universal constant c.\nConsequently, with high enough precision one can infer that the Doppler e\u000bect is relativistic, i.e. the\nfrequency!and the wave vector kform an entity that is seen di\u000berently by di\u000berent inertial observers,\nand passing from!\nc;kto!0\nc;k0is the Lorentz transformation.\nBoth e\u000bects, proving the relativistic formulae\n!0=!\u0000Vkq\n1\u0000V2\nc2; k0=k\u0000V\nc2!q\n1\u0000V2\nc2;\nhave been checked experimentally by Ives and Stilwell in 1937, then con\frmed in many more precise\nexperiences. Reliable experimental con\frmations of the validity of Lorentz transformations concern\nmeasurable quantities such as charges, currents, energies (frequencies) and momenta (wave vectors)\nmuch more than the less intrinsic quantities which are the di\u000berentials of the space-time variables. In\nprinciple, the Lorentz transformations could have been established by very precise observations of the\nDoppler e\u000bect alone.\nIt should be stressed that had we only the light at our disposal, i.e. massless photons propagating\nwith the same velocity c, we would infer that the general symmetry of physical phenomena is the\nConformal Group , and not the Poincar\u0013 e group. To the observations of light must be added the the\nprinciple of inertia , i.e. the existence of massive bodies moving with speeds lower than c, and constant\nif not sollicited by external in\ruence.\nTranslated into the modern language of particles and \felds this means that besides the massless\nphotons massive particles must exist, too. The distinctive feature of such particles is their inertial\nmass , equivalent with their energy at rest, which can be measured classically via Newton's law, whose\nfundamental equation a=1\nmF:relates the only observable quantity (using clocks and light rays as\nmeasuring rods), the acceleration a, with a combination of less evidently de\fned quantities, mass and\nforce, which is interpreted as a causality relation , the force being the cause, and acceleration the e\u000bect.\nIt turned out soon that the force Fmay symbolize the action of quite di\u000berent physical phenomena\nlike gravitation, electricity or inertia, and is not a primary cause, but rather a manner of intermediate\nbookkeeping. The more realistic sources of acceleration - or rather of the variation of energy and momenta\n- are the intensities of electric, magnetic or gravitational \felds. The di\u000berential form of the Lorentz force,\ncombined with the energy conservation of a charged particle under the in\ruence of electromagnetic \feld\ndp\ndt=qE+qv\nc^BdE\ndt=qE\u0001v (2)\nis also Lorentz-invariant:\ndp\u0016=q\nmcF\u0016\n\u0017p\u0017; (3)\nwherep\u0016= [p0;p] is the four-momentum and F\u0016\n\u0017is the Maxwell-Faraday tensor.\nThese are the fundamental physical quantities that impose the Lorentz-Poincar\u0013 e group of transfor-\nmations, which are imprinted on the dual space which we call space and time variables.\n3 Combinatorics and covariance\nSince the advent of quantum theory the discrete view of phenomena observed on microscopic level took\nover the continuum view prevailing in the nineteenth century physics. The dichotomy between discrete\nand continuous symmetries has become a major issue in quantum \feld theory, of which the fundamental\nspin and statistics theorem provides the best illustration. It stipulates that \felds describing particles\nwhich obey the Fermi-Dirac statistics, called fermions , transform under the half-integer representations\n3of the Lorentz group, whereas \felds describing particles which obey the Bose-Einstein statistics, bosons ,\nmust transform under the integer representations of the Lorentz group.\nThe fundamental principle ensuring the existence of electron shells and the Periodic Table is the\nexclusion principle formulated by Pauli: fermionic operators must satisfy the anti-commutation relations\n\ta\tb=\u0000\tb\tawhich means that two electrons cannot coexist in the same state [6].\nWith two possible values of spin for the electron in each state, the total number of states corresponding\nto each shell (i.e. principal quantum number n) becomes 2 n2. which is the basis of Mendeleev's periodical\nsystem, and of resulting stability of matter [7].\nQuantum Mechanics started as a non-relativistic theory, but very soon its relativistic generalization\nwas created. As a result, the wave functions in the Schroedinger picture were required to belong to one\nof the linear representations of the Lorentz group, which means that they must satisfy the following\ncovariance principle :\n~ (~x) =~ (\u0003(x)) =S(\u0003) (x):\nThe nature of the representation S(\u0003) determines the character of the \feld considered: spinorial,\nvectorial, tensorial... As in many other fundamental relations, the seemingly simple equation\n~ (~x) =~ (\u0003(x)) =S(\u0003) (x):\ncreates a bridge between two totally di\u000berent realms: the space-time accessible via classical macroscopic\nobservations, and the Hilbert space of quantum states. It can be interpreted in two opposite ways,\ndepending on which side we consider as the cause, and which one as the consequence.\nA question can be asked, what is the cause, and what is the e\u000bect, not only in mathematical terms,\nbut also in a deeper physical sense. In other words, is the macroscopically observed Lorentz symmetry\nimposed on the micro-world of quantum physics, or maybe it is already present as symmetry of quantum\nstates, and then implemented and extended to the macroscopic world in classical limit ? In such a case,\nthe covariance principle should be written as follows:\n\u0003\u00160\n\u0016(S)j\u0016=j\u00160( 0) =j\u00160(S( ));\nIn the above formula j\u0016=\u0016 \r\u0016 is the Dirac current,  is the electron wave function.\nIn view of the analysis of the causal chain, it seems more appropriate to write the same transforma-\ntions with \u0003 depending on S:\n 0(x\u00160) = 0(\u0003\u00160\n\u0017(S)x\u0017) =S (x\u0017) (4)\nThis form of the same relation suggests that the transition from one quantum state to another, rep-\nresented by the transformation Sis the primary cause that implies the transformation of observed\nquantities such as the electric 4-current, and as a \fnal consequence, the apparent transformations of\ntime and space intervals measured with classical physical devices.\nThe Pauli exclusion principle gives a hint about how it might work. In its simplest version, it\nintroduces an anti-symmetric form on the Hilbert space describing electron's states:\n\u000f\u000b\f=\u0000\u000f\f\u000b; \u000b;\f = 1;2;\u000f12= 1;\nNow, if we require that Pauli's principle must apply independently of the choice of a basis in Hilbert\nspace, i.e. that after a linear transformation we get\n\u000f\u000b0\f0=S\u000b0\n\u000bS\f0\n\f\u000f\u000b\f=\u0000\u000f\f0\u000b0; \u000f1020= 1;\nthen the matrix S\u000b0\n\u000bmust have the determinant equal to 1, which de\fnes the SL(2;C) group.\nThe existence of two internal degrees of freedom had to be taken into account in fundamental equation\nde\fning the relationship between basic operators acting on electron states. To acknowledge this, Pauli\nproposed the simplest equation expressing the relation between the energy, momentum and spin:\nE =mc2 +\u001b\u0001p : (5)\n4The existence of anti-particles (in this case the positron), suggests the use of the non-equivalent\nrepresentation of SL(2;C) group by means of complex conjugate matrices. along with the time reversal,\nthe Dirac equation can be now constructed. It is invariant under the Lorentz group.\nE +=mc2 ++\u001b\u0001p \u0000;\u0000E \u0000=mc2 \u0000+\u001b\u0001p + (6)\nAlthough mathematically the two formulations are equivalent, it seems more plausible that the Lorentz\ngroup resulting from the averaging of the action of the SL(2;C) in the Hilbert space of states contains\nless information than the original double-valued representation which is a consequence of the particle-\nanti-particle symmetry, than the other way round. In what follows, we shall draw physical consequences\nfrom this approach, concerning the strong interactions in the \frst place.\nIn purely algebraical terms Pauli's exclusion principle amounts to the anti-symmetry of wave functions\ndescribing two coexisting particle states. The easiest way to see how the principle works is to apply\nDirac's formalism in which wave functions of particles in given state are obtained as products between\nthe \\bra\" and \\ket\" vectors. Consider the wave function of a particle in the state jx>,\n\b(x) =< jx>: (7)\nA two-particle state of ( jx>;jy) is a tensor product\nj >=X\n\b(x;y) (jx>\njy>): (8)\nIf the wave function \b( x;y) is anti-symmetric, i.e. if it satis\fes\n\b(x;y) =\u0000\b(y;x); (9)\nthen \b(x;x) = 0 and such states have vanishing probability.\nConversely, suppose that \b( x;x) does vanish. This remains valid in any basis provided the new basis\njx0>;jy0>was obtained from the former one via unitary transformation.\nLet us form an arbitrary state being a linear combination of jx>andjy>,\njz>=\u000bjx>+\fjy>; \u000b;\f2C;\nand let us form the wave function of a tensor product of such a state with itself:\n\b(z;z) =< j(\u000bjx>+\fjy>)\n(\u000bjx>+\fjy>); (10)\nwhich develops as follows:\n\u000b2< jx;x> +\u000b\f < jx;y>\n+\f\u000b < jy;x> +\f2< jy;y> =\n=\u000b2\b(x;x) +\u000b\f\b(x;y) +\f\u000b\b(y;x) +\f2\b(y;y): (11)\nNow, as \b( x;x) = 0 and \b( y;y) = 0, the sum of remaining two terms will vanish if and only if (9) is\nsatis\fed, i.e. if \b( x;y) is anti-symmetric in its two arguments.\nAfter second quantization, when the states are obtained with creation and annihilation operators\nacting on the vacuum, the anti-symmetry is encoded in the anti-commutation relations\n (x) (y) + (y) (x) = 0 (12)\nwhere (x)j0>=jx>.\nAccording to present knowledge, the ultimate undivisible and undestructible constituents of matter,\ncalled atoms by ancient Greeks, are in fact the QUARKS, carrying fractional electric charges and baryonic\nnumbers, two features that appear to be undestructible and conserved under any circumstances.\n5Taking into account that quarks evolve inside nucleons as almost point-like objects, one may wonder\nhow the notions of space and time still apply in these conditions ? Perhaps in this case, too, the Lorentz\ninvariance can be derived from some more fundamental discrete symmetries underlying the interactions\nbetween quarks ? If this is the case, then the symmetry Z3must play a fundamental role.\nIn Quantum Chromodynamics quarks are considered as fermions, endowed with spin1\n2. Only three\nquarks or anti-quarks can coexist inside a fermionic baryon (respectively, anti-baryon), and a pair quark-\nantiquark can form a meson with integer spin. Besides, they must belong to di\u000berent colors , also a\nthree-valued set. There are two quarks in the \frst generation, uandd(\\up\" and \\down\"), which may\nbe considered as two states of a more general object, just like proton and neutron in SU(2) symmetry\nare two isospin components of a nucleon doublet.\nThis suggests that a convenient generalization of Pauli's exclusion principle would be that no three\nquarks in the same state can be present in a nucleon.\nLet us require then the vanishing of wave functions representing the tensor product of three (but not\nnecessarily two) identical states. That is, we require that \b( x;x;x ) = 0 for any state jx >. As in the\nformer case, consider an arbitrary superposition of three di\u000berent states, jx>;jy>andjz>,\njw>=\u000bjx>+\fjy>+\rjz>\nand apply the same criterion, \b( w;w;w ) = 0.\nWe get then, after developing the tensor products,\n\b(w;w;w ) =\u000b3\b(x;x;x ) +\f3\b(y;y;y ) +\r3\b(z;z;z )\n+\u000b2\f[\b(x;x;y ) + \b(x;y;x ) + \b(y;x;x )] +\r\u000b2[\b(x;x;z ) + \b(x;z;x ) + \b(z;x;x )]\n+\u000b\f2[\b(y;y;x ) + \b(y;x;y ) + \b(x;y;y )] +\f2\r[\b(y;y;z ) + \b(y;z;y ) + \b(z;y;y )]\n+\f\r2[\b(y;z;z ) + \b(z;z;y ) + \b(z;y;z )] +\r2\u000b[\b(z;z;x ) + \b(z;x;z ) + \b(x;z;z )]\n+\u000b\f\r[\b(x;y;z ) + \b(y;z;x ) + \b(z;x;y ) + \b(z;y;x ) + \b(y;x;z ) + \b(x;z;y )] = 0:\nThe terms \b( x;x;x );\b(y;y;y ) and \b(z;z;z ) do vanish by virtue of the original assumption; in what\nremains, combinations preceded by various powers of independent numerical coe\u000ecients \u000b;\fand\r, must\nvanish separately.\nThis is achieved if the following Z3symmetry is imposed on our wave functions:\n\b(x;y;z ) =j\b(y;z;x ) =j2\b(z;x;y ):\nwithj=e2\u0019i\n3; j3= 1; j+j2+ 1 = 0:\nNote that the complex conjugates of functions \b( x;y;z ) transform under cyclic permutations of their\narguments with j2=\u0016jreplacingjin the above formula\n\t(x;y;z ) =j2\t(y;z;x ) =j\t(z;x;y ):\nInside a hadron, not two, but three quarks in di\u000berent states (colors) can coexist.\nAfter second quantization, when the \felds become operator-valued, an alternative cubic commutation\nrelations seems to be more appropriate:\nInstead of \ta\tb= (\u00001) \tb\tawe can introduce \u0012A\u0012B\u0012C=j\u0012B\u0012C\u0012A=j2\u0012C\u0012A\u0012B;withj=e2\u0019i\n3\n4 Quark algebra\nOur aim now is to derive the space-time symmetries from minimal assumptions concerning the properties\nof the most elementary constituents of matter, and the best candidates for these are quarks.\nTo do so, we should explore algebraic structures that would privilege cubic orternary relations, in\nother words, \fnd appropriate cubic or ternary algebras re\recting the most important properties of quark\n6states. The minimal requirements for the de\fnition of quarks at the initial stage of model building are\nthe following:\ni) The mathematical entities representing the quarks form a linear space over complex numbers, so\nthat we could form their linear combinations with complex coe\u000ecients.\nii) They should also form an associative algebra, so that we could form their multilinear combinations;\niii) There should exist two isomorphic algebras of this type corresponding to quarks and anti-quarks,\nand the conjugation that maps one of these algebras onto another, A! \u0016A.\niv) The three quark (or three anti-quark) and the quark-anti-quark combinations should be distin-\nguished in a certain way, for example, they should form a subalgebra in the enveloping algebra spanned\nby the generators.\nThe fact that hadrons obeying the Fermi statistics (protons and neutrons, to begin with) are com-\nposed of three quarks raises naturally the question how their quantum states respond to permutations\nbetween these elementary components.\nThe symmetric group S3containing all permutations of three di\u000berent elements is a special case\namong all symmetry groups SN. It is the \frst in the row to be non-abelian, and the last one that possesses\na faithful representation in the complex plane C1. It contains six elements, and can be generated with\nonly two elements, corresponding to one cyclic and one odd permutation, e.g. ( abc)!(bca), and\n(abc)!(cba). All permutations can be represented as di\u000berent operations on complex numbers as\nfollows.\nLet us denote the primitive third root of unity by j=e2\u0019i=3.\nThe cyclic abelian subgroup Z3contains three elements corresponding to the three cyclic permuta-\ntions, which can be represented via multiplication by j,j2andj3= 1 (the identity).\n\u0012ABC\nABC\u0013\n!1;\u0012ABC\nBCA\u0013\n!j;\u0012ABC\nCAB\u0013\n!j2; (13)\nOdd permutations must be represented by idempotents, i.e. by operations whose square is the identity\noperation. We can make the following choice:\n\u0012ABC\nCBA\u0013\n!(z!\u0016 z);\u0012ABC\nBAC\u0013\n!(z!^ z);\u0012ABC\nCBA\u0013\n!(z!z\u0003); (14)\nHere the bar ( z!\u0016 z) denotes the complex conjugation, i.e. the re\rection in the real line, the hat z!^ zdenotes\nthe re\rection in the root j2, and the star z!z\u0003the re\rection in the root j. The six operations close in a\nnon-abelian group with six elements. However, if it acts on three objects out of which two are identical, e.g.\n(AAB ), then odd permutations give the same result as even ones, so that only the Z3cyclic abelian group is\noperating, With this in mind, let us de\fne the following Z3-graded algebra introducing Ngenerators spanning\na linear space over complex numbers, satisfying the following cubic relations:\n\u0012A\u0012B\u0012C=j\u0012B\u0012C\u0012A=j2\u0012C\u0012A\u0012B; (15)\nwithj=e2i\u0019=3, the primitive root of 1. We have obviously 1 + j+j2= 0 and\u0016j=j2.\nWe shall also introduce a similar set of conjugate generators, \u0016\u0012_A,_A;_B;::: = 1;2;:::;N , satisfying\nsimilar condition with j2replacingj:\n\u0016\u0012_A\u0016\u0012_B\u0016\u0012_C=j2\u0016\u0012_B\u0016\u0012_C\u0016\u0012_A=j\u0016\u0012_C\u0016\u0012_A\u0016\u0012_B; (16)\nLet us denote this algebra by A.\nWe shall endow it with a natural Z3grading, considering the generators \u0012Aas grade 1 elements, their\nconjugates \u0016\u0012_Abeing of grade 2. The grades add up modulo 3; the products \u0012A\u0012Bspan a linear subspace\nof grade 2, and the cubic products \u0012A\u0012B\u0012Care of grade 0.\nSimilarly, all quadratic expressions in conjugate generators, \u0016\u0012_A\u0016\u0012_Bare of grade 2 + 2 = 4 mod3= 1,\nwhereas their cubic products are again of grade 0, like the cubic products od \u0012A's.\n7Combined with the associativity, these cubic relations impose \fnite dimension on the algebra gener-\nated by the Z3graded generators. As a matter of fact, cubic expressions are the highest order that does\nnot vanish identically. The proof is immediate:\n\u0012A\u0012B\u0012C\u0012D=j\u0012B\u0012C\u0012A\u0012D=j2\u0012B\u0012A\u0012D\u0012C=j3\u0012A\u0012D\u0012B\u0012C=j4\u0012A\u0012B\u0012C\u0012D; (17)\nand because j4=j6= 1, the only solution is \u0012A\u0012B\u0012C\u0012D= 0:\nThe total dimension of the algebra de\fned via the cubic relations (15) is equal to N+N2+(N3\u0000N)=3:\ntheNgenerators of grade 1, the N2independent products of two generators, and ( N3\u0000N)=3 independent\ncubic expressions, because the cube of any generator must be zero by virtue of (15), and the remaining\nN3\u0000Nternary products are divided by 3, also by virtue of the constitutive relations (15).\nThe conjugate generators \u0016\u0012_Bspan an algebra \u0016Aisomorphic withA.\nIf we want the products between the generators \u0012Aand the conjugate ones \u0016\u0012_Bto be included into the\ngreater algebra spanned by both types of generators, we should consider all possible products, between\nboth types of generators, which will span the resulting algebra A\n \u0016A.\nThe fact that the conjugate generators are endowed with grade 2 could suggest that they behave\njust like the products of two ordinary generators \u0012A\u0012B. However, such a choice does not enable one to\nmake a clear distinction between the conjugate generators and the products of two ordinary ones, and\nit would be much better, to be able to make the di\u000berence.\nDue to the binary nature of the products, another choice is possible, namely, to require the following\ncommutation relations:\n\u0012A\u0016\u0012_B=\u0000j\u0016\u0012_B\u0012A; \u0016\u0012_B\u0012A=\u0000j2\u0012A\u0016\u0012_B; (18)\nIn fact, introducing the \\minus\" sign, i.e. the multiplication by \u00001, we extend the discrete symmetry\ngroup acting on our algebra to the product Z3\u0002Z2. It is easy to prove that this product is isomorphic\nwith the cyclic group Z6. The choice of commutation relations (18) leads to the anticommutation\nproperty between the conjugate cubic monomials:\n\u0000\n\u0012A\u0012B\u0012C\u0001\u0010\n\u0016\u0012_D\u0016\u0012_E\u0016\u0012_F\u0011\n=\u0000\u0010\n\u0016\u0012_D\u0016\u0012_E\u0016\u0012_F\u0011\u0000\n\u0012A\u0012B\u0012C\u0001\n; (19)\ncharacteristic for the fermions. This is another hint towards the possibility of forming anti-commuting\nfermionic variables with cubic combinations of our \\quark\" operators.\n5 Two-generator algebra and its invariance group\nThe three quarks constituting hadrons (the latter behaving as fermions) are found in two states, \\up\"\nand \\down\", designed by uandd, endowed with fractional electric charges, +2\n3for theu-quark and\u00001\n3\nfor thed-quark. Therefore the product state uudwill represent a proton (electric charge +1), whilst the\ncombination uddhaving zeo electric charge represents a neutron. We shall therefore reduce the number\nof generators of our Z3-graded algenra representing quark operators, to the minimal number, i.e. two\ngenerators only.\nLet us consider the simplest case of cubic algebra with two generators, A;B;::: = 1;2. Its grade 1\ncomponent contains just these two elements, \u00121and\u00122; its grade 2 component contains four indepen-\ndent products, \u00121\u00121; \u00121\u00122; \u00122\u00121;and\u00122\u00122:Finally, its grade 0 component (which is a subalgebra)\ncontains the unit element 1 and the two linearly independent cubic products, \u00121\u00122\u00121=j\u00122\u00121\u00121=\nj2\u00121\u00121\u00122and\u00122\u00121\u00122=j\u00121\u00122\u00122=j2\u00122\u00122\u00121:with similar two independent combinations of conjugate\ngenerators \u0016\u0012_A:\nLet us consider multilinear forms de\fned on the algebra A\n \u0016A. Because only cubic relations are\nimposed on products in Aand in \u0016A, and the binary relations on the products of ordinary and conjugate\nelements, we shall \fx our attention on tri-linear and bi-linear forms, conceived as mappings of A\n \u0016A\n8into certain linear spaces over complex numbers. Consider a tri-linear form \u001a\u000b\nABC. We shall call this\nformZ3-invariant if we can write, by virtue of (15):\n\u001a\u000b\nABC\u0012A\u0012B\u0012C=1\n3\u0014\n\u001a\u000b\nABC\u0012A\u0012B\u0012C+\u001a\u000b\nBCA\u0012B\u0012C\u0012A+\u001a\u000b\nCAB\u0012C\u0012A\u0012B\u0015\n=\n=1\n3\u0014\n\u001a\u000b\nABC\u0012A\u0012B\u0012C+\u001a\u000b\nBCA(j2\u0012A\u0012B\u0012C) +\u001a\u000b\nCABj(\u0012A\u0012B\u0012C)\u0015\n;\nFrom this it follows that we should have\n\u001a\u000b\nABC\u0012A\u0012B\u0012C=1\n3\u0014\n\u001a\u000b\nABC +j2\u001a\u000b\nBCA +j\u001a\u000b\nCAB\u0015\n\u0012A\u0012B\u0012C; (20)\nfrom which we get the following properties of the \u001a-cubic matrices:\n\u001a\u000b\nABC =j2\u001a\u000b\nBCA =j\u001a\u000b\nCAB: (21)\nEven in this minimal and discrete case, there are covariant and contravariant indices: the lower and the\nupper indices display the inverse transformation property. If a given cyclic permutation is represented\nby a multiplication by jfor the upper indices, the same permutation performed on the lower indices is\nrepresented by multiplication by the inverse, i.e. j2, so that they compensate each other.\nSimilar reasoning leads to the de\fnition of the conjugate forms \u0016 \u001a_\u000b\n_C_B_Asatisfying the relations similar\nto (21) with jreplaced be its conjugate, j2:\n\u0016\u001a_\u000b\n_A_B_C=j\u0016\u001a_\u000b\n_B_C_A=j2\u0016\u001a_\u000b\n_C_A_B(22)\nIn the simplest case of two generators, the j-skew-invariant forms have only two independent components:\n\u001a1\n121=j\u001a1\n211=j2\u001a1\n112; \u001a2\n212=j\u001a2\n122=j2\u001a2\n221;\nand we can set\n\u001a1\n121= 1; \u001a1\n211=j2; \u001a1\n112=j; \u001a2\n212= 1; \u001a2\n122=j2; \u001a2\n221=j:\nThe constitutive cubic relations between the generators of the Z3graded algebra can be considered as\nintrinsic if they are conserved after linear transformations with commuting (pure number) coe\u000ecients,\ni.e. if they are independent of the choice of the basis.\nLetUA0\nAdenote a non-singular N\u0002Nmatrix, transforming the generators \u0012Ainto another set of\ngenerators, \u0012B0=UB0\nB\u0012B.\nWe are looking for the solution of the covariance condition for the \u001a-matrices:\nS\u000b0\n\f\u001a\f\nABC =UA0\nAUB0\nBUC0\nC\u001a\u000b0\nA0B0C0: (23)\nNow,\u001a1\n121= 1, and we have two equations corresponding to the choice of values of the index \u000b0equal to\n1 or 2. For \u000b0= 10the\u001a-matrix on the right-hand side is \u001a10\nA0B0C0, which has only three components,\n\u001a10\n102010= 1; \u001a10\n201010=j2; \u001a10\n101020=j;\nwhich leads to the following equation:\nS10\n1=U10\n1U20\n2U10\n1+j2U20\n1U10\n2U10\n1+jU10\n1U10\n2U20\n1=U10\n1(U20\n2U10\n1\u0000U20\n1U10\n2);\nbecausej2+j=\u00001.\nFor the alternative choice \u000b0= 20the\u001a-matrix on the right-hand side is \u001a20\nA0B0C0, whose three non-\nvanishing components are\n\u001a20\n201020= 1; \u001a20\n102020=j2; \u001a20\n202010=j:\n9The corresponding equation becomes now:\nS20\n1=U20\n1U10\n2U20\n1+j2U10\n1U20\n2U20\n1+jU20\n1U20\n2U10\n1=U20\n1(U10\n2U20\n1\u0000U10\n1U20\n2);\nThe remaining two equations are obtained in a similar manner. We choose now the three lower indices\non the left-hand side equal to another independent combination, (212). Then the \u001a-matrix on the left\nhand side must be \u001a2whose component \u001a2\n212is equal to 1. This leads to the following equation when\n\u000b0= 10:\nS10\n2=U10\n2U20\n1U10\n2+j2U20\n2U10\n1U10\n2+jU10\n2U10\n1U20\n2=U10\n2(U10\n2U20\n1\u0000U10\n1U20\n2);\nand the fourth equation corresponding to \u000b0= 20is:\nS20\n2=U20\n2U10\n1U20\n2+j2U10\n2U20\n1U20\n2+jU20\n2U20\n1U10\n2=U20\n2(U10\n1U20\n2\u0000U20\n1U10\n2):\nThe determinant of the 2 \u00022 complex matrix UA0\nBappears everywhere on the right-hand side.\nS20\n1=\u0000U20\n1[det(U)]; (24)\nThe remaining two equations are obtained in a similar manner, resulting in the following:\nS10\n2=\u0000U10\n2[det(U)]; S20\n2=U20\n2[det(U)]: (25)\nThe determinant of the 2 \u00022 complex matrix UA0\nBappears everywhere on the right-hand side. Taking\nthe determinant of the matrix S\u000b0\n\fone gets immediately\ndet (S) = [det (U)]3: (26)\nHowever, the U-matrices on the right-hand side are de\fned only up to the phase, which due to the\ncubic character of the covariance relations (5 - 25), and they can take on three di\u000berent values: 1, j\norj2, i.e. the matrices jUA0\nBorj2UA0\nBsatisfy the same relations as the matrices UA0\nBde\fned above.\nThe determinant of Ucan take on the values 1 ; jorj2ifdet(S) = 1 But for the time being, we have\nno reason yet to impose the unitarity condition. It can be derived from the conditions imposed on the\ninvariance and duality.of binary relations between \u0012Aand their conjugates \u0016\u0012_B.\nIn the Hilbert space of spinors the SL(2;C) action conserved naturally two anti-symmetric tensors,\n\"\u000b\fand\"_\u000b_\fand their duals \"\u000b\fand\"_\u000b_\f:\nSpinorial indeces thus can be raised or lowered using these fundamental SL(2;C) tensors:\n \f=\u000f\u000b\f \u000b;  _\u000e=\"_\u000e_\f _\f:\nIn the space of quark states similar invariant form can be introduced, too. Theere is only one\nalternative: either the Kronecker delta, or the anti-symmetric 2-form \". Supposing that our cubic\ncombinations of quark states behave like fermions, there is no choice left: if we want to de\fne the duals\nof cubic forms \u001a\u000b\nABC displaying the same symmetry properties, we must impose the covariance principle\nas follows:\n\u000f\u000b\f\u001a\u000b\nABC =\"AD\"BE\"CG\u001aDEG\n\f:\nThe requirement of the invariance of tensor \"AB,A;B = 1;2 with respect to the change of basis of quark\nstates leads to the condition det U= 1, i.e. again to the SL(2;C) group.\nA similar covariance requirement can be formulated with respect to the set of 2-forms mapping\nthe quadratic quark-anti-quark combinations into a four-dimensional linear real space. As we already\nsaw, the symmetry (18) imposed on these expressions reduces their number to four. Let us de\fne two\nquadratic forms, \u0019\u0016\nA_Band its conjugate \u0016 \u0019\u0016\n_BA\n\u0019\u0016\nA_B\u0012A\u0016\u0012_Band \u0016\u0019\u0016\n_BA\u0016\u0012_B\u0012A: (27)\n10The Greek indices \u0016;\u0017::: take on four values, and we shall label them 0 ;1;2;3. The four tensors \u0019\u0016\nA_Band\ntheir hermitina conjugates \u0016 \u0019\u0016\n_BAde\fne a bi-linear mapping from the product of quark and anti-quark\ncubic algebras into a linear four-dimensional vector space, whose structure is not yet de\fned. Let us\nimpose the following invariance condition:\n\u0019\u0016\nA_B\u0012A\u0016\u0012_B= \u0016\u0019\u0016\n_BA\u0016\u0012_B\u0012A: (28)\nIt follows immediately from (18) that\n\u0019\u0016\nA_B=\u0000j2\u0016\u0019\u0016\n_BA: (29)\nSuch matrices are non-hermitian, and they can be realized by the following substitution:\n\u0019\u0016\nA_B=j2i\u001b\u0016\nA_B;\u0016\u0019\u0016\n_BA=\u0000ji\u001b\u0016\n_BA(30)\nwhere\u001b\u0016\nA_Bare the unit 2 matrix for \u0016= 0, and the three hermitian Pauli matrices for \u0016= 1;2;3.\nAgain, we want to get the same form of these four matrices in another basis. Knowing that the lower\nindicesAand _Bundergo the transformation with matrices UA0\nBand \u0016U_A0\n_B, we demand that there exist\nsome 4\u00024 matrices \u0003\u00160\n\u0017representing the transformation of lower indices by the matrices Uand \u0016U:\n\u0003\u00160\n\u0017\u0019\u0017\nA_B=UA0\nA\u0016U_B0\n_B\u0019\u00160\nA0_B0; (31)\nThis de\fnes the vector (4 \u00024) representation of the Lorentz group. The system (31) contains four groups\nof four equations each, fgollowing the choice of values for indices \u00160on one side, and the indices Aand\nB. We shall show explicitly only the \frst four equations relating the 4 \u00024 real matrices \u0003\u00160\n\u0017with the\n2\u00022 complex matrices UA0\nBand \u0016U_A0\n_B, corresponding to the value \u00160= 00:\n\u000300\n0+ \u000300\n3=U10\n1\u0016U_10\n_1+U20\n1\u0016U_20\n_1;\u000300\n0\u0000\u000300\n3=U10\n2\u0016U_10\n_2+U20\n2\u0016U_20\n_2;\n\u000300\n0\u0000i\u000300\n2=U10\n1\u0016U_10\n_2+U20\n1\u0016U_20\n_2;\u000300\n0+i\u000300\n2=U10\n2\u0016U_10\n_1+U20\n2\u0016U_20\n_1(32)\nThe next three groups of four equations are similar to the above.\nWith the invariant \\spinorial metric\" in two complex dimensions, \"ABand\"_A_Bsuch that\"12=\n\u0000\"21= 1 and\"_1_2=\u0000\"_2_1, we can de\fne the contravariant components \u0019\u0017 A_B. It is easy to show that\nthe Minkowskian space-time metric, invariant under the Lorentz transformations, can be de\fned as\ng\u0016\u0017=1\n2\u0014\n\u0019\u0016\nA_B\u0019\u0017 A_B\u0015\n=diag(+;\u0000;\u0000;\u0000) (33)\nTogether with the anti-commuting spinors  \u000bthe four real coe\u000ecients de\fning a Lorentz vector, x\u0016\u0019\u0016\nA_B,\ncan generate now the supersymmetry via standard de\fnitions of super-derivations.\nLet us then choose the matrices S\u000b0\n\fto be the usual spinor representation of the SL(2;C) group,\nwhile the matrices UA0\nBwill be de\fned as follows:\nU10\n1=jS10\n1;U10\n2=\u0000jS10\n2;U20\n1=\u0000jS20\n1;U20\n2=jS20\n2; (34)\nthe determinant of Ubeing equal to j2. Obviously, the same reasoning leads to the conjugate cubic\nrepresentation of the same symmetry group SL(2;C) if we require the covariance of the conjugate tensor\n\u0016\u001a_\f\n_D_E_F=j\u0016\u001a_\f\n_E_F_D=j2\u0016\u001a_\f\n_F_D_E;\nby imposing the equation similar to (23)\n\u0016S_\u000b0\n_\f\u0016\u001a_\f\n_A_B_C= \u0016\u001a_\u000b0\n_A0_B0_C0\u0016U_A0\n_A\u0016U_B0\n_B\u0016U_C0\n_C: (35)\n11The matrix \u0016Uis the complex conjugate of the matrix U, and its determinant is j. Moreover, the\ntwo-component entities obtained as images of cubic combinations of quarks,  \u000b=\u001a\u000b\nABC\u0012A\u0012B\u0012Cand\n\u0016 _\f= \u0016\u001a_\f\n_D_E_F\u0016\u0012_D\u0016\u0012_E\u0016\u0012_Fshould anti-commute, because their arguments do so, by virtue of (18):\n(\u0012A\u0012B\u0012C)(\u0016\u0012_D\u0016\u0012_E\u0016\u0012_F) =\u0000(\u0016\u0012_D\u0016\u0012_E\u0016\u0012_F)(\u0012A\u0012B\u0012C)\nWe have found the way to derive the covering group of the Lorentz group acting on spinors via the usual\nspinorial representation. The spinors are obtained as the homomorphic image of tri-linear combination\nof three quarks (or anti-quarks). The quarks transform with matrices U(or\u0016Ufor the anti-quarks), but\nthese matrices are not unitary: their determinants are equal to j2orj, respectively. So, quarks cannot\nbe put on the same footing as classical spinors; they transform under a Z3-covering of the Lorentz group.\n6 A Z3generalization of Dirac's equation\nLet us \frst underline the Z2symmetry of Maxwell and Dirac equations, which implies their hyperbolic\ncharacter, which makes the propagation possible. Maxwell's equations in vacuo can be written as follows:\n1\nc@E\n@t=r^B;\u00001\nc@B\n@t=r^E: (36)\nThese equations can be decoupled by applying the time derivation twice, which in vacuum, where\ndivE= 0 anddivB= 0 leads to the d'Alembert equation for both components separately:\n1\nc2@2E\n@t2\u0000r2E= 0;1\nc2@2B\n@t2\u0000r2B= 0:\nNevertheless, neither of the components of the Maxwell tensor, be it EorB, can propagate separately\nalone. It is also remarkable that although each of the \felds EandBsatis\fes a second-order propagation\nequation, due to the coupled system (36) there exists a quadratic combination satisfying the forst-order\nequation, the Poynting four-vector:\nP\u0016=\u0002\nP0;P\u0003\n; P0=1\n2\u0000\nE2+B2\u0001\n;P=E^B;with@\u0016P\u0016= 0:\nThe Dirac equation for the electron displays a similar Z2symmetry, with two coupled equations which\ncan be put in the following form:\ni~@\n@t +\u0000mc2 +=i~\u001b\u0001r \u0000;\u0000i~@\n@t \u0000\u0000mc2 \u0000=\u0000i~\u001b\u0001r +; (37)\nwhere +and \u0000are the positive and negative energy components of the Dirac equation; this is visible\neven better in the momentum representation:\n\u0002\nE\u0000mc2\u0003\n +=c\u001b\u0001p \u0000;\u0002\n\u0000E\u0000mc2\u0003\n \u0000=\u0000c\u001b\u0001p +: (38)\nThe same e\u000bect (negative energy states) can be obtained by changing the direction of time, and putting\nthe minus sign in front of the time derivative, as suggested by Feynman.\nEach of the components satis\fes the Klein-Gordon equation, obtained by successive application of\nthe two operators and diagonalization:\n\u00141\nc2@2\n@t2\u0000r2\u0000m2\u0015\n \u0006= 0\nAs in the electromagnetic case, neither of the components of this complex entity can propagate by itself;\nonly all the components can.\n12Apparently, the two types of quarks, uandd, cannot propagate freely, but can form a freely propa-\ngating particle perceived as a fermion, only under an extra condition: they must belong to three di\u000berent\nspecies called colors ; short of this they will not form a propagating entity.\nTherefore, quarks should be described by three \felds satisfying a set of coupled linear equations,\nwith theZ3-symmetry playing a similar role of the Z2-symmetry in the case of Maxwell's and Dirac's\nequations. Instead of the \\-\" sign multiplying the time derivative, we should use the cubic root of unity\njand its complex conjugate j2according to the following scheme:\n@\n@tj >=^H12j\u001e>; j@\n@tj\u001e>=^H23j\u001f>; j2@\n@tj\u001f>=^H31j >: (39)\nWe do not specify yet the number of components in each state vector, nor the character of the hamiltonian\noperators on the right-hand side; the three \felds j >,j\u001e>andj\u001f>should represent the three colors,\nnone of which can propagate by itself.\nThe quarks being endowed with mass, we can suppose that one of the main terms in the hamiltonians\nis the mass operator ^ m; and let us suppose that the remaining parts are the same in all three hamiltonians.\nThis will lead to the following three equations:\n@\n@tj >\u0000^mj >=^Hj\u001e>; j@\n@tj\u001e>\u0000^mj\u001e>=^Hj\u001f>; j2@\n@tj\u001f>\u0000^mj\u001f>=^Hj >:\nSupposing that the mass operator commutes with time derivation, by applying three times the left-hand\nside operators, each of the components satis\fes the same common third order equation:\n\u0014@3\n@t3\u0000^m3\u0015\nj >=^H3j >: (40)\nThe anti-quarks should satisfy a similar equation with the negative sign for the Hamiltonian operator.\nThe fact that there exist two types of quarks in each nucleon suggests that the state vectors j >,j\u001e>\nandj\u001f>should have two components each. When combined together, the two postulates lead to the\nconclusion that we must have three two-component functions and their three conjugates:\n\u0012 1\n 2\u0013\n;\u0012\u0016 _1\u0016 _2\u0013\n;\u0012'1\n'2\u0013\n;\u0012\u0016'_1\n\u0016'_2\u0013\n;\u0012\u001f1\n\u001f2\u0013\n;\u0012\u0016\u001f_1\n\u0016\u001f_2\u0013\n;\nwhich may represent three colors, two quark states (e.g. \\up\" and \\down\"), and two anti-quark states\n(with anti-colors, respectively). Finally, in order to be able to implement the action of the SL(2;C)\ngroup via its 2\u00022 matrix representation de\fned in the previous section, we choose the Hamiltonian ^H\nequal to the operator \u001b\u0001r, the same as in the usual Dirac equation. The action of the Z3symmetry is\nrepresented by factors jandj2, while theZ2symmetry between particles and anti-particles is represented\nby the \\-\" sign in front of the time derivative. The di\u000berential system that satis\fes all these assumptions\nis as follows:\n\u0000i~@\n@t \u0000mc2 =\u0000i~c(\u001b\u0001r) \u0016'; i ~@\n@t\u0016'\u0000jmc2\u0016'=\u0000i~c(\u001b\u0001r)\u001f;\u0000i~@\n@t\u001f\u0000j2mc2\u001f=\u0000i~c(\u001b\u0001r)\u0016 ;\ni~@\n@t\u0016 \u0000mc2\u0016 =\u0000i~c(\u001b\u0001r)';\u0000i~@\n@t'\u0000j2mc2'=\u0000i~c(\u001b\u0001r)\u0016\u001f; i ~@\n@t\u0016\u001f\u0000jmc2\u0016\u001f=\u0000i~c(\u001b\u0001r) ;(41)\nHere we made a simplifying assumption that the mass operator is just proportional to the identity\nmatrix, and therefore commutes with the operator \u001b\u0001r.\nThe functions  ,'and\u001fare related to their conjugates via the following third-order equations:\n\u0014\n\u0000i@3\n@t3\u0000m3c6\n~3\u0015\n =\u0000i(\u001b\u0001r)3\u0016 = [\u0000i\u001b\u0001r] (\u0001\u0016 );\n\u0014\ni@3\n@t3\u0000m3c6\n~3\u0015\n\u0016 =\u0000i(\u001b\u0001r)3 = [\u0000i\u001b\u0001r] (\u0001 ); (42)\n13and the same, of course, for the remaining wave functions 'and\u001f.\nThe overall Z2\u0002Z3symmetry can be grasped much better if we use the matrix notation, encoding the\nsystem of linear equations (41) as an operator acting on a single vector composed of all the components.\nThen the system (41) can be written with the help of the following 6 \u00026 matrices composed of blocks\nof 3\u00023 matrices as follows:\n\u00000=\u0012I0\n0\u0000I\u0013\n; B =\u0012B10\n0B2\u0013\n; P =\u00120Q\nQT0\u0013\n; (43)\nwithIthe 3\u00023 identity matrix, and the 3 \u00023 matricesB1; B2andQde\fned as follows:\nB1=0\n@1 0 0\n0j0\n0 0j21\nA; B 2=0\n@1 0 0\n0j20\n0 0j1\nA; Q =0\n@0 1 0\n0 0 1\n1 0 01\nA:\nThe matrices B1andQgenerate the algebra of traceless 3 \u00023 matrices with determinant 1, introduced\nby Sylvester and Cayley under the name of nonion algebra . With this notation, our set of equations (41)\ncan be written in a very compact way:\n\u0000i~\u00000@\n@t\t = [Bm\u0000i~Q\u001b\u0001r] \t; (44)\nHere \t is a column vector containing the six \felds, [  ;';\u001f; \u0016 ;\u0016';\u0016\u001f];in this order.\nBut the same set of equations can be obtained if we dispose the six \felds in a 6 \u00026 matrix, on which\nthe operators in (44) act in a natural way:\n\t =\u00120X1\nX20\u0013\n;withX1=0\n@0 0\n0 0\u001e\n\u001f0 01\nA; X 2=0\n@0 0 \u0016\u001f\n\u0016 0 0\n0 \u0016'01\nA (45)\nBy consecutive application of these operators we can separate the variables and \fnd the common equation\nof sixth order that is satis\fed by each of the components:\n\u0000~6@6\n@t6 \u0000m6c12 =\u0000~6\u00013 : (46)\nIdentifying quantum operators of energy and momentum, \u0000i~@\n@t!E;\u0000i~r! p;we can write (46)\nsimply as follows:\nE6\u0000m6c12=jpj6c6: (47)\nThis equation can be factorized showing how it was obtained by subsequent action of the operators of\nthe system, (41):\nE6\u0000m6c12= (E3\u0000m3c6)(E3+m3c6) =\n(E\u0000mc2)(jE\u0000mc2)(j2E\u0000mc2)(E+mc2)(jE+mc2)(j2E+mc2) =jpj6c6:\nThe equation (46) can be solved by separation of variables; the time-dependent and the space-dependent\nfactors have the same structure:\nA1e!t+A2ej!t+A3ej2!t; B 1ek:r+B2ejk:r+B3ej2k:r\nwith!andksatisfying the following dispersion relation:\n!6\nc6=m6c6\n~6+jkj6; (48)\nwhere we have identi\fed E=~!andp=~k. The relation (48) is invariant under the action of\nZ2\u0002Z3=Z6symmetry, because to any solution with given real !andkone can add solutions with\n14!replaced by j!orj2!,jkorj2k, as well as\u0000!; there is no need to introduce also \u0000kinstead of\nkbecause the vector kcan take on all possible directions covering the unit sphere. The nine complex\nsolutions can be displayed in two 3 \u00023 matrices as follows:\n0\nB@e!t\u0000k\u0001re!t\u0000jk\u0001re!t\u0000j2k\u0001r\nej!t\u0000k\u0001rej!t\u0000jk\u0001rej!t\u0000j2k\u0001r\nej2!t\u0000k\u0001rej2!t\u0000k\u0001rej2!t\u0000j2k\u0001r1\nCA;0\nB@e\u0000!t\u0000k\u0001re\u0000!t\u0000jk\u0001re\u0000!t\u0000j2k\u0001r\ne\u0000j!t\u0000k\u0001re\u0000j!t\u0000jk\u0001re\u0000j!t\u0000j2k\u0001r\ne\u0000j2!t\u0000k\u0001re\u0000j2!t\u0000k\u0001re\u0000j2!t\u0000j2k\u0001r1\nCA\nand their nine independent products can be represented in a basis of real functions as\n0\nB@e!t\u0000k\u0001re!t+k\u0001r\n2cos(k\u0001\u0018)e!t+k\u0001r\n2sin(k\u0001\u0018)\ne\u0000!t\n2\u0000k\u0001rcos!\u001c e\u0000!t\n2+k\u0001r\n2cos(!\u001c\u0000k\u0001\u0018)e\u0000!t\n2+k\u0001r\n2cos(!\u001c+k\u0001\u0018)\ne\u0000!t\n2\u0000k\u0001rsin!\u001c e\u0000!t\n2+k\u0001r\n2sin(!\u001c+k\u0001\u0018)e\u0000!t\n2+k\u0001r\n2sin(!\u001c\u0000k\u0001\u0018)1\nCA\nwhere\u001c=p\n3\n2tand\u0018=p\n3\n2kr; similarly for the conjugate solutions (with \u0000!instead of!).\nThe functions displayed in the matrix do not represent a wave; however, one can produce a propa-\ngating solution by forming certain cubic combinations, e.g.\ne!t\u0000k\u0001re\u0000!t\n2+k\u0001r\n2cos(!\u001c\u0000k\u0001\u0018)e\u0000!t\n2+k\u0001r\n2sin(!\u001c\u0000k\u0001\u0018) =1\n2sin(2!\u001c\u00002k\u0001\u0018):\nWhat we need now is a multiplication scheme that would de\fne triple products of non-propagating\nsolutions yielding propagating ones, like in the example given above, but under the condition that the\nfactors belong to three distinct subsets b(which can be later on identi\fed as \\colors\").\nThis can be achieved with the 3 \u00023 matrices of three types, containing the solutions displayed in\nthe matrix, distributed in a particular way, each of the three matrices containing the elements of one\nparticular line of the matrix:\n[A] =0\nB@0 A12e!t\u0000k\u0001r0\n0 0 A23e!t+k\u0001r\n2cosk\u0001\u0018\nA31e!t+k\u0001r\n2sink\u0001\u0018 0 01\nCA (49)\n[B] =0\n@0 B12e\u0000!\n2t+k\u0001r\n2cos(\u001c+k\u0001\u0018) 0\n0 0 B23e\u0000!\n2t\u0000k\u0001rsin\u001c\nB31e!t\u0000k\u0001rcos\u001c 0 01\nA (50)\n[C] =0\nB@0 C12e\u0000!\n2t+k\u0001r\n2cos(u) 0\n0 0 C23e\u0000!\n2t+k\u0001r\n2sin(v)\nC31e\u0000!\n2t+k\u0001r\n2cos(u) 0 01\nCA (51)\nwhere we have set u=\u001c+k\u0001\u0018; v =\u001c\u0000k\u0001\u0018\nNow it is easy to check that in the product of the above three matrices, ABC all real exponentials\ncancel, leaving the periodic functions of the argument \u001c+k\u0001r. The trace of this triple product is equal\ntoTr(ABC ) = [sin\u001ccos(k\u0001r) + cos\u001csin(k\u0001r)] cos(\u001c+k\u0001r) + cos(\u001c+k\u0001r) sin(\u001c+k\u0001r);\nrepresenting a plane wave propagating towards \u0000k. Similar solution can be obtained with the\nopposite direction. From four such solutions one can produce a propagating Dirac spinor. This model\nmakes free propagation of a single quark impossible, (except for a very short distances due to the damping\nfactor), while three quarks can form a freely propagating state.\nAcknowledgments\nThe author expresses his deep gratitude to Michel Dubois-Violette for his enlightning suggestions\nand remarks.\n15References\n[1] Mermin D, \\What is bad about this habit\", Physics Today , p.8-9, May 2009\n[2] Verlinde E, On the Origin of Gravity and the Laws of Newton , arXiv:1001.0785 [hep-th] (2010).\n[3] Born M, Jordan R, Zeitschrift fur Physik 34858-878 (1925); ibidW. Heisenberg, 879-890 (1925)\n[4] J. von Neumann, Mathematical Foundations of Quantum Mechanics , Princeton Univ. Press (1996).\n[5] M. Dubois-Violette, R. Kerner, J. Madore, Journ. Math. Phys. 31, 316-322 (1990); ibid,31, 323-331\n(1990).\n[6] Pauli W. 1940 The Connection Between Spin and Statistics ,Phys. Rev. 58, pp. 716-722\n[7] Dyson F.J. 1967 Journ. Math. Phys. 8, pp.1538-1545\n[8] Gelfand I M, Kapranov M M, Zelevinsky A V 1994 Multidimensional Determinants, Discriminants\nand Resultants (book), Birkhauser, Boston\n[9] Vainerman L, Kerner R 1996 Journal of Mathematical Physics 37(5) pp. 2553-2665\n[10] Kerner R 1991 Comptes Rendus Acad. Sci. Paris. 101237\n[11] Kerner R 1992 Journal of Mathematical Physics ,33(1) pp.403-4011\n[12] Abramov V, Kerner R, Le Roy B 1997 Journal of Mathematical Physics ,38(3) pp. 1650-1669\n[13] Kerner R 1997 Classical and Quantum Gravity 14(1A) pp. A203-A225\n[14] Kerner R 2001 Proceedings of the 23-rd ICGTMP colloquium, Dubna 2000 arXiv:math-ph/0011023\n[15] Kerner R Suzuki O 2012 Int. J. Geom. Methods Mod. Phys. 09, 1261007\n[16] Feynman R P, Phys. Rev. 76, pp.69-789 (1949)\n[17] Cayley A, Cambridge Math. Journ. ,4, (1845), p.1\n[18] Sylvester J J., Johns Hopkins Circ. Journ. ,3, (1883), p.7.\n[19] Nambu Y 1973. Physical Review D 7(8) pp.2405-2412\n16"    },    {        "title": "1509.04217v1.Beliaev_damping_in_quasi_2D_dipolar_condensates.pdf",        "content": "Beliaev damping in quasi-2D dipolar condensates\nRyan M. Wilson1and Stefan Natu2\n1Department of Physics, The United States Naval Academy, Annapolis, MD 21402, USA and\n2Condensed Matter Theory Center and Joint Quantum Institute,\nDepartment of Physics, University of Maryland, College Park, MD 20742, USA\nWe study the e\u000bects of quasiparticle interactions in a quasi-two dimensional (quasi-2D), zero-\ntemperature Bose-Einstein condensate of dipolar atoms, which can exhibit a roton-maxon feature in\nits quasiparticle spectrum. Our focus is the Beliaev damping process, in which a quasiparticle collides\nwith the condensate and resonantly decays into a pair of quasiparticles. Remarkably, the rate for\nthis process exhibits a highly non-trivial dependence on the quasiparticle momentum and the dipolar\ninteraction strength. For weak interactions, the low energy phonons experience no damping, and the\nhigher energy quasiparticles undergo anomalously weak damping. In contrast, the Beliaev damping\nrates become anomalously large for stronger dipolar interactions, as rotons become energetically\naccessible as \fnal states. Further, we \fnd a qualitative anisotropy in the damping rates when the\ndipoles are tilted o\u000b the axis of symmetry. Our study reveals the unconventional nature of Beliaev\ndamping in dipolar condensates, and has important implications for ongoing studies of equilibrium\nand non-equilibrium dynamics in these systems.\nThe quasiparticle picture of \ructuations and excited\nstates in condensed matter systems is a fundamental\nmodern paradigm. Early investigations in this direc-\ntion focused on super\ruid4He, which hosts very low en-\nergy quasiparticles at intermediate wave vectors, termed\n\\rotons\" [1{3]. Rotons were \frst observed in neutron\nscattering experiments with4He [1, 4{6], and are now\nunderstood to emerge in strongly interacting super\ruids\ndue to strong, longer-range two-body correlations [7{9].\nBose-Einstein condensates (BECs) of atoms with large\nmagnetic dipole moments, such as Cr, Er, or Dy, are\nunique in that they are predicted to support roton quasi-\nparticles when con\fned to highly oblate, quasi-two di-\nmensional (quasi-2D) geometries, despite remaining ex-\ntremely dilute and weakly interacting compared to super-\n\ruid4He [10{13]. Thus, mean-\feld theories typically pro-\nvide good descriptions of these systems [14, 15], despite\ntheir treatment of quasiparticles as free, non-interacting\nexcitations. Here, we systematically step beyond the\nmean-\feld approximation, and study the e\u000bect of quasi-\nparticle interactions on the damping of collective excita-\ntions in quasi-2D dipolar condensates, \fnding non-trivial\ne\u000bects beyond the free quasiparticle picture.\nIn 1958, Beliaev \frst presented a theory of the Bose-\ncondensed state that includes quasiparticle interactions,\nshowing how they manifest as e\u000bective condensate-\nmediated processes [16, 17]. An important consequence\nof such interactions is the damping of quasiparticle mo-\ntion, resulting in \fnite lifetimes for collective condensate\nexcitations. Beliaev specialized to the case of isotropic,\nshort-range (contact) interactions, which is relevant for\nalkali atom condensates [18]. A number of subsequent\nworks have following along these lines [19{24], and there\nis notable agreement with experimental work [25, 26].\nHowever, despite a growing interest in the experimen-\ntal study of quantum many-body physics with dipolar\natoms [27{33] and polar molecules [34{38], a systematictheoretical understanding of beyond mean-\feld e\u000bects,\nsuch as quasiparticle damping, is lacking for these sys-\ntems.\nIn this Letter, we present a theory describing these ef-\nfects in a quasi-2D dipolar BEC, and \fnd a number of\nstriking results. When the dipolar interactions are weak,\nthe damping rates are anomalously small, being signi\f-\ncantly less than those of a gas with purely contact inter-\nactions of equal strength. In contrast, when the dipolar\ninteractions are stronger and rotons begin to emerge in\nthe quasiparticle spectrum, the Beliaev damping rates ac-\nquire anomalously large values, though the rotons them-\nselves remain undamped. For all interaction strengths,\nthe low energy phonon modes are immune to damping\n[39]. Additionally, the dipolar interactions can be made\nstrongly anisotropic in the quasi-2D geometry [40]. In\nthis case, the Beliaev damping rates acquire qualitatively\ndi\u000berent character depending on the direction of quasi-\nparticle propagation; this feature has no analog in con-\nventional super\ruids. Our results mark an important\nstep towards understanding the physics of dipolar con-\ndensates beyond the mean-\feld approximation, and have\nimportant implications for both the equilibrium and non-\nequilibrium properties of these novel super\ruids.\nIn the grand canonical ensemble, the Bose gas Hamil-\ntonian is\n^H=Z\ndr^ y(r)\u0012p2\n2m+U(r)\u0000\u0016\u0013\n^ (r)\n+1\n2Z\ndrZ\ndr0^ y(r)^ y(r0)V(r\u0000r0)^ (r0)^ (r):(1)\nHere,mis the atomic mass, U(r) is the external po-\ntential,\u0016is the chemical potential of the gas, and ^ (r)\n(^ y(r)) is the Bose annihilation (creation) operator. For\nfully polarized dipoles with dipole moments d, the two-\nbody interaction potential is V(r) =d2(1\u00003 cos2\u0012)=jrj3,\nwhere\u0012is the angle between randd.arXiv:1509.04217v1  [cond-mat.quant-gas]  14 Sep 20152\nAt ultracold temperatures, the dilute Bose gas can be\ndescribed by a mean-\feld theory with a condensate or-\nder parameter \u001e(r) =h^ (r)i, which evolves under the\nequation of motion,\ni~_\u001e(r) =\u0012p2\n2m+U(r)\u0000\u0016\u0013\n\u001e(r)\n+Z\ndr0V(r\u0000r0)h^\ty(r0)^\t(r0)^\t(r)i:(2)\nUnder the decomposition ^\t(r) =\u001e(r)+ ^'(r), where ^'(r)\nannihilates non-condensed atoms, h^\ty(r0)^\t(r0)^\t(r)i '\nn(r0)\u001e(r) + ~n(r0;r)\u001e(r0), wheren(r) =j\u001e(r)j2+ ~n(r;r) is\nthe total density of the gas and ~ n(r0;r) =h^'y(r0) ^'(r)i\nis the non-condensate density matrix. We work in the\nPopov approximation, and omit the anomalous density\nmatrix ~m(r0;r) =h^'(r0) ^'(r)ifrom the theory [41]. In the\nperturbative framework we employ, the Beliaev damping\nrates are insensitive to this approximation [20, 21].\nSmall amplitude condensate oscillations can be mod-\neled as perturbations \u000e\u001e(r) about the stationary state\nof Eq. (2), denoted \u001e0(r). We obtain equations of mo-\ntion for these condensate oscillations by inserting \u001e(r) =\n\u001e0(r) +\u000e\u001e(r) into Eq. (2) and linearizing about \u000e. If the\ncouplings between \u000e\u001e(r) and the non-condensate density\n~nare ignored, this procedure reproduces the Bogoliubov\nfree-quasiparticle description of small amplitude conden-\nsate oscillations. This description, however, is inadequate\nto describe the damping of condensate oscillations. To\ncorrect this, we couple the condensate oscillations, which\ntake the form of Bogoliubov quasiparticles, to the non-\ncondensate atoms perturbatively in \u000e, following the pro-\ncedures of Refs. [20, 21, 39, 42]. We obtain eigenfrequen-\ncies!0=!+\u000e!, where!are the bare (non-interacting)\nquasiparticle frequencies and \u000e!are frequency shifts that\narise due to quasiparticle interactions. The imaginary\npart of\u000e!corresponds to a damping rate for conden-\nsate oscillations. At T= 0, this is a Beliaev process,\nwhich involves the resonant decay of a quasiparticle into a\npair of quasiparticles under the constraints of energy and\nmomentum conservation [16, 17]. The relevant damping\nrate is thus \rB= Im[\u000e!]T=0. This perturbative scheme\nremains valid for large damping rates, as long as the non-\ncondensate density remains small.\nWe restrict our study to the quasi-2D regime, where\nthe atoms are free to move in-plane but are tightly\ncon\fned in the axial direction by an external potential\nU(r) =m!2\nzz2=2. If ~!zis the dominant energy scale\nin the system, to a good approximation all atoms oc-\ncupy the single-particle ground state in the z-direction,\n\u001f(z) = exp[\u0000z2=2l2\nz]=p\u0019l1=4\nz, wherelz=p\n~=m! z. An\ne\u000bective quasi-2D theory is obtained by separating all\nbosonic \felds into this axial wave function and inte-\ngrating the z-coordinate from the theory [43]. Below,\nwe rescale all lengths in units of lzand all energies in\nunits of ~!z. The condensate order parameter becomes\n0\n1\n2 01200.511.5pxpy(a)ω\n0\n1\n2 012pxpy(b)FIG. 1: (color online). Energetics of Beliaev damping for a\nquasiparticle with momentum (a) plz= 0:9^x, a maxon (black\n+), and (b) plz= 2:2^x, in the free-particle part of the spec-\ntrum (black circle), in a quasi-2D dipolar condensate with\nn0gd= 1:7 and\u000b= 0. The vertical panels show the quasi-\nparticle spectrum (red lines). The dark blue lines show the\nmanifold of decay channels allowed by energy and momentum\nconservation. The light blue lines show the allowed momenta\nonly.\n\u001e(r) =pn0\u001f(z) wheren0is the uniform areal conden-\nsate density, and the condensate oscillations take the\nform\u000e\u001e(r) =\u001f(z)P\np(upei(p\u0001\u001a\u0000!0\npt)+v\u0003\npe\u0000i(p\u0001\u001a\u0000!0\npt)),\nwhere \u001aandpare in-plane spatial and momentum co-\nordinates, respectively. The coe\u000ecients upandvpare\nthe Bogoliubov quasiparticle amplitudes, given by up=p\n\"p=2!p+ 1 andvp=\u0000sgn[~V(p)]p\n\"p=2!p\u00001, where\n\"p=p2=2 +n0gd~V(p). The bare quasiparticle spectrum\nis\n!p=s\np2\n2\u0012p2\n2+ 2gdn0~V\u0012pp\n2\u0013\u0013\n; (3)\nwheregd=p\n8\u0019d2=3 is the quasi-2D dipolar interac-\ntion strength and ~V(p) =F?(p) cos2\u000b+Fk(p) sin2\u000b\nis the quasi-2D momentum-space dipolar interaction po-\ntential, with F?(p) = 2\u00003p\u0019pep2erfc[p] andFk(p) =\n\u00001+3p\u0019(p2\ny=p)ep2erfc[p]. Here, erfc[ p] is the complimen-\ntary error function and \u000bis the polarization tilt angle\nbetween d=d(^zcos\u000b+ ^ysin\u000b) and thez-axis. The Be-\nliaev damping rate for a quasiparticle with momentum p\nis found to be\n\rB;p=2\u0019\n~X\nkqj\u0016Ap\nkqj2\u000e(!p\u0000(!k+!q)); (4)\nwhere \u0016Ap\nkq=Ap\nkq+Ap\nqk, andAp\nkqhas matrix elements\nAp\nkq=\u0019pn0h\nup\u0010\n~V(k)(u\u0003\nku\u0003\nq+v\u0003\nku\u0003\nq) +~V(k+q)v\u0003\nku\u0003\nq\u0011\n+vp\u0010\n~V(k)(v\u0003\nkv\u0003\nq+u\u0003\nkv\u0003\nq) +~V(k+q)u\u0003\nkv\u0003\nq\u0011i\n\u000ep;k+q:\n(5)\nWe take the thermodynamic limit, and evaluate Eq. (4)\nnumerically.\nWe \frst consider a quasi-2D dipolar condensate that\nis polarized perpendicular to the 2D plane ( \u000b= 0). In3\nthis case, an expansion of the small-momentum, phonon\npart of the quasiparticle spectrum gives !p'cdp(1\u0000p\n9\u0019=32p+:::), wherecd=p2n0gdis the phonon speed.\nThis downward curvature prohibits the Beliaev damping\nof phonons, due to the impossibility of simultaneous en-\nergy and momentum conservation. Thus, phonons do\nnot damp in quasi-2D dipolar condensates [39]. This\nis in contrast to quasi-2D condensates with repulsive,\nisotropic contact interactions, which host quasiparticle\nspectra with upward curvature at small momenta, re-\nsulting in Beliaev damping rates /p3at smallp[44].\nAt larger momenta, a local \\roton\" minimum with an\nenergy gap \u0001 rdevelops in the quasiparticle spectrum for\ndipolar interaction strengths n0gd&1:15, which ulti-\nmately softens to \u0001 r= 0 at a momentum pr'1:62 when\nn0gd'1:72. This is accompanied by a local \\maxon\"\nmaximum at p'0:74. An example roton-maxon spec-\ntrum forn0gd= 1:7 is shown in the vertical panels of\nFig. 1 and by the red curve in Fig. 2(d).\nAs the roton minimum develops, the density of quasi-\nparticle states grows signi\fcantly. Near the minimum,\nthe spectrum can be expanded about p\u0018prto give\n!p'\u0001r+ (p\u0000pr)2=2mrwheremris the e\u000bective ro-\nton mass. The density of states near the roton minimum\nis thus\u001ar(!) = 2\u0019mr(1 +pr=p\n2mr(!\u0000!r)). The di-\nvergence of this expression at !=!rcontributes to an\nanomalously large density of states in this vicinity. It\nis instructive to note that the expression for the Beliaev\ndamping rate in Eq. (4) is reminiscent of Fermi's Golden\nRule, which describes the scattering of a quantum state\ninto other \fnal states at a rate proportional to the den-\nsity of available \fnal states. Indeed, the evaluation of\nthe Dirac-delta function in Eq. (4) produces a factor re-\nsembling the density of \fnal quasiparticle states; we thus\nexpect large damping rates for quasiparticles that can\ndecay into rotons.\nIn Fig. 1, we illustrate the manifold of available \f-\nnal quasiparticle states for n0gd= 1:7, which supports\na prominent roton-maxon feature. In panel (a), we con-\nsider a quasiparticle with momentum p= 0:9^x(shown\nby the black + sign in the px-pyplane), which is in the\nmaxon part of the spectrum. Whenever the maxon en-\nergy exceeds 2\u0001 r, it is energetically possible to decay\ninto a pair of rotons. The blue lines, which show the\nenergy and momenta of the available \fnal quasiparti-\ncle states, are centered about the roton minima in the\n+yand\u0000ydirections. This indicates that maxons un-\ndergo Beliaev damping by decaying into a pair of nearly\ncounter-propagating rotons that travel transverse to the\ninitial quasiparticle direction. In panel (b) of Fig. 1, we\nconsider a quasiparticle with momentum p= 2:2^x(black\ncircle), which is in the higher energy, free particle-like\npart of the spectrum. A number of \fnal states are avail-\nable to these quasiparticles; they can decay into rotons,\nmaxons, and phonons. In the latter process, the quasi-\nparticle \\sheds\" low-energy phonons and loses a corre-\n−3−2−1012log10γB(a)\n0 1 200.511.52pω\n  \n(c)1234(b) roton pairs\n  \n0 1 2 3p  \n(d)n0gd=0.1n0gd=0.5n0gd=1.0n0gd=1.5n0gd=1.7FIG. 2: (color online). (a) Beliaev damping rate for weak\ndipolar interactions, where no roton is present in the quasi-\nparticle spectrum; the corresponding spectra are shown in\n(c). (b) Rate for stronger dipolar interactions, where a roton\nis present; the corresponding spectra are shown in (d).\nspondingly small amount of energy and momentum. In\nthe former processes, many combinations are \fnal states\nare possible. Roton-maxon pairs can be produced, or\npairs of forward-propagating quasiparticles; the momenta\nof these \fnal states are shown by the detached blue loop\nin Fig. 1(b).\nWe plot the Beliaev damping rates for a range of dipo-\nlar interaction strengths in Fig. 2; the rates are scaled\nby the axial trap frequency !z. Panel (a) shows rates\nfor quasiparticle spectra that lack roton-maxon features\n(shown in panel (c)). The downward curvature of the\nquasiparticle spectrum forbids phonon damping below\na critical momentum pcrit; this is apparent in all cases\nshown. For small p,pcrit=p\n9\u0019c2\nd, andpcrit\u0018cd\nfor largerp[39]. Additionally, we see evidence that as\nn0gdincreases, the downward curvature of the quasi-\nparticle spectrum becomes more pronounced, and pcrit\nincreases correspondingly. As p!pcritfrom above,\nthe damping rate becomes anomalously large. The only\navailable mechanism for Beliaev damping in this small\nrange of momenta near pcritis the shedding of low energy\nphonons. We attribute these anomalously large damping\nrates to the unique curvature of the quasiparticle spec-\ntrum, which produces a factor resembling a large density\nof phonon states in the evaluation of Eq (4).\nNote that for n0gd= 0:1, the Beliaev damping rates\nare very small, remaining much less than !zin the range\nofpshown. These rates are nearly an order of magnitude\nsmaller than those of a quasi-2D non-dipolar BEC with\nan equivalent chemical potential. As n0gdincreases, the\ndamping rates increase signi\fcantly across the range of\np, which we expect due to the proportionality \rB/g2\nd.\nThe rates at larger pbecome more comparable to those\nof a system with purely contact interactions as n0gd!1.\nPanel (b) of Fig. 2 shows Beliaev damping rates for\nlarger dipolar interaction strengths, which support spec-4\ntra with pronounced roton-maxon features (shown in\npanel (d)). We consider two distinct cases; for n0gd=\n1:5, the maxon energy is less than 2\u0001 r(blue curve), and\nforn0gd= 1:7, the maxon energy is greater than 2\u0001 r\n(red curve). In the former case, it is energetically forbid-\nden for a maxon to decay into a pair of rotons. Thus, all\nlow-energy quasiparticles (phonons, maxons, and rotons)\nremain undamped, and Beliaev damping only occurs for\np&2. For the latter case, a maxon can damp into a pair\nof transverse, counter-propagating rotons. Notice that\nthe red curve in panel (b) is separated into three distinct\nparts. The two left-most parts correspond to Beliaev\ndamping into roton pairs only. These damping rates are\nanomalously large, achieving values well over 100 !zfor\nsome values of p. This is due to the large density of\nstates near the roton minimum. Additionally, the Beli-\naev damping rate vanishes for a range of pnear the roton\nminimum, re\recting the fact that rotons are undamped ,\ndue to their anomalously low energy and large momenta.\nThe black + sign and circle show the Beliaev damping\nrates for quasiparticles with p= 0:9 andp= 2:2 respec-\ntively, corresponding to the discussion of Fig. 1. Though\nthe non-condensate density grows as \u0001 rsoftens [11], it\nremains dilute for the cases we consider here. We thus\nexpect our perturbation theory to remain valid for these\nlarge damping rates.\nBy tilting the external polarizing \feld o\u000b axis ( \u000b6= 0),\nthe dipolar interactions can be made strongly anisotropic.\nIt is predicted that anisotropic dipolar interactions will\nproduce a quasiparticle spectrum with correspondingly\nstrong anisotropies, supporting rotons for only a narrow\nrange of propagation directions [40]. Such anisotropic\nspectra have important consequences for the damping of\nquasiparticles in these systems.\nWe plot the quasiparticle spectra and Beliaev damp-\ning rates for a condensate with n0gd= 1:3 and a tilt\nangle\u000b=\u0019=8 in Figs. 3(b) and 3(c), respectively. In\nthis case, the spectrum for quasiparticles propagating in\nthex-direction (?, red line) exhibits roton-maxon char-\nacter, while the spectrum in the y-direction (k, blue line)\ndoes not. Above, we noted that maxons can only un-\ndergo Beliaev damping by decaying into a pair of nearly\ncounter propagating rotons when \u000b= 0. Here, no ro-\ntons exist in the transverse direction, and maxons are\nconsequently undamped despite the fact that the maxon\nenergy exceeds 2\u0001 r. Quasiparticles propagating in the\nx-direction begin to damp near p= 1:7^x, shown by the\nblack circle(s) in Fig. 3. The momenta of the available\n\fnal states are shown by the red line in panel (a), and\nthe corresponding damping rates are shown in panel (c).\nThe onset of damping is due to the shedding of phonons\nnear this momentum. Interestingly, the damping rate is\nnot anomalously large near this onset, unlike the \u000b= 0\ncase; this is due to the anisotropy of the spectrum, which\nskews its curvature unfavorably.\nAlthough no roton-maxon feature exists in the y-\n−1.5 −1 −0.5 0 0.5 1 1.5 200.51\npxpy\n  \n⊥∥(a)\n0 1 200.20.40.60.8(b) ∥ ⊥\npωp=0.7ˆyp=1.7ˆx\n0 1 2 30510γB\np(c)FIG. 3: (color online). Beliaev damping of a quasi-2D dipo-\nlar condensate with n0gd= 1:3,\u000b=\u0019=8, and\u0012= 0 (dipoles\ntilted in the y-direction). (a) Manifold of allowed decay chan-\nnels for a quasiparticle with momentum p= 0:7^y(black +)\nshown by the blue lines, and momentum p= 1:7^x(black cir-\ncle) shown by the red lines. (b) Spectrum for quasiparticles\npropagating in the y-direction (blue line) and the x-direction\n(red line). (c) The corresponding Beliaev damping rates.\ndirection, quasiparticles propagating in this direction can\ndamp by decaying into transverse roton pairs, as illus-\ntrated by the blue lines in Fig. 3(a), which show the\nmomenta of the available \fnal states for a quasiparticle\nwith p= 1:7^x(shown by the black + sign). The damp-\ning rates for this process are shown by the two left-most\nblue line segments in panel (c). For larger momenta, the\nquasiparticles begin to shed phonons in the y-direction.\nThe damping rates for these momenta are shown by the\nright-most blue line in panel (c). Thus, the anisotropic\ndipolar interactions not only lead to anisotropic damping\nrates, but rather qualitatively di\u000berent damping mecha-\nnisms depending on the direction of quasiparticle propa-\ngation.\nOur predictions have important consequences for on-\ngoing experiments with ultracold dipolar atoms. For\nexample, in experiments measuring the dynamic struc-\nture factor S(p;!) of the condensate via, for exam-\nple, optical Bragg scattering [45], these rates will ap-\npear as spectral widths [46]. Further, our results can\nbe used to predict the short-time non-equilibrium dy-\nnamics of these systems, as the Beliaev mechanism is\nresponsible for the redistribution of quasiparticles near\nT= 0. Take, for example, the anisotropic case dis-\ncussed above. If an oblate dipolar condensate is pre-\npared with n0gd= 1:3 and\u000b=\u0019=8, and modes with\np= 0:5^xare excited, they should undergo coherent dy-\nnamics for long times. On the other hand, the excitation\nof modes with p= 0:5^ywill result in the nearly imme-\ndiate redistribution of energy into transverse rotons. In\nthis sense, the anisotropic Beliaev damping should re-5\nsult in strongly anisotropic relaxation dynamics. Exper-\nimentally, the limit of a deep roton ( n0gd= 1:7) can\nbe achieved, for example, with164Dy [29] in an oblate\ntrap with axial frequency !z= 2\u0019\u0002103Hz and a mean\n3D density \u0016 n3D\u00183\u00021014cm\u00003. For strongly dipolar\nmolecules, much smaller densities are required.\nAcknowledgements | We thank John Corson for his\ninvaluable correspondence. RW acknowledges partial\nsupport from the O\u000ece of Naval Research under Grant\nNo. N00014115WX01372, and from the National Sci-\nence Foundation under Grant No. PHY-1516421. SN\nacknowledges support from LPS-CMTC, LPS-MPO-\nCMTC, AROs Atomtronics MURI, and the AFOSRs\nQuantum Matter MURI.\n[1] L. D. Landau, J. Phys. USSR 11, 91 (1947).\n[2] R. P. Feynman, Phys. Rev. 94, 262 (1954).\n[3] R. P. Feynman and M. Cohen, Phys. Rev. 102, 1189\n(1956).\n[4] M. Cohen and R. P. Feynman, Phys. Rev. 107, 13 (1957).\n[5] D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121,\n1266 (1961).\n[6] O. W. Dietrich, E. H. Graf, C. H. 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A 86,\n021604(R) (2012)."    },    {        "title": "1610.09960v1.Looking_for_Lorentz_Violation_in_Short_Range_Gravity.pdf",        "content": "arXiv:1610.09960v1  [gr-qc]  28 Oct 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nLooking for Lorentz Violation in Short-Range Gravity\nRui Xu\nPhysics Department, Indiana University,\nBloomington, IN 47405, USA\nGeneral violations of Lorentz symmetry can be described by t he Standard-\nModel Extension (SME) framework. The SME predicts modificat ions to ex-\nisting physics and can be tested in high-precision experime nts. By looking for\nsmall deviations from Newton gravity, short-range gravity experiments are ex-\npected to be sensitive to possible gravitational Lorentz-v iolation signals. With\ntwo group’s short-range gravity data analyzed recently, no nonminimal Lorentz\nviolation signal is found at the micron distance scale, whic h gives stringent con-\nstraints on nonminimal Lorentz-violation coefficients in th e SME.\n1. Pure-gravity sector in the SME framework\nLorentz symmetry is a built-in element of both General Relativity and\nthe Standard Model. To describe nature using them, we need to tes t this\nsymmetry precisely. Also, if we seek a unified theory combining Gener al\nRelativity and the Standard Model, we also need to consider possible v io-\nlations of Lorentz symmetry that could emerge from the underlying theory,\ncausing suppressed signals at attainable energy levels.1\nThe SME framework is an approach to describing Lorentz violation\nusing effective field theory,2where a series of terms that break Lorentz\nsymmetry spontaneously in Lagrange density can be constructed . These\nterms are couplings between Lorentz-violation coefficients and kno wn fields\nsuch as the gravity field, photon field, and fermion fields. For examp le,\nin the pure-gravity sector, the Lorentz-violation couplings in the L agrange\ndensity are written as3\nLLV=e/parenleftBig/bracketleftbig\n(kR)αβγδ+(kR)αβγδλDλ+(kR)αβγδλσDλDσ+.../bracketrightbig\nRαβγδ\n+/bracketleftbig\n(kRR)αβγδµνκρ+.../bracketrightbig\nRαβγδRµνκρ+.../parenrightBig\n, (1)\nwhere ( kR)αβγδ,(kR)αβγδλ,(kR)αβγδλσ,(kRR)αβγδµνκρ,...are Lorentz-\nviolation coefficients. According to the mass dimensions of the coeffic ients,Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\nthe term with ( kR)αβγδis called the minimal Lorentz-violation coupling,\nand all the other terms are nonminimal Lorentz-violation couplings.\nCombining with the Lagrange densities for General Relativity and the\nStandard Model, as well as taking the fact of spontaneous Lorent z sym-\nmetry breaking into consideration, we can formally write down the fu ll\nLagrange density for pure-gravity sector SME,3and hence the modified\nEinstein field equation.\n2. Weak-field approximation\nThe essential problem is that the dynamics of the Lorentz-violation coeffi-\ncients is unknown. To solve this problem we have to assume both grav ity\nandLorentz-violationcoefficientsareweakfields.4Namely, foranyLorentz-\nviolation coefficient k, its constant background value ¯kis much larger than\nits fluctuation ˜k.\nThen, by making the further assumptions that the modified Einstein\nfield equation is diffeomorphism invariant and that the conventional m atter\nenergy-momentum tensor is conserved, the leading-order contr ibution from\nthedynamicsoftheLorentz-violationcoefficientsisactuallyfixedwit hsome\nparameters that depend on the unknown dynamics model.3,4\n3. Nonrelativistic solution\nIn the weak-field approximation, the modified Einstein field equation t urns\nout to give the modified Poisson equation3,4\n/vector▽2φ= 4πGρ+(¯k(4)\neff)jk∂j∂kφ+(¯k(6)\neff)jklm∂j∂k∂l∂mφ+...,(2)\nwhere (¯k(4)\neff)jkis the trace of ( ¯kR)αβγδ, and (¯k(6)\neff)jklminvolves more\ncomplicated combinations of ( ¯kR)αβγδλσand (¯kRR)αβγδµνκρ. Notice the\nLorentz-violation coefficients that have odd mass dimensions, for e xample\n(kR)αβγδλ, do not appear in the modified Poisson equation. This is a result\nof conservation of momentum.\nTreating the Lorentz-violation terms in the modified Poisson equatio n\nperturbatively, the solution is a modified Newton potential3,4\nφ=−GM\nr/bracketleftBig\n1+(¯k(4)\neff)jkrjrk\n2r2\n+/parenleftBig15\n2(¯k(6)\neff)jklmrjrkrlrm\nr6−9(¯k(6)\neff)jkllrjrk\nr4+3\n2(¯k(6)\neff)jkjk1\nr2/parenrightBig/bracketrightBig\n.(3)\nwhich shows violation of rotation symmetry due to the direction depe n-\ndence, and hence violation of Lorentz symmetry.Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\n4. Relationship to short-range gravity experiments\nShort-range gravity experiments are designed to test small devia tions from\nNewton gravity at short distance scales, ranging from microns to m illime-\nters in different experiments. By comparing the experiment result w ith\nthe predicted result from a modified Newton gravity, the paramete rs in the\nmodified theory can be determined within uncertainties. As the unce rtain-\nties are usually larger than the values, the uncertainties are also re garded\nas constraints on the parameters.\nIn the case of Lorentz-violation gravity, discrete Fourier analysis for\nthe experiment data is required to compare the experimental data and the\ntheoretical result because the Lorentz-violation corrections in t he modified\nNewton potential indicate sidereal-variation signals. The reason fo r the\nsidereal variations is that the Lorentz-violation backgrounds ( ¯k(4)\neff)jkand\n(¯k(6)\neff)jklm, which are constant in inertial frames such as the conventional\nSun-centered frame,5vary in laboratories due to the Earth’s rotation.\nSo far the experimental data analyzed are from the IU and HUST\ngroups.6–9Both groups adopt planar tungsten as test masses. The differ-\nence is that the IU experiment detects the force between two tes t masses,\nwhile the HUST experiment detects the torque produced by the for ce with\na torsion-pendulum design. The planar geometry concentrates as much\nmass as possible at the scale of interest. However, it is insensitive to the\n1/r2force. In the modified Newton potential, the ( ¯k(4)\neff)jkterm gives a 1 /r2\nforce modification. Thus, both IU and HUST experiments areinsens itive to\n(¯k(4)\neff)jk. As for the nonminimal Lorentz-violation background ( ¯k(6)\neff)jklm,\ncombining both experiments gives consistent constraints around 1 0−9m2.\nReferences\n1. V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 39, 683 (1989).\n2. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998); V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n3. Q.G. Bailey, V.A. Kosteleck´ y, and R. Xu, Phys. Rev. D 91, 022006 (2015).\n4. Q.G. Bailey and V.A. Kosteleck´ y, Phys. Rev. D 74, 045001 (2006).\n5. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 66, 056005 (2002).\n6. D. Bennett, V. Skavysh and J.C. Long, arXiv:1008.3670.\n7. J.C. Long and V.A. Kosteleck´ y, Phys. Rev. D 91, 092003 (2015).\n8. C.-G. Shao, Y.-J. Tan, W.-H. Tan, S.-Q. Yang, J. Luo, and M. E. Tobar, Phys.\nRev. D91, 102007 (2015).\n9. C.-G. Shao et al., Phys. Rev. Lett. 117, 071102 (2016)."    },    {        "title": "1310.5533v1.A_Critical_History_of_Renormalization.pdf",        "content": "1\n\n\nACriticalHistoryofRenormalization1\n\nKersonHuang\n\nMassachusetts InstituteofTechnology, \nCambridge MA,USA02139\nand\nInstituteofAdvanced Studies,NanyangTechnological University \nSingapore 639673\n\n\nAbstract\n\nThehistoryofrenormalization isreviewed withacriticaleye,startingwith\nLorentz'stheoryofradiationdamping, throughperturbative QEDwithDyson,\nGell‐Mann&Low,andothers,toWilson'sformulation andPolchinski's functional \nequation, andapplications to\"triviality\", anddarkenergyincosmology. \n\n1. Dedication \n\nRenormalization, thatastounding mathematical trickthatenabledonetotame\ndivergences inFeynmandiagrams, ledtothetriumphofquantum\nelectrodynamics. KenWilsonmadeitphysics,byuncovering itsdeepconnection \nwithscaletransformations. Theideathatscaledetermines theperception ofthe\nworldseemsobvious.Whenoneexamines anoilpainting,forexample,whatone\nseesdependsontheresolution oftheinstrument oneusesfortheexamination. \nAtresolutions ofthenakedeye,oneseesart,perhaps,butupongreaterand\ngreatermagnifications, oneseespigments, thenmolecules andatoms,andso\nforth.Whatisnon‐trivialistoformulate thismathematically, asaphysicaltheory,\nandthisiswhatKenWilsonhadachieved. Toremember him,Irecallsome\neventsatthebeginning ofhisphysicscareer.\nIfirstmetKenaround1957,whenIwasafreshassistantprofessor atM.I.T.,\nandKenaJuniorFellowatHarvard's SocietyofFellows.Hehadjustgottenhis\nPh.D.fromCal.Tech.underGell‐Mann'ssupervision. Inhisthesis,heobtained\n \n1TothememoryofKennethG.Wilson(1936‐‐2013)\n2\nexactsolutionsoftheLowequation, whichdescribes π‐mesonscattering froma\nfixed‐sourcenucleus.(Hedescribed himselfasan\"aficionado\" oftheequation.) I\nhadoccasiontorefertothisthesisyearslater,whenFrancisLowandIproved\nthattheequationdoesnotpossessthekindof\"bootstrap solution\"that\nGeoffreyChewadvocated [2,3].\nWhileattheSocietyofFellows,KenspentmostofhistimeatM.I.T.usingthe\ncomputing facilities.Hewasfrequently seendashingaboutwithstacksofIBM\npunchedcardsusedthenforFortranprogramming. \nHeusedtoplaytheoboeinthosedays,andIplayedtheviolin,andwehad\ntalkedaboutgettingtogethertoplaytheBachconcertoforoboeandviolinwith\napianist(forwedarenotcontemplate anorchestra), butwenevergotaroundto\nthat.Ihadhimoverfordinneratourapartment onWendellStreetinCambridge, \nandreceivedathank‐youpostcardafewdayslater,withanitemizedlistofthe\ndishesheliked.\nAtthetime,myM.I.T.colleague KenJohnsonwasworkingonnon‐\nperturbative QED,onwhichKenWilsonhadstrongopinions.Oneday,when\nFrancisLowandIwentbyJohnson's officetopickhimupforlunch,wefoundthe\ntwooftheminviolentargument attheblackboard. SoFrancissaid,\"We'llgoto\nlunch,andleaveyoutwoscorpions tofightitout.\"\nThatwasquiteawhileago,andKenwentontodogreatthings,includingthe\ntheoryofrenormalization thatearnedhimtheNobelPrizeof1982.Inthisarticle,\nIattempttoputmyselfintheroleofa\"physicscritic\"onthissubject.Iwill\nconcentrate onideas,andrefertechnicaldetailsto[4,5].\nWhileKen'sworkhasastrongimpactonthetheoryofcriticalphenomena, I\nconcentrate hereonparticlephysics.\n\n2. Lorentz:electronself‐forceandradiationdamping\n\nAfterJ.J.Thomson discovered thefirstelementary particle,theelectron[6],\nthequestionnaturallyaroseaboutwhatitwasmadeof.Lorentzventuredinto\nthesubjectbyregarding theelectronasauniformchargedistribution ofradiusa,\nheldtogetherbyunknown forces.Asindicated inFig.1,thechargeelements of\nthisdistribution exertCoulombforcesoneachother,buttheydonotcancelout,\nduetoretardation. Thus,thereisanet\"self‐force\",andLorentzobtaineditin\nthelimit 0a[7]:\n 2\nself self 32\n3eFm xx O ac  \n   (1)\nInternalCoulombinteractions giverisetoa\"self‐mass\":\nmselfc2dqdq′\n|r−r′|a→0O1a\n  (2)\n3\nwhichdivergeslinearlywhen 0a .Thiswasthefirstoccurrence ofthe\n\"ultraviolet catastrophe\", whichbefallsanyonetoyingwiththeinnerstructureof\nelementary particles.\n\n  \n\nFig1.Modeling theclassicalelectronaschargedistribution ofradiusa.TheCoulombic \nforcesbetweenchargeelements donotadduptozero,becauseofretardation: \n|' | | ' |rr c tt  .Consequently, thereisa\"self‐force\",featuringa\"self‐mass\"that\ndivergesinthelimit 0a ,butcanbeabsorbed intothephysicalmass.Thefinite\nremainder givestheforceofradiationdamping. \n\nOnenoticeswithgreatrelievethattheself‐masscanbeabsorbed intothe\nphysicalmassintheequationofmotion\n\nm0ẍFselfFext\nm0mselfẍ2e2\n3…xFextOa\n  (3)\n\nwherem₀isthebaremass.Onecantakesthephysicalmassfromexperiments, \nandwrite\nmẍ2e2\n3c3…xFext  (4)\nwithm=m₀+m\nself.Oneimaginesthatthedivergence ofmselfiscancelled bym₀,\nwhichcomesfromtheunknown forcesthatholdtheelectrontogether. Thisis\ntheearliestexampleof\"massrenormalization\". Thus,thexterm,thefamous\nradiationdamping, isexactinthelimita→0withintheclassicaltheory.Of\ncourse,whenaapproaches theelectronCompton wavelength, thismodelmust\nbereplacedbyaquantum‐mechanical one,andthisleadsustoQED(quantum \nelectrodynamics). \n\n\n4\n3. ThetriumphofQED\n\nModernQEDtookshapesoonaftertheadventoftheDiracequationin1928\n[8],andtheholetheoryin1930[9].Thesetheoriesmakethevacuuma\ndynamical mediumcontaining virtualelectron‐positronpairs.Weisskopf [10]was\nthefirsttoinvestigate theelectronself‐energyinthislight,andfoundthat\nscreening byinducedpairsreducesthelineardivergence intheLorentztheoryto\nalogarithmic one2[11,12].Heisenberg, Dirac,andothers[13‐16]studiedthe\nelectron's chargedistribution dueto\"vacuumpolarization\", i.e.,momentary \nchargeseparation intheDiracvacuum.Theunscreened \"barecharge\"wasfound\ntobedivergent, againlogarithmically. Asketchofthechargedistribution ofthe\nelectronisshowninFig.2.Themildnessofthelogarithmic divergence playedan\nimportant roleinthesubsequent renormalization ofQED.Butitwasdelayedfor\nadecadebecauseofWorldWarII.\n\n  \n\nFig2.Chargedensityofthebareelectron(left)andthatofthephysicalelectron,which\nis\"dressed\" byvirtualpairsinducedintheDiracvacuum(vacuumpolarization). The\nbarechargeislogarithmically divergent. \n\nThebreakthrough inQEDcamein1947,withthemeasurements oftheLamb\nshift[17]andtheelectronanomalous moment[18].Inthefirstpost‐warphysics\nconference atShelterIsland,LI,NY,June2‐4,1947,participants thrashedout\nQEDissues.(Fig.3showsagrouppicture.)Bethe[19]madeanestimateofthe\nLambshiftimmediately aftertheconference, (reportedly onthetrainbackto\nIthaca,N.Y.,)byimplementing chargerenormalization, inadditiontoLorentz's\n \n2Weisskopf wasthenPauli'sassistant. According tohisrecollection (private\ncommunication), hemadeanerrorinhisfirstpaperandgotaquadratic \ndivergence. Onday,hegotaletterfrom\"anobscurephysicistatHarvard\"bythe\nnameofWendellFurry,whopointedoutthatthedivergence shouldhavebeen\nlogarithmic. Greatlydistressed, Weisskopf showedPaulitheletter,andasked\nwhetherheshould\"quitphysics\".TheusuallyacerbicPaulibecamequite\nrestraintatmoments likethis,andmerelyhuffed,\"Inevermakemistakes!\" \n5\nmassrenormalization. Thispointedthewaytothesuccessful calculation ofthe\nLambshift[17‐‐19]inlowest‐orderperturbation theory.\nAsfortheelectronanomalous moment,Schwinger [23]calculated itto\nlowestorderasα/2π,whereαisthefine‐structureconstant, without\nencountering divergences. \n\n\n\n\nFig.3ShelterIslandConference onQED,June2‐4,1947.Participants: 1I.Rabi,2L.\nPauling,3J.VanVleck,4W.Lamb,5G.Breit,6LocalSponsor,7K.Darrow,8G.\nUhlenbeck, 9J.Schwinger, 10E.Teller,11B.Rossi,12A.Nordsiek, 13J.v.Neumann, 14\nJ.Wheeler,15H.Bethe,16R.Serber,17R.Marshak, 18A.Pais,19J.Oppenheimer, 20\nD.Bohm,21R.Feynman, 22V.Weisskopf, 23H.Feshbach. \n\n\n4. Dyson:subtraction=multiplication, orthemagicof\nperturbative renormalization \n\nDysonmadeasystematic studyofrenormalization inQEDinperturbation \ntheory[24].Thedynamics inQEDcanbedescribed intermsofscattering \nprocesses. Inperturbation theory,oneexpandsthescattering amplitude as\npowerseriesintheelectronbarechargee₀,(thechargethatappearsinthe\nLagrangian). Termsinthisexpansion areassociated withFeynmangraphs,which\ninvolvemomentum ‐spaceintegralsthatdivergeattheupperlimit.Toworkwith\nthem,oneintroduces ahigh‐momentum cutoffΛ.Dysonshowsthatshowing\nthatmassandchargerenormalization removealldivergences, toallordersof\nperturbation theory.\n6\nThedivergences canbetracedtooneofthreebasicdivergent elements in\nFeynmangraphs,contained inthefullelectronpropagator S′,thefullphoton\npropagator D′,andthefullvertexΓ.Theycanbereducedtothefollowing \nforms:\nS′pp−m0p−1\nD′k2−k−21−e02k2−1\nΓp1,p2∗p1,p2\n  (5)\nwhichmustberegardedaspowerseriesexpansions ine₀².Thedivergent \nelements areΣ,Π,Λ*,calledrespectively theself‐energy,thevacuumpolarization, \nandthepropervertexpart.TheFeynmangraphsforthesequantities areshown\ninFig.4,andtheyarealllogarithmically divergent. Thus,onesubtraction will\nsufficetorenderthemfinite.\n\n  \nFig4.Thebasicdivergent elements inFeynmangraphs.Theyarealllogarithmically \ndivergent. \n\nThedivergent partofΣcanabsorbed intothebaremassm₀,asintheLorentz\ntheory.Whatisnewisthedivergent subtracted partofΠcanbeconverted into\nmultiplicative chargerenormalization, wherebye₀isreplacedbythe\nrenormalized chargee=Ze₀.Thedivergence inΛ*canbesimilarlydisposedof.\nWeillustratehowthishappentolowestorder.\nTheelectronchargecanbedefinedviatheelectron‐electronscattering \namplitude, whichisgiveninQEDbytheFeynmangraphsinFig.5.Thetwo\nelectrons exchange aphoton.Wecanwritethepropagator intheform\n\n\n2\n2\n2\n2\n22\n0()\n1\n1dkDkk\ndk\nek\n\n  (6)\n7\nwherekisthe4‐momentum transfer.\n\n \nFig5.Electron‐electronscattering. Thelimitofzero4‐momentum transferk→0defines\ntheelectroncharge.\n\nTolowestorder,thevacuumpolarization isgivenby\nk2−1\n122lnmRk2Oe02\n   (7)\nwhereΛisthehigh‐momentum cutoff,andmistheelectronmass.(Tothis\norderitdoesnotmatterwhetheritisthebaremassorrenormalized mass.)The\nfirsttermislogarithmically divergent whenΛ→∞ ,andthetermRisconvergent. \nOnesubtraction atsomemomentum μmakesΠconvergent: \nck2≡k2−2\n (8)\nWenowwrite\ne02d′k2e02\nZ−12−e02ck2e02Z2\n1−e02Z2ck2 \n (9)\nwhere\nZ−12≡1−e022\n  (10)\nBothZandZ⁻¹arepowerserieswithdivergent coefficients, andbothdiverge\nwhenΛ→∞ .Thecombination e₀²Z(μ²)givesarenormalized fine‐structure\nconstant\n2e02Z2\n  (11)\nandthephysicalfine‐structureconstantcorresponds tozeromomentum \ntransfer:\n≡0≈137. 036−1\n  (12)\n\nWeseethatthesubtraction ofΠ(μ²)in(8)hasbeenturnedintoamultiplication \nbyZ(μ²)in(11);butonlytoordere₀⁴inperturbation theory.Dysonprovesthe\nseemingmiracle,thatthisisvalidorderbyorder,toallordersofperturbation \ntheory.\n\n8\n5. Gell‐Mann&Low:it'sallamatterofscale\n\nGell‐MannandLow[25]reformulates Dyson'srenormalization program,using\nafunctional approach, inwhichthedivergent elements Σ,Π,Λ*areregardedas\nfunctionals ofoneother,andfunctional equations forthemcanbederivedfrom\ngeneralproperties ofFeynmangraphs.Thedivergent partsofthesefunctionals \ncanbeisolatedviasubtractions, andthesubtracted partscanbeabsorbed into\nmultiplicative renormalization constant, byvirtueofthebehaviors ofthe\nfunctionals underscaletransformations. \n\n  \nFig6.Thedegreesoffreedomofthesystemathighermomenta thanthecutoffΛare\nomittedfromthetheory,bydefinition. ThedegreesoffreedombetweenΛandthe\nslidingrenormalization pointμare\"hidden\"intherenormalization constants. Thus,μis\naneffectivecutoff,representing thescaleatwhichoneisobserving thesystem.\n\nOneseesthecutoffΛinanewlight,asascaleparameter. Infact,itistheonly\nscaleparameter inaself‐contained theory.Whenoneperformsasubtraction at\nmomentum μ,andabsorbstheΛ‐dependent partintorenormalization constants, \noneeffectively lowersthescalefromΛtoμ.ThedegreesoffreedombetweenΛ\nandμarenotdiscarded, buthiddenintherenormalization constants; the\nidentityofthetheoryispreserved. Thesituationisillustrated inFig.6.\nTherenormalized chargetoorderα²,andfor|k|²≫m²,isgivenby\nk22\n3ln|k|2\nm2\n  (13)\nThisiscalleda\"runningcouplingconstant\", becauseitdependsonthe\nmomentum scalek.Ithasbeenmeasured atahighmomentum [26]:\n9\nk02≈127. 944−1\n  (14)\nwherek₀≈91.2GeV.TheFouriertransform ofα(k²)givestheelectrostatic \npotentialofanelectron[27].Asexpected, itapproaches theCoulombpotential\ner⁻¹asr→∞ ,whereeisthephysicalcharge.Forr≪/mc,itisgivenby\nVr≈er12\n3lnr0rO2\n  (15)\nwherer₀=(/mc)(e5/6γ)⁻¹,γ≈1.781.Weseethethebarechargee₀ofthe\nelectron,namelythatresidingatthecenter,divergeslikeln(1 / ) r.\nGell‐MannandLow[25]givethefollowing physicalinterpretation ofcharge\nrenormalization: \n\nAtestbodyof\"barecharge\"q₀polarizesthevacuum,surrounding itselfbya\nneutralcloudofelectrons andpositrons; someofthese,withanetchargeδq,\nofthesamesignasq₀,escapetoinfinity,leavinganetcharge‐δqinthepart\nofthecloudwhichiscloselyboundtothetestbody(withinadistanceof\n/mc).Ifweobservethebodyfromadistancemuchgreaterthan/mc,we\nseeaneffectivechargeq=q₀‐δq,therenormalized charge.However, aswe\ninspectmorecloselyandpenetrate throughthecloudtothecoreofthetest\ncharge,thechargethatweseeinsideapproaches thebarechargeq₀\nconcentrated atapointatthecenter.\n\n6. Asymptotic freedom\n\nTherunningcouplingconstant\"runs\"ataratedescribed bytheβ‐function\n(introduced asψbyGell‐MannandLow):\n22∂2\n∂2\n  (16)\nForQEDwecancalculatethisfrom(13)tolowestorderinα:\nQED2\n3  (17)\nThatthisispositivemeansthatαincreases withthemomentum scale.Butithas\ntheoppositesigninQCD(quantum chromodynamics) [28,29]:\nQCD−2\n633\n2−Nf\n  (18)\nwhereαhereistheanalogofthefine‐structureconstant, andNf=6isthe\nnumberofquarkflavors.Thus,QCDapproaches afreetheoryinthehigh‐\nmomentum limit.Thisiscalled\"asymptotic freedom\". \nQCDisagaugetheorylikeQED,butthereare8\"color\"charges,and8gauge\nphotons,calledgluons,and.Unlikethephoton,whichisneutral,thegluonscarry10\ncolorcharge.Whenabareelectronemitsorabsorbsaphoton,itscharge\ndistribution doesnotchange,becausethephotonisneutral.Incontrast,whena\nquarkemitsorabsorbsagluon,itschargecenterisshifted,sincethegluonis\ncharged.Consequently, the\"dressing\" ofabarequarksmearsoutitschargetoa\ndistribution withoutacentralsingularity. Asonepenetrates thecloudofvacuum\npolarization ofadressedquark,oneseelessandlesschargeinside,andfinally\nnothingatthecenter.Thisisthephysicaloriginofasymptotic freedom.Fig.7\nshowsacomparison betweenthedressedelectronandthedressedquark,with\nrelevantFeynmangraphsthatcontribute tothedressing.\nInthestandardmodelofparticlephysics,thereare3forcesstrong,\nelectromagnetic andweak,whosestrengths canbecharacterized respectively by\nαQCD,αQED,αWeak,withstrengthstandingatlowmomenta intheapproximate \nratio10:10⁻²:10⁻⁵.WhileαQCDisasymptotically free,theothertwoarenot.\nConsequently αQCDwilldecreasewithmomentum scale,whereastheothertwo\nincrease.Extrapolation ofpresenttrendindicatetheywouldallmeetatabout\n10¹⁷GeV,asindicated inFig.8.Thisunderlies thesearchfora\"grandunified\ntheory\"atthatscale.\n\n \n\nFig7.Comparison betweenadressedelectronandadressedquark.Thereisapoint\nchargeatthecenterofthedressedelectron,butnoneinthedressedquark,forithas\nbeensmearedoutbythegluons,whicharethemselves charged.Lowerpanelsshowthe\nrelevantFeynmangraphs.Forthequark,therearetwoextragraphsarisingfromgluon‐\ngluoninteractions. \n\n11\n\n\nFig8.Extrapolation oftherunningcouplingconstants forthestrong,electromagnetic, \nandweakinteractions indicatethattheywouldmeetatamomentum k≈10¹⁷GeV,\ngivingrisetospeculations ofa\"grandunification\". \n\n7. Therenormalization group(RG)\n\nThetransformations ofthescaleμformformagroup,andtherunning\ncouplingconstantα(μ²)givesarepresentation ofthisgroup,whichwasnamed\nRG(therenormalization group)byBogoliubov [30].Theβ‐functionisa\"tangent\nvector\"tothegroup.Byintegrating (16),weobtaintherelation\nln2\n02\n022d\n    (19)\nAsμ→∞ ,theleftsidediverges,andtherefore 2musteitherdiverge,or\napproach azeroof.ThelatterisafixedpointofRG,atwhichthesystem\nisscale‐invariant. Thisshowsthatthescaleμisdetermined bythevalueof\n2.\n12\n  \n\nFig9.Thesignofβ(α)determines thedirectionofarrowsontheαaxis,indicating the\ndirectionofrunningasthemomentum scalekincreases. Fixedpointsarewhereβ(α)=0,\natthesystemisscale‐invariant, and.TheoriginisanIR(infrared) fixedpointforQED,for\nthesystemgoestowardsitwhenk→0.ItisanUV(ultraviolet) fixedpointforQCD,\nwhichapproaches itwhenk→∞ .\n\nFig.9showsplotsoftheβ‐functionforQEDandQCD.Asthemomentum scale\nkincreases, α(k²)runsalongthedirectionofthearrowsdetermined bythesign\nofβ.ForQED,αincreases withk,andsinceperturbation theorybecomesinvalid\nathighk,welosecontroloverhigh‐energyQED.ForQCD,ontheotherhand,α\nrunstowardstheUVfixedpointatzero,perturbation theorybecomes\nincreasingly accurate, andwehaveagoodunderstanding inthisregime.The\nothersideofthecoinisthatQCDbecomesahardproblematlowenergies,\nwhereitexhibitsquarkconfinement. \n\nTheplotsclarifytherelationbetweenthecutoffscaleΛthatdefinesthebare\nsystem,andtheeffectivescaleμ,whichdefinestherenormalized system.\nWenowhaveabetterunderstanding ofwhatcanbedonewiththeoriginal\ncutoffΛ.Beingascaleparameter, Λisdetermined by(19)andthelimitΛ→∞\ncanbeachievedonlybymoving2()toafixedpoint,andinQCDthismeans\n2()=0.Andthereisnoproblemwiththis.\n13\nInQED,ontheotherhand,thereisnoknownfixedpointexcepttheoneatthe\norigin.Inpractice,onekeepsΛfinite,whosevalueisnotimportant. Inthisway,\nonecanperformcalculations thatagreewithexperiments toonepartin10¹²,in\nthecaseoftheelectronanomalous moment[29,30].IfoneinsistsonmakingΛ\ninfinite,onemustmake2()=0,butthatmakes2()=0forallμ<Λ,and\nonehasatrivialfreetheory.Wewillexpandonthis\"trivialityproblem\" later.\nParticletheoristshaveapeculiarsensitivity tothecutoff,becausetheyregard\nitasastigmathatexposesanimperfect theory.Intheearlydaysof\nrenormalization, whenthecutoffwasputoutofsightbyrenormalization, some\nleapedtodeclarethatthecutoffhasbeen\"senttoinfinity\".That,ofcourse,\ncannotbedonebyfiat.OnlyinQCDcanoneachievethat,owingtoasymptotic \nfreedom. \nAmoregeneralstatement ofrenormalization referstoanycorrelation \nfunctionG’:\nG′p;,e02Z∗/,e02Gp;,2\n  (20)\nwherepcollectively denotesalltheexternalexternalmomenta, Z*isa\ndimensionless renormalization constantthatdivergeswhenΛ→∞ ,μisan\narbitrarymomentum scalelessthanΛ,α(μ²)isgivenby(11),andGisa\nconvergent correlation function.Sincetheleftsideisindependent ofμ,wehave\n  2\n0 /, ;, 0dZe G pd \n (21)\nwhichleadstotheCallan‐Symanzik equation[32,33]\n∂\n∂∂\n∂Gp;,0\n (22)\nwhereβisdefinedby(16),andγ(μ)=μ(∂/∂μ)lnZ*(Λ/μ,α₀)iscalledthe\n\"anomalous dimension\". Thisshowshowrenormalization accompanies ascale\ntransformation, soastopreservethebasicidentityofthetheory.\n\n8. TheLandaughost\n\nBetweenthegreattriumphofquantumfieldtheoryinQEDin1947,andthe\nemergence ofthestandardmodelofparticlephysicsaround1975,particle\ntheoristswandered likeMosesinsomedesert,fornearlythreedecades.During\nthattimetheygetdisenchanted withquantumfieldtheory,becausethegreat\nhopetheyhadpinnedonthetheorytoexplainthestronginteractions didnot\nmaterialize3.Thewasafeelingthatsomething crazywascalledfor,likequantum\n \n3Irecallthat,inthelatefifties,E.FermiandF.J.Dysonseparately gavethe\nMorrisLoebLectureatHarvardUniversity. Fermitalkedaboutanewly\ndiscovered pio‐nucleon\"33resonance\", withspin3/2andisospin3/2(now14\nmechanics4,ormaybethe\"bootstrap\" [2,3].Landauthoughthehasatleast\ndisposedofquantumfieldtheorybyexposingafatalflaw.\nSubstituting (16)into(19)andperforming theintegration, oneobtains\nk2\n1−/3lnk2/m2\n  (23)\nThisissupposed tobeanimprovement on(13),equivalent tosummingacertain\nclassofFeynmangraphs‐‐‐theso‐called\"leadinglogs\"withtermsoftheform\n(e₀²lnΛ)ⁿ.Landau[34]pointedoutthatthereisapolewithnegativeresidue:\nk2≈−3kghost2\nk2−kghost2\nkghost2\nm2exp3  (24)\nThisrepresents aphotonexcitedstate,whosewavefunctionhasnegative\nsquaredmodulus, andiscalleda\"ghoststate\".Itsmassisoforder10³⁰⁰m.Itcan\nbeshownthatΛ<kghost,andthustheghostoccursonlyifwecontinuethetheory\ntobeyondthepresetcutoff.However, ifoneinsistsonmakingΛ→∞ ,onemust\npushtheghosttoinfinity,andthismeansα→ 0,leadingtoatrivialtheory.\nLandausaidthatthispossibility exposesafundamental flawinquantumfield\ntheory5,which\"shouldbeburiedwithhonors\".\nThetrivialityproblemalsooccursinothertheories,forexamplethescalarφ⁴\nHiggsfieldinthestandardmodel.Earlier,itwasfoundintheLeemodel[35],an\nexactlysolublemodelofmesonscattering. KällénandW.Pauli[36]showedthat\n   \ncalledthedeltabaryon),andsaid,\"Iwillnotunderstand thisinmylifetime.\" \nDysontalkedabouttheso‐called\"Tamm‐Dancoffapproximation\" forpion‐\nnucleonscattering, andsaid,\"Wewillnotunderstand thisprobleminahundred\nyears.\"\n\n4In1958,Heisenberg andPauliproposed a\"unifiedfieldtheory\".Pauligavea\nseminaratColumbia University withNielsBohrinattendance. Whentheseminar\nbegan,Bohrsaid,\"Toberight,thetheoryhadbetterbecrazy\".Paulisaid,\"It's\ncrazy!Youwillsee.It'scrazy!\"Thetheoryturnsoutbeaversionofthefour‐\nfermioninteraction. \n5Apparently, Landauconsidered theghoststateahallmarkofquantumfield\ntheories.Hereportedly calculated theβ‐functionofYang‐Millstheory(onwhich\nQCDisbased),butmadeasignerror,andmissedasymptotic freedom.\n15\ntheghoststaterenderstheS‐matrixnon‐unitary,andthispathology cannotbe\ncuredbyredefining Hilbertspacetoadmitnegativenorms.6\nWeshallseethatthetrivialityproblemisageneralpropertyofIRfixedpoints.\nThemoralis:togetinfinitecutoff,getyourselfaUVfixedpoint!\nQuantum fieldtheorydidnotdie,butbouncedbackwithavengeance, inthe\nformofYang‐Millsgaugetheoryinthestandardmodel.\n\n9. Renormalizability \n\nRenormalization inperturbation theoryhingesonthedegreeofdivergence K\nofFeynmangraphs,whichisdetermined viaapower‐countingprocedure. It\ndependsontheformofcoupling‐‐‐howmanylinesmeetatavertex,etc.\nRenormalization inQEDreliesonthefacttheinteraction AgivesK=0\n(logarithmic divergence). Onecanimagineinteractions thatwouldgiveK>0,and\nthatwouldbenon‐renormalizable. Anexampleisthe4‐fermioninteraction \n2().Thereisthusacriterionofrenormalizability: underascalechange,the\nexistingcouplingconstantundergorenormalization, andnonewcouplingshould\narise.Inotherwords,thesystemshouldbeself‐similar.\nSuchconsiderations arebasedonthepresumption thateachnewcoupling\nbringinitsownscale.Inaself‐contained system,however, thecutoffΛsetsthe\nonlyscale,andallcouplingconstants mustbeproportional toanappropriate \npowerofΛ.Whenthisistakenintoaccountinthepowercounting, whatwas\nconsidered anon‐renormalizable interaction canbecomerenormalizable. Ifall\ncouplingconstants aremadedimensionless inthismanner,thentheycould\nfreelyariseunderscaletransformations, andthesystemneednotbeself‐similar\ntoberenormalizable. \nAsillustration, considerscalarfieldtheorywithaLagrangian densityofthe\nform(with=c=1)\nL1\n2∂2−V\nVg22g44g66  (25)\nThetheoryiscalledφMtheory,whereMisthehighestpowerthatoccurs.Each\ncouplinggncorresponds toavertexinaFeynmangraph,atwhichnlinesmeet,\nandeachlinecarriesmomentum. Themomenta oftheinternallinesare\nintegrated over,andproducedivergences. Thus,eachFeynmangraphis\nproportional toΛK,withadegreeofdivergence Kthatcanbefoundbya\ncountingprocedure. TherelationbetweenKandtopological properties of\n \n6Thereotherghoststatesinquantumfieldtheory,arisingfromgauge‐fixing,\nsuchtheFadeev‐Popovghost[5].Butthesearemathematical devicesthathave\nnophysicalconsequences. 16\nFeynmangraphs,suchasthenumberofverticesandinternallines,determines \nrenormalizability. \nItwassaidconventionally thatonlytheφ⁴theoryisrenormalizable. This\ndetermination, however, assumesthatthegnarearbitraryparameters. The\ndimensionality ofgnind‐dimensional space‐timeis\ngnLengthnd/2−n−d\n  (26)\nTreatingthemasindependent willmeansthateachgnbringintothesysteman\nindependent scale.Buttheonlyintrinsiclengthscaleinaself‐contained system\nistheinversecutoffΛ⁻¹.Thuseachgnshouldbescaledwithappropriate powers\nofΛ:\ngnunnd−nd/2\n  (27)\nsothatunisdimensionless. Whenthisisdone,thecutoffdependence ofgn\nentersintothepowercounting, andallφKtheoriesbecomerenormalizable [37].\nWiththescaling(27),onecanconstruct anasymptotically freescalarfield,one\nthatisfreefromthetrivialityproblem.ForanN‐component scalarfieldind=1,\nV(φ)isuniquelygivenbytheHalpern‐Huangpotential[38]\nVc4−bM−2b/2,N/2,z−1\nz82\n2∑\nn1N\nn2\n  (28)\nwherec,barearbitraryconstants, andM(a,b;z)istheKummerfunction,which\nhasexponential behaviorforlargefields:\nMp,q,z≈ΓqΓ−1pzp−qexpz\n  (29)\nThetheoryisasymptotically freeforb>0.Thishasapplications intheHiggs\nsectorofthestandardmodelandincosmology, tobediscussed later.\nNotalltheoriesarerenormalizable, evenwiththescalingofcouplingconstants. \nThereisatruespoiler,namely,the\"axialanomaly\" infermionic theories.It\narisesfromthefactthattheclassically conserved axialvectorcurrentbecomes\nnon‐conserved inquantumtheory,duetotheexistence oftopological charges.\n(See[2,4]).ThisleadstoFeynmangraphswiththe\"wrong\"scalingbehavior, and\ntheonlywaytogetridofdivergences arisingfromsuchFeynmangraphsisto\ncancelthemwithsimilargraphs.Thepracticalconsequence isthatquarksand\nleptonsinthestandardmodelmustoccurinafamily,suchthattheiranomalies \ncancel.Weknowofthreefamilies:{u,d,e,νe},{s,c,μ,νμ},{t,b,τ,ντ}.Ifanewquark\norleptonisdiscovered, itshouldbringwithitanentirefamily.\n\n17\n10. Wilson'srenormalization theory\n\nWilsonreformulates renormalization independent ofperturbation theory,and\nputsscaletransformations attheforefront. Hewasconcerned withcritical\nphenomena inmatter,wherethereisanaturalcutoff,theatomiclatticespacing\na.WhenonewritesdownaHamiltonian, adoesnotexplicitlyappear,becauseit\nonlysuppliesthelengthscale.Thescaling(27)ofcouplingconstants isnatural\nandautomatic. Thisisanimportant psychological factorinone'sapproach tothe\nsubject. \nThefirsthintofhowtodorenormalization onaspatiallatticespacecomes\nfromKadanoff's \"blockspin\"transformations [39].Thisisacoarse‐graining\nprocess,asillustrated inFig.10.Spinswithonlyup‐downstatesarerepresented \nbytheblackdots,withnearest‐neighbor(nn)interactions. Inthefirstlevelof\ncoarse‐graining,spinsaregroupedintoblocks,indicated bythesolidenclosures. \nTheoriginalspinsarereplacedbyasingleaveragedspinatthecenter.Thelattice\nspacingbecomes2,butisrescaledbackto1.Theblock‐blockinteractions now\nhaverenormalized couplingconstants; however, newcouplings arise,forthe\nblockingprocessgenerates nnnandlonger‐rangedinteractions. Kadanoff\nconcentrates onthefixedpointsofiterativeblocking,andignoresthenew\ncouplings forthispurpose.Wilsontakethenewcouplings intoaccount,by\nproviding \"hooks\"forthemfromthebeginning. Thatis,thecoupling‐constant\nspaceisenlargedtoincludeallpossiblecouplings: nnn,nnnn,etc.Inthe\nbeginning, whentherewereonlythenncouplings, oneregardedtherestas\npotentially present,butnegligible. Thecouplings cangrowordecreasein\nsuccessive blockingtransformations. \n\n  \n\nFig10.Block‐spintransformations. Inthespinlattice,theup‐downspins,represented \nbytheblackdots,interactwitheachothervianearest‐neighbor(nn)interactions. Inthe\nfirstlevelofcoarse‐graining,theyaregroupedintoblocksof4,indicated bythesolid\nenclosures, andreplacedbyasingleaveraged spinatthecenter.Theoriginallattice\nspacinganowbecomes2a,butisrescaledbacktoa.Inthenextlevel,theseblocksare\ngroupedintohigherblocksindicated bythedottedenclosure, andsoforth.However,the \nblock‐blockinteractions willincludennn,nnnninteractions, andsoforth.\n\n18\nWilsonimplements renormalization usingtheFeynmanpathintegral,as\nfollows.Aquantumfieldtheorycanbedescribed throughitscorrelation \nfunctions. Forascalarfield,forexample,thesearethefunctional averages\n<φφ>,<φφφ >,<φφφφ >,…,andtheycanbeobtainedfromthegenerating \nfunctional \nWJNDexpiS,−J,\n  (30)\n\nbyrepeatedfunctional differentiation withrespecttotheexternalcurrentJ(x).\nHere[]dSd xListheclassicalaction,whereListheclassicalLagrangian \ndensity(25),and∫Dφdenotesfunctional integration. Thereisashort‐distance\ncutoffΛ⁻¹,whichisonlyscaleinS[φ].Ofcourse,Jintroduces anscale,butthatis\nexternalratherthanintrinsic.Forsimplicity wesetJ≡0inthisdiscussion. \nBymakingthetimepureimaginary (Euclidean time,inthelanguageof\nrelativistic quantumfieldtheory)onecanregardW[J]asthepartitionfunction\nforathermalsystemdescribed byanorderparameter φ(x),andtheimaginary \ntimecorresponds totheinversetemperature. Inthisway,aresultfromquantum\nfieldtheorycanbetranslated intothatinstatistical mechanics, andviceversa.\nThefunctional integration ∫Dφextendsoverallpossiblefunctional formsof\nφ(x).Itmaybecarriedoutbydiscretizing xasaspatiallattice,andintegrating \noverthefieldateachsite.Alternatively onecanintegrateoverallFourier\ntransforms inmomentum space,madediscretebyenclosing thesystemina\nlargespatialbox.Herewechoosethelatterroute:\nD\n|k|\n−\ndk\n  (31)\nwhereφkdenotesaFouriercomponent ofthefield,andΛisthehigh‐\nmomentum cutoff.Welowertheeffectivecutofftoμby\"hiding\"thedegreesof\nfreedombetweenΛandμ,asindicated inFig.6.Todothis,weintegrateoverthe\nmomenta inthisinterval,andputtheresultintheformofaneweffectiveaction.\nThatis,wewrite\nN\n|k|\n−\ndkexpiSN\n|k|\n−\ndk\n|k|\n−\ndk′expiS\nN′\n|k|\n−\ndkexpiS′\n(32)\nTheintegrations inthebracketsdefinethenewactionS′,which\ncontainsonlydegreesoffreedombelowmomentum μ7.Fromthis,wecan\n \n7ThecutoffΛactuallydoesnotappearinanyoftheformulas, becauseitmerely\nsuppliesascale.LoweringthecutofffromΛtoμactuallymeansloweringitfrom\n1toμ/Λ.See[4]fordetails.19\nobtainanewLagrangian densityL′,whichcontainsnewcouplings un′thatare\nfunctions oftheoldonesun8.This,inanutshell,isWilson'srenormalization \ntransformation. \nSuccessive renormalization transformations giveaseriesofeffective\nLagrangians: \nL→L′→L′′→L′′′→\n  (33)\nwhichdescribehowtheappearance ofthesystemchangesundercoarse‐\ngraining.Theidentityofthesystemispreserved, becausethegenerating \nfunctional Wisnotchanged.Weallowforallpossiblecouplings un,andthusthe\nparameter spaceisthatofallpossibleLagrangians. Renormalization generates a\ntrajectory inthatspace‐‐‐theRGtrajectory. Couplings thatwereoriginally\nnegligible cangrow,andsothetrajectory canbreakoutintonewdimensions, as\nillustrated inFig.11.Thereisnorequirement thatthetheorybeself‐similar,and\nthusitappearsthatalltheoriesarerenormalizable9.\nThatthismethodofrenormalization reducestothatinperturbation theory\ncanbeprovenbyderiving(20)withthisapproach [4].\n\n\n \n\nFig11.Byrendering allcouplingconstantdimensionless throughscalingwith\nappropriate powersofthecutoffmomentum, thesystemcanbreakoutintoanew\ndirectioninparameter spaceunderrenormalization. Thetrajectories sketchedhere\nrepresent RGtrajectories withvariousinitialconditions. \n\n\n   \n\n8SomerescalingneedtobedonetoputthenewLagrangian inastandardform.\nTheseoperations canaffectthenormalization constantN.\n\n9Notableexceptions, asmentioned previously, arefermionic theoriesexhibiting \ntheaxialanomaly.IntermsoftheFeynmanpathintegral,certainscaling\noperations fail,owingtonon‐invariance oftheintegration measure. See[2,4].\n\n20\n\n\n11. InthespaceofallpossibleLagrangians \n\nUnderthecoarse‐grainingsteps,theeffectiveLagrangian tracesouta\ntrajectory inparameter space,theRGtrajectory10.Withdifferentinitial\nconditions, onegoesondifferenttrajectories, andthewholeparameter isfilled\nwiththem,likestreamslinesinahydrodynamic flow.Therearesourcesand\nsinksintheflow,andthesearefixedpoints,wherethesystemremainsinvariant\nunderscalechanges.Thecorrelation lengthbecomesinfiniteatthesefixed\npoints.Thismeansthatthelatticeapproaches acontinuum: a→0,orΛ→∞ .\nLetusdefinethedirectionofflowalonganRGtrajectory tobethecoarse‐\ngrainingdirection, ortowardslowmomentum. Ifitflowsoutofafixedpoint,\nthenthefixedpointappearstobeaUVfixedpoint,foritistobereachedby\ngoingoppositetotheflow,towardsthehigh‐momentum limit.Suchatrajectory \niscalledaUVtrajectory. Ififflowsintoafixedpoint,itiscalledanIRtrajectory, \nalongwhichthefixedpointappearstobeanIRfixedpoint.Thisisillustrated in\nFig.12.\n\n \n\nFig12.FixpointsaresourcesandsinksofRGtrajectories, atwhichthecutoffisinfinite.\n\nActually,ΛisinfinitealongtheentireIRtrajectory, becausethisissoatthe\nfixedpoint,andΛcanonlydecreaseuponcoarse‐graining.Thus,onecannot\nplaceasystemonanIRtrajectory, butonlyonanadjacenttrajectory. Whenwe\ngetcloserandclosertotheIRtrajectory, Λ→∞ ,andsystemmoreclosely\nresemble thatattheIRfixedpoint.Itismostcommontohaveafreetheoryat\n \n10Thecoarse‐grainingproceedsonlyinonedirection; butthatisamatterof\ndefiningtheRGtrajectory. Oncedefined,onecantravelbackandforthalongthe\ntrajectory. \n\n21\nthefixedpoint,sinceitisscale‐invariant, andthisgivesamorephysical\nunderstanding ofthetrivialityproblem.\nTheflowvelocityalonganRGtrajectory canbemeasured bythearclength\ncoveredinacoarse‐grainingstep.Itslowsdownintheneighborhood ofafixed\npoint,andspeedupbetweenfixedpoints.Thus,itdartsfromfixedpointtothe\nnext,likeashipsailingbetweenportsofcall.Somecouplings growasit\napproaches afixedpoint,andthesearecalled\"relevant\" interactions. Theone\nthatdieoutarecalled\"irrelevant\", andmaybeneglected. Thus,eachport\ncorresponds toacharacteristic setofinteractions, andthesystemputsona\ncertainfaceatthatport.Thisisillustrated inFig.13.\n\n \n\nFig13.Differentphysicaltheoriesgoverndifferentlengthscales.Eachtheorycan\nberepresented byafixedpointinthespaceofLagrangians. Theworldislikea\nshipnavigating thisspace,andthefixedpointsaretheportsofcall.Asthescale\nchanges,theworldsailsfromporttoport,andlingersforawhileateachport.\n\n12. Polchinski's functional equation\n\nWehaveusedasharpmomentum cutoffin(31).Inasignificant improvement, \nPolchinski [41]generalizes themethodtoanarbitrarycutoff,andderivea\nfunctional equationfortherenormalized action.Thecutoffisintroduced by\nmodifying thefreepropagator k⁻²ofthefieldtheory,byreplacingitby\n\n22\nΔk2fk2/2\nk2\nfz\nz→→0\n   (34)\nwherethedetailedformofthecutofffunctionf(k²/Λ²)isnotimportant. Whatis\nimportant isthatΛistheonlyscaleinthetheory.Theregulated propagator in\nconfigurational spacewillbedenotedbyK(x,Λ).\nTheactioniswrittenas\nS,S0,S′,\n  (35)\nwherethefirsttermcorresponds tothefreefield,andthesecondterm\nrepresents theinteraction. Wehave\nS0,1\n2ddxddyxK−1x−y,y \n (36)\nwhereK⁻¹(x‐y,Λ)istheinverseofthepropagator K(x‐y,Λ),inanoperatorsense.It\ndiffersfromtheLaplacian operatorsignificantly onlyinaneighborhood of\n|x‐y|=0,ofradiusΛ⁻¹.Thegenerating functional iswrittenas\nWJ,NDe−S,−J,\n    (37)\nwherethenormalization constantNmaydependonΛ.\nIntheWilsonmethod,oneintegrates outmodebetweenΛandμtolower\ntheeffectivecutofftoμ.AmoregeneralpointofviewisthatanychangeinΛis\ncompensated byachangeinS’[φ,Λ],inordertopreservethebasicidentityof\nthetheory:\ndWJ,\nd0\n   (38)\nThisisthegeneralization of(21)inperturbative renormalization. \nTheremarkable thingisthatPolchinski solves(38)byfindingafunctional \nintegro‐differential equationforS’[φ,Λ].ForJ≡0,itreads11\ndS′\nd−1\n2dxdy∂Kx−y,\n∂2S′\nxy−S′\nxS′\ny \n (39)\nPeriwal[42]showshowonecanusethistoderivetheHalpern‐Huangpotential\nin\"twolines\".(Theoriginalderivation involvessummingone‐loopFeynman\ngraphs[38]).Assuming thattherearenoderivative couplings12,wecanwriteS′\nastheintegralofalocalpotential: \n \n11See[4]foraproof.\n12Theoriginalderivation didnotassumethis,butshowedthatnoderivative \ncouplings arisefromrenormalization, ifnonewerepresentoriginally. \n23\n\nS′,dddxUx,\nx1−d/2x  (40)\nwhereUisadimensionless function,andϕisadimensionless field.Thescalar\npotentialisgivenbyV=ΛdU.\nNeartheGaussianfixedpoint,whereS’≡0,onecanlinearize(39),andobtain\nalineardifferential equationforU(ϕ,Λ):\n\n∂U\n∂\n2U′′1−d\n2U′Ud0\n   (41)\nwhereaprimedenotepartialderivative withrespecttoϕ,andκ=Λ3‐d∂K(0,Λ)/∂Λ.\nNowweseekeigenpotentials Ub(ϕ,Λ)withtheproperty\n\n∂Ub\n∂−bUb  (42)\nInthelanguageofperturbative renormalization theory,therightsideisthe\nlinearapproximation totheβ‐function., andthesolutionisasymptotically free\nforb>0.Substitution intothepreviousequationleadstothedifferential \nequation\n\n\n2d2\nd2−1\n2d−2d\ndd−bUb0\n   (43)\nSincethisequationdoesnotdependonΛ,theΛ‐dependence ofthepotentialis\ncontained inamultiplicative factor.Inviewof(42),thefactorisΛ\n‐b.\nFord≠2,(42)canbetransformed intoKummer's equation: \n\nzd2\ndz2q−zd\ndz−pUb0\n    (44)\nwhere\nq1/2\npb−d\nd−2\nz2−1d−22\n  (45)\nThesolutionis\nUbzc−bMp,q,z−1\n  (46)24\n\nwherecisanarbitraryconstant, andMistheKummerfunction.Wehave\nsubtracted 1tomakeUb(0)=0.Thisispermissible, sinceitmerelychangesthe\nnormalization ofthegenerating functional. Ford=2,(43)leadstotheso‐called\nsine‐Gordontheory.\n\n13. Whytrivialityisnotaproblem\n\nThemasslessfreescalarisscale‐invariant, andcorresponds totheGaussian\nfixedpoint.Whenthelengthscaleincreases fromzero,andweimaginethe\nsystembeingdisplaced infinitesimally fromthisfixedpoint,itwillsailalongsome\nRGtrajectory, alongsomedirectioninparameter space,thefunctionspace\nspannedbypossibleformsofV(φ).Eq.(42)describes theproperties associated \nwithvariousdirections. Alongdirections withb>0,thesystemwillbeonaUV\ntrajectory. Withb<0,thesystemisonanIRtrajectory, andbehaveasifithad\nneverleftthefixedpoint.Thisisillustrated inFig.14.\n\n \n\nFig14.TheGaussianfixedpoint.RJtrajectories emanatealongallpossibledirections in\nparameter space.Arrowsdenotedirectionofincreasing lengthscale.Non‐trivial\ndirections correspond totheorieswithasymptotic freedom. Trivialdirectionsignifies\nthatthetheoryremainsatthefixedpointunderscalechange.\n\nTheexistence ofUVdirections suggestsapossiblesolutiontothetriviality\nproblem,asillustrated inFig.15.ConsidertwoGaussianfixedpointsAandB.A\nscalarfieldleavesAalongaUVtrajectory, andcrossesovertoaneighborhood B,\nskirtinganIRtrajectory ofB.Atpoint1,thepotentialisHalpern‐Huang,butat2\nitbecomesφ⁴,withallhighercouplingbecoming irrelevant. TheoriginalcutoffΛ\nisinfinite,beingpushedintoA.Theeffectivecutoffat2isarenormalization \npointμ,andthereisnoreasontomakeitinfinite.\nInthecaseofQED,thefixedpointBwouldcorrespond toourQEDLagrangian, \nandAcouldrepresent someasymptotically freeYang‐Millgaugetheory.\n25\n\n \n\nFig15.How\"triviality\" mayarise,andwhyitisnotaproblem.Here,AandBrepresent \ntwoGaussianfixedpoints.Thesystemat1isonaUVtrajectory andasymptotically free.\nItcrossesovertoaneighborhood ofB,skirtinganIRtrajectory. Atpoint2itresembles a\ntrivialφ⁴theory,becausehighercouplings havebecomeirrelevant. \n\n\n14. Asymptotic freedomandthebigbang\n\nThevacuumcarriescomplexscalarfields.ThereisatleasttheHiggsfieldofthe\nstandardmodel,whichgenerates massforgaugebosonsintheweaksector.\nGrandunifiedtheoriescallformorescalarfields.Acomplexscalarfieldservesas\norderparameter forsuperfluidity, andfromthispointofviewtheentire\nuniverseisasuperfluid. Inarecenttheory,darkenergyanddarkmatterinthe\nuniversearisefromthissuperfluid. Briefly,darkenergyistheenergydensityof\nthesuperfluid, anddarkmatteristhemanifestation ofdensityfluctuations of\nthesuperfluid fromitsequilibrium vacuumvalue[43‐45].\nAtthebigbang,thescalarfieldisassumedtoemergefromtheGaussianfixed\npointalongsomedirectioninparameter space,asindicated inFig.14.Ifthe\nchosendirectioncorresponds toanIRtrajectory, thenthesystemneverleftthe\nfixedpoint,andnothinghappens.IfitisaUVtrajectory, however, itwilldevelop\nintoaHalpern‐Huangpotential, andspawnapossibleuniverse.Weassumethat\nwasonlyonescaleatthebigbang,theradiusoftheuniversea(t)inthe\nRobertson ‐Walkermetric.Thus,itmustbeidentified withthecutoffΛofthe\nscalarfield:\n \nat\n  (47)\nThisrelationcreatesadynamical feedback: thescalarfieldgenerates gravity,\nwhichsuppliesthecutofftothefield.Einstein's equationthenleadstoapower‐\nlawexpansion oftheform\n atexpt1−p\n  (48)\nwherep<1.Thisdescribes auniversewithaccelerated expansion, thushaving\ndarkenergy.Theequivalent cosmological constantdecaysintimeliket‐2p,\n26\ncircumventing theusual\"fine‐tuningproblem\". Vortexactivitiesinthesuperfluid \ncreatesquantumturbulence, inwhichallmatterwascreatedduringainitial\n\"inflation era\".Manyobserved phenomena, suchasdarkmasshalosaround\ngalaxies,canbeexplained. \n\n\nReferences \n\n[1]K.G.Wilson,\"AnInvestigation oftheLowEquationandtheChew‐\nMandelstam Equations\", Ph.D.thesis,Cal.Tech.(1961).\n[2]K.HuangandF.E.Low,J.Math.Phys .6,795(1965);\n[3]K.HuangandA.H.Mueller,Phys.Rev.Lett.14,396(1965).\n[4]K.Huang,Quantum FieldTheory,fromOperators toPathIntegrals ,2nded\n(Wiley‐VCH,Weinheim, Germany, 2010).\n[5]K.Huang,Quarks,Leptons,andGaugeFields ,2nded(WorldScienific,\nSingapore, 1992).\n[6]J.J.Thomson, TheElectrician ,39,104(1897).\n[7]H.A.Lorentz,TheTheoryoftheElectron ,2nded(Doverpublications, New\nYork,1952).\n[8]P.A.M.Dirac,Proc.Roy.Soc.A117(778):610(1928).\n[9]P.A.M.Dirac,Proc.Roy.Soc.A126(801):360(1930).\n[10]V.F.Weisskopf, Z.Physik ,89,27(1934);Z.Physik,90,817(1934).\n[11]V.F.Weisskopf, Phys.Rev.56,72(1939).\n[12]K.Huang,Phys.Rev.101,1173(1956).\n[13]W.Heisenberg, Z.Physik ,90,209(1934).\n[14]P.A.M.Dirac,Proc.Camb,Phil.Soc.30,150(1934).\n[15]R.Serber,Phys.Rev.48,49(1935).\n[16]A.E.Uehling,Phys.Rev.48,55(1935).\n[17]W.E.LambandR.C.Retherford, Phys.Rev.72,241(1947).\n[18]P.KuschandH.M.Foley,Phys.Rev.74,250(1948)..\n[19]H.A.Bethe,Phys.Rev.72,339(1947).\n[20]J.Schwinger, Phys.Rev.73,1439(1948).\n[21]R.P.Feynman, Phys.Rev.74,1430(1948).\n[22]B.FrenchandV.F.Weisskopf, Phys.Rev.75,1240(1949).\n[23]J.Schwinger, Phys.Rev.73,416L(1948).\n[24]F.J.Dyson,Phys.Rev.75,486(1949).\n[25]M.Gell‐MannandF.E.Low,Phys.Rev.95,1300(1954).\n[26]J.Beringeretal(PDG)Phys.Rev.D86,016001(2012).\n[27]J.Schwinger, Phys.Rev.75,651(1949).\n[28]D.J.GrossandF.Wilczek,Phys.Rev.Lett.30,1343(1973).\n[29]H.D.Politzer,Phys.Rev.Lett.30,1346(1973).\n[30]N.N.Bogoliubov andD.V.Shirkov,Introduction tothetheoryofQuantum \nFields(Wiley‐Interscience, NewYork,1959).27\n[31]G.Gabrielse, T.Kinoshita etal,Phys.Rev.Lett.97,030802(2006).\n[32]C.G.Callan,Phys.Rev.D12,1541 (1970).\n[33]K.Symanzik, Comm.Math.Phys.18,27(1970).\n[34]L.D.LandauinNielsBohrandtheDevelopment ofPhysics ,edW.Pauli\n(McGraw ‐Hill,NewYork,1955).\n[35]T.D.Lee,Phys.Rev.95,1329(1954).\n[36]G.KällénandW.Pauli,Dan.Mat.Fys.Medd .30,no.7(1955).\n[37]K.HalpernandK.Huang,Phys.Rev.Lett.74,3526(1995).\n[38]K.HalpernandK.Huang,Phys.Rev.53,3252(1996).\n[39]L.P.Kadanoff, Physics(LongIslandCity,N.Y.)2,263(1966).\n[40]K.G.Wilson,Rev.Mod.Phys .55,583(1983).\n[41]J.Polchinski, Nucl.Phys. ,B231,269(1984).\n[42]V.Periwal,Mod.Phys.Lett.A11,2915(1996).\n[43]K.Huang,H.B.Low,andR.S.Tung,Class.Quantum Grav .29,155014\n(2012);arXiv:1106.5282. \n[44]K.Huang,H.B.Low,andR.S.Tung,Int.J.Mod.Phys .A27,1250154(2012);\narXiv:1106.5283. \n[45]K.Huang,C.Xiong,andX.Zhao,\"Scalar‐fieldtheoryofdarkmatter\",\narXiv:1304.1595 (2013)."    },    {        "title": "2307.00094v1.A_finite_element_method_to_compute_the_damping_rate_of_oscillating_fluids_inside_microfluidic_nozzles.pdf",        "content": "AFINITE ELEMENT METHOD TO COMPUTE THE DAMPING RATE\nOF OSCILLATING FLUIDS INSIDE MICROFLUIDIC NOZZLES\nA P REPRINT\nSøren Taverniers∗, Svyatoslav Korneev, Christoforos Somarakis, and Adrian J. Lew\nPalo Alto Research Center (PARC), 3333 Coyote Hill Road, Palo Alto, CA 94304, USA\nJuly 4, 2023\nABSTRACT\nWe introduce a finite element method for computing the damping rate of fluid oscillations in nozzles\nof drop-on-demand (DoD) microfluidic devices. Accurate knowledge of the damping rates for the\nleast-damped oscillation modes following droplet ejection is paramount for assessing jetting stabil-\nity at higher jetting frequencies, as ejection from a non-quiescent meniscus can result in deviations\nfrom nominal droplet properties. Computational fluid dynamics (CFD) simulations often struggle to\naccurately predict meniscus damping in the limit of low viscosity and high surface tension. More-\nover, their use in design loops aimed at optimizing the nozzle geometry for stable jetting is slow\nand computationally expensive. The faster alternative we adopt here is to compute the damping rate\ndirectly from the eigenvalues of the linearized problem. Starting from a variational formulation of\nthe linearized governing equations, we obtain a generalized eigenvalue problem for the oscillation\nmodes, and approximate its solutions with a finite element method that uses Taylor-Hood elements.\nWe solve the matrix eigenvalue problem with a sparse, parallelized implementation of the Krylov-\nSchur algorithm. The spatial shape and temporal evolution (angular frequency and damping rate) of\nthe set of least-damped oscillation modes are obtained in a matter of minutes, compared to days for\na CFD simulation. We verify that the method can reproduce an analytical benchmark problem, and\nthen determine numerical convergence rates on two examples with axisymmetric geometry. We also\nprove that the method is free of spurious modes with zero or positive damping rates. The method’s\nability to quickly generate accurate estimates of fluid oscillation damping rates makes it suitable for\nintegration into design loops for prototyping microfluidic nozzles.\nKeywords Generalized Eigenvalue Problem ·Capillary Oscillations ·Additive Manufacturing ·3D Printing ·Nozzle\n1 Motivation\nDroplet-based microfluidics [1] is an area of fluid dynamics with applications in droplet generation, micro fabrication\n[2], and manufacturing of core-shell particles [3]. In particular, microfluidic technologies are promising in addi-\ntive manufacturing (AM) through drop-on-demand (DoD) 3D printing of parts, which involves pulsed generation of\ndroplets [4]. In DoD 3D printing, the material (e.g., a metal alloy) is first melted and then led through a nozzle to\ngenerate and eject droplets that are deposited onto a substrate, layer by layer, to produce the desired part. The resulting\nparts typically consist of hundreds of thousands or even millions of coalesced droplets. Part quality (e.g., porosity and\nstructural integrity) and part-to-part consistency are therefore intimately linked to droplet jetting conditions. Heuristi-\ncally speaking, the more stable the generation of liquid droplets at the nozzle level, the more consistent the deposition\nof these droplets on the substrate.\nThe coherency of jetting (how similar ejected droplets are) in DoD 3D printing is strongly associated with the pulsed\ndroplet generation and ejection mechanism, as well as the dynamics of the residual material that remains in the nozzle\n∗Corresponding author\nEmail address: stavernier@parc.com (Søren Taverniers)arXiv:2307.00094v1  [physics.comp-ph]  30 Jun 2023arXiv Template A P REPRINT\nfollowing the ejection event. As a rule of thumb, the more quiescent the liquid material in the nozzle at the moment\nof droplet ejection, the more coherent the jetting. Therefore, the jetting coherency is strongly affected by the rate\nat which the kinetic energy of the liquid in the nozzle gets dissipated during the time interval between subsequent\nejection events.\nRecently, the significant impact of the rate of damping of an oscillating meniscus on jetting of inks and liquid metals\nin electrohydrodynamic [5] and magnetohydrodynamic [6] DoD 3D printers was highlighted. In these contributions,\nthe role of the interplay between nozzle geometry and material properties in meniscus damping is emphasized. Specif-\nically, [5] uses a linear mass-spring-damper model developed in [7] to estimate the angular frequency and damping\nrate of small-amplitude oscillations of the ink inside the nozzle. The estimated damping rates are not compared to\nexperimental measurements, but higher damping rates are found to be advantageous for high-frequency DoD jetting.\n1.1 Challenges in Accurate, Efficient Computation of Microfluidic Nozzle Damping Rates\nComputing the damping rate using computational fluid dynamics (CFD) is challenging in the limit of small fluid\nviscosity and high interfacial surface tension, as is the case for microfluidic problems [6]. Detailed CFD models can\nhave a hard time converging to a damping rate value within a reasonable tolerance, since both the meniscus dynamics\nand the boundary layers need to be resolved. Nicolás [8] and Kidambi [9] pointed out three sources of damping: (a)\nviscous dissipation in the boundary layers near the container walls and in the liquid’s interior; (b) capillary damping\ndue to meniscus effects and a moving contact line; and (c) contamination of the free surface. For a detailed review of\npast efforts elucidating these damping mechanisms on this problem we refer the reader to these two papers.\nCFD solvers become prohibitively expensive when there is a need for repeated nozzle re-design as part of a prototyping\noptimization loop. In this work, we develop a finite-element based solver for computing the frequencies and damping\nrates of a small set of least-damped modes. This solver is orders of magnitude faster (minutes vs. ∼a day) than\nhigh-fidelity CFD runs executed on comparable hardware.\n1.2 Related Literature\nNumerous experimental, analytical and numerical studies have been conducted to analyze the dynamics of free-surface\noscillations inside channels and rigid or deformable containers. Among the earliest works is the theoretical and\nexperimental analysis by Benjamin and Scott [10] in 1970 for an inviscid liquid’s meniscus with pinned contact line\nin a narrow open channel, which was later extended by Graham-Eagle [11] who also considered more complicated\ngeometries. Following experimental investigations by Cocciaro et al. [12] into the static and dynamic properties of\nsurface waves in cylinders making a static wall contact angle of 62◦, Nicolás [8] computed the damping rate and\nfrequency of the oscillation modes of capillary-gravity waves of a slightly viscous fluid in a rigid brimful cylinder with\nflat equilibrium meniscus. Later, he also considered a static contact angle of 62◦for both a pinned and a moving contact\nline [13]. Following studies by Gavrilyuk et al. [14] and Chantasiriwan [15] of the linearized free-surface oscillations\ninside rigid containers using fundamental solutions, Kidambi [16] studied the effects of a curved equilibrium meniscus\non the frequency and damping rate of free-surface oscillation modes in a rigid brimful cylinder by solving an eigenvalue\nproblem with complex eigenvalues.\nAnother line of research, driven mostly by engineering needs, involves the analysis of sloshing dynamics in partially\nfilled containers undergoing accelerated motion. Sloshing is encountered in problems ranging from ground-supported\nstorage tanks impacted by earthquakes to fuel tanks in aircraft or spacecraft carrying out sharp maneuvers [17]. A\nfew authors used computational fluid dynamics (CFD) simulations based on the volume of fluid (V oF) method to\nsimulate liquid sloshing with and without anti-sloshing baffles, or a combination of CFD and non-CFD techniques\n[18]. Cruchaga et al. [19] performed finite element computations using a monolithic solver that included a stabilized\nformulation and a Lagrangian tracking technique for updating the free surface. Choudhary et al. [20] employed\nanalytical techniques as well as finite element and boundary element methods to study liquid oscillations in rigid\ncircular cylindrical shells with internal flexible membranes or covered by membranes. The majority of efforts in\ncharacterizing liquid sloshing, however, were focused on linearizing the governing Navier-Stokes equations around\nan equilibrium meniscus and using finite elements to discretize a variational formulation of the resulting problem\n([21], [17], [22], [23], [24], [25], and [26]); the latter approach results in a generalized matrix eigenvalue problem\nthat is suitable for a modal analysis yielding the eigenmodes (velocity, pressure and meniscus surface deformation\nprofiles) and corresponding eigenvalues (yielding the modes’ angular frequencies for inviscid liquids, and both their\nangular frequencies and damping rates for viscous liquids). These finite element-based efforts typically account for\ngravity, and vary in their inclusion of viscosity and surface tension, and in their consideration of arbitrary container\ngeometries. None of them simultaneously account for the effects of viscosity and surface tension. They are part\nof a broader research thrust to deploy model order reduction techniques for performing modal analyses of complex\n2arXiv Template A P REPRINT\nflows, which also include proper orthogonal decomposition (POD), balanced POD, and dynamic mode decomposition\n(DMD), among others [27].\n1.3 Our Contribution\nWe consider free-surface oscillations of an incompressible fluid in rigid axisymmetric nozzles relevant to microfluidics .\nBesides departing from the typical target application of related finite element-based works mostly focused on liquid\nsloshing in containers undergoing acceleration, our approach also differs in the fact that it simultaneously accounts\nfor viscosity and surface tension . Unlike these related works, we opted to ignore gravity and perform the linearization\nof the equations around a flat equilibrium meniscus. This choice simplified the analysis, but it is relatively straight-\nforward to modify it and start from a curved meniscus instead. Performing a Laplace transform in time and deriving\na variational formulation (i.e., weak form), we arrive at a continuous complex eigenvalue problem. We show that\nthe damping rates of all oscillation modes in this problem are non-negative, as expected. We then introduce a finite\nelement discretization to arrive at a generalized matrix eigenvalue problem that we solve using the Krylov-Schur iter-\native method. By selecting a combination of velocity and pressure finite element spaces that satisfy a suitable inf-sup\ncondition, in our case Taylor-Hood elements, we show that the damping rates of the oscillation modes in this discrete\nsetting are also non-negative. Therefore, no modes with zero or positive damping rate arise as a result of numerical\nartifacts.\nWhile CFD simulations performed on comparable hardware may take days, the method we introduce here can deter-\nmine the damping rates of the least-damped meniscus oscillation modes in minutes.\n1.4 Paper Structure\nIn Section 2, we detail the problem setup and governing set of partial differential equations, as well as the ansatz we\nmake to obtain solutions that are oscillation modes. We demonstrate how the resulting weak form of the governing\nequations yields a continuous generalized eigenvalue problem (GEP). In Section 3, we show how a finite element\ndiscretization of the domain of interest yields a matrix GEP with complex eigenvectors, and present a procedure\nfor numerically solving this GEP using the Krylov-Schur algorithm. In Section 4, we first verify our method and\nimplementation on a planar viscous capillary wave, and then use it to predict the meniscus dynamics for a brimful\ncylinder and a liquid-filled container with a generic axisymmetric shape. We offer concluding remarks and a future\noutlook in Section 5.\n2 Problem Setup and Governing Equations\nIn the following, we state the linearized governing equations for the motion of an incompressible fluid inside a nozzle\nwith a free surface, directly in their non-dimensional form. To this end, we select a characteristic length Rto non-\ndimensionalize every length in the problem. The length scale then defines a characteristic time T=p\nρR3/σ, speed\nU=R/T =p\nσ/(ρR), and pressure or stress P=σ/R, where σis the surface tension, ρis the mass density and µ\nis the dynamic viscosity of the fluid. These quantities are adopted to non-dimensionalize time, velocity, and pressure,\nrespectively. The Reynolds number is defined as Re=p\nρRσ/µ2.\nConsider a nozzle full of an incompressible fluid occupying an open, bounded, connected domain Ω⊂R3with smooth\nboundary ∂Ω. We partition the boundary into three disjoint sets, each of non-zero area, ∂Ω =Γm∪Γw∪Γtsuch that\nΓm∩Γt=∅,Γmhas a bounded positive area (measure), i.e., |Γm|>0, and Γm’s manifold boundary is Γm∩Γw.\nIn this work, we consider the set Γmto be planar, and without loss of generality, adopt a Cartesian coordinate system\nOxyzsuch that Γm⊂ {z= 0}and the unit exterior normal to ΩonΓmpoints towards the negative z-axis, see Fig.\n1. The fluid inside the nozzle is assumed to be in contact with the (rigid) walls of the nozzle on the set Γw, in contact\nwith the atmosphere on Γm, and in contact with the walls of the nozzle on the boundary of Γm, forming a meniscus\ntherein. To be able to assume that the meniscus is pinned and the contact line does not move, we will additionally\nrequire that the contact line Γm∩Γwdefines a sharp corner. Finally, the fluid in the nozzle is in contact with more\nof the same fluid on Γt. For simplicity, we imposed a stress-free condition on Γt, consistent with the fact that we are\nneglecting the effect of gravity in the fluid.\nWe are interested in finding the late-time oscillation modes of the fluid in the nozzle, once the non-linear convective\neffects become negligible and the amplitude of oscillations of the meniscus in Γmis small. Under these conditions,\nwe will parameterize the deformation of the meniscus surface with the displacement from Γmalong the z-direction,\nξ: Γm×R+→R:t7→ξ(x, y, t ), for(x, y)∈Γmand for any time t≥0, so that the shape of the meniscus surface\nin time is {(x, y, ξ (x, y, t ))|(x, y)∈Γm, t≥0}, as shown in Fig. 1.\n3arXiv Template A P REPRINT\nFigure 1: Sectional view of a typical nozzle design, illustrating interior and boundary domains as they are defined in\nEq. (1).\nDenoting by u: Ω×R+→R3the velocity field, by p: Ω×R+→Rthe pressure field, and by ezthe unit vector in\nthe positive zdirection, the linearized equations of motion for the fluid inside the nozzle are:\n∂u\n∂t=−∇p+1\nRe∇ ·\u0000\n∇u+ (∇u)T\u0001\ninΩ (1a)\n∇ ·u= 0 inΩ (1b)\nu= 0 onΓw (1c)\n∂ξ\n∂t=u·ez onΓm (1d)\n−pez+1\nRe(∇u+ (∇u)T)·ez=−∆Sξez onΓm (1e)\n−pez+1\nRe(∇u+ (∇u)T)·ez=0 onΓt (1f)\nξ= 0 onΓm∩Γw. (1g)\nHere\n∆Sξ=∂2ξ\n∂x2+∂2ξ\n∂y2\nis the surface Laplacian on Γm. The external pressure on ΓtandΓmis assumed to be the same, and set to zero here.\nHence, these equations represent a linearization of the problem around a hydrostatic equilibrium at zero pressure with\na flat meniscus. No initial conditions are specified because we are interested in the late-time dynamics of the nozzle.\nTo find the oscillation modes, we will seek solutions to (1) according to the ansatz2u(x, y, z, t ) = exp( λt)ˆu(x, y, z ),\np(x, y, z, t ) = exp( λt)ˆp(x, y, z )andξ(x, y, t ) = exp( λt)ˆξ(x, y), where λ∈Cis a complex frequency and ˆu: Ω→\n2Or, equivalently, by applying a Laplace transform in time.\n4arXiv Template A P REPRINT\nC3,ˆp: Ω→C, and ˆξ: Γm→Care unknown functions for which (1) becomes:\nλˆu=−∇ˆp+1\nRe∇ ·\u0000\n∇ˆu+ (∇ˆu)T\u0001\ninΩ (2a)\n∇ ·ˆu= 0 inΩ (2b)\nˆu= 0 onΓw (2c)\nλˆξ=ˆu·ez onΓm (2d)\n−ˆpez+1\nRe(∇ˆu+ (∇ˆu)T)·ez=−∆Sˆξez onΓm (2e)\n−ˆpez+1\nRe(∇ˆu+ (∇ˆu)T)·ez=0 onΓt (2f)\nˆξ= 0 onΓm∩Γw (2g)\nThe complex frequency λdefines the damping rate η=−Re(λ), and the angular frequency ω= Im( λ).\nWe show in 6.3 that the only solution of (2) when λ= 0 is the trivial one, so we henceforth only consider the case\nλ̸= 0. After obtaining the weak form, we show that (2) defines a generalized eigenvalue problem (GEP) with complex\neigenvalues.\n2.1 Weak Form of the Problem\nThe solution of problem (2) satisfies the following weak form, which is derived in 6.2. Let\nΞ ={ζ∈H1(Γm,C)|ζ= 0onΓm∩Γw}, (3a)\nV={v∈[H1(Ω,C)]2|v·ez|Γm∈Ξ,v=0onΓw}, (3b)\nP=L2(Ω,C), (3c)\nandvz=v·ezforv∈ V. Find{ˆu,ˆp,ˆξ} ∈ V × P × Ξ,ˆu̸=0, and λ∈C,λ̸= 0, such that for all v∈ V, allq∈ P,\nand all ζ∈Ξ,\n0 =Z\nΓm∇Sˆξ· ∇SvzdS (4a)\n+Z\nΩλˆu·v−ˆp∇ ·v+1\nRe(∇ˆu+ (∇ˆu)T):∇vdΩ\n0 =Z\nΩq∇ ·ˆudΩ (4b)\n0 =Z\nΓm(∇Sˆuz−λ∇Sˆξ)· ∇Sζ dS (4c)\nWe note the weak compatibility condition between ˆξandˆuzexpressed in terms of the equality of the gradients, instead\nof equality of the functions themselves. As we will see below, this is necessary to obtain a symmetric GEP.\nThis weak form can be more succinctly expressed through the following bilinear forms defined for any u,v∈ V,\np, q∈ P andξ, ζ∈Ξ:\na(u,v) =Z\nΩu·vdΩ (5a)\nb(u, q) =Z\nΩ−q∇ ·udΩ (5b)\nc(u,v) =Z\nΩ1\nRe(∇u+ (∇u)T):∇vdΩ (5c)\ns(ξ, ζ) =Z\nΓm∇Sξ· ∇Sζ dS, (5d)\nand\ng((u, p, ξ),(v, q, ζ)) =c(u,v) +b(v, p) +s(ξ, vz) +b(u, q) +s(uz, ζ) (5e)\nh((u, p, ξ),(v, q, ζ)) =−a(u,v) +s(ξ, ζ). (5f)\n5arXiv Template A P REPRINT\nWith these, equations (4) can be restated as\n0 =λa(ˆu,v) +c(ˆu,v) +b(v,ˆp) +s(ˆξ, vz) (6a)\n0 =b(ˆu, q). (6b)\n0 =s(ˆuz, ζ)−λs(ˆξ, ζ), (6c)\nor, more succinctly, as\n0 =g((ˆu,ˆp,ˆξ),(v, q, ζ))−λ h((ˆu,ˆp,ˆξ),(v, q, ζ)). (6d)\nWhen written in this way, it is clear that the weak form defines a GEP in which λis the eigenvalue and (ˆu,ˆp,ˆξ)is the\neigenvector. Setting ˆu=0,ˆp= 0 andˆξ= 0 satisfies these equations for any λ∈C. However, we are looking for\nnon-trivial solutions, and those are obtained for selected values of λ.\nNotice that both bilinear forms gandhare symmetric, so (6d) defines a symmetric GEP.\nIt is shown in 6.3 that all eigenvalues need to have a negative real part, i.e., Re(λ)<0, so the amplitude of each\nmode decays in time. Additionally, it is trivial to see that if λis an eigenvalue and (ˆu,ˆp,ˆξ)is its eigenvector, then its\ncomplex conjugate λis also an eigenvalue with (ˆu,ˆp,ˆξ)as its eigenvector. Therefore, oscillatory modes of the fluid\ncome in pairs with frequencies of opposite signs (imaginary parts of λandλ) and the same relaxation time (real parts\nofλandλ)3.\nIn general, we will be interested in finding all modes whose eigenvalue λsatisfy ητ=−Re(λ)τ < 1, for some\ncharacteristic time τ >0we choose. This corresponds to seeking all modes for which |Re(λ)|is smaller than 1/τ,\ni.e., which will take longer to decay than the chosen relaxation time τ.\n2.2 Restriction to Two-Dimensional Problems\nIn the following, we consider domains Ωand flows that can be represented in two dimensions, namely, either planar\nor axisymmetric flows.\nFor planar flows we assume that Ω ={(x, y, z )∈R3|(x, z)∈˘Ω, y∈[0, L]}, where ˘Ω⊂R2andL >0, and that\nΓi=γi×[0, L]withγi⊂∂˘Ωandi=b, torm. Additionally, we assume that neither ˆu,ˆp, nor ˆξdepend on y, and\nthatˆu= ˆuxex+ ˆuzezwhere exis the unit vector along the xaxis.\nFor axisymmetric flows, let (r, θ, z )denote the standard cylindrical coordinates. We assume that Ω =\n{(rcosθ, rsinθ, z)∈R3|(r, z)∈˘Ω, θ∈[0,2π)}, where ˘Ω⊂R+×R, and that Γi={(rcosθ, rsinθ, z)|\n(r, z)∈γi, θ∈[0,2π)}withγi⊂∂˘Ωandi=b, torm. Additionally, we assume that neither ˆu,ˆporˆξdepend on θ,\nand that ˆu= ˆurer+ ˆuzez, where eris the unit vector in the direction in which rgrows and θandzare constant.\nFor concreteness, we will set ˘Ω = Ω ∩ {(x,0, z)∈R3}in both cases, so that we can embed the two-dimensional\ndomain ˘ΩinR3. Equivalently, ˘Ωis the set of points in Ωwithy= 0 orθ= 0; see Fig. 2 for the case of an\naxisymmetric geometry. Additionally, because ˆu,ˆpandˆξchange only over ˘Ω, we will indistinctly treat them as\nfunctions over Ωor˘Ω.\n3 Finite Element Discretization\nWe discretize the two-dimensional restriction of the problem in §2.2. We mesh ˘Ωwith triangular elements such that\nγm,γtandγware each meshed by a distinct collection of edges in the mesh, and such that every triangle has at least\none node in the interior of ˘Ω4. On curved boundaries, a collection of edges in the mesh interpolates the boundary at\nthe vertices, as customary. We did not implement an isoparametric construction to more accurately approximate the\ndomain.\nOver any such mesh, we construct the following finite element spaces for planar flows :\nΞh={ζ∈Ξ|ζ|e∈P2(e,C),in all mesh edges eonγm}\nVh={v∈ V | (vx, vz)|K∈[P2(K,C)]2,in all mesh elements Kin˘Ω}\nPh={p∈ P | p|K∈P1(K,C),in all mesh elements Kin˘Ω}.\n3Herexindicates the complex conjugate of complex number x.\n4This is a sufficient requirement to satisfy the inf-sup condition, see 6.4\n6arXiv Template A P REPRINT\nFigure 2: Domain ˘Ωfor a planar (left) and an axisymmetric (right) geometry.\nHere Pk(β,C)is the space of complex-valued polynomials of degree k∈Nover the domain β⊂R2. Because of the\nplanar flow assumption, v=vxex+vzezinΩforv∈ Vh, and it is enough to define the fields over ˘Ω, since they are\nextended to Ωusing that ζ∈Ξh,p∈ Phandv∈ Vhdo not depend on y. The construction of finite element spaces\nforaxisymmetric flows is obtained in exactly the same way, after replacing xforrandyforθin the definition above.\nNote that by the inclusion Ξh⊂Ξ, any function ζh∈Ξhneeds to satisfy ζh= 0onΓm∩Γw, and hence on γm∩γw.\nSimilarly, vh∈ Vhneeds to satisfy that vh=0onΓw, and hence on γw. By construction, vh·ez∈Ξhfor any\nvh∈ Vh.\nThe spaces VhandPhfor the approximation of the velocity and pressure fields are constructed with Taylor-Hood ele-\nments [28, 29], see Fig. 3, and they just involve continuous piecewise quadratic functions for each velocity component\nand continuous piecewise linear functions for the pressure. Taylor-Hood elements are standard finite elements used in\nthe simulation of incompressible fluid flow, since under mild conditions on the mesh, they satisfy the inf-sup stability\ncondition for the incompressibility constraint for both planar [30, 31] and axisymmetric [32] flows.\nFigure 3: Section of the computational domain for the axisymmetric geometry considered in Section 4.3. The triangles\nare Taylor-Hood finite elements, with filled circles indicating velocity nodes and empty circles denoting pressure\nnodes.\nThe numerical approximation is obtained from the weak form (4) by substituting the infinite-dimensional function\nspaces Ξ,VandPby their finite-dimensional (finite-element) counterparts Ξh,VhandPh, respectively. Precisely, we\n7arXiv Template A P REPRINT\nseek(ˆuh,ˆph,ˆξh)∈ Vh× Ph×Ξhandλ∈Csuch that\n0 =g((ˆuh,ˆph,ˆξh),(ˆvh,ˆqh,ˆζh))−λ h((ˆuh,ˆph,ˆξh),(ˆvh,ˆqh,ˆζh)) (7)\nfor all (ˆvh,ˆqh,ˆζh)∈ Vh× Ph×Ξh. Henceforth, we will refer to the approximate solutions computed with this finite\nelement discretization as the finite element analysis (FEA) solutions.\n3.1 Matrix Formulation\nProblem (7) leads to a symmetric matrix GEP. To this end, we let {Na}denote the standard basis function of P2-\nelements, where index aruns over all corner and midnodes in the mesh except for those on Γw, and{Mβ}the\nstandard basis function for P1-elements, where index βruns over all corner nodes in the mesh. With them, we can\nwrite (ˆuh,ˆph,ˆξh)∈ Vh× Ph×Ξhas\nˆuh=X\navaNa,ˆph=X\nβpβMβ,ˆξh=X\na∈ΓmξaNa,\nwhere we abused notation and by a∈Γmindicated all nodes on Γmexcept for those on Γm∩Γw. Here va∈C2,\npβ∈Candζa∈Cfor each aandβ, and they can be grouped in column matrices U,PandZ, respectively. The\nrelevant matrices in the problem follow from\nAab=a(Na, Nb), B aβ=b(Na, Mβ), C ab=c(Na, Nb)\nand\nSab=s(Na, Nb) a, b∈Γm.\nThe symmetric GEP can then be stated as finding X= [U P Z ]Tandλ∈Csuch that\n\nC B S\nBT0 0\nS 0 0\n\n|{z }\nG\"U\nP\nZ#\n=λ\"−A0 0\n0 0 0\n0 0 S#\n|{z}\nH\"U\nP\nZ#\n(8a)\nor\nGX=λHX. (8b)\nWe show in 6.4 that Re(λ)<0, and hence that matrix Gis invertible. This also implies that the method is free from\nspurious (as an artifact of the discretization) modes with zero or positive damping rate. The size of the system in\n(8b) is given by the total number of degrees of freedom for the velocity components, pressure and meniscus surface\ndisplacement in ˘Ω, and hereafter we will denote it by n.\nSince GandHare real symmetric matrices, if λis a complex eigenvalue with eigenvector X, then then λis also\nan eigenvalue with eigenvector X. This also implies that if λ∈R, then Xis a real eigenvector. Each pair of\ncomplex eigenvectors XandXdefines complex-valued evolutions in time Xexp(λt)andXexp(λt) =Xexp(λt).\nA two-dimensional subspace of real-valued velocities, pressures and meniscus surface displacements follows as linear\ncombinations of X+(t) = Re( Xexp(λt))andX−(t) = Im( Xexp(λt)). Explicitly, if λ=−η+iω, then\nX+(t) = exp( −ηt) [Re( X) cos( ωt)−Im(X) sin(ωt)],\nX−(t) = exp( −ηt) [Re( X) cos( ωt−π/2)−Im(X) sin(ωt−π/2)].\nBoth X+andX−have the same damping rate ηand angular frequency ω, but a different phase.\n3.2 Implementation and Solution of the Matrix GEP\nWe implemented the finite element discretization in the open-source framework FEniCS [33], which provides matrices\nGandHof the GEP in (8b). We either used built-in meshing tools from FEniCS (for the planar and cylindrical\ndomains) or Gmsh [34] (for the generic axisymmetric domain) to generate the finite element mesh.\nGiven that we are only interested in a small subset of pairs of eigenvalues and eigenvectors (i.e., those with the small-\nest values of |Re(λ)|), an iterative solver that converges only a selected set of eigenpairs is most appropriate. To\nthis end, we used the Krylov-Schur method [35, 36, 37] with a random vector to initialize the Krylov subspace. This\nis a projection-based algorithm that uses a Krylov subspace, and is an improvement over the traditional Arnoldi and\nLanczos schemes in that it incorporates an effective and robust restarting technique, generally resulting in improved\n8arXiv Template A P REPRINT\nconvergence. Given that His singular, in order to apply Krylov-Schur we needed to perform a shift-and-invert trans-\nform (see [38]).\nThe matrix Gcan display poor conditioning which leads to problems with the convergence of some components of\nthe eigenvectors. To address this, we left-multiplied both sides of (8b) by a diagonal preconditioner Λwith diagonal\ncomponents given by Λii= 1/max j=1,...,n|Gij|fori= 1, . . . , n , where nis the dimension (8b). This operation does\nnot change the eigenvectors or eigenvalues of (8b), but it does transform the matrices of the system into non-symmetric\nones. It is possible to keep the symmetry by preconditioning also on the right, but we did not do this here.\nFor the computations in Section §4, the size nof the GEP (8b) can approach O(106). It can be seen from (8a), G\nandHare highly sparse matrices, and thus can be efficiently represented using PETSc [39] to overcome memory\nlimitations. In turn, this enabled us to deploy the PETSc -based eigensolver SLEPc [38], which has the Krylov-Schur\nalgorithm. Additionally, we used MUMPS [40] for LU factorization of large sparse matrices. The combination of\nthese resulted in a highly parallelizable eigensolver for this problem.\nInSLEPc , we can set flags to control the number of requested eigenvalues, nev, and to set the convergence criteria,\ni.e., maximum number of iterations to perform Niter,max and the tolerance ϵbelow which we would like the error of\nthe computed eigenvalues to fall. We set Niter,max = 5000 andϵ= 10−16, which we found to be appropriate for our\nproblems of interest. The maximum dimension of the Krylov subspace has a default value of twice nev, which we\nused in our computations. Typically, we only needed to request SLEPc to converge nev= 70 eigenvalues to obtain\nthe least-damped modes of interest. Finally, we note that the use of the governing equations in their dimensionless\nform, (1), was critical to obtain significantly improved robustness of the eigensolver with respect to the choice of the\nshift value.\nAccounting for the Dirichlet boundary conditions on ΓwIn order to implement the boundary conditions on Γw\n(slip or no-slip), it is sufficient to zero out the rows of GandHcorresponding to the constrained nodal degrees of\nfreedom on Γw, and set the diagonal components of Gin these rows to be 1. To keep the symmetry of GandHit\nwould have been necessary to zero out the corresponding columns, but because we only used a left preconditioner, we\ndid not do this and worked with the non-symmetric matrices.\n4 Results\nWe showcase the performance of the method and compute the least-damped modes of oscillation of a fluid in a nozzle,\nincluding the accuracy and convergence rate of the FEA solutions. To this end, we perform a suite of numerical tests\naimed at verifying the code and the method on an analytical benchmark problem (planar capillary wave), and analyzing\nthe convergence as the mesh size is decreased on cylindrical and arbitrary axisymmetric domains.\n4.1 Analytical Benchmark Problem: 2D Inviscid and Viscous Capillary Wave\nIn this example, we consider a planar flow problem that involves oscillations of a capillary wave in a two-dimensional\ndomain ˘Ω ={(x, z)|x∈[0,1]andz∈[0, H]}, for some H≫1, with γm= [0,1]× {0},γw={0,1} ×[0, H]∪\n[0,1]× {H}, and γt=∅. To obtain an analytical solution, we first consider the inviscid case and find the modes of\noscillation. We then use these solutions to approximate the damping rate and angular frequency for each of the modes\nwhen the viscosity is small but not zero.\n4.1.1 Inviscid Case\nWhen there is no viscosity and the flow is assumed to be irrotational, the system of equations (2) can be restated in\nterms of a velocity potential ϕ: Ω→Rso that ∇ϕ=ˆu, taking advantage that ϕcan be chosen so that p+∂ϕ/∂t = 0\ninΩfor any time t[41]. Since the flow is planar, neither ξnorϕdepend on y, and using the ansatz ϕ(x, z, t ) =\n9arXiv Template A P REPRINT\nˆϕ(x, z) exp( λt)we can write ˆp+λˆϕ= 0in lieu of (2a). In terms of ˆϕandˆξ, (2b) to (2f) become\n∆ˆϕ= 0 inΩ (9a)\n∂ˆϕ\n∂x\f\f\f\n(0,z)=∂ˆϕ\n∂x\f\f\f\n(1,z)= 0 z∈[0, H] (9b)\n∂ˆϕ\n∂z\f\f\f\n(x,H)= 0 x∈[0,1] (9c)\nλˆξ=∂ˆϕ\n∂z\f\f\f\n(x,0)x∈[0,1] (9d)\nλˆϕ\f\f\f\n(x,0)=−∂2ˆξ\n∂x2x∈[0,1]. (9e)\nBecause it is inviscid flow, the no-slip boundary condition (2c) was replaced by the no-penetration conditions (9b) and\n(9c). Also, since it is potential flow, we cannot impose (2g).\nWe can seek exact solutions of the form\nˆϕn(x, z) = cos( nπx) exp(−nπz), ˆξn(x) =−nπ\nλnˆϕn(x,0), (10)\nforn∈N, since in this way (9a), (9b),(9d) and (9e) are satisfied. From (9e), it follows that\nλn=±i(nπ)3/2,\nand from ˆp+λˆϕ= 0we get that\nˆp(x, z) =∓i(nπ)3/2ˆϕn(x, z).\nClearly, (10) cannot satisfy (9c). However, if H≫1, then∂ˆϕn\n∂z(x, H)≈0and (9c) is approximately satisfied. We\nuseH= 4in our numerical experiments.\nThe time evolution of these capillary waves is then given by\nϕ±\nn(x, z, t ) = cos( knx) exp(−knz±iωnt),\nξ±(x, t) =±iω−1/3\nnϕ±\nn(x,0, t),\np±(x, z, t ) =∓iωnϕ±\nn(x,0, t),(11)\nwhere\nkn=nπ, ω n=k3/2\nn= (nπ)3/2. (12)\nThe velocity field u±\nnfollows as the spatial gradient of ϕ±\nn.\nWe compare the analytical frequencies from (12) to the frequencies we obtain from the FEA solutions. Because the\nFEA formulation needs a nonzero viscosity for Gto be invertible, we compute solutions with progressively smaller\nvalues of dynamic viscosity, or progressively larger values of the Reynolds number Re. The expectation is that the\nangular frequencies computed with FEA will converge to their analytical counterparts (12) as the value of Retends to\ninfinity.\nFrom (12), it follows that the expected (dimensionless) angular frequencies are ω1= 5.5683 ,ω2= 15 .7496 and\nω3= 28.9339 . We perform FEA simulations for a series of Reynolds numbers ranging from 1004 to 8034. Figure 4\ndemonstrates convergence of the angular frequencies computed with FEA and varying viscosity values to the analytical\nvalues for the inviscid case, (12), when n= 1,2,3.\nFinally, Figure 5 compares the profiles of the velocity components from the FEA solutions with their analytical coun-\nterparts given by ˆun=∇ˆϕn, with ˆϕngiven by (10), for n= 1. Note that for this example ˆu1can be chosen to have\nall real components. In contrast, the FEA solutions are computed with a small value of Re−1, so it is not necessarily\npossible to make all components of the computed velocity field real. However, the FEA solution can be multiplied by\na complex scalar to make the real part of the components much larger than the imaginary part (in this case two orders\nof magnitude larger), and normalized so that the analytic and numerical solution have the same maximum value for\nsome field, in this case, the absolute value of the real part of the velocity component in the x-direction. We show the\nreal part of the resulting velocities in the figure.\n10arXiv Template A P REPRINT\n(a) Mode n= 1\n (b) Mode n= 2\n (c) Mode n= 3\nFigure 4: Comparison of the angular frequencies obtained from the FEA solutions for the three lowest-angular-\nfrequency modes of the viscous capillary wave with their counterparts computed via (15) for (a) mode n= 1, (b)\nmode n= 2, and (c) mode n= 3. For Re−1approaching zero, the angular frequencies from the FEA solutions\nconverge to the analytical expressions for the inviscid capillary wave given by (12). The values computed from the\nFEA solutions are largely converged, as the comparison between the two levels of refinements suggests.\n(a) Analytical values of ˆux.\n (b) Computed values of ˆuxfrom the FEA solution.\n(c) Analytical values of ˆuz.\n (d) Computed values of ˆuzfrom the FEA solution.\nFigure 5: Comparison of the analytical and computed values of ˆuxandˆuzof a planar inviscid capillary wave for\nn= 1. The analytical values followed from ∇ˆϕnforˆϕnin (10).\n11arXiv Template A P REPRINT\n(a) Mode n= 1\n (b) Mode n= 2\n (c) Mode n= 3\nFigure 6: Comparison of the damping rates obtained from the FEA solutions for the three lowest-angular-frequency\nmodes of a planar viscous capillary wave with their counterparts computed via (15) for (a) mode 1, (b) mode 2, and (c)\nmode 3. As in Fig. 4, the values computed from the FEA solutions are largely converged, as the comparison between\nthe two levels of refinements suggests.\n4.1.2 Viscous Case\nThe inviscid flow modes can be used to estimate the angular frequency and damping rate for the viscous flow modes\nfor small enough viscosity values. To do this, we will assume that the inviscid flow modes are good approximations\nto the viscous flow modes, and use them to solve for λ. Specifically, consider an incompressible mode (ˆu,ˆp,ˆξ)with\ncomplex frequency λ, whose meniscus surface displacement satisfies ˆξ=uz/λ, from (2d). Then, setting v=ˆuin\n(6a) implies that λshould satisfy\n0 =λa(ˆu,ˆu) +c(ˆu,ˆu) +1\nλs(ˆuz,ˆuz), (13)\nwhere we used that b(ˆu, p) = 0 because of the incompressibility of ˆu. This same equation can alternatively be obtained\ndirectly from a mechanical energy balance.\nTo estimate the damping rate, we just need to solve (13) for λby replacing\nˆu=∇ˆϕn, (14)\nwhere ˆϕnis given by (10). In the limit of infinite channel height ( H→ ∞ ), we obtain\nλn=−2k2\nn\nRe±is\nk3n\u0012\n1−4kn\nRe2\u0013\n. (15)\nHere, the (negative) real part corresponds to the damping rate, while the absolute value of the imaginary part cor-\nresponds to the angular frequency. For weak damping ( Re≫k1/2\nn), the damping rate is ηn=−2k2\nn/Reand the\nangular frequency will approach k3/2\nn, which corresponds to ωnfor the inviscid case (see (12)).\nFigure 4 compares the angular frequencies from (15) to estimates from the FEA solutions for n= 1,2,3. Figure 6\nshows a similar comparison for the damping rates. For both quantities, we find a good agreement between the FEA\nestimates and the values obtained from (15), with deviations becoming larger as Re−1increases and the use of inviscid\nflow modes is no longer justified in deriving (15).\n4.2 Mesh Convergence on a Cylindrical Domain\nNext, we compute the least-damped eigenmodes for capillary wave oscillations in a brimful cylinder, and evaluate the\nconvergence of the computed eigenvalues and eigenvectors as the mesh size is decreased. Specifically, we consider\nan axisymmetric problem in the two-dimensional domain ˘Ω ={(r, z)|r∈[0,1]andz∈[0, H]}, forH= 2.4, with\nγm= [0,1]× {0},γw={1} ×[0, H], and γt= [0,1]× {H}. We set the Reynolds number to Re= 710 , which\nwas inspired by the case of ˘Ωhaving a dimensional length of R= 5·10−4m (acting as the characteristic length) and\nbeing filled with liquid aluminum with density ρ= 2435 kg /m3and dynamic viscosity ν= 4.16×10−7m2/s, and\nthe gas below γmto be argon at atmospheric conditions such that the surface tension at γmisσ= 0.85 N/m.\nFigure 7 shows the computed magnitude of ˆur, or(ˆurˆur)1/2, for the three least-damped modes. For notational\nconvenience, we will refer to them below as modes 1, 2 and 3, in order of increasing damping rate.\n12arXiv Template A P REPRINT\n(a)|ˆur|for mode 1\n (b)|ˆur|for mode 2\n (c)|ˆur|for mode 3\nFigure 7: Magnitude of ˆurfor the three least-damped modes in a cylindrical nozzle.\nsecond order third order\nFigure 8: Convergence curves for the computed angular frequency ω(left) and damping rate η(right) for the three\nleast-damped modes in a cylindrical domain.\nTo evaluate the convergence of the method in this example, we consider a sequence of uniform meshes with decreasing\nelement size, and examine the convergence of the eigenvalues (angular frequencies and damping rates) and associated\neigenvectors (velocity, pressure and meniscus surface displacement) over this sequence. Figure 8 shows the relative\nerrors (ω−ωfinest)/ωfinestand(η−ηfinest)/ηfinestfor the three least-damped modes as a function of mesh resolution,\nindicated by the total number of finite elements in the computational domain. Here ωfinest andηfinest are the values\ncomputed with the mesh that has the highest number of elements. The convergence of these differences is strong\nevidence that that the computed values converge for all three modes. For both ωandη, the convergence rate seems to\nbe between second and third order for mode 1, and faster than third order for modes 2 and 3.\nNext, we investigate mesh convergence for the eigenvectors. To easily compare eigenvectors from different meshes,\nwe define a regular Cartesian mesh Tcartthat covers the entire domain of the problem, and whose mesh size is finer\nthan the finest finite element mesh we use. Part of the boundary of this Cartesian mesh exactly meshes γm. Given\na function f:˘Ω→Corf:γm→C, we define the vector of interpolated values Ifwhose i-th component is the\nvalue of fat the i-th vertex of Tcart. Iffis not defined on the i-th vertex, the i-th component of Ifis set to zero.\nAs a result, we define the i-th component of Iˆξhas equal to zero if the i-th vertex is not on γm. Therefore, given\na computed eigenvector (ˆuh,ˆph,ˆξh)over a finite element mesh, we compute the vector of interpolated values Xh=\n(I(ˆur)h,I(ˆuz)h,Iˆph,Iˆξh), and normalize it so that XT\nhXh= 1. For simplicity of notation, for any such interpolated\neigenvector Xh, we denote the components associated to each one of the fields as Xr\nh=I(ˆur)h,Xz\nh=I(ˆuz)h,\nXp\nh=Iˆph, and Xξ\nh=Iˆξh. Each one of these vectors has a length equal to the number of vertices in the Cartesian\nmesh. Finally, we can compute the Euclidean norm of any of these vectors as, for example, |Xr\nh|= ((Xr\nh)TXr\nh)1/2,\nand so on.\nComplex eigenvectors are defined up to a complex scalar, i.e., if Xis an eigenvector, so is βXfor any β∈C. To\ncompare two interpolated eigenvectors X1andX2for convergence studies, we find βas\nβ(X1, X2) = arg min\nβ∈C,|β|=1|X1−βX2|2=XT\n2X1/|XT\n2X1|, (16)\n13arXiv Template A P REPRINT\nFigure 9: Convergence curves for the velocity ( eh\nuin the left column), pressure ( eh\npin the middle column) and meniscus\nsurface displacement ( eh\nξin the right column) for modes 1 (top row), 2 (middle row) and 3 (bottom row) in a cylindrical\ndomain.\nand compute the difference between the two as\n∆X(X1, X2) =X1−β(X1, X2)X2.\nThe computation of β(X1, X2)is shown in §6.5.\nAs a proxy for the exact solution, we compute errors with respect to the interpolation of the eigenvector obtained on\nthe finest finite element mesh we used, Xf. We compute the relative errors\neh\nu=\"\n|∆X(Xr\nh, Xr\nf)|2+|∆X(Xz\nh, Xz\nf)|2\n|Xr\nf|2+|Xz\nf|2#1/2\n,\neh\np=|∆X(Xp\nh, Xp\nf)|\n|Xp\nf|,\neh\nξ=|∆X(Xξ\nh, Xξ\nf)|\n|Xξ\nf|,\nfor modes 1, 2 and 3 in Figure 9 as a function of the mesh size. These are approximations of the relative L2-errors\nof each one the fields. Third-order convergence is observed for the velocities and the meniscus displacement, while\nsecond-order convergence is observed for the pressure field.\n4.3 Mesh Convergence on an Arbitrary Axisymmetric Domain\nThe last example of the paper demonstrates the use of the algorithm in axisymmetric domains with more complex\ngeometry. In this example we compute the least-damped modes for the domain ˘Ωshown in Figure 10. We constructed\n14arXiv Template A P REPRINT\nFigure 10: Domain ˘Ωused to demonstrate the performance of the method in axisymmetric geometries.\n(a)|ˆur|for mode 1\n (b)|ˆur|for mode 2\n (c)|ˆur|for mode 3\nFigure 11: Magnitude of ˆurfor the three least-damped modes in the nozzle shown in Figure 10.\nthe geometry and mesh in Gmsh by modifying the “chess pawn.geo\" file obtained from [42]. We again set the\nReynolds number to Re= 710 as in Section 4.2. Figure 11 shows the computed magnitude of ˆur, or(ˆurˆur)1/2, for\nthe three least-damped modes.\nTo evaluate mesh convergence, we again consider a sequence of meshes with decreasing element size5, and examine\nthe convergence of the eigenvalues and associated eigenvectors of these modes over this sequence. Figure 12 shows\nthe relative errors (ω−ωfinest)/ωfinestand(η−ηfinest)/ηfinestfor the three least-damped modes as a function of mesh\nresolution, indicated by the inverse of the grid size parameter used in Gmsh , where ωfinest andηfinest are defined as\nbefore. We can see that the method is mesh-convergent in both the angular frequency and damping rate for all three\nmodes. The convergence rate is between 2 and 3 in all cases, with the damping rate apparently converging at a faster\nrate for mode 3.\nWe observe similar results for the eigenvectors, after computing relative errors as described in Section 4.2. Figure\n13 shows the relative errors for velocity, pressure and meniscus surface deformation. Compared to the cylindrical\ndomain in Section 4.2, we observe a similar third order (or higher) convergence for the meniscus displacement, while\nthe velocities now display a convergence rate between second and third order. It is not possible to state the order of\nconvergence of the pressure field in this example, but it is definitively second-order of higher.\n5Unlike for the cylindrical domain in Section 4.2, due to the irregular shape of γwthe mesh will be mostly, but not entirely,\nuniform as it needs to conform to γw.\n15arXiv Template A P REPRINT\nsecond order third order\nFigure 12: Convergence curves for the computed angular frequency ω(left) and damping rate η(right) for the three\nleast-damped modes in the domain of Figure 10.\nFigure 13: Convergence curves for the velocity ( eh\nuin the left column), pressure ( eh\npin the middle column) and\nmeniscus surface displacement ( eh\nξin the right column) for modes 1 (top row), 2 (middle row) and 3 (bottom row) in\nthe domain of Figure 10.\n16arXiv Template A P REPRINT\n5 Summary and Outlook\nWe introduced a fast eigensolver for computing the late-time oscillation modes of the fluid in the nozzles of drop-on-\ndemand (DoD) microfluidic devices. Starting from the linearized Navier-Stokes equations, we demonstrated how their\nweak form leads to a generalized eigenvalue problem (GEP) whose eigenfunctions describe the spatial dependence of\nthe liquid’s oscillations, and whose eigenvalues contain the angular frequencies and damping rates of these modes. To\ncompute these, we discretized the computational domain via Taylor-Hood finite elements to obtain a matrix GEP.\nThrough a suite of numerical tests, we verified the accuracy of the finite element analysis (FEA) solutions both in\nterms of the computed oscillation modes (eigenvectors) and the corresponding angular frequencies and damping rates\n(complex eigenvalues):\n1. The FEA solutions correctly approximate the analytical modal angular frequencies and damping rates of a\nviscous capillary wave in the limit of small viscosity.\n2. In a cylindrical domain, the FEA solution achieves at least second-order convergence in angular frequency and\ndamping rate. The convergence rate of the eigenvectors coincides with that expected from the approximation\nproperties of Taylor-Hood elements: second order in the pressure field and third order in the velocities and\nmeniscus surface displacement.\n3. For a more generic axisymmetric domain, the FEA solution achieves at least second-order convergence in\nangular frequency and damping rate of the three least-damped modes. For the oscillation modes, the conver-\ngence rate is at least third order in the meniscus surface displacement, but between second and third order in\nthe velocity field. The observed pressure field convergence rate is mixed, but at least second order or higher.\nIn future work, we plan to extend the current framework to handle arbitrary, non-axisymmetric nozzle shapes, and\nincorporate the effects of gravity by allowing the equilibrium meniscus shape to be curved. Moreover, we intend\nto integrate our linearized solver into a constrained optimization loop to speed up nozzle prototyping, where the\nconstraints may come from droplet specifications (e.g., user-defined mass or velocity), and minimum liquid capacity\ninside the nozzle, among others. Such a solver would facilitate the optimal design of nozzles in, for example, DoD\nprinting devices for additive manufacturing.\nAcknowledgments\nThis research was developed with funding from the Xerox Corporation. The views, opinions and/or findings expressed\nare those of the authors and should not be interpreted as representing the official views or policies of the Xerox\nCorporation.\nThe authors would like to thank Prof. Jose E. Roman for his very helpful comments regarding the use of the SLEPc\neigensolver. Furthermore, they would like to thank Prof. John Burkardt for his advice on using and modifying routines\nfrom his library of open-source finite element MATLAB codes.\n6 Appendix\n6.1 Preliminaries\nForu∈ V andξ∈Ξ, it holds that\nc(u,u) = 0⇐⇒u=0 (17a)\ns(ξ,ξ) = 0⇐⇒ξ= 0, (17b)\nwhere the bilinear forms candswere defined in (5). To see that (17a) holds, notice that c(u,u) = 0 implies that\n∇u+ (∇u)T= 0. Since u= 0 onΓwfrom (3b) and Ωis connected, Korn’s inequality [43] helps us conclude that\n∇u= 0, and hence that u= 0. The converse is trivial.\nTo see that (17b) holds, notice that s(ξ,ξ) = 0 implies that ∇Sξ= 0 onΓm, or that ξis constant on Γm. Because\nξ= 0onΓm∩Γw, we conclude that ξ= 0. Again, the converse is trivial.\nFinally, since c(u,u)≥0, (17a) implies that\nc(u,u)>0⇐⇒u̸=0. (17c)\n17arXiv Template A P REPRINT\n6.2 Derivation of the Weak Form\nLetˆΣ =−ˆpI+1\nRe(∇ˆu+ (∇ˆu)T)denote the stress tensor, so that (2a), (2e) and (2f) become\nλˆu=∇ ·ˆΣ inΩ, (18a)\nˆΣ·ez=−∆Sˆξez onΓm, (18b)\nˆΣ·ez= 0 onΓt. (18c)\nTo obtain (4a), we compute the dot product of (18a) with any v∈ V and integrate over Ω, to get\n0 =Z\nΩλˆu·v−(∇ ·ˆΣ)·vdΩ\n=Z\nΓmv·ˆΣ·(−n)dS+Z\nΩλˆu·v+ˆΣ:∇vdΩ (19)\n=Z\nΓm−(∆Sˆξ)vzdS+Z\nΩλˆu·v+ˆΣ:∇vdΩ\n=Z\nΓm∇Sˆξ· ∇SvzdS+Z\nΩλˆu·v+ˆΣ:∇vdΩ, (20)\nwhere we applied the divergence theorem over Ω, (18b) and (18c) in (19), and the divergence theorem over Γmin (20)\ntogether with the fact that v= 0onΓw, and in particular, v= 0onΓw∩Γm. The weak form of the incompressibility\ncondition (4b) is obtained by multiplying (2b) by any q∈ P and integrating over Ω.\nFinally, the weak compatibility between meniscus displacements and normal velocities (4c) is obtained by computing\nthe surface gradient ∇Son both sides of (2d), multiplying by a test function ζ∈Ξand integrating over Γm. The\ncompatibility condition (2d) is recovered from the weak statement (4c) by first noticing that λˆξ−ˆuz∈Ξ, and then\nsetting ζ=λˆξ−ˆuzin (4c) to conclude from (17b) that λˆξ= ˆuzonΓm.\n6.3 Sign of the Real Part of the Eigenvalues of the Continuous Problem\nWe formally show next that for any non-trivial solution ˆu,ˆp,ˆξof the weak form (4), the eigenvalue λneeds to have\na negative real part, i.e., Re(λ)<0. To see that Re(λ)<0, notice first that for a solution (u, p, ξ)6withu̸=0,\na(u,u)>0,c(u,u)>0(from (17c)), and s(ξ, ξ)≥0, while 0 =b(u,p) =b(u, p)from (6b). Therefore,\nRe\u0002\ng((u, p, ξ),(u,p,−ξ))\u0003\n=c(u,u) + Re\u0002\ns(ξ,uz)−s(uz,ξ) +b(u, p) +b(u,p)\u0003\n=c(u,u), (21)\nh((u, p, ξ),(u,p,−ξ)) =−a(u,u)−s(ξ,ξ)<0,\nand from (6d), it follows that\nRe(λ) = Re\u0014g((u, p, ξ),(u,p,−ξ))\nh((u, p, ξ),(u,p,−ξ))\u0015\n=−c(u,u)\na(u,u) +s(ξ,ξ)<0.\nNext, if (0, p, ξ)is a solution ( u=0), we show that ξ= 0 andp= 0, so there are no non-trivial solutions of weak\nform (6b). We can then conclude that Re(λ)<0for any eigenvalue λ.\nAssuming that u=0, the fact that ξ= 0 andp= 0 can be obtained from (6d) as well using the inf-sup condition\nfor Stokes problem (see, e.g. [44, 45]), but for the sake of simplicity we provide a formal argument here based on (2).\nSince u=0, (2a) implies that ∇p= 0inΩand (2f) implies that p= 0onΓt, from where we conclude that p= 0in\nΩ. Similarly, (2e) implies that ∆Sξ= 0onΓm, and hence that ξis an affine function on Γmthat is equal to zero on\nΓm∩Γw, from where we conclude that ξ= 0.\n6.4 Sign of the Real Part of the Eigenvalues of the Discrete Problem\nWe prove next that if λis an eigenvalue of the generalized eigenvalue problem (7), then Re(λ)<0. In particular, this\nimplies that the matrix Gis invertible.\n6For simplicity of notation, we drop the ˆ·notation in this section.\n18arXiv Template A P REPRINT\nThis result is a consequence of the fact that if X= [U P Z ]Tcorresponds to (uh, ph, ξh)∈ Vh× Ph×Ξhwith\nuh̸=0, then as in (21),\nRe\u0002\ng((uh, ph, ξh),(uh,ph,−ξh))\u0003\n=c(uh,uh)>0, (22)\nh((uh, ph, ξh),(uh,ph,−ξh)) =−a(uh,uh)−s(ξh,ξh)<0.\nFrom (7), it follows that\nRe(λ) = Re\u0014g((uh, ph, ξh),(uh,ph,−ξh))\nh((uh, ph, ξh),(uh,ph,−ξh))\u0015\n=−c(uh,uh)\na(uh,uh) +s(ξh,ξh)<0.\nNext, we see that if uh=0, then ph= 0andξh= 0. This result does not follow directly from that in §6.3 because an\ninf-sup condition for the discrete problem is needed, as we show next.\nBy selecting any vh∈ Vhsuch that vh·ez= 0onΓm, we conclude from (7) that\n0 =g((0, ph, ξh),(vh,0,0)) = b(vh, ph). (23)\nTo conclude that ph= 0, we need to take advantage of inf-sup conditions in both planar and axisymmetric flows.\nSpaces VhandPhsatisfy the following inf-sup condition\ninf\n0̸=ph∈Re(Ph,0)sup\n0̸=vh∈Re(Vh,0)b(vh, ph)\n∥vh∥1,2∥ph∥0,2> c > 0 (24)\nfor planar [31, Lemma 4.23] or axisymmetric[32] flows, where ∥·∥1,2and∥·∥0,2denote the H1(Ω,R2)andL2(Ω,R)\nnorms, and where Vh,0={v∈ Vh|v|∂Ω=0}andPh,0={p∈ Ph|R\nΩp dV = 0}. To show that ph= 0, we will\nfirst show that the real and the imaginary parts are constants in Ω. To this end, let pav∈Cbe the average of ph, i.e.\n|Ω|pav=R\nΩphdV, and let ˜ph=ph−pav∈ Ph,0. Notice that for vh∈ Vh,0\nb(vh, pav) =−pavZ\n∂Ωvh·ndS= 0 =⇒b(vh, ph) =b(vh,˜ph).\nNext, from (23) and (24) we can write that for any vh∈Re(Vh,o)⊂ Vh,Re(vh)̸=0,\n0 =Re [b(vh, ph)]\n∥vh∥1,2=Re [b(vh,˜ph)]\n∥vh∥1,2=b(vh,Re(˜ph))\n∥vh∥1,2> c∥Re(˜ph)∥0,2,\nfrom where we conclude that Re(˜ph) = 0 . Similarly, we conclude that Im(˜ph) = 0 and hence that ph=pavfrom\n0 =Re [b(−ivh, ph)]\n∥vh∥1,2=Re [b(−ivh,˜ph)]\n∥vh∥1,2=b(vh,Im(˜ph))\n∥vh∥1,2> c∥Im(˜ph)∥0,2.\nTo see that pav= 0, and conclude that ph= 0, it suffices to select vh∈ Vhsuch thatR\n∂Ωvh·ndS̸= 0 in (23);\nnamely,\n0 =b(vh, pav) =−pavZ\n∂Ωvh·ndS.\nTo show that ξh= 0, we select v∈ Vhsuch that v·ez=ξhonΓm, and write\n0 =GX=⇒0 =g((0,0, ξh),(v,0,0)) = s(vz, ξh) =s(ξh, ξh). (25)\nThen (17b) implies that ξh= 0, and hence that X= [0 0 0]T.\n6.5 Distance between Eigenvectors\nWe show next the computation of β(X1, X2)in (16). From (16) we have\nd(X1, X2) = min\nβ∈C,|β|=1|X1−βX2|2\n= min\nβ∈C,|β|=1(XT\n1−βXT\n2)(X1−βX2)\n= min\nβ∈C,|β|=12−2 Re( βXT\n2X1),\nwhere we assumed that XT\niXi= 1 fori= 1,2. The minimum is attained when Re(βXT\n2X1)is maximized, or for\nβ=XT\n2X1/|XT\n2X1|, which is (16). Finally, by replacing we obtain\nd(X1, X2) = 2(1 − |XT\n2X1|). (26)\n19arXiv Template A P REPRINT\nReferences\n[1] Ralf Seemann, Martin Brinkmann, Thomas Pfohl, and Stephan Herminghaus. Droplet based microfluidics.\nReports onprogress inphysics, 75(1):016601, 2011.\n[2] W. J. Seeto, Y . Tian, S. Pradhan, D. Minond, and E. A. Lipke. Droplet microfluidics-based fabrication of\nmonodisperse poly(ethylene glycol)-fibrinogen breast cancer microspheres for automated drug screening ap-\nplications. ACS Biomater. Sci.Eng., 8(9):3831–3841, 2022.\n[3] S. Y . Kashani, A. Afzalian, F. Shirinichi, and M. K. Moraveji. Microfluidics for core-shell drug carrier particles\n- a review. RSC Adv., 11(1):229–249, 2020.\n[4] I. 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Springer Science &\nBusiness Media, 2012.\n22"    },    {        "title": "1304.6484v2.Aspects_of_Electrodynamics_Modified_by_Some_Dimension_five_Lorentz_Violating_Interactions.pdf",        "content": "Aspects of Electrodynamics Modi\fed by Some\nDimension-\fve Lorentz Violating Interactions\nShan-quan Lan1and Feng Wu1,\u0003\n1Department of Physics, Nanchang University, 330031 Nanchang, China\n(Dated: March 2, 2022)\nAssuming Lorentz symmetry is broken by some \fxed vector background, we study\nthe spinor electrodynamics modi\fed by two dimension-\fve Lorentz-violating inter-\nactions between fermions and photons. The e\u000bective polarization and magnetization\nare identi\fed from the modi\fed Maxwell equations, and the theoretical consequences\nare investigated. We also compute the corrections to the relativistic energy levels\nof hydrogen atom induced by these Lorentz-violating operators in the absence and\npresence of uniform external \felds in \frst-order perturbation theory. We \fnd that\nthe hydrogen spectrum is insensitive to the breakdown of Lorentz boost symmetry.\n\u0003Electronic address: fengwu@ncu.edu.cnarXiv:1304.6484v2  [hep-th]  13 Jun 20132\nI. INTRODUCTION\nThe breaking of Lorentz symmetry is an extensively studied topic. Although no departure\nfrom Lorentz invariance has yet been detected experimentally, there is no reason to believe\nthat Lorentz invariance would be intact at all energies. As a matter of fact, there are reasons\nto suspect its exactness in the context of string theory. For example, the potential instability\nof string vacuum would induce spontaneous breakdown of Lorentz symmetry [1]. Also, high-\nenergy \feld theories constructed on a Moyal space, viewed as low-energy e\u000bective theories\nfrom open string theory with a constant NS-NS B\feld, explicitly spoil Lorentz invariance\n[2].\nOne direction in the study of Lorentz violation is to regard Lorentz symmetry breaking\nas a possible extension of the standard model in particle physics. The \frst work in this con-\ntext is the investigation of the Carroll-Field-Jackiw term [3], which has inspired numbers of\nstudies in recent years. Without the criterion of Lorentz symmetry, one may construct new\nadditive terms to the minimal standard model. Note that since CPT invariance is necessary\nbut not su\u000ecient for Lorentz invariance of an interacting quantum \feld theory [4], Lorentz\nviolating (LV) terms can be either CPT even orCPT odd. LV terms of renormalizable\ndimensions have been systematically constructed in [5], known as the standard model exten-\nsion, and many related issues have been discussed [6{10]. However, experimental data put\nvery stringent constraints on the renormalizable LV terms and indicate that they must be\nextremely small. To avoid the subtle \fne-tuning problem [11], we assume that the symmetry\nof the underlying theory prohibits the generation of the renormalizable LV operators and\nexplore the nonrenormalizable LV operators in this paper.\nRenormalizability was considered to be an axiom when constructing the standard model.\nQuantum corrections to a renormalizable theory will generate UV divergences only to op-\nerators whose mass dimensions are less than \fve and this fact assures the predictiveness of\nthe theory. However, a modern point of view is that reliable predictions could still be made\nfrom a nonrenormalizable theory within the framework of e\u000bective \feld theories. Thus,\nhigher-dimensional LV operators should also be considered, and several studies involving\ndimension-\fve LV operators have been carried out [12{16].\nIn this paper, assuming a \fxed vector background v\u0016to be the only source that induces\nthe breaking of Lorentz symmetry, we shall consider the spinor electrodynamics modi\fed3\nby some dimension-\fve LV interactions between fermions and photons. Higher-dimensional\ninteractions are best classi\fed in terms of derivative expansion. Dimension-\fve interactions\nare quadratic in gauge covariant derivative D\u0016, which is given by D\u0016=@\u0016+iA\u0016. Recall\nthat [D\u0016;D\u0017] =iF\u0016\u0017, whereF\u0016\u0017is the electromagnetic tensor. If we restrict ourself to\nconsider only terms linear in the vector background v\u0016and the photon \feld A\u0016, then the\nmost general dimension-\fve interactions are of the form\nLv=v\u0016\tK\u0016\u0017\u000b\f\r\u0017\tF\u000b\f(1)\nwhereK\u0016\u0017\u000b\f =1\n2(a1+b1\r5)\u000f\u0016\u0017\u000b\f+ (a2+b2\r5)g\u0016\u000bg\u0017\fwithaiandbibeing dimensionless\nconstants. The factor 1 =2 inK\u0016\u0017\u000b\f is introduced for later convenience. One can easily see\nthat the mass dimension of the background vector v\u0016is [v\u0016] =\u00001. Note that each term in (1)\nviolatesCPT . Whileaiterms preserve Cparity (and thus violate PT),biterms violated\nit (and thus preserve PT). Constraints from the electric dipole moments of paramagnetic\natoms put very stringent limits on biterms [13]. Therefore, we will not discuss C-violating\nterms in this paper and simply set bi= 0 from now on.\nThe modi\fed QED, after rescaling the background v\u0016by absorbing the parameter a1,\nthen reads\nL=\u00001\n4e2F\u0016\u0017F\u0016\u0017+\t\u0010\niD =\u0000m\u0000\r\u0016v\u0017\u0010\n~F\u0016\u0017+aF\u0016\u0017\u0011\u0011\n\t (2)\nwhere ~F\u0016\u0017is the dual electromagnetic tensor, ~F\u0016\u0017\u00111\n2\u000f\u0016\u0017\u000b\fF\u000b\f. The Lorentz symmetry\nSO(3;1) is broken by the irrelevant dimension-\fve operators to its subgroup SO(2), which\nadmits the background vector v\u0016as an invariant tensor. At low energies, e\u000bects due to\nnonrenormalizable couplings are suppressed at least by powers of 1 =M, withMbeing some\nfundamental large mass scale in the underlying theory. In the limit M!1 , the symmetry of\nthe Lagrange density (2) is enhanced to the Lorentz group, along with spacetime translations.\nThe two LV terms in (2) have been considered in Refs. [14{16]. The crucial di\u000berence be-\ntween the existing works related to these two terms and the Lagrange density (2) considered\nin this paper is that the dimensionless coupling constant ein (2) is the unique gauge coupling\nconstant determining the strength of the electromagnetic interaction. Thus, di\u000berent from\nother works, we restrict our consideration to the case where electrically neutral particles will\nnot interact with photons at tree level. This can be seen even more transparently by letting\nA\u0016!eA\u0016so that (2) becomes\nL=\u00001\n4F\u0016\u0017F\u0016\u0017+\t (iD =\u0000m) \t\u0000j\u0016v\u0017\u0010\n~F\u0016\u0017+aF\u0016\u0017\u0011\n(3)4\nwhere the gauge covariant derivative now takes the form D\u0016=@\u0016+ieA\u0016and the 4-vector\nj\u0016\u0011e\t\r\u0016\t = (\u001a;~j) is the current density. Apparently Lreduces to the free theory for\nneutral particles.\nThe rest of the paper is organized into three parts. In Sec. II, we examine the QED\nmodi\fed by the dimension-\fve LV operator, j\u0016v\u0017~F\u0016\u0017. The theoretical consequences of the\nmodi\fed Maxwell and Dirac equations are studied. In particular, we compute the corrections\nto the hydrogen spectrum by applying the perturbation theory to the exactly solved Dirac\nequation. To our knowledge, the corrections to the hydrogen spectrum induced by the LV\noperatorj\u0016v\u0017~F\u0016\u0017were calculated only in the nonrelativistic limit in the literature [15]. The\ne\u000bect on the spectral lines of hydrogen atom due to the presence of a static external electric\n\feld and a static external magnetic \feld is also considered. In Sec. III, we present similar\nanalysis on the QED modi\fed by another dimension-\fve LV operator, j\u0016v\u0017F\u0016\u0017. We give our\nconclusions in the \fnal section.\nII. MODEL I\nOur starting point is the following modi\fed QED Lagrange density:\nL1=\u00001\n4F\u0016\u0017F\u0016\u0017+\t\u0010\niD =\u0000m\u0000e\r\u0016v\u0017~F\u0016\u0017\u0011\n\t: (4)\nThe \feld equations derived from L1read1\n@\u0017F\u0017\u0016=j\u0016+\u000f\u0016\u0017\u000b\fv\f@\u0017j\u000b: (5)\nThe continuity equation @\u0016j\u0016= 0 follows from (5) as a result of gauge symmetry. The \feld\nequations (5) can be rewritten in terms of components as the familiar form of inhomogeneous\nMaxwell equations. Together with the homogeneous Maxwell equations coming from the\ngauge invariance of the system, we have\n\u0000 !r\u0001\u0000 !B= 0; (6)\n@\u0000 !B\n@t+\u0000 !r\u0002\u0000 !E= 0; (7)\n\u0000 !r\u0001\u0000 !D=\u001a; (8)\n\u0000 !r\u0002\u0000 !H\u0000@\u0000 !D\n@t=\u0000 !j : (9)\n1The convention for the metric in this paper has the signature(+ ;\u0000;\u0000;\u0000)5\nHere the e\u000bective displacement \feld\u0000 !Dand the e\u000bective magnetic \feld\u0000 !Hare de\fned as\n\u0000 !D=\u0000 !E+\u0000 !Pand\u0000 !H=\u0000 !B\u0000\u0000 !M, respectively, where the e\u000bective polarization\u0000 !Pand the\ne\u000bective magnetization\u0000 !M, de\fned by\u0000 !P= (\u0000 !j\u0002\u0000 !v) and\u0000 !M= (\u001a\u0000 !v\u0000v0\u0000 !j), respectively,\nare the components of the rank-2 object M\u0016\u0017\u0011j[\u0016v\u0017]:\nM\u0016\u0017=0\nBBBBB@0\u0000M1\u0000M2\u0000M3\nM10P3\u0000P2\nM2\u0000P30P1\nM3P2\u0000P101\nCCCCCA: (10)\nLorentz symmetry of the Maxwell equations is spoiled by the nonzero e\u000bective polarization\nor the nonzero e\u000bective magnetization. In the presence of stationary sources (such that\n~r\u0001~j= 0), the solution of the gauge \feld A\u0016= (\u001e(~ r);~A(~ r)) is\n\u001e(~ r) =1\n4\u0019Z\nd3r0\u001a(~ r0)\u0000~r\u0001~P(~ r0)\nj~ r\u0000~ r0j(11)\nand\n~A(~ r) =1\n4\u0019Z\nd3r0~j(~ r0) +~r0\u0002~M(~ r0)\nj~ r\u0000~ r0j: (12)\nConsequently, one can see that a static electric \feld can arise from stationary and neutral\nsources (\u001a= 0) as long as the e\u000bective polarization is not divergence-free. Also, even for\nthe steady and irrotational current density (such that both the divergence and the curl of ~j\nvanish), a nonvanishing magnetic \feld ~Bmay still arise from ~B=~r\u0002~Awith~Agiven by\n~A(~ r) =1\n4\u0019Z\nd3r0(~r0\u001a(~ r0))\u0002~ v\nj~ r\u0000~ r0j: (13)\nWe now switch to the fermion sector in (4). The equation of motion for the fermion \t\nfollowing fromL1is\u0010\niD =\u0000m\u0000e\r\u0016v\u0017~F\u0016\u0017\u0011\n\t = 0: (14)\nMultiplying on the left by the Dirac matrix \r0, we can identify the Hamiltonian operator of\none-particle quantum mechanics:\nH=\r0\u0010\n~ \r\u0001~ p+eA =+m+ev0~ \r\u0001~B\u0000e\r0~ v\u0001~B\u0000e~ \r\u0001(~ v\u0002~E)\u0011\n=H0+\u000eH (15)\nwhereH0=\r0(~ \r\u0001~ p+eA =+m) is the Dirac Hamiltonian and \u000eH=e\r0(v0~ \r\u0001~B\u0000\r0~ v\u0001~B\u0000~ \r\u0001\n(~ v\u0002~E)) is the LV perturbation. It is well known that hydrogen atom can be solved exactly6\nin Dirac's theory and the \fne structure of the hydrogen spectrum comes out naturally from\nit. Using degenerate perturbation theory, we are able to compute the \frst-order correction\nto the hydrogen spectrum induced by \u000eH.\nSince the degenerate unperturbed states are the stationary state vectors jn;j;l;m jiof\nthe Dirac Hamiltonian H0for a \fxednandj, in the absence of external \felds we need to\ncalculate the following matrix elements of the perturbation:\nhn;j;l0;m0\njj\u000eHjn;j;l;m ji=\u0000ehn;j;l0;m0\njj\r0~ \r\u0001(~ v\u0002~E)jn;j;l;m ji (16)\nwhere the Coulomb \feld ~Eis given by ~E=\u0000e\n4\u0019^r\nr2. The term\u0000e\r0~ \r(~ v\u0002~E) in\u000eHorigi-\nnates from the CP-even operator jivk~FikinL1. We note in passing that the energy shifts\nare independent of the time component v0of the background vector, indicating that the\nhydrogen spectrum is insensitive to the breakdown of invariance under Lorentz boosts. It is\neasy to show that the matrix elements (16) of the perturbation between state vectors with\ndi\u000berent eigenvalues for the square of the orbital angular momentum L2or thezcomponent\nJzof the total angular momentum all vanish. Indeed, by judiciously choosing a coordinate\nsystem such that ~ v=j~ vj^z, we have, in Dirac representation,\n[Jz;\r0~ \r\u0001(~ v\u0002~E)]/[\u0000i@\n@\u001e1 +1\n20\n@^\u001bz0\n0 ^\u001bz1\nA;0\n@0 sin \u001e^\u001bx\u0000cos\u001e^\u001by\nsin\u001e^\u001bx\u0000cos\u001e^\u001by 01\nA]\n=\u0000i0\n@0 cos \u001e^\u001bx+ sin\u001e^\u001by\ncos\u001e^\u001bx+ sin\u001e^\u001by 01\nA\n+1\n20\n@0 sin \u001e[^\u001bz;^\u001bx]\u0000cos\u001e[^\u001bz;^\u001by]\nsin\u001e[^\u001bz;^\u001bx]\u0000cos\u001e[^\u001bz;^\u001by] 01\nA= 0: (17)\nAlso, the unperturbed states jn;j;l;m jiare simultaneous eigenstates of H0andJz. It follows\nthen that the matrix elements (16) vanishes unless l=l0andmj=m0\nj.\nTo evaluate the expectation value in the unperturbed state of the perturbation, we recall\nthat the unperturbed wave functions in Dirac representation take the form\nhx\u0016jn;j;l =j\u00061\n2;mji=e\u0000i\u000ft\u0012iF\u0000(\u0006\u0014jr)Yj;m j(j\u00061\n2;1\n2j^r)\nF+(\u0006\u0014jr)Yj;m j(j\u00071\n2;1\n2j^r)\u0013\n: (18)\nHere the radial wave functions F\u0006(\u0014jr) are given by\nF\u0006(\u0014jr) =\u0007N\u0006(\u0014)(2\u0016r)\r\u00001e\u0000\u0016rf[(n0+\r)me\n\u000f\u0000\u0014]F(\u0000n0;2\r+1; 2\u0016r)\u0006n0F(1\u0000n0;2\r+1; 2\u0016r)g\n(19)7\nwhere\nN\u0006(\u0014) =(2\u0016)3\n2\n\u0000(2\r+ 1)s\n(me\u0007\u000f)\u0000(2\r+n0+ 1)\n4me(n0+\r)me\n\u000f((n0+\r)me\n\u000f\u0000\u0014)n0!;\n\u0016=p\n(me\u0000\u000f)(me+\u000f);\n\u000f=meq\n1 +\u000b2\n(n0+\r)2;\n\r=r\n(j+1\n2)2\u0000\u000b2;\nn0=n\u0000\u0014;\n\u0014=j+1\n2; (20)\n\u000bis the \fne structure constant given by \u000b=e2=4\u0019, andmeis the electron mass. The\nspin-angular functions Yj;m j(l;1\n2j^r) are of the form\nYj;m j(l;1\n2j^r) =\u0012(\u00001)l\u0000j+1\n2r\nl+1\n2+(\u00001)l\u0000j+1\n2mj\n2l+1Ymj\u00001\n2\nl (\u0012;')\nr\nl+1\n2+(\u00001)l\u0000j+3\n2mj\n2l+1Ymj+1\n2\nl (\u0012;')\u0013\n: (21)8\nIt follows that\n\u0000ehn;j;l0;m0\njj\r0~ \r\u0001(~ v\u0002~E)jn;j;l;m ji=\u0000\u000ell0\u000emjm0\njehn;j;l;m jj\r0~ \r\u0001(~ v\u0002~E)jn;j;l;m ji\n=\u0000\u000ell0\u000emjm0\nj\u000bj~ vjZ\nd3r1\nr2\u0012\n\u0000iF\u0000(\u0006\u0014jr)Yy\nj;m j(j\u00061\n2;1\n2j^r);F+(\u0006\u0014jr)Yy\nj;m j(j\u00071\n2;1\n2j^r)\u00130\n@0\u0000 !\u001b\n\u0000 !\u001b01\nA\u0001\n(sin\u0012sin'^x\u0000sin\u0012cos'^y)\u0012iF\u0000(\u0006\u0014jr)Yj;m j(j\u00061\n2;1\n2j^r)\nF+(\u0006\u0014jr)Yj;m j(j\u00071\n2;1\n2j^r)\u0013\n=\u0007\u000ell0\u000emjm0\nj\u000bj~ vjZ\ndrdcos\u0012F\u0000(\u0006\u0014jr)F+(\u0006\u0014jr) sin\u0012[(j\u0000mj+ 1)!\n(j+mj)!Pmj\u00001\n2\nj+1\n2(cos\u0012)Pmj+1\n2\nj\u00001\n2(cos\u0012)\n+(j\u0000mj)!\n(j+mj\u00001)!Pmj+1\n2\nj+1\n2(cos\u0012)Pmj\u00001\n2\nj\u00001\n2(cos\u0012)]\n=\u0007\u000ell0\u000emjm0\nj\u000bj~ vjZ\ndrdcos\u0012F\u0000(\u0006\u0014jr)F+(\u0006\u0014jr)[\u0000(j\u0000mj+ 1)!\n(j+mj)!(j\u0000mj+ 1)(j\u0000mj)\n2j(Pmj\u00001\n2\nj+1\n2(cos\u0012))2\n+(j\u0000mj)!\n(j+mj\u00001)!1\n2j(Pmj+1\n2\nj+1\n2(cos\u0012))2]\n=\u0007\u000ell0\u000emjm0\nj\u000bj~ vjmj(2j+ 1)\nj(j+ 1)Z1\n0drF\u0000(\u0006\u0014jr)F+(\u0006\u0014jr)\n=\u0006\u000ell0\u000emjm0\nj\u000bj~ vjmj(2j+ 1)\nj(j+ 1)N\u0000(\u0006\u0014)N+(\u0006\u0014)Z1\n0dr(2\u0016r)2\r\u00002e\u00002\u0016r\nf[(n0+\r)me\n\u000f\u0007\u0014]2F2(\u0000n0;2\r+ 1; 2\u0016r)\u0000n02F2(1\u0000n0;2\r+ 1; 2\u0016r)g\n=\u0006\u000ell0\u000emjm0\nj\u000bj~ vj(m2\ne\u0000\u000f2)3\n2mj(2j+ 1)\n4m2\nej(j+ 1)\r(\r2\u00001\n4)(n+\r\u0000j\u00001\n2)\n0\n@(n+\r\u0000j)\u0012\n(n+\r\u0000j\u00001\n2)me\u0007(j+1\n2)\u000f\u0013\n\u0000\u0010\u0000\nn+\r\u0000j\u00001\n2\u00012\u0000\r2\u0011\n(n+\r\u0000j\u00001)\u000f2\n\u0000\nn+\r\u0000j\u00001\n2\u0001\nme\u0007(j+1\n2)\u000f1\nA(22)\nforl=j\u00061\n2. In deriving the above result, we have used the following formula for con\ruent\nhypergeometric functions:\nZ1\n0d\u0018\u00182l\u00001e\u0000\u0018F2(\u0000n+l+ 1;2l+ 2;\u0018) =n\u00002(2l+ 2)\u0000(n\u0000l)\n4l(l+1\n2)(l+ 1)\u0000(n+l+ 1): (23)\nExpanding (22) in powers of the \fne structure constant, we obtain the energy shifts\nproduced by \u000eH:\n\u000eEnjlm j=\u0000j~ vjm2\ne\u000b4mj\nn3j(j+ 1)(l+1\n2)+ O(\u000b6): (24)\nThe degeneracy of the \fne structure in landmjhas been removed by the LV perturbation\n\u000eH. Figure 1 shows the low-lying energy levels of the hydrogen atom. Note that the energy\nshifts\u000eEnjlm jis of order ( mej~ vj)me\u000b4, where (mej~ vj) is a dimensionless product. This is a9\nm\nj\n=\n-\n3/\n2\nm\nj\n=\n-\n1/\n2\nm\nj\n=\n1/\n2\nm\nj\n=\n3/\n2\nm\nj\n=\n-\n1/\n2\nm\nj\n=\n1/\n2\nm\nj\n=\n-\n1/\n2\nm\nj\n=\n1/\n2\nm\nj\n=\n-\n1/\n2\nm\nj\n=\n1/\n2\nl\n=\n1\nl\n=\n0\nn\n=\n2\nn\n=\n1\nFIG. 1: The low-lying energy levels of the hydrogen atom, including the \frst-order LV correction\n(not to scale).\ntiny e\u000bect in comparison with the Lamb shift, which is of order me\u000b5, and the hyper\fne\nsplitting, which is of order ( me=mp)me\u000b4withmpbeing the mass of the proton, since the\nirrelevant LV operator j\u0016v\u0017~F\u0016\u0017is highly suppressed by some large fundamental mass scale\nMmentioned in the introduction.\nWe are now in a position to consider the shift of the hydrogen energy levels in the presence\nof a uniform external magnetic \feld ~Bext, assuming that its strength is weak in comparison\nwith the \feld produced by the proton. The unperturbed Hamiltonian is taken to be the\nDirac Hamiltonian H0in the absence of the magnetic \feld. The term e\r0~ \r\u0001~AinH0is\nthus treated as a perturbation and responsible for the well-known Zeeman e\u000bect in the\nnonrelativistic limit. With the LV perturbation \u000eH, we now also need to consider the e\u000bect\ninduced by the terms e(v0\r0~ \r\u0001~Bext\u0000~ v\u0001~Bext) in\u000eH. It is straightforward to show that\nfor any constant vector ~ a, the matrix elements hn;j;l;m jj\r0~ \r\u0001~ ajn;j;l0;m0\njibetween the\nstates of the same unperturbed energy vanish. Indeed, from the explicit form of the angular\npart of the matrix elements of the operator \r0~ \r\u0001~ abetween the degenerate unperturbed wave\nfunctions, one can easily see that hn;j;l;m jj\r0~ \r\u0001~ ajn;j;l0;m0\njivanish for eitherjmj\u0000m0\njj6= 0\nor 1, orjl\u0000l0j6= 1. However, when jmj\u0000m0\njj= 0 or 1, andjl\u0000l0j= 1, the radial integral10\nofhn;j;l;m jj\r0~ \r\u0001~ ajn;j;l0;m0\njivanishes. The constant term \u0000e~ v\u0001~Bextin\u000eHjust shifts\neach energy level by the same amount. Therefore, we conclude that, in the presence of the\nuniform magnetic \feld ~Bext, the hydrogen spectrum is not altered by the LV \u000eHin \frst-order\nperturbation theory.\nWe can also consider the change to the hydrogen energy levels in the presence of a uniform\nelectric \feld ~Eext. Again, we assume that the external electric \feld ~Eextis weak so that the\nunperturbed Hamiltonian is the Dirac Hamiltonian of the hydrogen atom. Besides the Stark\ne\u000bect which mixes the 2 sand 2pstates, we also need to calculate the matrix elements of\nthe LV perturbation between the degenerate unperturbed states: \u0000ehn;j;l0;m0\njj\r0~ \r\u0001(~ v\u0002\n~Eext)jn;j;l;m ji. Since~ v\u0002~Eextis a constant vector, we know that \u0000ehn;j;l0;m0\njj\r0~ \r\u0001\n(~ v\u0002~Eext)jn;j;l;m ji= 0 by the same reasoning as before, and therefore the interaction\n\u0000e\r0~ \r\u0001(~ v\u0002~Eext) from the LV perturbation \u000eHdoes not add any new e\u000bect on the hydrogen\nenergy levels.\nIII. MODEL II\nWe now turn to another model constructed from QED modi\fed by another dimension-\fve\nLV operator j\u0016v\u0017F\u0016\u0017, so that the Lagrange density is given by\nL2=\u00001\n4F\u0016\u0017F\u0016\u0017+\t (iD =\u0000m\u0000e\r\u0016v\u0017F\u0016\u0017) \t: (25)\nThe \feld equations which follows from (25) are\n@\u0017F\u0017\u0016= (1 +v\u0017@\u0017)j\u0016: (26)\nIn terms of components, we obtain\n~r\u0001~E=\u0012\n1 +v0@\n@t+~ v\u0001~r\u0013\n\u001a; (27)\n~r\u0002~B\u0000@~E\n@t=\u0012\n1 +v0@\n@t+~ v\u0001~r\u0013\n~j: (28)\nUsing the continuity equation, we have\n\u0010\nv0@0+~ v\u0001~r\u0011\n\u001a=~r\u0001\u0010\n\u001a~ v\u0000v0~j\u0011\n(29)\nand\u0010\nv0@0+~ v\u0001~r\u0011\n~j=~r\u0002\u0010\n~j\u0002~ v\u0011\n\u0000@0\u0010\n\u001a~ v\u0000v0~j\u0011\n; (30)11\nand thus the inhomogeneous Maxwell equations (27) and (28) can be expressed as\n~r\u0001~D=\u001a; (31)\n~r\u0002~H\u0000@~D\n@t=~j; (32)\nwhere the e\u000bective displacement \feld ~Dand the e\u000bective magnetic \feld ~Hare given, re-\nspectively, by\n~D=~E+\u0010\nv0~j\u0000\u001a~ v\u0011\n\u0011~E+\u0000 !~P; (33)\nand\n~H=~B\u0000\u0010\n~j\u0002~ v\u0011\n\u0011~B\u0000\u0000 !~M: (34)\nCompared with the model (4) in Sec. II, we see that the e\u000bective polarization\u0000 !~Pand the\ne\u000bective magnetization\u0000 !~Msatisfy\u0000 !~P=\u0000\u0000 !Mand\u0000 !~M=\u0000 !P. This is not surprising, since the\ndimension-\fve operator j\u0016v\u0017~F\u0016\u0017in model (4) can be written as\nj\u0016v\u0017~F\u0016\u0017=1\n2\u000f\u0016\u0017\u000b\fM\u0016\u0017F\u000b\f\u0011~M\u0016\u0017F\u0016\u0017; (35)\nand the duality between (\u0000 !P;\u0000 !M) and (\u0000 !~M;\u0000\u0000 !~P) follows immediately. Together with the\nhomogeneous Maxwell equations, one can show that, in the presence of stationary sources,\nthe gauge \feld A\u0016is given by\nA\u0016(~ r) =1\n4\u0019Z\nd3r0(1 +~ v\u0001~r0)j\u0016(~ r0)\nj~ r\u0000~ r0j: (36)\nIt follows from (36) that, di\u000berent from the consequence of the model L1, a nonvanishing\nelectric \feld cannot arise from neutral sources and a nonvanishing magnetic \feld cannot\narise from steady and irrotational current density.\nThe modi\fed Dirac equation following from L2reads\n(iD =\u0000m\u0000e\r\u0016v\u0017F\u0016\u0017) \t = 0: (37)\nAgain, we can easily identify from the above equation the Hamiltonian operator ~Hof one-\nparticle quantum mechanics:\n~H=\r0\u0010\n~ \r\u0001~ p+eA =+m\u0000ev0~ \r\u0001~E+e\r0~ v\u0001~E\u0000e~ \r\u0001(~ v\u0002~B)\u0011\n=H0+\u000e~H (38)\nwhere the Dirac Hamiltonian H0is the same as before and \u000e~H=\u0000e(v0\r0~ \r\u0001~E\u0000~ v\u0001~E+\n\r0~ \r\u0001(~ v\u0002~B)) is the LV perturbation. We note that \u000e~Hcan be obtained from \u000eHin (15) by12\nchanging~E!~Band~B!\u0000~E. This is again due to the dual relation between the operator\nj\u0016v\u0017~F\u0016\u0017inL1and the operator j\u0016v\u0017F\u0016\u0017inL2.\nTo consider the \frst-order energy shift in the states of hydrogen atom induced by \u000e~Hin\nthe absence of external \felds, we need to calculate the following matrix elements:\nhn;j;l0;m0\njj\u000e~Hjn;j;l;m ji=\u0000ehn;j;l0;m0\njj(v0\r0~ \r\u0000~ v)\u0001~Ejn;j;l;m ji (39)\nwhere~Eis the Coulomb \feld. Once again, without loss of generality, we can choose a\ncoordinate system in which the zaxis is in the direction of ~ v. Since it is easy to show that\nJzcommutes with ( v0\r0~ \r\u0000~ v)\u0001~E, by the same argument as given in Sec. II we know that\nthe matrix elements (39) vanish for di\u000berent states of the same unperturbed energy. As for\nthe diagonal matrix elements, a straightforward calculation gives\n\u0000ehn;j;l;m jj(v0\r0~ \r\u0000~ v)\u0001~Ejn;j;l;m ji\n=\u000bZ\ndrdcos\u0012d'\u0010\n\u0000iF\u0000(\u0006\u0014jr)Yy\nj;m j(l;1\n2j^r); F+(\u0006\u0014jr)Yy\nj;m j(2j\u0000l;1\n2j^r)\u0011\n[v00\n@0\u0000 !\u001b\n\u0000 !\u001b01\nA\u0001(sin\u0012cos'^x+ sin\u0012sin'^y+ cos\u0012^z)\u0000j~ vjcos\u0012]\u0012iF\u0000(\u0006\u0014jr)Yj;m j(l;1\n2j^r)\nF+(\u0006\u0014jr)Yj;m j(2j\u0000l;1\n2j^r)\u0013\n=\u000bZ\ndrdcos\u0012d'f\u0000iv0F\u0000(\u0006\u0014jr)F+(\u0006\u0014jr)(sin\u0012cos'^x+ sin\u0012sin'^y+ cos\u0012^z)\u0001\n[Yy\nj;m j(l;1\n2j^r)\u0000 !\u001bYj;m j(2j\u0000l;1\n2j^r)\u0000Yy\nj;m j(2j\u0000l;1\n2j^r)\u0000 !\u001bYj;m j(l;1\n2j^r)]\u0000j~ vjcos\u0012[\nF2\n\u0000(\u0006\u0014jr)Yy\nj;m j(l;1\n2j^r)Yj;m j(l;1\n2j^r) +F2\n+(\u0006\u0014jr)Yy\nj;m j(2j\u0000l;1\n2j^r)Yj;m j(2j\u0000l;1\n2j^r)]g\n=\u0007i\u000bv0Z\ndrdcos\u0012d'F\u0000(\u0006\u0014jr)F+(\u0006\u0014jr)(\u00001)j\u0000l+1\n2f\n[\u0000s\n(j\u0000mj+ 1)(j+mj)\n(2j)(2j+ 2)cos\u0012Y\u0003mj\u00001\n2\nj+1\n2Ymj\u00001\n2\nj\u00001\n2\u0000s\n(j\u0000mj+ 1)(j\u0000mj)\n(2j)(2j+ 2)sin\u0012e\u0000i'Y\u0003mj\u00001\n2\nj+1\n2Ymj+1\n2\nj\u00001\n2\n+s\n(j+mj+ 1)(j+mj)\n(2j)(2j+ 2)sin\u0012ei'Y\u0003mj+1\n2\nj+1\n2Ymj\u00001\n2\nj\u00001\n2\u0000s\n(j+mj+ 1)(j\u0000mj)\n(2j)(2j+ 2)cos\u0012Y\u0003mj+1\n2\nj+1\n2Ymj+1\n2\nj\u00001\n2]\n\u0000[\u0000s\n(j\u0000mj+ 1)(j+mj)\n(2j)(2j+ 2)cos\u0012Y\u0003mj\u00001\n2\nj\u00001\n2Ymj\u00001\n2\nj+1\n2+s\n(j+mj+ 1)(j+mj)\n(2j)(2j+ 2)sin\u0012e\u0000i'Y\u0003mj\u00001\n2\nj\u00001\n2Ymj+1\n2\nj+1\n2\n\u0000s\n(j\u0000mj+ 1)(j\u0000mj)\n(2j)(2j+ 2)sin\u0012ei'Y\u0003mj+1\n2\nj\u00001\n2Ymj\u00001\n2\nj+1\n2\u0000s\n(j+mj+ 1)(j\u0000mj)\n(2j)(2j+ 2)cos\u0012Y\u0003mj+1\n2\nj\u00001\n2Ymj+1\n2\nj+1\n2]g\n= 0 (40)\nforl=j\u00061\n2, in which we have used the facts that terms proportional to j~ vjare odd functions13\nof cos\u0012and terms proportional to v0cancel each other out. Thus the energy levels of the\nhydrogen atom are not shifted by the LV perturbation \u000e~H.\nThe shift of the hydrogen energy levels induced by \u000e~Hin the presence of uniform external\n\felds can be easily analyzed in the same way as we did in Sec. II, assuming the external \felds\n(denoted again by ~Eextand~Bext) are weak. Since v0~Eextand~ v\u0002~Bextare constant vectors,\nusing the fact that for any constant vector ~ athe matrix elements hn;j;l0;m0\njj\r0~ \r\u0001~ ajn;j;l;m ji\nvanish, and knowing that the interaction e~ v\u0001~Eextshifts each energy level by the same amount,\nwe can conclude that, in the presence of uniform external \felds, the LV perturbation \u000e~H\nstill produces no e\u000bect on the hydrogen spectrum in \frst-order perturbation theory.\nIV. CONCLUSION\nIn this paper, QED modi\fed by dimension-\fve LV operators j\u0016v\u0017~F\u0016\u0017andj\u0016v\u0017F\u0016\u0017has\nbeen studied separately. In both cases, we have identi\fed the e\u000bective polarization and\nmagnetization, which are components of the rank-2 object j[\u0016v\u0017], from the \feld equations of\nmotion. In particular, we \fnd that, with the LV interaction j\u0016v\u0017~F\u0016\u0017, any charged spinor has\na spin-independent magnetic dipole moment density \u001a~ v, along with the one associated with\nits spin. Also, a static electric \feld can arise from stationary and neutral sources. These\nnovel properties do not come up from the other interaction j\u0016v\u0017F\u0016\u0017.\nWe have computed the shift in the energies of the states of a hydrogen atom in \frst-order\nperturbation theory. Our result shows that only the CP-even operator jivk~Fikproduces\nthe energy shifts, given by (22), and the degeneracy of each level is completely removed.\nInterestingly, the breakdown of Lorentz boost symmetry, induced by the v0terms, in these\ntwo models plays no role in determining the atomic energy spectrum. 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Plus 127, 102 (2012)."    },    {        "title": "1102.4000v1.Lorentz_Harmonics__Squeeze_Harmonics__and_their_Physical_Applications.pdf",        "content": "arXiv:1102.4000v1  [math-ph]  19 Feb 2011Lorentz Harmonics, Squeeze Harmonics, and\ntheir Physical Applications\nYoung S. Kim1\nCenter for Fundamental Physics, University of Maryland,\nCollege Park, Maryland 20742, U.S.A.\nMarilyn E. Noz2\nDepartment of Radiology, New York University,\nNew York, New York 10016, U.S.A.\nAbstract\nAmong the symmetries in physics, the rotation symmetry is mo st\nfamiliar to us. It is known that the spherical harmonics serv e useful\npurposes when the world is rotated. Squeeze transformation s are also\nbecoming more prominent in physics, particularly in optica l sciences\nand in high-energy physics. As can be seen from Dirac’s light -cone\ncoordinate system, Lorentz boosts are squeeze transformat ions. Thus\nthe squeeze transformation is one of the fundamental transf ormations\nin Einstein’s Lorentz-covariant world. It is possible to de fine a com-\nplete set of orthonormal functions defined for one Lorentz fr ame. It is\nshown that the same set can be used for other Lorentz frames. T rans-\nformationpropertiesarediscussed. Physicalapplication sarediscussed\nin both optics and high-energy physics. It is shown that the L orentz\nharmonics provide the mathematical basis for squeezed stat es of light.\nIt is shown also that the same set of harmonics can be used for u nder-\nstanding Lorentz-boosted hadrons in high-energy physics. It is thus\npossible to transmit physics from one branch of physics to th e other\nbranch using the mathematical basis common to them.\n1email: yskim@umd.edu\n2email: marilyne.noz@gmail.com\n11 Introduction\nInthispaper, weareconcernedwithsymmetrytransformationsin twodimen-\nsions, and we are accustomed to the coordinate system specified b yxandy\nvariables. On the xyplane, we know how to make rotations and translations.\nThe rotation in the xyplane is performed by the matrix algebra\n/parenleftbiggx′\ny′/parenrightbigg\n=/parenleftbiggcosθ−sinθ\nsinθcosθ/parenrightbigg/parenleftbiggx\ny/parenrightbigg\n, (1)\nbut we are not yet familiar with\n/parenleftbiggz′\nt′/parenrightbigg\n=/parenleftbiggcoshηsinhη\nsinhηcoshη/parenrightbigg/parenleftbiggz\nt/parenrightbigg\n. (2)\nWe see this form when we learn Lorentz transformations, but ther e is a ten-\ndency in the literature to avoid this form, especially in high-energy ph ysics.\nSince this transformation can also be written as\n/parenleftbiggu′\nv′/parenrightbigg\n=/parenleftbiggexp(η) 0\n0 exp( −η)/parenrightbigg/parenleftbiggu\nv/parenrightbigg\n, (3)\nwith\nu=z+t√\n2, v=z−t√\n2, (4)\nwhere the variables uandvare expanded and contracted respectively, we\ncall Eq.(2) or Eq.(3) squeeze transformations [1].\nFrom the mathematical point of view, the symplectic group Sp(2) con-\ntains both the rotation and squeeze transformations of Eqs. (1) and (2),\nand its mathematical properties have been extensively discussed in the liter-\nature [1, 2]. This group has been shown to be one of the essential to ols in\nquantum optics. From the mathematical point of view, the squeeze d state in\nquantumopticsisaharmonicoscillatorrepresentationofthis Sp(2)group[1].\nWe are interested in this paper in “squeeze transformations” of loc alized\nfunctions. We are quite familiar with the role of spherical harmonics in\nthree dimensional rotations. We use there the same set of harmon ics, but\nthe rotated function has different linear combinations of those har monics.\nLikewise, we are interested in a complete set of functions which will se rve the\nsame purpose for squeeze transformations. It will be shown that harmonic\noscillator wave functions can serve the desired purpose. From the physical\npoint of view, squeezed states define the squeeze or Lorentz har monics.\n2In 2003, Giedke et al.used the Gaussian function to discuss the entan-\nglement problems in information theory [3]. This paper allows us to use\nthe oscillator wave functions to address many interesting current issues in\nquantum optics and information theory. In 2005, the present aut hors noted\nthat the formalism ofLorentz-covariant harmonic oscillators leads to a space-\ntime entanglement [4]. We developed the oscillator formalism to deal w ith\nhadronic phenomena observed in high-energy laboratories [5]. It is r emark-\nable that the mathematical formalism of Giedke et al.is identical with that\nof our oscillator formalism.\nWhile quantum optics or information theory is a relatively new branch\nof physics, the squeeze transformation has been the backbone o f Einstein’s\nspecial relativity. While Lorentz, Poincar´ e, and Einstein used the t ransfor-\nmation of Eq.(2) for Lorentz boosts, Dirac observed that the sam e equation\ncan be written in the form of Eq.(3) [6]. Unfortunately, this squeeze aspect\nof Lorentz boosts has not been fully addressed in high-energy phy sics dealing\nwith particles moving with relativistic speeds.\nThus, we can call the same set of functions “squeeze harmonics” a nd\n“Lorentzharmonics” inquantum opticsandhigh-energyphysics re spectively.\nThis allows us to translate the physics of quantum optics or informat ion\ntheory into that of high-energy physics.\nThe physics of high-energy hadrons requires a Lorentz-covarian t localized\nquantum system. This description requires one variable which is hidde n in\nthe present form of quantum mechanics. It is the time-separation variable\nbetween two constituent particles in a quantum bound system like th e hy-\ndrogen atom, where the Bohr radius measures the separation bet ween the\nproton and the electron. What happens to this quantity when the h ydrogen\natom is boosted and the time-separation variable starts playing its r ole? The\nLorentz harmonics will allow us to address this question.\nIn Sec. 2, it is noted that the Lorentz boost of localized wave funct ions\ncan be described in terms of one-dimensional harmonic oscillators. T hus,\nthose wave functions constitute the Lorentz harmonics. It is also noted that\nthe Lorentz boost is a squeeze transformation.\nIn Sec. 3, we examine Dirac’s life-long effortsto make quantum mecha nics\nconsistent with special relativity, and present a Lorentz-covaria nt form of\nbound-state quantum mechanics. In Sec. 4, we construct a set o f Lorentz-\ncovariant harmonic oscillator wave functions, and show that they c an be\ngiven a Lorentz-covariant probability interpretation.\nIn Sec. 5, the formalism is shown to constitute a mathematical basis for\n3squeezed states of light, and for quantum entangled states. In S ec. 6, this\nformalism can serve as the language for Feynman’s rest of the unive rse [7].\nFinally, in Sec. 7, we show that the harmonic oscillator formalism can be\napplied to high-energy hadronic physics, and what we observe ther e can be\ninterpreted in terms of what we learn from quantum optics.\n2 Lorentz or Squeeze Harmonics\nLet us start with the two-dimensional plane. We are quite familiar with rigid\ntransformations such as rotations and translations in two-dimens ional space.\nThings are different for non-rigid transformations such as a circle b ecoming\nan ellipse.\nWe start with the well-known one-dimensional harmonic oscillator eige n-\nvalue equation\n1\n2\n−/parenleftBigg∂\n∂x/parenrightBigg2\n+x2\nχn(x) =/parenleftbigg\nn+1\n2/parenrightbigg\nχn(x). (5)\nFor a given value of integer n,the solution takes the form\nχn(x) =/bracketleftBigg1√π2nn!/bracketrightBigg1/2\nHn(x)exp/parenleftBigg−x2\n2/parenrightBigg\n, (6)\nwhereHn(x) is the Hermite polynomial of the n-th degree. We can then\nconsider a set of functions with all integer values of n. They satisfy the\northogonality relation/integraldisplay\nχn(x)χn′(x) =δnn′. (7)\nThis relation allows us to define f(x) as\nf(x) =/summationdisplay\nnAnχn(x), (8)\nwith\nAn=/integraldisplay\nf(x)χn(x)dx. (9)\nLet us next consider another variable addedto Eq.(5), andthediffe rential\nequation\n1\n2\n\n\n−/parenleftBigg∂\n∂x/parenrightBigg2\n+x2\n+\n−/parenleftBigg∂\n∂y/parenrightBigg2\n+y2\n\n\nφ(x,y) =λφ(x,y),(10)\n4This equation can be re-arranged to\n1\n2\n\n−/parenleftBigg∂\n∂x/parenrightBigg2\n−/parenleftBigg∂\n∂y/parenrightBigg2\n+x2+y2\n\nφ(x,y) =λφ(x,y),(11)\nThisdifferential equationisinvariantunder therotationdefinedinEq .(1).\nIn terms of the polar coordinate system with\nr=/radicalBig\nx2+y2,tanθ=/parenleftbiggy\nx/parenrightbigg\n(12)\nthis equation can be written:\n1\n2/braceleftBigg\n−∂2\n∂r2−1\nr∂\n∂r−1\nr2∂2\n∂θ2+r2/bracerightBigg\nφ(r,θ) =λφ(r,θ), (13)\nand the solution takes the form\nφ(r,θ) =e−r2/2Rn,m(r){Amcos(mθ)+Bnsin(mθ)}.(14)\nThe radial equation should satisfy\n1\n2/braceleftBigg\n−∂2\n∂r2−1\nr∂\n∂r+m2\nr2+r2/bracerightBigg\nRn,m(r) = (n+m+1)Rn,m(r).(15)\nIn the polar form of Eq.(14), we can achieve the rotation of this fun ction by\nchanging the angle variable θ.\nOn the other hand, the differential equation of Eq.(10) is separable in the\nxandyvariables. The eigen solution takes the form\nφnx,ny(x,y) =χnx(x)χny(y), (16)\nwith\nλ=nx+ny+1. (17)\nIf a function f(x,y) is sufficiently localized around the origin, it can be\nexpanded as\nf(x,y) =/summationdisplay\nnx,nyAnx,nyχnx(x)χny(y), (18)\nwith\nAnx,ny=/integraldisplay\nf(x,y)χnx(x)χny(y)dx dy. (19)\n5If we rotate f(x,y) according to Eq.(1), it becomes f(x∗,y∗), with\nx∗= (cosθ)x−(sinθ)y, y∗= (sinθ)x+(cosθ)y (20)\nThis rotated function can also be expanded in terms of χnx(x) andχny(y):\nf(x∗,y∗) =/summationdisplay\nnx,nyA∗\nnx,nyχnx(x)χny(y), (21)\nwith\nA∗\nnx,ny=/integraldisplay\nf(x∗,y∗)χnx(x)χny(y)dx dy. (22)\nNext, let us consider the differential equation\n1\n2\n\n−/parenleftBigg∂\n∂z/parenrightBigg2\n+/parenleftBigg∂\n∂t/parenrightBigg2\n+z2−t2\n\nψ(z,t) =λψ(z,t).(23)\nHere we use the variables zandt, instead of xandy. Clearly, this equation\ncan be also separated in the zandtcoordinates, and the eigen solution can\nbe written as\nψnz,nt(z,t) =χnz(z)χnt(z,t), (24)\nwith\nλ=nz−nt. (25)\nThe oscillator equation is not invariant under coordinate rotations o f the\ntypegiveninEq.(1). Itishowever invariantunderthesqueezetran sformation\ngiven in Eq.(2).\nThe differential equation of Eq.(23) becomes\n1\n4/braceleftBigg\n−∂\n∂u∂\n∂v+uv/bracerightBigg\nψ(u,v) =λψ(u,v). (26)\nBothEq.(11)andEq.(23)aretwo-dimensionaldifferentialequation s. They\nare invariant under rotations and squeeze transformations resp ectively. They\ntake convenient forms in the polar and squeeze coordinate system s respec-\ntively as shown in Eq.(13) and Eq.(26).\nThe solutions of the rotation-invariant equation are well known, bu t the\nsolutions of the squeeze-invariant equation are still strange to th e physics\ncommunity. Fortunately, both equations are separable in the Cart esian co-\nordinate system. This allows us to study the latter in terms of the fa miliar\n6rotation-invariant equation. This means that if the solution is sufficie ntly\nlocalized in the zandtplane, it can be written as\nψ(z,t) =/summationdisplay\nnz,ntAnz,ntχnz(z)χnt(t), (27)\nwith\nAnz,nt=/integraldisplay\nψ(z,t)χnz(z)χnt(t)dz dt. (28)\nIf we squeeze the coordinate according to Eq.(2),\nψ(z∗,t∗) =/summationdisplay\nnz,ntA∗\nnz,ntχnz(z)χnt(t), (29)\nwith\nA∗\nnz,nt=/integraldisplay\nψ(z∗,t∗)χnz(z)χnt(t)dz dt. (30)\nHere again both the original and transformed wave functions are lin ear com-\nbinations of the wave functions for the one-dimensional harmonic o scillator\ngiven in Eq.(6).\nThe wave functions for the one-dimensional oscillator arewell know n, and\ntheyplayimportantrolesinmany branches ofphysics. It isgratifyin g tonote\nthat they could play an essential role in squeeze transformations a nd Lorentz\nboosts. We choose to call them Lorentz harmonics or squeeze har monics.\nTable 1: Cylindrical and hyperbolic equations. The cylindrical equatio n is\ninvariant under rotation while the hyperbolic equation is invariant und er\nsqueeze transformation\nEquation Invariant under Eigenvalue\nCylindrical Rotation λ=nx+ny+1\nHyperbolic Squeeze λ=nx−ny\n73 The Physical Origin of Squeeze Transfor-\nmations\nPaul A. M. Dirac made it his life-long effort to combine quantum mechan ics\nwith special relativity. We examine the following four of his papers.\n•In 1927 [8], Dirac pointed out the time-energy uncertainty should b e\ntaken into consideration for efforts to combine quantum mechanics and\nspecial relativity.\n•In 1945 [9], Dirac considered four-dimensional harmonic oscillator wa ve\nfunctions with\nexp/braceleftbigg\n−1\n2/parenleftBig\nx2+y2+z2+t2/parenrightBig/bracerightbigg\n, (31)\nand noted that this form is not Lorentz-covariant.\n•In 1949 [6], Dirac introduced the light-cone variables of Eq.(4). He als o\nnoted that the construction of a Lorentz-covariant quantum me chanics\nis equivalent to the construction of a representation of the Ponca r´ e\ngroup.\n•In 1963 [10], Dirac constructed a representation of the (3 + 2) deS itter\ngroup using two harmonic oscillators. This deSitter group contains\nthree (3 + 1) Lorentz groups as its subgroups.\nIn each of these papers, Dirac presented the original ingredients which\ncan serve as building blocks for making quantum mechanics relativistic . We\ncombine those elements using Wigner’s little groups [11] and and Feynm an’s\nobservation of high-energy physics [12, 13, 14].\nFirst of all, let us combine Dirac’s 1945 paper and his light-cone coordi-\nnate system given in his 1949 paper. Since xandyvariables are not affected\nby Lorentz boosts along the zdirection in Eq.(31), it is sufficient to study\nthe Gaussian form\nexp/braceleftbigg\n−1\n2/parenleftBig\nz2+t2/parenrightBig/bracerightbigg\n. (32)\nThis form is certainly not invariant under Lorentz boost as Dirac not ed. On\nthe other hand, it can be written as\nexp/braceleftbigg\n−1\n2/parenleftBig\nu2+v2/parenrightBig/bracerightbigg\n, (33)\n8t\nz\nzzt\ntDirac 1927Dirac 1949c-number \nTime-energy \nUncertainty\n      Heisenberg \n      UncertaintyQuantum Mechanics Lorentz Covariance\nLorentz-covariant Quantum Mechanics \nFeynman's proposal allows \nus to combine Dirac's \nquantum mechanics and \nLorentz covariance to \ngenerate Lorentz-squeezed \nhadrons.  \nFigure1: Space-timepicture ofquantum mechanics. Inhis1927pap er, Dirac\nnoted that there is a c-number time-energy uncertainty relation, in addition\nto Heisenberg’s position-momentum uncertainty relations, with qua ntum ex-\ncitations. This idea is illustrated in the first figure (upper left). In his 1949\npaper, Dirac produced his light-cone coordinate system as illustrat ed in the\nsecondfigure(upper right). Itisthennotdifficult toproducethet hirdfigure,\nfor a Lorentz-covariant picture of quantum mechanics. This Lore ntz-squeeze\nproperty is observed in high-energy laboratories through Feynma n’s parton\npicture discussed in Sec. 7.\n9whereuandvare the light-cone variables defined in Eq.(4). If we make the\nLorentz-boost or Lorentz squeeze according to Eq.(3), this Gau ssian form\nbecomes\nexp/braceleftbigg\n−1\n2/parenleftBig\ne−2ηu2+e2ηv2/parenrightBig/bracerightbigg\n. (34)\nIf we write the Lorentz boost as\nz′=z+βt√1−β2, t′=t+βz√1−β2, (35)\nwhereβis the the velocity parameter v/c, thenβis related to ηby\nβ= tanh(η). (36)\nLet us go back to the Gaussian form of Eq.(32), this expression is co n-\nsistent with Dirac’s earlier paper on the time-energy uncertainty re lation [8].\nAccording to Dirac, this is a c-number uncertainty relation without e xcita-\ntions. The existence of the time-energy uncertainty is illustrated in the first\npart of Fig. 1.\nIn his 1927 paper, Dirac noted the space-time asymmetry in uncert ainty\nrelations. Whiletherearenotime-likeexcitations, quantummechanic sallows\nexcitations along the zdirection. How can we take care of problem?\nIf we suppress the excitations along the tcoordinate, the normalized\nsolution of this differential equation, Eq.( 24), is\nψ(z,t) =/parenleftbigg1\nπ2nn!/parenrightbigg1/2\nHn(z)exp/braceleftBigg\n−/parenleftBiggz2+t2\n2/parenrightBigg/bracerightBigg\n. (37)\nIfweboostthecoordinatesystem, theLorentz-boostedwave f unctionsshould\ntake the form\nψn\nη(z,t) =/parenleftbigg1\nπ2nn!/parenrightbigg1/2\nHn(zcoshη−tsinhη)\n×exp/braceleftBigg\n−/bracketleftBigg(cosh2η)(z2+t2)−4(sinh2η)zt\n2/bracketrightBigg/bracerightBigg\n.(38)\nThese are the solutions of the phenomenological equation of Feynm anet\nal.[12] for internal motion of the quarks inside a hadron. In 1971, Fey nman\net al.wrote down a Lorentz-invariant differential equation of the form\n1\n2\n\n−/parenleftBigg∂\n∂xµ/parenrightBigg2\n+x2\nµ\n\nψ(xµ) = (λ+1)ψ(xµ), (39)\n10wherexµis for the Lorentz-covariant space-time four vector. This oscillat or\nequation is separable in the Cartesian coordinate system, and the t ransverse\ncomponents can be seprated out. Thus, the differential of Eq.(23 ) contains\nthe essential element of the Lorentz-invariant Eq.( 39).\nHowever, the solutions contained in Ref. [12] are not normalizable an d\ntherefore cannot carry physical interpretations. It was shown later that there\nare normalizable solutions which constitute a representation of Wign er’s\nO(3)-like little group [5, 11, 15]. The O(3) group is the three-dimensional\nrotation group without a time-like direction or time-like excitations. T his\naddresses Dirac’s concern about the space-time asymmetry in unc ertainty\nrelations [8]. Indeed, the expression of Eq.(37) is considered to be t he rep-\nresentation of Wigner’s little group for quantum bound states [11, 1 5]. We\nshall return to more physical questions in Sec. 7.\n4 Further Properties of the Lorentz Harmon-\nics\nLet us continue our discussion of quantum bound states using harm onic os-\ncillators. We are interested in this section to see how the oscillator so lution\nof Eq.(37) would appear to a moving observer.\nThe variable zandtare the longitudinal and time-like separations be-\ntween the two constituent particles. In terms of the light-cone va riables\ndefined in Eq.(4), the solution of Eq.(37) takes the form\nψn\n0(z,t) =/bracketleftbigg1\nπn!2n/bracketrightbigg1/2\nHn/parenleftBiggu+v√\n2/parenrightBigg\nexp/braceleftBigg\n−/parenleftBiggu2+v2\n2/parenrightBigg/bracerightBigg\n,(40)\nand\nψn\nη(z,t) =/bracketleftbigg1\nπn!2n/bracketrightbigg1/2\nHn/parenleftBigge−ηu+eηv√\n2/parenrightBigg\nexp/braceleftBigg\n−/parenleftBigge−2ηu2+e2ηv2\n2/parenrightBigg/bracerightBigg\n,(41)\nfor the rest and moving hadrons respectively.\nIt is mathematically possible to expand this as [5, 16]\nψn\nη(z,t) =/parenleftBigg1\ncoshη/parenrightBigg(n+1)/summationdisplay\nk/bracketleftBigg(n+k)!\nn!k!/bracketrightBigg1/2\n(tanhη)kχn+k(z)χn(t),(42)\n11whereχn(z) is then-th excited state oscillator wave function which takes the\nfamiliar form\nχn(z) =/bracketleftBigg1√π2nn!/bracketrightBigg1/2\nHn(z)exp/parenleftBigg−z2\n2/parenrightBigg\n, (43)\nasgiven inEq.(6). This isanexpansion of theLorentz-boostedwave function\nin terms of the Lorentz harmonics.\nIf the hadron is at rest, there are no time-like oscillations. There ar e\ntime-like oscillations for a moving hadron. This is the way in which the\nspace and time variable mix covariantly. This also provides a resolution of\nthe space-time asymmetry pointed out by Dirac in his 1927 paper [8]. We\nshall return to this question in Sec. 6. Our next question is whether those\noscillator equations can be given a probability interpretation.\nEven though we suppressed the excitations along the tdirection in the\nhadronic rest frame, it is an interesting mathematical problem to st art with\nthe oscillator wave function with an excited state in the time variable. This\nproblem was adressed by Rotbart in 1981 [17].\n4.1 Lorentz-invariant Orthogonality Relations\nLet us consider two wave functions ψn\nη(z,t).If two covariant wave functions\nare in the same Lorentz frame and have thus the same value of η, the orthog-\nonality relation /parenleftBig\nψn′\nη,ψn\nη/parenrightBig\n=δnn′ (44)\nis satisfied.\nIf those two wave functions have different values of η, we have to start\nwith /parenleftBig\nψn′\nη′,ψn\nη/parenrightBig\n=/integraldisplay/parenleftBig\nψn′\nη′(z,t)/parenrightBig∗ψn\nη(z,t)dzdt. (45)\nWithout loss of generality, we can assume η′= 0 in the system where η= 0,\nand evaluate the integration. The result is [18]\n/parenleftBig\nψn′\n0,ψn\nη/parenrightBig\n=/integraldisplay/parenleftBig\nψn′\n0(z,t)/parenrightBig2ψn\nη(z,t)dxdt=/parenleftbigg/radicalBig\n1−β2/parenrightbigg(n+1)\nδn,n′.(46)\nwhereβ= tanh(η),as given in Eq.(36). This is like the Lorentz-contraction\nproperty of a rigid rod. The ground state is like a single rod. Since we o btain\nthefirstexcitedstatebyapplyingastep-upoperator,thisstate shouldbehave\n12Figure 2: Orthogonality relations for the covariant harmonic oscillat ors. The\northogonality remains invariant. For the two wave functions in the o rthog-\nonality integral, the result is zero if they have different values of n. If both\nwave functions have the same value of n, the integral shows the Lorentz\ncontraction property.\n13like a multiplication of two rods, and a similar argument can be give to n\nrigid rods. This is illustrated in Fig. 2.\nWith these orthogonality properties, it is possible to give quantum pr ob-\nability interpretation in the Lorentz-covariant world, and it was so s tated in\nour 1977 paper [19].\n4.2 Probability Interpretations\nLet us study the probability issue in terms of the one-dimensional os cillator\nsolution of Eq.(6) whose probability interpretation is indisputable. Le t us\nalso go back to the rotationally invariant differential equation of Eq.( 11).\nThen the product\nχnx(x)χny(y) (47)\nalsohasaprobability interpretationwiththeeigenvalue( nx+ny+1).Thus\nthe series of the form [1, 5]\nφn\nη(x,y) =/parenleftBigg1\ncoshη/parenrightBigg(n+1)/summationdisplay\nk/bracketleftBigg(n+k)!\nn!k!/bracketrightBigg1/2\n(tanhη)kχn+k(x)χn(y) (48)\nalso has its probability interpretation, but it is not in an eigen state. E ach\nterm in this series has an eigenvalue (2 n+k+1). The expectation value of\nEq.(11) is\n/parenleftBigg1\ncoshη/parenrightBigg2(n+1)/summationdisplay\nk(2n+k+1)(n+k)!\nn!k!(tanhη)2k. (49)\nIf we replace the variables xandybyzandtrespectively in the above\nexpression of Eq.(48), it becomes the Lorentz-covariant wave fu nction of\nEq.(42). Each term χn+k(z)χk(t) in the series has the eigenvalue n. Thus\nthe series is in the eigen state with the eigenvalue n.\nThis difference does not prevent us from importing the probability int er-\npretation from that of Eq.(48).\nIn the present covariant oscillator formalism, the time-separation variable\ncan be separated from the rest of the wave function, and does no t requite\nfurther interpretation. For a moving hadron, time-like excitations are mixed\nwith longitudinal excitations. Is it possible to give a physical interpre tation\nto those time-like excitations? To address this issue, we shall study in Sec. 5\ntwo-mode squeezed states also based on the mathematics of Eq.(4 8). There,\nboth variables have their physical interpretations.\n145 Two-mode Squeezed States\nHarmonic oscillators play the central role also in quantum optics. The re\nthenthexcited oscillator state corresponds to the n-photon state |n>. The\ngroundstate meansthezero-photonorvacuum state |0>. The single-photon\ncoherent state can be written as\n|α>=e−αα∗/2/summationdisplay\nnαn\n√\nn!|n>, (50)\nwhich can be written as [1]\n|α>=e−αα∗/2/summationdisplay\nnαn\nn!/parenleftBig\nˆa†/parenrightBign|0>=/braceleftBig\ne−αα∗/2/bracerightBig\nexp/braceleftBig\nαˆa†/bracerightBig\n|0>.(51)\nThis aspect of the single-photon coherent state is well known. Her e we are\ndealing with one kind of photon, namely with a given momentum and polar -\nization. The state |n>means there are nphotons of this kind.\nLet us next consider a state of two kinds of photons, and write |n1,n2>\nas the state of n1photons of the first kind, and n2photons of the second\nkind [20]. We can then consider the form\n1\ncoshηexp/braceleftBig\n(tanhη)ˆa†\n1ˆa†\n2/bracerightBig\n|0,0>. (52)\nThe operator ˆ a†\n1ˆa†\n2was studied by Diracinconnection with his representation\nof the deSitter group, as we mentioned in Sec. 3. After making a Tay lor\nexpansion of Eq.(52), we arrive at\n1\ncoshη/summationdisplay\nk(tanhη)k|k,k>, (53)\nwhich is the squeezed vacuum state or two-photon coherent stat e [1, 20].\nThis expression is the wave function of Eq.(48) in a different notation . This\nform is also called the entangled Gaussian state of two photons [3] or the\nentangled oscillator state of space and time [4].\nIf we start with the n-particle state of the first photon, we obtain\n/bracketleftBigg1\ncoshη/bracketrightBigg(n+1)\nexp/braceleftBig\n(tanhη)ˆa†\n1ˆa†\n2/bracerightBig\n|n,0>\n=/bracketleftBigg1\ncoshη/bracketrightBigg(n+1)/summationdisplay\nk/bracketleftBigg(n+k)!\nn!k!/bracketrightBigg1/2\n(tanhη)k|k+n,k>, (54)\n15which is the wave function of Eq.(42) in a different notation. This is the\nn-photon squeezed state [1].\nSince the two-mode squeezed state and the covariant harmonic os cillators\nshare the same set of mathematical formulas, it is possible to trans mit physi-\ncal interpretations from one to the other. For two-mode squeez ed state, both\nphotons carry physical interpretations, while the interpretation is yet to be\ngiven to the time-separation variable in the covariant oscillator form alism.\nIt is clear from Eq. (42) and Eq. (54) that the time-like excitations a re like\nthe second-photon states.\nWhat would happen if the second photon is not observed? This inter-\nesting problem was addressed by Yurke and Potasek [21] and by Eke rt and\nKnight [22]. They used the density matrix formalism and integrated ou t the\nsecond-photon states. This increases the entropy and tempera ture of the\nsystem. We choose not to reproduce their mathematics, because we will be\npresenting the same mathematics in Sec. 6.\n6 Time-separationVariableinFeynman’sRest\nof the Universe\nAs was noted in the previous section, the time-separation variable h as an\nimportant role in the covariant formulation of the harmonic oscillator wave\nfunctions. It should exist wherever the space separation exists. The Bohr\nradiusisthemeasureoftheseparationbetween theprotonandele ctroninthe\nhydrogen atom. If this atom moves, the radius picks up the time sep aration,\naccording to Einstein [23].\nOn the other hand, the present form of quantum mechanics does n ot\ninclude this time-separation variable. The best way we can interpret it at\nthe present time is to treat this time-separation as a variable in Feyn man’s\nrest of the universe [24]. In his book on statistical mechanics [7], Fey nman\nstates\nWhen we solve a quantum-mechanical problem, what we really\ndo is divide the universe into two parts - the system in which\nwe are interested and the rest of the universe. We then usuall y\nact as if the system in which we are interested comprised the\nentire universe. To motivate the use of density matrices, le t us\n16see what happens when we include the part of the universe outs ide\nthe system.\nThe failure to include what happens outside the system results in an in -\ncrease of entropy. The entropy isa measure of our ignorance and is computed\nfrom the density matrix [25]. The density matrix is needed when the ex per-\nimental procedure does not analyze all relevant variables to the ma ximum\nextentconsistentwithquantummechanics[26]. Ifwedonottakeint oaccount\nthe time-separation variable, the result is an increase in entropy [2 7, 28].\nFor the covariant oscillator wave functions defined in Eq. (42), the pure-\nstate density matrix is\nρn\nη(z,t;z′,t′) =ψn\nη(z,t)ψn\nη(z′,t′), (55)\nwhich satisfies the condition ρ2=ρ:\nρn\nη(z,t;x′,t′) =/integraldisplay\nρn\nη(z,t;x”,t”)ρn\nη(z”,t”;z′,t′)dz”dt”.(56)\nHowever, in the present form of quantum mechanics, it is not possib le to\ntake into account the time separation variables. Thus, we have to t ake the\ntrace of the matrix with respect to the t variable. Then the resultin g density\nmatrix is\nρn\nη(z,z′) =/integraldisplay\nψn\nη(z,t)ψn\nη(z′,t)dt\n=/parenleftBigg1\ncoshη/parenrightBigg2(n+1)/summationdisplay\nk(n+k)!\nn!k!(tanhη)2kψn+k(z)ψ∗\nn+k(z′).(57)\nThe trace of this density matrix is one, but the trace of ρ2is less than one,\nas\nTr/parenleftBig\nρ2/parenrightBig\n=/integraldisplay\nρn\nη(z,z′)ρn\nη(z′,z)dzdz′\n=/parenleftBigg1\ncoshη/parenrightBigg4(n+1)/summationdisplay\nk/bracketleftBigg(n+k)!\nn!k!/bracketrightBigg2\n(tanhη)4k,(58)\nwhich is less than one. This is due to the fact that we do not know how\nto deal with the time-like separation in the present formulation of qu antum\nmechanics. Our knowledge is less than complete.\n17The standard way to measure this ignorance is to calculate the entr opy\ndefined as\nS=−Tr(ρln(ρ)). (59)\nIf we pretend to know the distribution along the time-like direction an d use\nthe pure-state density matrix given in Eq.(55), then the entropy is zero.\nHowever, if we do not know how to deal with the distribution along t, then\nwe should use the density matrix of Eq.(57) to calculate the entropy , and the\nresult is\nS= 2(n+1)/braceleftBig\n(coshη)2ln(coshη)−(sinhη)ln(sinhη)/bracerightBig\n−/parenleftBigg1\ncoshη/parenrightBigg2(n+1)/summationdisplay\nk(n+k)!\nn!k!ln/bracketleftBigg(n+k)!\nn!k!/bracketrightBigg\n(tanhη)2k.(60)\nIn terms of the velocity vof the hadron,\nS=−(n+1)/braceleftBigg\nln/bracketleftBigg\n1−/parenleftbiggv\nc/parenrightbigg2/bracketrightBigg\n+(v/c)2ln(v/c)2\n1−(v/c)2/bracerightBigg\n−/bracketleftBigg\n1−/parenleftbigg1\nv/parenrightbigg2/bracketrightBigg/summationdisplay\nk(n+k)!\nn!k!ln/bracketleftBigg(n+k)!\nn!k!/bracketrightBigg/parenleftbiggv\nc/parenrightbigg2k\n.(61)\nLet us go back to the wave function given in Eq.(41). As is illustrated\nin Fig. 3, its localization property is dictated by the Gaussian factor w hich\ncorresponds to the ground-state wave function. For this reaso n, we expect\nthat much of the behavior of the density matrix or the entropy for thenth\nexcited state will be the same as that for the ground state with n= 0.For\nthis state, the density matrix and the entropy are\nρ(z,z′) =/parenleftBigg1\nπcosh(2η)/parenrightBigg1/2\nexp/braceleftBigg\n−1\n4/bracketleftBigg(z+z′)2\ncosh(2η)+(z−z′)2cosh(2η)/bracketrightBigg/bracerightBigg\n,\n(62)\nand\nS= 2/braceleftBig\n(coshη)2ln(coshη)−(sinhη)2ln(sinhη)/bracerightBig\n, (63)\nrespectively. The quark distribution ρ(z,z) becomes\nρ(z,z) =/parenleftBigg1\nπcosh(2η)/parenrightBigg1/2\nexp/parenleftBigg−z2\ncosh(2η)/parenrightBigg\n. (64)\n18β = 0\nzt\nMeasurableNot Measurableβ = 0.8 Feynman's \nRest of the \nUniverset\nz\n             First PhotonSecond photon\nFigure 3: Localization property in the ztplane. When the hadron is at\nrest, the Gaussian form is concentrated within a circular region spe cified\nby (z+t)2+ (z−t)2= 1.As the hadron gains speed, the region becomes\ndeformed to e−2η(z+t)2+e2η(z−t)2= 1.Since it is not possible to make\nmeasurements along the tdirection, we have to deal with information that\nis less than complete.\nThe width of the distribution becomes√coshη, andbecomes wide-spread\nas the hadronic speed increases. Likewise, the momentum distribut ion be-\ncomes wide-spread [5, 29]. This simultaneous increase in the momentu m and\nposition distribution widths is called the parton phenomenon in high-en ergy\nphysics [13, 14]. The position-momentum uncertainty becomes cosh η. This\nincrease in uncertainty is due to our ignorance about the physical b ut un-\nmeasurable time-separation variable.\nLetusnextexaminehowthisignorancewillleadtotheconcept oftem per-\nature. For the Lorentz-boosted ground state with n= 0, the density matrix\nof Eq.(62) becomes that of the harmonic oscillator in a thermal equilib rium\nstate if (tanh η)2is identified as the Boltzmann factor [29]. For other states,\nit is very difficult, if not impossible, to describe them as thermal equilibr ium\nstates. Unlike the case of temperature, the entropy is clearly defi ned for all\nvalues ofn. Indeed, the entropy in this case is derivable directly from the\nhadronic speed.\nThetime-separationvariableexistsintheLorentz-covariantworld , butwe\npretend not to know about it. It thus is in Feynman’s rest of the univ erse.\nIf we do not measure this time-separation, it becomes translated in to the\n19zp\n Quantum \nUncertainty Statistical \nUncertainty\nFigure 4: The uncertainty from the hidden time-separation coordin ate. The\nsmall circle indicates the minimal uncertainty when the hadron is at re st.\nMore uncertainty is added when the hadron moves. This is illustrated by a\nlarger circle. The radius of this circle increases by/radicalBig\ncosh(2η).\nentropy.\nWe can see the uncertainty in our measurement process from the W igner\nfunction defined as\nW(z,p) =1\nπ/integraldisplay\nρ(z+y,z−y)e2ipydy. (65)\nAfter integration, this Wigner function becomes\nW(z,p) =1\nπcosh(2η)exp/braceleftBigg\n−/parenleftBiggz2+p2\ncosh(2η)/parenrightBigg/bracerightBigg\n. (66)\nThis Wigner phase distribution is illustrated in Fig. 4. The smaller inner cir -\ncle corresponds to theminimal uncertainty ofthe single oscillator. T he larger\ncircle is for the total uncertainty including the statistical uncerta inty from\nour failure to observe the time-separation variable. The two-mode squeezed\nstate tells us how this happens. In the two-mode case, both the fir st and\nsecond photons are observable, but we can choose not to observ e the second\nphoton.\n207 Lorentz-covariant Quark Model\nThehydrogenatomplayedthepivotalrolewhilethepresent formof quantum\nmechanics was developed. At that time, the proton was in the absolu te\nGalilean frame of reference, and it was thinkable that the proton co uld move\nwith a speed close to that of light.\nAlso, at that time, both the proton and electron were point particle s.\nHowever, the discovery of Hofstadter et al.changed the picture of the pro-\nton in 1955 [30]. The proton charge has its internal distribution. With in\nthe framework of quantum electrodynamics, it is possible to calculat e the\nRutherford formula for the electron-proton scattering when bo th electron\nand proton are point particles. Because the proton is not a point pa rticle,\nthere is a deviation from the Rutherford formula. We describe this d evia-\ntion using the formula called the “proton form factor” which depend s on the\nmomentum transfer during the electron-proton scattering.\nIndeed, the study of the proton form factor has been and still is o ne\nof the central issues in high-energy physics. The form factor dec reases as\nthe momentum transfer increases. Its behavior is called the “dipole cut-off”\nmeaning an inverse-square decrease, and it has been a challenging p roblem in\nquantum field theory and other theoretical models [31]. Since the e mergence\nof the quark model in 1964 [32], the hadrons are regarded as quant um bound\nstates of quarks with space-time wave functions. Thus, the quar k model is\nresponsible for explaining this form factor. There are indeed many p apers\nwritten on this subject. We shall return to this problem in Subsec. 7 .2.\nAnother problem in high-energy physics is Feynman’s parton picture [13,\n14]. If the hadron is at rest, we can approach this problem within the frame-\nwork of bound-state quantum mechanics. If it moves with a speed c lose to\nthat of light, it appears as a collection of an infinite number of parton s,\nwhich interact with external signals incoherently. This phenomenon raises\nthe question of whether the Lorentz boost destroys quantum co herence [33].\nThis leads to the concept of Feynman’s decoherence [34]. We shall dis cuss\nthis problem first.\n7.1 Feynman’s Parton Picture and Feynman’s Deco-\nherence\nIn 1969, Feynman observed that a fast-moving hadron can be reg arded as a\ncollection of many “partons” whose properties appear to be quite d ifferent\n21Energy\r\ndistributionβ=0.8 β=0\nzt\nzBOOST\nSPACE-TIME\r\n\r\nDEFORMATIONWeaker spring\r\nconstant\r\nQuarks become\r\n(almost) free\nTime dilationTIME-ENERGY UNCERTAINTYt\n( (\nβ=0.8 β=0\nqzqo\nqzBOOST\nMOMENTUM-ENERGY\r\n\r\nDEFORMATIONParton momentum\r\ndistribution\r\nbecomes widerqo\n( (\n((QUARKS PARTONS\nFigure 5: Lorentz-squeezed space-time and momentum-energy w ave func-\ntions. As the hadron’s speed approaches that of light, both wave f unctions\nbecome concentrated along their respective positive light-cone ax es. These\nlight-cone concentrations lead to Feynman’s parton picture.\n22from those of the quarks [5, 14]. For example, the number of quar ks inside\na static proton is three, while the number of partons in a rapidly movin g\nproton appears to be infinite. The question then is how the proton lo oking\nlike a bound state of quarks to one observer can appear different t o an ob-\nserver in a different Lorentz frame? Feynman made the following sys tematic\nobservations.\na. The picture is valid only for hadrons moving with velocity close to tha t\nof light.\nb. The interaction time between the quarks becomes dilated, and pa rtons\nbehave as free independent particles.\nc. The momentum distribution of partons becomes widespread as th e\nhadron moves fast.\nd. The number of partons seems to be infinite or much larger than th at\nof quarks.\nBecause the hadron is believed to be a bound state of two or three q uarks,\neach of the above phenomena appears as a paradox, particularly b ) and c)\ntogether. How can a free particle have a wide-spread momentum dis tribu-\ntion?\nIn order to address this question, let us go to Fig. 5, which illustrate s\nthe Lorentz-squeeze property of the hadron as the hadron gain s its speed.\nIf we use the harmonic oscillator wave function, its momentum-ener gy wave\nfunction takes the same form as the space-time wave function. As the hadron\ngains its speed, both wave functions become squeezed.\nAs the wave function becomes squeezed, the distribution becomes wide-\nspread, the spring constant appear to become weaker. Consequ ently, the\nconstituent quarks appear to become free particles.\nIf the constituent particles are confined in the narrow elliptic region , they\nbecome like massless particles. If those massless particles have a wid e-spread\nmomentumdistribution, it islike ablack-bodyradiationwithinfinitenumb er\nof photon distributions.\nWe have addressed this question extensively in the literature, and c on-\ncluded Gell-Mann’s quark model and Feynman’s partonmodel are two differ-\nent manifestations of the same Lorentz-covariant quantity [19, 35, 36]. Thus\ncoherent quarks and incoherent partons are perfectly consiste nt within the\n23framework of quantum mechanics and special relativity [33]. Indeed , this\ndefines Feynman’s decoherence [34].\nMore recently, we were able to explain this decoherence problem in te rms\nof the interaction time among the constituent quarks and the time r equired\nfor each quark to interact with external signals [4].\n7.2 Proton Form Factors and Lorentz Coherence\nAs early as in 1970, Fujimura et al.calculated the electromagnetic form\nfactoroftheprotonusing thewave functionsgiven inthispaper an dobtained\nthe so-called “dipole” cut-off of the form factor [37]. At that time, t hese\nauthors did not have a benefit of the differential equation of Feynm an and\nhis co-authors [12]. Since their wave functions can now be given a bo na-\nfide covariant probability interpretation, their calculation could be p laced\nbetween the two limiting cases of quarks and partons.\nEven before the calculation of Fujimura et al.in 1965, the covariant\nwave functions were discussed by various authors [38, 39, 40]. In 1 970, Licht\nand Pagnamenta also discussed this problem with Lorentz-contrac ted wave\nfunctions [41].\nIn our 1973 paper [42], we attempted to explain the covariant oscillat or\nwave function in terms of the coherence between the incoming signa l and the\nwidth of the contracted wave function. This aspect was explained in terms\nof the overlap of the energy-momentum wave function in our book [5 ].\nInthispaper, wewould like togoback tothecoherence problemwe ra ised\nin 1973, and follow-up on it. In the Lorentz frame where the moment um of\nthe proton has the opposite signs before and after the collision, th e four-\nmomentum transfer is\n(p,E)−(−p,E) = (2p,0), (67)\nwhere the proton comes along the zdirection with its momentum p, and its\nenergy√p2+m2.\nThen the form factor becomes\nF(p) =/integraldisplay\ne2ipz(ψη(z,t))∗ψ−η(z,t)dz dt. (68)\nIf we use the ground-state oscillator wave function, this integral becomes\n1\nπ/integraldisplay\ne2ipzexp/braceleftBig\n−cosh(2η)/parenleftBig\nz2+t2/parenrightBig/bracerightBig\ndz dt. (69)\n24After thetintegration, this integral becomes\n1/radicalBig\nπcosh(2η)/integraldisplay\ne2ipzexp/braceleftBig\n−z2cosh(2η)/bracerightBig\ndz. (70)\nThe integrand is a product of a Gaussian factor and a sinusoidal osc illation.\nThe width of the Gaussian factor shrinks by 1 //radicalBig\ncosh(2η), which becomes\nexp(−η) asηbecomes large. The wave length of the sinusoidal factor is\ninversely proportional to the momentum p. The wave length decreases also\nat the rate of exp( −η).Thus, the rate of the shrinkage is the same for both\nthe Gaussian and sinusoidal factors. For this reason, the cutoff r ate of the\nform factor of Eq.(68) should be less than that for\n/integraldisplay\ne2ipz(ψ0(z,t))∗ψ0(z,t)dz dt=1√π/integraldisplay\ne2ipzexp/parenleftBig\n−z2/parenrightBig\ndz,(71)\nwhich corresponds to the form factor without the squeeze effect on the wave\nfunction. The integration of this expression lead to exp( −p2),which corre-\nsponds to an exponential cut-off as p2becomes large.\nLet us go back to the form factor of Eq.(68). If we complete the int egral,\nit becomes\nF(p) =1\ncosh(2η)exp/braceleftBigg−p2\ncosh(2η)/bracerightBigg\n. (72)\nAsp2becomes large, the Gaussian factor becomes a constant. Howeve r, the\nfactor 1/cosh(2η) leads the form factor decrease of 1 /p2, which is a much\nslower decrease than the exponential cut-off without squeeze eff ect.\nThere still is a gap between this mathematical formula and the obser ved\nexperimental data. Before looking at the experimental curve, we have to\nrealizethattherearethreequarksinsidethehadronwithtwo oscilla tormode.\nThis will leadto a(1 /p2)2cut-off, which iscommonly called the dipolecut-off\nin the literature.\nThere is still more work to be done. For instance, the effect of the q uark\nspin should be addressed [43, 44]. Also there are reports of deviatio ns from\nthe exact dipole cut-off [45]. There have been attempts to study th e form\nfactors based on the four-dimensional rotation group [46], and als o on the\nlattice QCD [47],\nYet, it is gratifying to note that the effect of Lorentz squeeze lead to\nthe polynomial decrease in the momentum transfer, thanks to the Lorentz\ncoherence illustrated in Fig. 6. We started our logic from the fundam ental\nprinciples of quantum mechanics and relativity.\n25Increasing Frequency Increasing Momentum Transfer \n   without \nContraction       with \nContraction     Lorentz \nCoherence    without\nCoherence \nFigure 6: Coherence between the wavelength and the proton size. As the\nmomentum transfer increases, the external signal sees Lorent z-contracting\nproton distribution. On the other hand, the wavelength of the sign al also\ndecreases. Thus, the cutoff is not as severe as the case where th e proton\ndistribution is not contracted.\n26Conclusions\nIn this paper, we presented one mathematical formalism applicable b oth to\ntheentanglement problemsinquantumoptics[3]andtohigh-energ yhadronic\nphysics [4]. Theformalismisbasedonharmonicoscillatorsfamiliartous. We\nhave presented a complete orthonormal set with a Lorentz-cova riant proba-\nbility interpretation.\nSince both branches of physics share the same mathematical base , it is\npossible to translate physics from one branch to the other. In this paper,\nwe have given a physical interpretation to the time-separation var iable as a\nhidden variable in Feynman’s rest of the universe, in terms of the two -mode\nsqueezed state where both photons are observable.\nThis paper is largely a review paper with an organization to suit the\ncurrent interest in physics. For instance, the concepts of entan glement and\ndecoherecne did not exist when those original papers were written . Further-\nmore, the probability interpretation given in Subsection 4.2 has not b een\npublished before.\nThe rotation symmetry plays its role in all branches of physics. We\nnoted that the squeeze symmetry plays active roles in two different subjects\nof physics. 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Few-Body Systems 2005,37,\n1-31.\n[47] Matevosyan, H. H.; Thomas, A. W.; Miller, G. A. Study of lattice QC D\nform factors using the extended Gari-Krumpelmann model. Phys.Rev.C\n2005,72, 065204-5.\n31"    },    {        "title": "1910.06372v3.Decay_rates_for_the_damped_wave_equation_with_finite_regularity_damping.pdf",        "content": "arXiv:1910.06372v3  [math.AP]  16 Jun 2021Decay rates for the damped wave equation with finite\nregularity damping\nPerry Kleinhenz\nAbstract\nDecay rates for the energy of solutions of the damped wave equat ion on the torus\nare studied. In particular, damping invariant in one direction and equ al to a sum of\nsquares of nonnegative functions with a particular number of deriv atives of regularity\nis considered. For such damping energy decays at rate 1 /t2/3. If additional regularity\nis assumed the decay rate improves. When such a damping is smooth t he energy\ndecays at 1 /t4/5−δ. The proof uses a positive commutator argument and relies on a\npseudodifferential calculus for low regularity symbols.\n1 Introduction\nLetWbe a bounded, nonnegative damping function on a compact Riem annian manifold\nM, and letvsolve\n/braceleftigg\n∂2\ntv−∆v+W(x)∂tv= 0 t>0,\n(v,∂tv)|t=0= (v0,v1)∈C∞(M)×C∞(M)t= 0.\nThe primary object of study in this paper is the energy\nE(v,t) =1\n2/integraldisplay\n|∇v|2+|∂tv|2dx.\nWhenWis continuous it is classical that uniform stabilization is equivalent to geometric\ncontrol by the positive set of the damping. That is E(t)≤Cr(t)E(0), withr(t)→0 as\nt→ ∞, if and only if there exists L, such that all geodesics of length at least Lintersect\n{W >0}. Furthermore, in this case the optimal r(t) is exponentially decaying in t.\nWhen the geometric control condition does not hold decay is i nstead of the form.\nE(t)1/2≤Cr(t)(||v0||H2+||v1||H1). (1)\nThen the optimal r(t) depends on the geometry of Mand{W >0}, as well as properties\nofWin a neighborhood of {W= 0}. This paper explores this dependence for translation\ninvariant damping functions on the torus, and proves decay o f the form\nE(t)1/2≤C(1+t)−α(||v0||H2+||v1||H1). (2)\n1Such decay is guaranteed on the torus with α= 1/2 when{W >0}is open and nonempty\nby [AL14].\nFirst, when the damping is a sum of squares of sufficiently regu lary-invariant functions\nthere is an improved decay rate.\nTheorem 1.1. LetMbe the torus (R/2πZ)x×(R/2πZ)y. SupposeW(x,y) =W(x)and\nsatisfies\n1. For some σ∈(0,π),Wis bounded below by apositive constant for x∈[−π,π]\\[−σ,σ],\n2. There exists σ1∈(0,π−σ)and there exist functions vj(x)≥0,vj∈W9,∞(−σ−\nσ1,σ+σ1), such that W(x) =/summationtext\njvj(x)2on(−σ−σ1,σ+σ1).\nThen there exists Csuch that (2)holds withα=2\n3.\nIf the damping is instead smooth and y-invariant there is an additional improvement.\nTheorem 1.2. LetMbe the torus (R/2πZ)x×(R/2πZ)y. SupposeW(x,y) =W(x)and\nsatisfies\n1. For some σ∈(0,π),Wis bounded below by apositive constant for x∈[−π,π]\\[−σ,σ],\n2.W∈C∞(R/2πZ).\nThen for all ε>0there exists Csuch that (2)holds withα=4\n5−ε.\nBoth of these theorems are actually consequences of the foll owing result. When the\ndamping is a sum of squares of functions with k0derivatives there is an improved decay\nrate which depends on k0.\nTheorem 1.3. LetMbe the torus (R/2πZ)x×(R/2πZ)y. SupposeW(x,y) =W(x)and\nsatisfies\n1. For some σ∈(0,π),Wis bounded below by apositive constant for x∈[−π,π]\\[−σ,σ],\n2. There exists k0≥9,σ1∈(0,π−σ)and there exist functions vj(x)≥0,vj∈\nWk0,∞(−σ−σ1,σ+σ1), such that W(x) =/summationtext\njvj(x)2on(−σ−σ1,σ+σ1).\nLetτmin>max/parenleftig\nk0+2\n2k0−4,7\nk0−1/parenrightig\nthen there exists Csuch that (2)holds withα=2\nτmin+2.\nRemarks\n•The two constraints for τminin terms of the regularity k0are needed to guarantee\nerror terms in composition expansions are small. In particu larτmin>k0+2\n2k0−4is needed\ntoensure(15)holdsand τmin>7\nk0−1isneededtoensure(19)holds. Theseconstraints\nare sharp on these inequalities, but are also used in other es timates in the proof.\n2•Theorem 1.1 is just Theorem 1.3 when k0= 9. Soτmincan be taken = 1 which gives\ndecay atα= 2/3.\n•On the other hand by [Bon05] if W∈C2k0(a,b) then there exist v1,v2∈Ck0(a,b)\nsuch thatW=v2\n1+v2\n2on (a,b). Therefore if W∈C2k0(−σ−σ1,σ+σ1) it satisfies\nhypothesis 2 of the theorem. Theorem 1.2 then follows from Th eorem 1.3 and the\nresult of Bony. In particular for any fixed k0there is an appropriate expansion and\nsoτmincan be taken arbitrarily close to 1 /2 which gives decay at α= 4/5−δ.\nThe equivalence of uniform stabilization and geometric con trol for continuous damping\nfunctions was proved by Ralston [Ral69], and Rauch and Taylo r [RT75] (see also [BLR92]\nand [BG97], where Mis also allowed to have a boundary). For some more recent finer\nresults concerning discontinuous damping functions, see B urq and G´ erard [BG18].\nDecay ratesoftheform(1)goback toLebeau[Leb96]. When W∈C(M)isnonnegative\nand{W >0}is open andnonempty, then decay of the form (1) holds with r(t) = 1/log(2+\nt) in [Bur98, Leb96]. Furthermore, this is optimal on spheres and some other surfaces of\nrevolution [Leb96]. At the other extreme, if Mis a negatively curved (or Anosov) surface,\nandW∈C∞(M),Wnonnegativeandnotidentically zero, then(1)holdswith r(t) =Ce−ct\n[DJN19].\nWhenMis a torus, these extremes are avoided and the best bounds are polynomially\ndecaying as in (2). Anantharaman and L´ eautaud [AL14] show ( 2) holds with α= 1/2\nwhenW∈L∞,W≥0, andW >0 on some open set, as a consequence of Schr¨ odinger\nobservability/control [Jaf90, Mac10, BZ12]. The more rece nt result of Burq and Zworski\non Schr¨ odinger observability and control [BZ19] weakens t he final requirement to merely\nW/\\⌉}atio\\slash≡0. Anantharaman and L´ eautaud [AL14] further show that if su ppWdoes not satisfy\nthe geometric control condition then (2) cannot hold for any α >1. They also show if\nthere exists C >0 such that Wsatisfies|∇W| ≤CW1−εforε<1/29 andW∈Wk0,∞for\nk0≥8 then (2) holds with α= 1/(1+4ε).\nNote that Theorem 1.3 improves the dependence between |∇W| ≤W1−εestimates and\ndecay rate with slightly different hypotheses. That is a dampi ng satisfying the hypotheses\nof Theorem 1.3 has |∇W| ≤CW1/2, which, if the [AL14] result applied to ε= 1/2, would\nonly give (2) with α= 1/3, no better than the generic upper bound, whereas Theorem 1. 3\ngives (2) with at least α= 2/3.\nAdditionally, becauseoftheresultin[Bon05], Theorem1.3 appliestosufficientlyregular\ndamping, which is invariant in one direction, without addit ional hypotheses. In particular\n[AL14] mention that their results do not give an improvement over the Schr¨ odinger observ-\nability bound for smooth damping vanishing like W=e−1/xsin(1/x)2, while Theorem 1.3\ndoes.\nFor earlier work on the square and partially rectangular dom ains see [LR05] and [BH07]\nrespectively, and for polynomial decay rates in the setting of a degenerately hyperbolic\nundamped set, see [CSVW14].\n3In [Kle19], it was shown that if W= (|x| −σ)β\n+near [−σ,σ], then (2) holds with\nα= (β+ 2)/(β+ 4) and cannot hold for all solutions with α >(β+ 2)/(β+ 3). In the\ncase of constant damping on a strip the result that (2) holds w ithα= 2/3 is due to\nStahn [Sta17], and the result that it does not hold for α >2/3 is due to Nonnenmacher\n[AL14]. In [DK19] it was shown that for W∼(|x| −σ)β\n+near [−σ,σ],(2) holds with\nα= (β+2)/(β+3), which is sharp when W= (|x|−σ)β\n+near [−σ,σ].\nThese results along with Theorem 1.3 suggest that sharp deca y rate on the torus could\nbe determined by the regularity of the damping at the boundar y of its support. The other\nlikely alternative is that the sharp decay rate is determine d by the value of εfor which\nWsatisfies |∇W| ≤CW1−ε. Although the sharp decay rate for polynomial damping\nW= (|x|−σ)β\n+dependson β, thisdoesnotdisambiguatebetweenthesecasesas W∈Wβ,∞\nandWsatisfies|∇W| ≤W1−1/β. A good candidate for distinguishing these is Wsmooth\nand vanishing like e−1/xsin(1/x)2, as it only satisfies |∇W| ≤CW1/2.\nIf regularity determines the sharp decay rate for any δ>0 such an oscillating damping\nshould decay at 1 /t1−δas there are other smooth dampings which decay this fast. As i n\n[AL14], a smooth damping vanishing like e−1/xsatisfies|∇W| ≤CW1−εfor anyε>0 and\nso for anyδ >0 decays at 1 /t1−δ. If on the other hand the derivative bound condition\n|∇W| ≤CW1−εdetermines the sharpdecay rate, the fact that W= (|x|−σ)2\n+also satisfies\n|∇W| ≤W1/2and has solutions which decay no faster than 1 /t4/5, means an oscillating\ndamping also should have solutions which decay no faster tha n 1/t4/5. Theorem 1.3 does\nnot guarantee or rule out either of these, so resolving this q uestion would be an interesting\narea for future work.\nAcknowledgements I would like to thank Jared Wunsch and Oran Gannot for helpful\nconversations and comments on early drafts. I would also lik e to thank the anonymous\nreferee for their helpful comments, which improved the clar ity of the paper and led to\nan improvement in the overall result. I would also like to tha nk Andras Vasy for helpful\nconversations while I was improving the result. I was partia lly supported by the National\nScience Foundation grant RTG: Analysis on Manifolds at Nort hwestern University.\n1.1 Outline of Proof\nBy a Fourier transform in time, it is enough to study the assoc iated stationary problem.\nMore precisely, by Theorem 2.4 of [BT10], as formulated in Pr oposition 2.4 of [AL14],\ndecay with α=2\nτmin+2follows from showing that there are constants C,q0>0 such that,\nfor anyq≥q0,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(−∆+iqW−q2)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2(T2)→L2(T2)≤Cq1/α−1=Cqτmin/2.\nBecause the damping Wdependsonly on xthis can be reduced to a 1 dimensional problem\nby expanding in a Fourier series in the yvariable. Let kbe the vertical Fourier mode, set\n4β=q2−k2,takef∈L2(R/2πZ) and consider u∈H2(R/2πZ) solving\n−u′′+iqWu−βu=f. (3)\nThen it is enough to show that there are C,q0>0,such that for any f,anyq≥q0and\nany realβ≤q2, ifusolves (3) then\n/integraldisplay\n|u|2≤Cqτmin/integraldisplay\n|f|2.\nHere, and below, all integrals are over R/2πZ. A more precise dependenceon βis obtained,\nfor anyε1,ε2>0 there exists a constant Csuch that\n/integraldisplay\n|u|2≤C/integraldisplay\n|f|2,whenβ <π2\n16(σ+ε1)2, q≥q0, (4)\nand /integraldisplay\n|u|2≤Cqτmin/integraldisplay\n|f|2,whenε2<β≤q2, q≥q0. (5)\nIt is clear that for ε1,ε2small enough, (4) and (5) cover all β≤q2. Thisβcan be thought\nof as the “horizontal energy” of the solution. The larger it i s, the larger uis relative to q\nin theξdirection in phase space.\nEquation (4) is the low horizontal energy case and is proved i n section 2. Equation\n(5) is the high horizontal energy case and is the main estimat e. It follows by an elliptic\nestimate and a positive commutator argument. Section 3 cont ains an outline of the proof\nof (5), and sections 4 and 5 contain proofs of the subsidiary e stimates in the proof of (5).\nAppendix A contains some important facts about pseudodiffere ntial operators with finite\nregularity symbols.\nThe following is a frequently invoked and important estimat e.\nLemma 1.4. For anyβ∈R,q>0andu,fsolving(3)\n/integraldisplay\nW|u|2≤q−1/integraldisplay\n|fu|. (6)\nProof.Multiply (3) by ¯ uthen take the imaginary part, integrating by parts to see tha t the\nterm/a\\}b∇ack⌉tl⌉{t∆u,u/a\\}b∇ack⌉t∇i}htis real.\n2 Proof of low horizontal energy estimate (4)\nProof.To prove (4) multiply (3) by ¯ uand a nonnegative function bε1∈C∞(R/2πZ) with\nbε1(x) =/braceleftigg\ncos/parenleftig\nπ\n2(σ+ε1)x/parenrightig\n|x|<σ+ε1/2,\n0 |x|>σ+ε1.\n5Then integrate and take the real part to obtain\n−Re/integraldisplay\nbε1u′′¯u−β/integraldisplay\nbε1|u|2=Re/integraldisplay\nbε1f¯u.\nIntegrating by parts once gives\n/integraldisplay\nbε1|u′|2+Re/integraldisplay\nub′\nε1¯u′−β/integraldisplay\nbε1|u|2=Re/integraldisplay\nbε1f¯u.\nIntegrating by parts the ub′\nε1¯u′term again and taking advantage of the Regives\n/integraldisplay\nbε1|u′|2+/integraldisplay/parenleftbigg\n−b′′\nε1\n2−βbε1/parenrightbigg\n|u|2=Re/integraldisplay\nbε1f¯u. (7)\nNow note that −b′′\nε1\n2=π2\n8(σ+ε1)2bε1for|x|<σ+ε1\n2. Thus forβ <π2\n16(σ+ε1)2\n−bε1\n2−βbε1>con|x|<σ+ε1\n2.\nSo adding a multiple of (6), the damping estimate, to (7) give s\n/integraldisplay\n|u|2≤/parenleftbigg\nCε1+1\nq/parenrightbigg/integraldisplay\n|fu| ≤Cε1/parenleftbigg/integraldisplay\n|f|2/parenrightbigg1/2/parenleftbigg/integraldisplay\n|u|2/parenrightbigg1/2\n.\nDividing both sides by/parenleftbig/integraltext\n|u|2/parenrightbig1/2gives exactly (4).\n3 Proof of high horizontal energy estimate (5)\nNow that the proof of (4) is complete for β <π2\n16(σ+ε1)2it remains to show (5) for ε2<\nβ≤q2. This estimate will actually be assembled from estimates on second microlocalized\nregions of phase space, in order to do so I take a semiclassica l rescaling. Let γ∈ {1,2},\nthen divide both sides of (3) by q2/γand seth=q−1/γ\nPu= (−h2∂2\nx+ih2−γW−h2β)u=h2f. (8)\nIn this rescaling the bounds ε2<β≤q2becomeε2<β≤h−2γ. Letτ∈[τmin,1].Takeσ1\nas specified by hypothesis 1 and divide phase space ( R/2πZ)x×Rξ=T∗S1into 3 regions:\n1. The set where the damping is nontrivial, {(x,ξ) :σ+σ1/4<|x|<π}\n2. Thehdependent elliptic set of P,{(x,ξ) :|ξ|>1.5h1−τ}\n3. The propagating region, {(x,ξ) :|x|<σ+σ1/2 and|ξ|<2h1−τ}.\n6Althoughγandτcan be adjusted freely, for this proof they will have a specifi c relation.\nIn particular, ( τ,γ) will only take values in ( τmin,2),(3τmin,2),(1,1).\nNote that in composition expansions involving symbols at sc aleh1−τeach additional\nterm is only hτsmaller than the previous one, rather than a full power of h. Regardless\nof the values of γandτthere is a fixed size error terms in the following calculation s must\nbe smaller than. Because of this the number of expansion term s taken (and the number of\nderivatives of regularity Wmust have) grows at least like1\nτ.τminis the smallest possible\nτsuch thatWhas enough regularity to achieve the desired error size.\nThis behavior also clarifies why τandγare separate parameters. In Proposition 3.5\nthe resolvent estimate is ||u||2\nL2≤Cq2τ\nγ\nβ2||f||2\nL2. Because of this a larger γproduces a better\nestimate without decreasing τ, so no additional regularity of Wis required. However γ\ncannot always be taken large because the estimate only appli es toβ <q2τ/γwhich will not\ninclude all of β <q2.\nNote that in the case γ=τ= 1, this is not a second microlocalization as there is no\nhdependence. The remainder of this section is the statement o f the estimates for these\nregions and then a proof of the high horizontal energy case, ( 5), using those estimates.\nThe damping estimate is immediate, the elliptic estimate is proved in section 4 and the\npropagation estimate is proved in section 5.\n3.1 Damping Estimate\nThis lemma gives an estimate for the size of uon the set where the damping is nontrivial.\nLemma 3.1. For anyβ∈R,h>0andu,fsolving(8)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤hγ||f||L2||u||L2. (9)\nThis follows immediately from the rescaling and (6).\n3.2 Elliptic Estimate\nThroughout the paper Op refers to the Weyl quantization on th e torus (see Appendix A\nfor more details). These lemmas gives an estimate for the siz e ofuon thehdependent\nelliptic set of P,{(x,ξ) :|ξ|>1.5h1−τ}. Note that in order for Pto be bounded away\nfrom zero on this set h2βmust be smaller than h2−2τ.\nBecause of a technicality in the proof there are separate ell iptic estimates on chτ−1<\nξ<2 and 1.5<ξ. The cause of this is that the low regularity composition res ult (Lemma\nA.6 which is used in the elliptic parametrix construction) r equires bounded symbols but\np=ξ2+ih2−γW−h2βis unbounded for large ξ.\nThis lemma provides the estimate on chτ−1< ξ <2. This estimate has additional\nimportance as it is used multiple times in the proof of the pro pagation estimate to provide\nadditional control over error terms.\n7Lemma 3.2. SupposeW∈Wk0,∞andτ∈[τmin,1]. Setz1∈C∞(R)with\nz1(ξ) =/braceleftigg\n0|ξ|<1.25\n1|ξ|>1.5,\nand setz2∈C∞\n0(R)with\nz2(ξ) =/braceleftigg\n1|ξ|<2\n0|ξ|>3,\nthen letz(ξ) =z1(hτ−1ξ)z2(ξ)andZ=Op(z(ξ)). There exist C,h0>0,such that for\nh≤h0,βsuch thath2β <h2−2τ, andu,fsolving(8)then\n||Zu||2\nL2≤Ch5τ−1||f||2\nL2+o(h2)||u||2\nL2. (10)\nThis lemma provides the estimate on 1 <ξ. It does not impose any regularity assump-\ntions onWnor does it have a size restriction on β\nLemma 3.3. Set/tildewidez∈C∞(R)with\n/tildewidez(ξ) =/braceleftigg\n0|ξ|<1\n1|ξ|>1.5,\nand let/tildewideZ=Op(/tildewidez). There exist C,h0>0such that for h≤h0andu,fsolving(8)then\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideZu/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Ch4||f||2\nL2+Ch4−γ||f||L2||u||L2. (11)\nLemmas 3.2 and 3.3 are proved in section 4.\n3.3 Propagation Estimate\nThis lemma gives an estimate for the size of uon the propagating region {(x,ξ) :|x|<\nσ+σ1/2 and|ξ|<2h1−τ}.\nDefineψ∈C∞\n0(R)\nψ(ξ) =/braceleftigg\n1 on|ξ|<2\n0 on|ξ|>3,\nandχ∈C∞\n0(−π,π)\nχ(x) =/braceleftigg\n1 on|x|<σ+σ1/2\n0 on|x|>σ+σ1,\nwhere both are chosen to have smooth square roots.\n8Lemma 3.4. Supposevj∈Wk0,∞and fixτ∈[τmin,1],ε2>0. SetJ=Op(χ1/2(x)ψ1/2(hτ−1ξ)).\nThere exist C,h0>0,such that if h≤h0andβsuch thath2ε2< h2β < h2−2τ, then for\nu,fsolving(8)\n||Ju||2\nL2≤C/parenleftbiggh−τ\nβ/parenrightbigg\n||f||L2||u||L2+Ch6τ−γ−1\nβ||f||2\nL2+o(1)||u||2\nL2.(12)\nLemma 3.4 is proved in section 5. h2β≤h2−2τis assumed in order to apply the elliptic\nregion estimate in the proof.\n3.4 Combination of Estimates\nThis subsection completes the proof of (5), the high horizon tal energy estimate, using the\nfollowing proposition on different regimes for β,τandγ.\nProposition 3.5. SupposeW∈Wk0,∞and fixτ∈[τmin,1],γ∈ {1,2}andε2>0. There\nexistC,q0>0,such that if q≥q0andβsatisfiesε2≤β≤q2τ/γthen foruandfsolving\n(3)\n||u||2\nL2≤Cq2τ\nγ\nβ2||f||2\nL2.\nThe form of this estimate heps show why τandγare taken as two separate parameters.\nTakingγ= 2 produces a better estimate for all values of τ, however the estimate then only\napplies toβ <qwhich does not cover the required range of β <q2.\nThis proposition will beproved using the estimates on the da mped region (Lemma 3.1),\nthe elliptic region (Lemmas 3.2 and 3.3) and the propagating region (Lemma 3.4).\nProof of Proposition 3.5. Note thatβ≤q2τ/γguarantees h2β≤h2−2τand taking q0large\nenough ensures that his small enough to apply the Lemmas. Add together (9), (10), ( 11)\nand (12),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+||Zu||2\nL2+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideZu/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+||Ju||2\nL2≤(hγ+Ch4−γ+Ch−τ\nβ)||f||L2||u||L2\n+C(h5τ−1+h4+h6τ−γ−1\nβ)||f||2\nL2\n+(o(h2)+o(1))||u||2\nL2.\nSinceγ≤2 andτ≥1/2, the (hγ+h4−γ)||f||L2||u||L2and (h4+h5τ−1)||f||L2terms on\nthe right hand side are automatically smaller than other ter ms and can be safely ignored.\nNow, rewrite the LHS as/angbracketleftig\n(W+Z2+/tildewideZ2+J2)u,u/angbracketrightig\nand use the fact that W(x) +\nz(ξ)2+/tildewidez(ξ)2+χ(ξ)ψ(hτ−1ξ) is strictly positive on T∗S1to bound that term from below\n9byc||u||2\nL2. Then forhsmall enough, absorb all the ||u||2\nL2terms from the right hand side\ninto the left\nc||u||2\nL2≤Ch−τ\nβ||f||L2||u||L2+Ch6τ−γ−1\nβ||f||2\nL2.\nNow use Young’s inequality on theh−τ\nβ||f||L2||u||L2term and group the resultant ||u||2\nL2\nonto the left hand side to obtain\n||u||L2≤C/parenleftbiggh−2τ\nβ2+h6τ−γ−1\nβ/parenrightbigg\n||f||L2≤Ch−2τ\nβ2||f||L2.\nWhere the second inequality follows because τ >τmin>1/2 andγ≤2 imply 6τ−γ−1>\n3−γ−1>0, so the second term goes to 0 as h→0 regardless of β.\nFinally the rescaling q= 1/hγgives the desired inequality.\nTo finish the proof of (5) it is necessary to consider different r egimes forβ,τandγin\norder to ensure that β≤q2τ/γand to obtain the best possible estimate. Suppose q≥q0\nand consider three cases for ε2≤β≤q2(recalling that 1 /2≤τmin≤1)\n1.ε2≤β≤qτmin\n2.1\n2qτmin≤β≤q3τmin\n3.q3τmin\n2≤β≤q2.\nIn case 1 choose τ=τmin,γ= 2.Then by Proposition 3.5 there exists C >0 such that\n||u||2\nL2≤Cqτmin||f||2\nL2.\nIn case 2 choose τ= 3τmin,γ= 2.Then by Proposition 3.5, since1\nβ≤2\nqτmin, there exists\nC >0 such that\n||u||2\nL2≤Cq3τmin\nq2τmin||f||2\nL2≤Cqτmin||f||2\nL2.\nIf 3τmin≥2, skip case 3 and for case 2 instead take τ= 1,γ= 1. Then by Proposition\n3.5, since1\nβ≤2\nqτmin, there exists C >0 such that\n||u||2\nL2≤Cq2\nq2τmin||f||L2≤Cq2−2τmin||f||2\nL2≤Cqτmin||f||2\nL2.\nwhere the final inequality follows since 3 τmin≥2 implies 2 −2τmin≤τmin.\nIn case 3 choose τ= 1,γ= 1. Then by Proposition 3.5, since1\nβ≤2\nq3τmin, there exists\nC >0 such that\n||u||2\nL2≤Cq2\nq6τmin||f||2\nL2≤Cq2−6τmin||f||2\nL2≤Cqτmin||f||2\nL2,\n10where the final inequality holds because τmin>1/2>2/7 implies 2 −6τmin≤τmin.\nSince allq,βsuch thatε2≤β≤q2are covered by these three cases this proves the\nhigh energy estimate (5). This along with the low energy esti mate (4) completes the proof\nof Theorem 1.3.\nSo it now remains to prove the elliptic estimates (Lemmas 3.2 and 3.3) and the propa-\ngating estimate (Lemma 3.4). They are proved in sections 4 an d 5, respectively.\n4 Proof of elliptic region estimates Lemmas 3.2 and 3.3\nIfWis smooth and τ−1 then a conventional semiclassical parametrix argument pr oduces\nthe desired elliptic estimate (see for example [DZ19] Propo sition E.32). Normally as part\nof that proof ξ2is composed with 1 /p. This becomes an issue when Wis not smooth as\nthe low regularity composition expansion (Lemma A.6) only w orks with bounded symbols.\nTo address this cutoff functions are used to split the estimat e into estimates on a bounded\nelliptic set (Lemma 3.2) and a standard elliptic set (Lemma 3 .3).\nTakingτ/\\⌉}atio\\slash= 1 produces additional issues. The bounded elliptic set is hdependent and\nξis only bounded from below by a power of h, rather than a constant. In order to ensure\nthatp=ξ2+ih2−γW−h2βis invertible on this set βmust satisfy h2β≤h2−2τ< cξ2.\nThereforepis only bounded from below by a power of hand every division by pcreates\nunfavorable powers of h. These unfavorable powers can be controlled, but this requi res\nadditional regularity of Wand the requirements grow as τapproaches 1 /2.\nIn this section I will first prove the estimate on the hdependent elliptic set (Lemma\n3.2) and then prove the estimate on the standard elliptic set (Lemma 3.3).\n4.1hdependent elliptic estimate, Lemma 3.2\nThe proof of Lemma 3.2 follows the conventional semiclassic al parametrix argument with\nadjustments made to handle the issues described above.\nThe first change is that the parametrix is constructed for a cu toff version of P, Op(χp),\nwhich is bounded.\nLetχ∈C∞\n0(R) have\nχ(ξ) =/braceleftigg\n1|ξ|<3.5\n0|ξ|>4.\nDefine\nq0(x,ξ) =h2−2τz(ξ)\nχ(ξ)(ξ2+ih2−γW−h2β)=h2−2τz(ξ)\nχ(ξ)p(x,ξ)=h2−2τz(ξ)\np(x,ξ),\nwhereχcan be replaced by 1 because χ≡1 onsuppz⊂ {1.5h1−τ<|ξ|<3}. Theχis\nincluded to simplify the composition with Op( χp).\n11Forj≥1,recursively define\nqj=−1\nχpj−1/summationdisplay\nl=0ql,j−l,\nwhereql,j−lis the (j−l)th term in the composition expansion Op( ql)Op(χp) and is given\nby\nql,j−l=Cj,lhj−l(∂j−l\nx(χp)∂j−l\nξql+(−1)j−l∂j−l\nξ(χp)∂j−l\nxql),\nforj−l≥1. Once again the χcan be replaced by 1 in the definitions of qjandql,j−l,\nsinceχis identically 1 on the support of zand thusqj.\nTheseqjare used to construct a parametrix for Op( χp), which in turn is used to control\nQjOp(χ)P. In particular, I will show that for Nlarge enough\nN/summationdisplay\nj=0QjOp(χp) =N/summationdisplay\nj=0QjOp(χ)P=h2−2τZ+o(h3−2τ).\nTo prove this, the qjare first shown to be in a particular symbol class (Lemmas 4.2 a nd\n4.3), which gives control of the size of Qjas operators on L2(Lemma 4.4). Then/summationtextQj\nand Op(χp) are composed in two different ways, producing error terms of t he appropriate\nsize (Lemmas 4.5 and 4.6). Finally, this composition formul a is applied to uwhich gives\nthe desired elliptic estimate.\nThe following symbol style estimates for pandW\npon suppz={1.5h1−τ<|ξ|<3}are\nneeded to prove the symbol estimates for qj.\nLemma 4.1.\n1. Form,j,v∈Nsuch that 2m≥2j\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂v\nξ(p(x,ξ)j)\np(x,ξ)m/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ch(τ−1)(2m−2j+v).\n2. Forα,j,t∈Nwithj≥t\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle/vextendsingleh(1−τ)2t∂α\nx(ih2−γW)j−t\npj/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ch(τ−1)α.\nProof.1) To begin the binomial expansion formula gives\n∂v\nξp(x,ξ)j=j/summationdisplay\nl=0Cl,j∂v\nξ(ξ2−h2β)l(ih2−γW(x))j−l.\n12Again using the binomial expansion formula\n∂v\nξ(ξ2−h2β)l=∂v\nξl/summationdisplay\nk=0Ck,lξ2k(−h2β)l−k=l/summationdisplay\nk=0Ck,v,lξ2k−v(−h2β)l−k.\nNow note that on supp z,|ξ|>1.5h1−τwhileh2β≤h2−2τand soh2β≤ξ2/2. Therefore\n|∂v\nξ(ξ2−h2β)l| ≤l/summationdisplay\nk=0C|ξ|2k−v|h2β|l−k≤l/summationdisplay\nk=0C|ξ|2l−v≤C|ξ|2l−v.\nNow split supp zinto two sets\n1.A={(x,ξ)∈suppz;h2−γW(x)≤ξ2}\n2.B={(x,ξ)∈suppz;h2−γW(x)≥ξ2}\nFor (x,ξ)∈ A\n|∂v\nξpj| ≤j/summationdisplay\nl=0|∂v\nξ(ξ2−h2β)l||h2−γW|j−l≤j/summationdisplay\nl=0|ξ|2l−v|ξ2|j−l≤C|ξ|2j−v.\nAlso|p|=/radicalbig\n(ξ2−h2β)2+(h2−γW)2≥ξ2−h2β≥cξ2.\nTherefore\nsup\n(x,ξ)∈A/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂v\nξpj\npm/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\n(x,ξ)∈AC|ξ|2j−v\n|p|m≤sup\n(x,ξ)∈AC|ξ|2j−v−2m≤Ch(1−τ)(2m−2j+v),\nwherethe last inequality follows since ξ>1.5h1−τon suppz. Thisis thedesired inequality.\nNow consider the second case when ( x,ξ)∈ Bi.e.h2−γW≥ξ2. Then∂v\nξp≤/summationtextj\nl=0|ξ|2l−v|h2−γW|j−l≤ |h2−γW|jand|p| ≥h2−γW. Therefore\nsup\n(x,ξ)∈B/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂v\nξpj\npm/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\n(x,ξ)∈BWj\n(h2−γW)m≤sup\n(x,ξ)∈B1\nξ2(m−j)≤h(τ−1)(2m−2j)≤h(τ−1)(2m−2j+v)\nsinceτ−1≤0 andξ>1.5h1−τon suppz. These two cases cover all of supp zand so the\ndesired inequality holds.\n2) When 2j−2t≤αthis is true as |p|j≥h−2j(τ−1)and so\nh(τ−1)(−2t)\n|p|j|∂α\nx(ih2−γW)j−t| ≤Ch(τ−1)(−2t+2j)≤Ch(τ−1)α.\n13So now assume 2 j−2t>α. Applying ∂α\nxto (ih2−γW)j−tproduces a sum of powers of\nderivatives of W. In particular letting j0,j1,...jα∈Nthen\n∂α\nxWj−t=/summationdisplay\nCj0,...,jαWj0(∂xW)j1(∂2\nxW)j2···(∂α\nxW)jα,\nwhere the sum is taken over j0,...,jαsuch thatj0+j1+···+jα=j−tandj1+2j2+\n···+αjα=α. These conditions guarantee that there are j−tfactors ofWon the right\nhand side and that each term in the sum has αtotal derivatives.\nRearranging the derivative equation and then substituting in a rearranged version of\ntheWpowers equation gives\nα−j1= 2j2+3j3+···+αjα≥2(j2+j3+···+jα) = 2(j−t−j0−j1).\nTherefore 0 <2(j−t)−α≤2j0+j1. That is the number of terms with no derivatives or\none derivative is somehow bounded from below.\nNow note that since p=ξ2+ih2−γW−h2βthen|p| ≥h2−γWand so/vextendsingle/vextendsingle/vextendsingleh2−γW\np/vextendsingle/vextendsingle/vextendsingle≤C.\nSimilarly since W=/summationtextv2\njand thevjare bounded and have bounded derivatives then\n|∂xW| ≤CW1/2and so/vextendsingle/vextendsingle/vextendsingleh2−γ∂xW\np1/2/vextendsingle/vextendsingle/vextendsingle≤h2−γW1/2\n|p|1/2≤C.\nTherefore, again taking the sum over j0,j1,...,jαsatisfying the derivative and powers\nofWconstraints, and using powers of Wand∂xWto cancel powers and half powers of p\nrespectively\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α\nx(ih2−γW)j−t\npj/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/summationdisplay\nCj0,...,jαh(2−γ)(j−t)Wj0|∂xW|j1|∂2\nxW|j2···|∂xW|jα\n|p|j≤/summationdisplayC\n|p|j−j0−j1\n2.\nNow using that |p| ≥cξ2and|ξ|>ch1−τon suppzthe above equation gives\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle/vextendsingleh(1−τ)2t∂α\nx(ih2−γW)j−t\npj/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\n(x,ξ)∈suppzCh(1−τ)2t\n|p|j−j0−j1\n2≤sup\n(x,ξ)∈suppzCh(1−τ)2t\n|ξ|2j−2j0−j1\n≤h(1−τ)2th(τ−1)(2j−2j0−j1)=Ch(τ−1)(2j−2t−2j0−j1)≤Ch(τ−1)α,\nwhere the final inequality follows from 2 j0+j1≥2(j−t)−α.\nThe proof of part 2 of this lemma is a key usage of the fact that |∇W| ≤W1/2in this\npaper. Because of this it is worth mentioning that this exact argument does not give a\nmeaningful improvement when |∇W| ≤CW1−εforε <1/2. With such an assumption\nthere is still no improvement from factors of Wwithout any derivatives, the improvement\ncan only come from factors of ∂xW. However when αis even there are always terms with\nj1= 0 with no improvement over the stated result.\nWith these symbol style estimates it is now possible to give t he symbol class for q0.\nUnlike other symbols in this paper, differentiating it in eith erξorxproduces factors of\nhτ−1.\n14Lemma 4.2.\nq0∈Wk0S1−τ,1−τ(T∗S1).\nProof.Since supp z⊂ {1.5h1−τ<|ξ|<3}it is enough to show for |ξ|<3 andθ∈N,|α| ≤\nk0that\n|∂α\nx∂θ\nξq0| ≤h(τ−1)(α+θ).\nTo begin, recall a classical fact about higher order derivat ives of a quotient.\n∂α\nx/parenleftbiggf(x)\ng(x)/parenrightbigg\n=α/summationdisplay\nk=0k/summationdisplay\nj=0(−1)j/parenleftbiggα\nk/parenrightbigg/parenleftbiggk+1\nj+1/parenrightbigg1\ngj+1∂α−k\nxf∂k\nxgj. (13)\nThis follows from the Leibniz rule and the Hoppe formula appl ied to 1/g, (for the Hoppe\nformula see [Joh02] (3.3))\nTherefore\n∂α\nxq0(x,ξ) =∂α\nx/parenleftbiggh2−2τz(ξ)\np(x,ξ)/parenrightbigg\n=α/summationdisplay\nj=0(−1)j/parenleftbiggα+1\nj+1/parenrightbiggh2−2τz(ξ)\npj+1∂α\nxpj.\nAnd so\n∂θ\nξ∂α\nxq0=α/summationdisplay\nj=0(−1)j/parenleftbiggα+1\nj+1/parenrightbigg\nh2−2τ∂θ\nξ/parenleftbiggz\npj+1∂α\nxpj/parenrightbigg\n.\nNow applying (13) to ∂θ\nξ/parenleftig\nz\npj+1∂α\nxpj/parenrightig\n∂θ\nξ∂α\nxq0=α/summationdisplay\nj=0θ/summationdisplay\nv=0v/summationdisplay\nw=0Cj,α,v,θ,wh2−2τ\np(j+1)(w+1)∂θ−v\nξ(z∂α\nxpj)∂v\nξp(j+1)w.\nSo it is sufficient to control each individual term in the sum, w hich is of the form\nCh2−2τ1\np(j+1)(w+1)∂θ−v\nξ(z∂α\nxpj)∂v\nξp(j+1)w, (14)\nwhere 0≤j≤α, 0≤v≤θand 0≤w≤v.\nWell by Lemma 4.1\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂v\nξ(pj+1)w\n(pj+1)w/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤O(h(τ−1)v).\nThis along with |p| ≥h2−2τgives\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle/vextendsingleh2−2τ1\np(j+1)(w+1)∂θ−v\nξ(z∂α\nxpj)∂v\nξp(j+1)w/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\n(x,ξ)∈suppzC/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\npj∂θ−v\nξ(z∂α\nxpj)h(τ−1)v/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n15Now use the product rule to expand out ∂θ−v\nξ(z∂α\nxpj)\nC/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\npj∂θ−v\nξ(z∂α\nxpj)h(τ−1)v/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤v/summationdisplay\nl=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\npj∂θ−v−l\nξz∂l\nξ∂α\nxpj/vextendsingle/vextendsingle/vextendsingle/vextendsingleh(τ−1)v.\nWell|∂θ−v−l\nξz| ≤Ch(τ−1)(θ−v−l)which gives\nv/summationdisplay\nl=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\npj∂θ−v−l\nξz∂l\nξ∂α\nxpjh(τ−1)v/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤v/summationdisplay\nl=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\npjh(τ−1)(θ−l)∂l\nξ∂α\nxpj/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nAgain use the binomial expansion to write\npj=j/summationdisplay\nt=0Cj,t(ξ2−h2β)t(ih2−γW)j−t,\nand so\n∂l\nξ∂α\nxpj=j/summationdisplay\nt=0Cj,t∂l\nξ(ξ2−h2β)t∂α\nx(ih2−γW)j−t.\nCombining this with the previous chain of inequalities and ( 14) gives\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle/vextendsingleh2−2τ\np(j+1)(w+1)∂θ−v\nξ(z∂α\nxpj)∂v\nξp(j+1)w/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤\nsup\n(x,ξ)∈suppzv/summationdisplay\nl=0j/summationdisplay\nt=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCh(τ−1)(θ−l)\npj∂l\nξ(ξ2−h2β)t∂α\nx(ih2−γW)j−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nBy the same argument used in part 1 of Lemma 4.1\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle∂l\nξ(ξ2−h2β)t/vextendsingle/vextendsingle/vextendsingle≤C(hτ−1)l−2t.\nTherefore\nsup\n(x,ξ)∈suppzv/summationdisplay\nl=0j/summationdisplay\nt=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCh(τ−1)(θ−l)\npj∂l\nξ(ξ2−h2β)t∂α\nx(ih2−γW)j−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤sup\n(x,ξ)∈suppzj/summationdisplay\nt=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleCh(τ−1)(θ−2t)\npj∂α\nx(ih2−γW)j−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nNow using part 2 of Lemma 4.1\nsup\n(x,ξ)∈suppz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleh(τ−1)θ\npjh(1−τ)2t∂α\nx(ih2−γW)j−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ch(τ−1)(θ+α),\nand combining this chain of inequalities gives the desired s tatement.\n16Next I show that qjis a symbol with the same behavior under differentiation by xand\nξbut with size hj(2τ−1).\nLemma 4.3.\nhj(1−2τ)qj∈Wk0−jS1−τ,1−τ(T∗S1)\nProof.Since supp z⊂ {1.5h1−τ<|ξ|<3}it is enough to show for |ξ|<3 andθ∈N,α≤\nk0−jthat\n|∂α\nx∂θ\nξqj| ≤Chj(2τ−1)h(τ−1)(α+θ).\nThis will be proved inductively in j. By Lemma 4.2, q0satisfies this. So assume |∂α\nx∂θ\nξql| ≤\nChl(2τ−1)h(τ−1)(α+θ)for all 0≤l≤kand it is enough to show for θ∈N,α≤k0−j\nsup\n|ξ|<3|∂α\nx∂θ\nξqk+1| ≤Ch(k+1)(2τ−1)h(τ−1)(α+θ).\nBy definition\nqk+1=−1\npk/summationdisplay\nl=0ql,k+1−l=−qk,1\np−1\npk−1/summationdisplay\nl=0ql,k+1.\nSince\nql,k+1−l=Chk+1−l(∂k+1−l\nxp∂k+1−l\nξql+(−1)k+1−l∂k+1−l\nξp∂k+1−l\nxql)\nthen by the inductive assumption\nsup\n|ξ|<3|ql,k+1−l| ≤sup\n|ξ|<3Ch(k+1−l)τ|ql| ≤Ch(k+1−l)τhl(2τ−1).\nTherefore /vextendsingle/vextendsingle/vextendsingle/vextendsingleql,k+1−l\np/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ch2τ−2h(k+1−l)τhl(2τ−1)=Ch2τ−2h(k+1)τhl(τ−1).\nNow for 0 ≤l≤k−1, sinceτ−1≤0.\nh2τ−2hτ(k+1)hl(τ−1)≤h2τ−2hτ(k+1)h(k−1)(τ−1).\nNote that\n2τ−2+τk+τ+kτ−τ−k+1 = 2τk+2τ−(k+1) = (2τ−1)(k+1).\nTherefore\n|qk+1| ≤ |qk,1\np|+Ch(k+1)(2τ−1).\nTheqk,1term requires separate treatment. Recall qk,1=h(∂xp∂ξqk−∂ξp∂xqk). Using the\narguments of part 2 of Lemma 4.1 |∂xp\np|=|∂xW\np| ≤Chτ−1and|∂ξp\np| ≤Chτ−1on suppz\nand by the inductive assumption |∂xqk|,|∂ξqk| ≤Ch(τ−1)hk(2τ−1). Therefore\nsup\n|ξ|<3/vextendsingle/vextendsingle/vextendsingle/vextendsingleqk,1\np/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\n|ξ|<3Ch/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂xp\np/vextendsingle/vextendsingle/vextendsingle/vextendsingle|∂ξqk|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ξp\np/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|∂xqk|/parenrightbigg\n≤Chhτ−1hτ−1hk(2τ−1)=Ch(k+1)(2τ−1).\n17It remains to be seen that qk+1has the correct behavior under differentiation. That is\n|∂α\nx∂θ\nξqk+1| ≤Ch(k+1)(2τ−1)h(τ−1)(α+θ). Well\n∂α\nx∂θ\nξqk+1=−k/summationdisplay\nl=0∂α\nx∂θ\nξql,k+1−l\np=−k/summationdisplay\nl=0∂α\nx∂θ\nξ/parenleftbigghk+1−l\np(∂k+1−l\nxp∂k+1−l\nξql+(−1)k+1−l∂k+1−l\nξp∂k+1−l\nxql)/parenrightbigg\nIfaderivativefallson ∂xp∂ξqor∂ξp∂xqthisonlyproducesanadditional hτ−1. Furthermore,\nbytheargumentofLemma4.2anyderivatives whichfallon1\npproduceonly hτ−1. Therefore\n|∂α\nx∂θ\nξqk+1| ≤Ch(τ−1)(α+θ)|qk+1| ≤Ch(τ−1)(α+θ)h(k+1)(2τ−1).\nThis is exactly the desired inductive statement, which comp letes the proof.\nWith these symbol estimates it is straightforward to contro l the size of Op( qj) onL2.\nLemma 4.4.\n/ba∇⌈blOp(qj)/ba∇⌈blL2→L2=Chj(2τ−1)hτ−1\nProof.This follows immediately from Lemma A.2 and Lemma 4.3. In par ticular\n/ba∇⌈blOp(qj)/ba∇⌈blL2→L2≤C/summationdisplay\nα,θ∈{0,1}hθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α\nx∂θ\nξqj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞≤/summationdisplay\nα,θ∈{0,1}Chj(2τ−1)h(τ−1)(α+θ)hθ\n=/summationdisplay\nα,θ∈{0,1}Chj(2τ−1)h(τ−1)αhθτ≤Chj(2τ−1)hτ−1.\nNow using these symbol and operator norm estimates it is poss ible to compute the\ncomposition of/summationtextQjwith Op(χp).\nLemma 4.5. IfW∈Wk0,∞andτ∈[τmin,1],\n\nk0−6/summationdisplay\nj=0Qj\nOp(χp) =h2−2τZ+o(h3−2τ).\nProof.Applying Lemma A.6 part 3 to the composition QjOp(χp), (withN=k0−jand\n˜N=k0−j−5) produces\nQjOp(χp) =k0−6/summationdisplay\nk=0Op(qj,k)+OL2→L2(hj(2τ−1)hτ(k0−j−5)h5(τ−1)).\nThe additional hj(2τ−1)in the remainder term comes from the fact that hj(1−2τ)qj∈\nWk0S1−τ,1−τ.\n18Now, to control the remainder term, since j≤k0−6 andτ−1<0\nj(2τ−1)+τ(k0−j−5)+5(τ−1) =τk0+j(τ−1)−5≥τk0+(k0−6)(τ−1)−5.\nFurthermore τ≥τmin>k0+2\n2k0−4andk0≥8≥8τand so\nτk0+(k0−6)(τ−1)−5 = (2k0+4)τ−10τ+1>k0+2−10τ+1 = 3−2τ+(k0−8τ)≥3−2τ,\n(15)\nand the remainder error term is always of size o(h3−2τ).\nNow summing these composition expansions from j= 0 toj=k0−6\n\nk0−6/summationdisplay\nj=0Qj\nOp(χp) =k0−6/summationdisplay\nj=0/parenleftiggk0−j−6/summationdisplay\nk=0Op(qj,k)/parenrightigg\n+o(h3−2τ)\n= Op\nq0,0+q0,1+q0,2+q0,3+···+q0,k0−6\n+q1,0+q1,1+q1,2+···+q1,k0−7\n+q2,0+q2,1+···+q2,k0−8\n+q3,0+···+q3,k0−9\n+···\n+qk0−6,0\n+o(h3−2τ).\nBy construction of the qj,k, all columns except for the first sum to zero leaving\n\nM−1/summationdisplay\nj=0Qj\nOp(χp) = Op(q0,0)+o(h3−2τ) = Op(q0χp)+o(h3−2τ).\nSinceq0=h2−2τz(ξ)\nχp,Op(q0χp) = Op(z(ξ)) =Zand this is the desired equality.\nThe composition of/summationtextQjwith Op(χp) can also be computed in a way that separates\nOp(χ) andP.\nLemma 4.6. IfW∈Wk0,∞andτ∈[τmin,1],\nk0−6/summationdisplay\nj=1QjOp(χp) =k0−6/summationdisplay\nj=1QjOp(χ)P+o(h3−2τ).\nProof.First, by Lemmas A.1 and A.6 part 1 (with /tildewideN= 3≤k0−5 andρ= 0))\nOp(χ)P=2/summationdisplay\nk=0(ih)k\n2kk!Op(∂k\nξχ∂k\nxp)+OL2→L2(h3)\n= Op(χp)+2/summationdisplay\nk=1CkhkOp(∂k\nξχ∂k\nxW)+OL2→L2(h3).\n19Therefore\nQjOp(χp) =QjOp(χ)P−2/summationdisplay\nk=1CkhkQjOp(∂k\nξχ∂k\nxW)+OL2→L2(h3),\nand it remains to control terms of the form QjOp(∂k\nξχ∂k\nxW). Well by Lemma A.6 part 3\n(since∂k\nxW∈Wk0−kS0andhj(1−2τ)Qj∈Wk0−jS1−τ,1−τtake/tildewideN=k0−max(j,k)−5)\nhkQjOp(∂k\nξχ∂k\nxW) =hk/tildewideN−1/summationdisplay\nl=0(ih)l\n2ll!(∂y∂ξ−∂x∂η)l(qj(x,ξ)∂k\nηχ(η)∂k\nyW(y))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ\n+OL2→L2(hkhj(2τ−1)hτ(k0−max(j,k)−5)h5(τ−1)).\nAll the terms in the sum are 0 because χ≡1 on suppz⊃suppqjand soχ(k)≡0 on\nsuppqj.\nThe size of the remainders can also be controlled. Since j≤k0−6\nk+j(2τ−1)+τ(k0−max(j,k)−5)+5(τ−1) =τk0+j(τ−1)−5+k+jτ−max(j,k)τ\n≥τk0+j(τ−1)−5\n≥τk0+(k0−6)(τ−1)−5>3−2τ,\nwhere the last inequality follows from (15). Therefore\nQjOp(χp) =QjOp(χ)P+o(h3−2τ),\nand so\nk0−6/summationdisplay\nj=1QjOp(χp) =k0−6/summationdisplay\nj=1QjOp(χ)P+o(h3−2τ),\nas desired.\nWith these two composition results the proof of the hdependent elliptic estimate can\nbe completed.\nProof of Lemma 3.2. By Lemma 4.5\nh2−2τZu=/parenleftig/summationdisplay\nQj/parenrightig\nOp(χp)u+o(h3−2τ)u.\nand by Lemma 4.6\nh2−2τZu=/parenleftig/summationdisplay\nQj/parenrightig\nOp(χp)u+o(h3−2τ)u=/parenleftig/summationdisplay\nQj/parenrightig\nOp(χ)Pu+o(h3−2τ)u\n=/parenleftig/summationdisplay\nQj/parenrightig\nOp(χ)h2f+o(h3−2τ)u.\n20Take theL2norm squared of both sides. Then by Lemma 4.4, the Qjare bounded by\nChτ−1onL2and Op(χ) is bounded on L2by Lemma A.2.\nh4−4τ||Zu||2\nL2≤h4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nQjOp(χ)f/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+o(h6−4τ)||u||2\nL2\n≤Ch4hτ−1||f||2\nL2+o(h6−4τ)||u||2\nL2.\nFinally multiply both sides by h4τ−4to obtain the desired inequality.\n4.2ξ > cElliptic estimate, Lemma 3.3\nThis proof follows the conventional semiclassical paramet rix argument with the caveat that\nWis treated as a perturbation. This allows the parametrix con struction to be exact, as it\ninvolves only Fourier multipliers. Because of this there ar e no compositions involving W\nand so the regularity of Wis not involved in this proof. This same construction can als o be\nusedtoprovean hdependentellipticestimate, however treating Wperturbativelyproduces\nan error term that weakens the estimate. This lessens the imp rovement the estimate makes\nwhen applied to error terms in the propagation argument and w ould weaken the overall\nconclusion.\nProof.Define\n/tildewideq0=/tildewidez(ξ)\nξ2−h2β.\nNoting that/tildewideq0∈S0\n0sinceξ >1 on supp/tildewidezandh2β <1. Now set/tildewiderQ0= Op(/tildewideq0) and let\np0=ξ2−h2β,P0=−h2∂2\nx−h2β. Sinceq0andp0both depend only on ξtheir composition\nis exact\n/tildewiderQ0P0= Op(/tildewideq0p0) =/tildewideZ.\nNow sinceP0+ih2−γW=P\n/tildewideZu=/tildewiderQ0Pu−/tildewiderQ0(ih2−γWu)\n=h2/tildewiderQ0f−/tildewiderQ0(ih2−γWu).\nTake theL2norm squared of both sides then use that /tildewiderQ0is bounded on L2by Lemma A.2\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideZu/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤h4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewiderQ0f/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2+h4−2γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewiderQ0Wu/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2\n≤Ch4||f||2\nL2+Ch4−2γ||Wu||2\nL2.\nFinally use that W2≤CWand (9), the damped region estimate, to obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideZu/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Ch4||f||2\nL2+Ch4−γ||f||L2||u||L2.\n215 Proof of propagating region estimate Lemma 3.4\nWith the elliptic and damped region estimates proved, it rem ains to prove the estimate for\nthe propagating region, that is Lemma 3.4. The plan for this s ection is as follows: first the\ncomputation of a commutator in two different ways, second the e stimation of terms in the\ncomputation using expansions of compositions of pseudodiffe rential operators.\nProof of Lemma 3.4. Seta=xχ(x)(ξhτ−1)ψ(ξhτ−1)andA= Op(a). NotethatbyLemma\nA.2,Ais bounded on L2. To begin, compute h1−τ(AP−P∗A) in two different ways\nh1−τ/angbracketleftbig\n[h2∂2\nx,A]u,u/angbracketrightbig\n+ih3−γ−τ/a\\}b∇ack⌉tl⌉{t(AW+WA)u,u/a\\}b∇ack⌉t∇i}ht=h1−τ/a\\}b∇ack⌉tl⌉{t(AP−P∗A)u,u/a\\}b∇ack⌉t∇i}ht= 2ih3−τIm/a\\}b∇ack⌉tl⌉{tf,Au/a\\}b∇ack⌉t∇i}ht.\n(16)\nThis equation is the basis of the proof. The right hand side is a term of the form\nCh3−τ||f||L2||u||L2, which is the primary term in the estimate.\nOnthelefthandside h3−γ−τ(AW+WA)will producea h3−τ||f||L2||u||L2andtwoerror\nterms:h2−γ+6τ||f||2\nL2ando(h3)||u||2\nL2. Theh1−τ[h2∂2\nx,A] term will provide the h3βJu\nterm as well as another h3−τ||f||L2||u||L2and error terms o(h3)||u||2\nL2andh2−γ+6τ||f||2\nL2.\nNote that most of these terms have a common factor of h3which will be divided out\nin order to obtain the final conclusion. Because of this throu ghout the section error terms\nmust be of size o(h3) to be negligible.\nI will firstcomputethe AWand[h2∂2\nx,A] terms and thenusethem to prove Lemma 3.4.\nSubsection 5.1 estimates the damping anti-commutator ( AW+WA), subsection 5.2 esti-\nmates theLaplacian commutator [ h2∂2\nx,A] andsubsection 5.3 synthesizes these tocomplete\nthe proof of Lemma 3.4.\nRemark In this section I write\na(j)(ξhτ−1) =hj(1−τ)∂j\nξ(a(x,ξhτ−1)) =hj(1−τ)xχ(x)∂j\nξ((ξhτ−1)ψ(ξhτ−1)).\nNote thata(j)∈S0\n1−τ(T∗S1), see Appendix A for the definition of Sm\nρ(T∗S1). The utility\nof this notation is that hj∂j\nξa=hjτa(j), which simplifies composition expansions. This\nagrees with the standard usage of the notation: if ψ(k)(ξhτ−1) is thekth derivative of ψ\nevaluated at ξh1−τthenhk(1−τ)∂k\nξ(ψ(ξhτ−1)) =ψ(k)(ξhτ−1).\nAlso in this section, recall that there is a fixed ε2>0 and it is assumed that h2ε2<\nh2β < h2−2τ. This assumption is needed in order to apply the elliptic est imate (Lemma\n3.2) in order to control the size of error terms.\n5.1 Damping Anti-commutator estimate\nInordertoestimate h3−γ−τ(AW+WA)Iwillwriteitas h3−γ−τvjAvjpluserrorterms. The\nh3−γ−τvjAvjtermcanbecontrolledusingthedampingestimateandisofsi zeh3−τ||f||L2||u||L2.\nThe error terms are either of size o(h3),which is small enough to be negligible or are\n22supported on the elliptic set of Pand can be further controlled by the elliptic esti-\nmate and Lemma A.7. The terms controlled with the elliptic es timate will produce the\nh2−γ+6τ||f||2\nL2. In particular in this subsection I will show\nh3−γ−τ|/a\\}b∇ack⌉tl⌉{t(AW+WA)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3−τ||f||L2||u||L2+Ch2+6τ−γ||f||2\nL2+o(h3)||u||2\nL2.(17)\nTo begin recall that W=/summationtextv2\njso\nh3−γ−τ(AW+WA) =h3−γ−τ/summationdisplay\njv2\njA+Av2\nj\n=/summationdisplay\njh3−γ−τvjAvj+/summationdisplay\njh2−γ[[h1−τA,vj],vj].\nTo control the first term use Lemma A.2 to see Ais bounded on L2\nh3−γ−τ/summationdisplay\nj|/a\\}b∇ack⌉tl⌉{tvjAvju,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3−γ−τ/summationdisplay\nj||vju||2\nL2.\nThen since v2\nj≤/summationtext\njv2\nj=Wuse (9)\nCh3−γ−τ/summationdisplay\nj||vju||2\nL2=Ch3−γ−τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Ch3−τ||f||L2||u||L2.\nCombining these inequalities gives\nh3−γ−τ|/a\\}b∇ack⌉tl⌉{t(AW+WA)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3−τ||f||L2||u||L2+/summationdisplay\njh2−γ|/angbracketleftbig\n[[h1−τA,vj],vj]u,u/angbracketrightbig\n|.\n(18)\nThe sum of double commutators will be error terms. Its size ca n be controlled using the\nelliptic estimate, to do so the double commutator must first b e computed.\nLemma 5.1. Ifvj∈Wk0,∞andτ∈[τmin,1], then\n[[h1−τA,vj],vj] =k0−6/summationdisplay\nl=1k0−6/summationdisplay\nk=1im+l\n2k+lk!l!hτ(k+l)+1−τ(1−(−1)k)(1−(−1)l)Op(∂k\nxvj∂l\nxvja(k+l))+o(h3)\nBefore proving this in the finite regularity case it is useful outline the proof when vj\nis smooth, as the argument is simpler but has the same structu re. FixM >⌈2\nτ+1⌉and\napply Lemma A.1 to compute the commutator\n[h1−τA,vj] =h1−τM−1/summationdisplay\nk=0(ih)k\n2kk!(1−(−1)k)Op(∂k\nxvj∂k\nξa)+OL2→L2(hMτ+1−τ).\n23Then apply Lemma A.1 again, to compute the double commutator\n[[h1−τA,vj],vj] =h1−τM−1/summationdisplay\nl=0M−1/summationdisplay\nk=0(ih)k+l\n2k+lk!l!(1−(−1)k)(1−(−1)l)Op(∂k\nxvj∂l\nxvj∂k+l\nξa)+OL2→L2(hMτ+1−τ).\nSinceM >⌈2\nτ+1⌉,Mτ+1−τ >3. Therefore\n[[h1−τA,vj],vj] =h1−τM−1/summationdisplay\nl=0M−1/summationdisplay\nk=0(ih)(k+l)\n2k+lk!l!(1−(−1)k)(1−(−1)l)Op(∂k\nxvj∂l\nxvj∂k+l\nξa)+o(h3).\nThe terms with l= 0 ork= 0 drop out because of the factors 1 −(−1)kor 1−(−1)l.\nThe final step is to substitute hk+l∂k+l\nξa=hτ(k+l)a(k+l), which gives an expansion of the\ndesired form.\nWhenWis not smooth there are two changes: the remainder term is lar ger and addi-\ntional care must be taken to track the exact number of derivat ives used. The computation\nof the size of the remainders in this proof are the reason τmin>7\nk0−1is required. Other\nremainder size calculations in this section involving τmake use of this relationship between\nτminandk0but are not sharp on it.\nProof of Lemma 5.1. Sincevj∈Wk0,∞usepart2ofLemmaA.6(with N=k0)tocompute\nthe commutator\n[h1−τA,vj] =k0−6/summationdisplay\nk=0(ih)k\n2kk!h1−τ(1−(−1)k)Op(∂k\nxvj∂k\nξa)+OL2→L2(h(k0−5)τ−5(1−τ)+1−τ).\nThere are two key differences between this and the smooth case. The expansion can only\nbe taken to the term k0−6 and the remainder term has an additional h−5(1−τ). These two\nchanges are connected; in order to show that the remainder te rm is a bounded operator\nonL2the symbol must be in W5,∞. Those derivatives of the symbol appear in the L2\noperator norm of the remainder and each ξderivative of aproduces a factor h1−τ. See\nAppendix A for a more detailed proof and discussion.\nThe relationship between the regularity of the damping, k0,andτguarantees that the\nremainder term is o(h3). In particular since τ≥τmin>7\nk0−1.\n(k0−5)τ−5(1−τ)+1−τ= (k0−5)τ−4(1−τ) = (k0−1)τ−4>(k0−1)7\nk0−1−4 = 3.(19)\nNow replacing hk∂k\nξa=hτka(k)gives\n[h1−τA,vj] =k0−6/summationdisplay\nk=0ikhkτ\n2kk!(1−(−1)k)h1−τOp(∂k\nxvja(k))+o(h3).\n24To finish computing [[ A,vj],vj] Lemma A.6 will be applied again, paying special attention\nto the terms in the sum that are not o(h3). In particular these terms will be supported on\nthe elliptic set of Pand so can be further estimated. However, these terms have de rivatives\nofvjin them and so their regularity must be carefully tracked.\nSincevj∈Wk0,∞and∂k\nxvja(k)∈Wk0−kSρ,I can apply Lemma A.6 part 1 with N=k0\n(ask0−k≥6>5) and obtain\nh1−τhkτ[Op(∂k\nxvja(k)),vj] =k0−6/summationdisplay\nl=0il\n2ll!h1−τhτ(k+l)(1−(−1)l)Op(∂k\nxvj∂l\nxvja(k+l))\n+OL2→L2(h(k0−5)τ−5(1−τ)+1−τ+kτ).\nWhere I have replaced hτkhl∂l\nξa(k)=hτ(k+l)a(k+l). The remainder term is of size ( k0−5+\nk)τ−5(1−τ)+1−τ. Sincek≥0, by the same argument as above, the remainder term\niso(h3).\nSo combining\nh1−τ[[A,vj],vj] =k0−6/summationdisplay\nk=0ik\n2kk!(1−(−1)k)h1−τhkτ[Op(∂k\nxvja(k)),vj]+o(h3)\n=k0−6/summationdisplay\nk=0k0−6/summationdisplay\nl=0ik+l\n2k+lk!l!(1−(−1)k)(1−(−1)l)h1−τh(k+l)Op(∂k\nxvj∂l\nxvja(k+l))+o(h3).\nThe terms with k= 0 orl= 0 again vanish because of the factors 1 −(−1)kand 1−(−1)l.\nThis gives the formula in the statement of the lemma.\nNow the size of [[ A,vj],vj] can be further controlled since Op( ∂k\nxvj∂l\nxvja(k+l)) has sup-\nport contained in {2h1−τ<|ξ|<3h1−τ}, which is contained in the elliptic set of P.\nBecause of this these terms can be further estimated using th e elliptic estimate.\nLemma 5.2. Ifvj∈Wk0,∞andτ∈[τmin,1]then\n|/angbracketleftbig\n[[h1−τA,vj],vj]u,u/angbracketrightbig\n| ≤Ch6τ||f||2\nL2+o(h3)||u||2\nL2. (20)\nProof.For this proof I will use the notation bk,l=∂k\nxvj∂l\nxvja(k+l). In this notation Lemma\n5.1 is\n[[h1−τA,vj],vj] =k0−6/summationdisplay\nk=1k0−6/summationdisplay\nl=1Ck,lhτ(k+l)h1−τOp(bk,l)+o(h3).\nThe elliptic estimate as written can’t be applied to Op( bk,l) directly because the symbol is\nnot smooth. Instead, I reintroduce the operator Zdefined in Lemma 3.2.\n25The key property here is that Z≡1 on{1.5h1−τ<|ξ|<2} ⊃supp (bk,l). This\nalong with bk,l∈Wk0−max(k,l)S1−τ(T∗S1), means Lemma A.7 can be applied with N=\nk0−max(k,l)\nOp(bk,l) =ZOp(bk,l)Z+OL2→L2(hτ(k0−max(k,l))−5).\nThat is conjugating Op( bk,l) byZis Op(bk,l) modulo an error term. Note that there is less\nregularity for larger l,kand so the error term is larger. However after reintroducing the\nh1−τh(k+l)τfrom the sum the error terms can be uniformly controlled\nh1−τh(k+l)τOp(bk,l) =h1−τh(k+l)τZOp(bk,l)Z+OL2→L2(h(k0−max(k,l))τ−5hτ(k+l)h1−τ).\nIn particular since τ≥τmin>7\nk0−1andk,l≥1\n(k0−max(k,l))τ−5+τ(k+l)+1−τ= (k0+min(k,l)−1)τ−4>7\nττ−4+τ= 3+τ,\nand the error term is o(h3).\nTherefore\nh1−τh(k+l)τOp(bk,l) =h1−τh(k+l)τZOp(bk,l)Z+oL2→L2(h3). (21)\nNow, apply (21) term by term to Lemma 5.1\n|/angbracketleftbig\n[[h1−τA,vj],vj]u,u/angbracketrightbig\n| ≤ |/angbracketleftbig\nZh1−τ[[A,vj],vj]Zu,u/angbracketrightbig\n|+o(h3)||u||2\nL2.\nThen use the self-adjointness of Zand the H¨ older inequality to write\n|/angbracketleftbig\nZh1−τ[[A,vj],vj]Zu,u/angbracketrightbig\n| ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleh1−τ[[A,vj],vj]Zu/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2||Zu||L2.\nNow note that for k,l≥1,\nh1−τh(k+l)τOp(bk,l) =OL2→L2(h1+τ),\nandh1−τ[[A,vj],vj] =/summationtext\nk,l≥1h1−τh(k+l)τOp(bk,l)+o(h3) =OL2→L2(h1+τ). Therefore\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleh1−τ[[A,vj],vj]Zu/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2||Zu||L2≤Ch1+τ||Zu||2\nL2.\nNow apply the elliptic estimate Lemma 3.2 to Zuto see\nCh1+τ||Zu||2\nL2≤Ch6τ||f||2\nL2+Ch1+τ+2||u||2\nL2.\nCombining this chain of inequalities gives (20).\nSo now use (20) to estimate the sum of double commutators in (1 8)\nh3−γ−τ|/a\\}b∇ack⌉tl⌉{t(AW+WA)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3−τ||f||L2||u||L2+Ch2+6τ−γ||f||2\nL2+o(h3)||u||2\nL2,\nwhich is exactly the desired inequality (17).\n265.2 Commutator Estimate of Aandh2∂2\nx\nThe Laplacian commutator estimate follows from writing [ h2∂2\nx,A] as a sum of a cutoff\nversion ofP,J= Op(χ1/2(x)ψ1/2(ξhτ−1)) and error terms. The error terms are supported\nin the elliptic or damped set and can be further estimated.\nIn particular in this subsection I will show\n2h3β||Ju||2\nL2≤/vextendsingle/vextendsingle/angbracketleftbig\n[h2∂2\nx,A]u,u/angbracketrightbig/vextendsingle/vextendsingle+Ch3−τ||f||L2||u||L2+Ch2+6τ−γ||f||2\nL2(22)\n+(Ch3+2τβ+o(h3))||u||2\nL2.\nTo begin note by Lemma A.1\n[h2∂2\nx,h1−τA] =2/summationdisplay\nk=0(ih)k\n2kk!(1−(−1)k)Op(∂k\nx(xχ(x))ξψ(ξhτ−1)∂k\nξξ2)\n= 2ihOp((xχ′+χ)ξ2ψ(ξh1−τ)),\nwhere there are no terms for k≥3 since∂k\nξξ2= 0 fork≥3, and thek= 0,2 terms cancel\nbecause 1 −(−1)k= 0 then.\nNow recall that p=ξ2+ih2−γW−h2βso\n2ih(xχ′+χ)ξ2ψ(ξhτ−1) = 2ihxχ′ξ2ψ+2ihχψ(ξhτ−1)(p−ih2−γW+h2β).(23)\nEach of the terms on the right hand side will be estimated in tu rn. Theh3βterm will\nproduce the h3J, the remaining terms produce errors.\n5.2.1 Estimate of Op (ξ2ψxχ′)\nTo estimate Op( ξ2ψxχ′) it is enough to use that χ′is supported only inside the damped\nset and apply (9). In order to do so the ξdependency of the operator must be eliminated,\nto do so the approach of Lemma 5.2 is adapted. Lemma A.7 is stil l used, but the localizing\nfunction now depends on x,andξ2ψxχ′is smooth in xso the error term is O(h∞).\nIn particular, define s∈C∞(−π,π) with\ns=/braceleftigg\n0|x|<σ\n1σ+σ1/2<|x|,\nand letSbe the operator of multiplication by s. Note supp ( χ′)⊂ {σ+σ1/2<|x|}so\ns≡1 on supp ( ξ2ψxχ′). Then by Lemma A.7 (since χ′∈C∞)\n/vextendsingle/vextendsingle/angbracketleftbig\nOp(ξ2ψ(ξhτ−1)xχ′(x))u,u/angbracketrightbig/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/angbracketleftbig\nSOp(ξ2ψ(ξhτ−1)xχ′(x))Su,u/angbracketrightbig/vextendsingle/vextendsingle+O(h∞)||u||2\nL2.\n27Use thatSis self adjoint and the H¨ older inequality to write\n/vextendsingle/vextendsingle/angbracketleftbig\nSOp(ξ2ψ(ξhτ−1)xχ′(x))Su,u/angbracketrightbig/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/angbracketleftbig\nOp(ξ2ψ(ξhτ−1)xχ′(x))Su,Su/angbracketrightbig/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleOp(ξ2ψ(ξhτ−1)xχ′(x))Su/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2||Su||L2.\nBy Lemma A.2, h2τ−2Op(ξ2ψ(ξhτ−1)xχ′(x)) is bounded on L2, so\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleOp(ξ2ψ(ξhτ−1)xχ′(x))Su/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2||Su||L2≤Ch2−2τ||Su||2\nL2.\nThen since s≤CW1/2and applying (9)\nh2−2τ||Su||2\nL2≤Ch2−2τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Ch2−2τ+γ||f||L2||u||L2.\nCombining this chain of inequalities and multiplying both s ides byhgives\n2h/vextendsingle/vextendsingle/angbracketleftbig\nOp(ξ2ψ(ξhτ−1)xχ′(x))u,u/angbracketrightbig/vextendsingle/vextendsingle≤Ch3−2τ+γ||f||L2||u||L2+O(h∞)||u||2\nL2.(24)\n5.2.2 Estimate of Op (h2βχψ)\nTo estimate Op( χψ) write it as J2plus an error term. By Lemma A.1\nOp(J)Op(J) = Op(χ1/2ψ1/2)Op(χ1/2ψ1/2) = Op(χψ)−h2τOp(r1),\nwherer1∈S0\n1−τ. Using that Jis self adjoint and Op( r1) is bounded on L2by Lemma A.2\n||Ju||2\nL2=/vextendsingle/vextendsingle/vextendsingle/angbracketleftig\nOp(χ1/2ψ1/2)Op(χ1/2ψ1/2)u,u/angbracketrightig/vextendsingle/vextendsingle/vextendsingle≤ |/a\\}b∇ack⌉tl⌉{tOp(χψ)u,u/a\\}b∇ack⌉t∇i}ht|+h2τ/a\\}b∇ack⌉tl⌉{tOp(r1)u,u/a\\}b∇ack⌉t∇i}ht\n≤ |/a\\}b∇ack⌉tl⌉{tOp(χψ)u,u/a\\}b∇ack⌉t∇i}ht|+Ch2τ||u||2\nL2.\nTherefore, multiplying through by h3β\nh3β|/a\\}b∇ack⌉tl⌉{tOp(χψ)u,u/a\\}b∇ack⌉t∇i}ht| ≥h3β||Ju||2\nL2−Cβh3+2τ||u||2\nL2. (25)\n5.2.3 Estimate of h Op (χψp)\nToestimate Op( χψp), writeit as Op( χψ)Ppluserrorterms. Theerrortermsaresupported\non the elliptic set of Por the damped region and are further estimated using Lemma 3. 2\nor (9) respectively. In particular the following inequalit y will be shown.\nh|/a\\}b∇ack⌉tl⌉{tOp(χψp)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3||f||L2||u||L2+Ch2+6τ−γ||f||2\nL2+o(h3)||u||2\nL2.(26)\nNote this term appears in (23) as hOp(χψ)p, but to simplify notation this extra factor of\nhis not carried through the intermediate calculations. Beca use of this remainders of size\no(h2) are acceptably small, instead of the o(h3) of other calculations.\n28To begin, Op( χψ)Pis computed, where special care must be taken with the regula rity\nof theWterms. Since W∈Wk0,∞by part 2 of Lemmas A.6 and A.1 (Lemma A.1 is used\nto compose Op( χψ) and−h2∂2\nx, as Lemma A.6 requires symbols to be bounded).\nOp(χψ)P= Op(χψ)(−h2∂2\nx+ih2−γW−h2β)\n= Op(χψp)+k0−6/summationdisplay\nk=1(ih)k\n2kk!Op/parenleftigg\n(∂y∂ξ−∂x∂η)kχ(x)ψ(ξh1−τ)(η2+ih2−γW(y))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ/parenrightigg\n+OL2→L2(h(k0−5)τ−5(1−τ)).\nAs in Lemma 5.1 since τ∈[τmin,1] withτmin>7\nk0−1\n(k0−5)τ−5(1−τ) =k0τ−5>7\nk0−1k0−5>2,\nwhich guarantees that the remainder term is of size o(h2).\nThe sum splits into two separate sums, where the first is only t aken tok= 2 because\n∂k\nξξ2= 0 fork≥3.\nOp(χψ)P= Op(χψp)+2/summationdisplay\nk=1(ih)k\n2kk!(−1)kOp/parenleftig\n∂k\nxχ(x)ψ(ξhτ−1)∂k\nξξ2/parenrightig\n+ih2−γk0−6/summationdisplay\nk=1(ih)k\n2kk!Op(χ(x)∂k\nξψ(ξh1−τ)∂k\nxW)+oL2→L2(h2)\n= Op(χψp)−Op(ihχ′ψξ+h2\n4χ′′ψ)+ih2−γk0−6/summationdisplay\nk=1ik\n2kk!hτkOp(χψ(k)∂k\nxW)+oL2→L2(h2).\n(27)\nThe two operators and the sum will each be estimated individu ally. The Op( χψ)Pterm\nis straightforward to control. The second term is supported inside the damped set and is\ncontrolled as in subsection 5.2.1. The sum will be controlle d by the elliptic estimate using\nthe same argument as Lemma 5.2.\nTo begin, using the boundedness of Op( χψ) onL2and thatPu=h2f\n|/a\\}b∇ack⌉tl⌉{tOp(χψ)Pu,u/a\\}b∇ack⌉t∇i}ht| ≤Ch2||f||L2||u||L2. (28)\nFor the second term, set g(x,ξ) =ihχ′ψξ+h2\n4χ′′ψ. Note that χ′andχ′′are supported\ninside the damping set and so an argument as in subsection 5.2 .1 will give an improvement.\nRecalls∈C∞(−π,π)\ns(x) =/braceleftigg\n0|x|<σ\n1σ+σ1/2<|x|,\n29andSis the operator of multiplication by s. Sinceg∈C∞ands≡1 on suppg, by Lemma\nA.7\n|/a\\}b∇ack⌉tl⌉{tOp(g)u,u/a\\}b∇ack⌉t∇i}ht|=|/a\\}b∇ack⌉tl⌉{tSOp(g)Su,u/a\\}b∇ack⌉t∇i}ht|+O(h∞)||u||2\nL2\nUsing that Sis self adjoint, along with the H¨ older inequality\n|/a\\}b∇ack⌉tl⌉{tSOp(g)Su,u/a\\}b∇ack⌉t∇i}ht| ≤ ||Op(g)Su||L2||Su||L2.\nNow by Lemma A.2, hτ−2Op(g) is bounded on L2, so\n||Op(g)Su||L2||Su||L2≤h2−τ||Su||2\nL2.\nThen since s≤CW1/2and applying (9)\nh2−τ||Su||2\nL2≤Ch2−τ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Ch2−τ+γ||f||L2||u||L2.\nCombining this chain of inequalities gives\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\nOp(ihχ′ψξ+h2\n4χ′′ψ)u,u/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ch2−τ+γ||f||L2||u||L2+O(h∞)||u||2\nL2.(29)\nNow to estimate the sum, note that χψ(k)∂k\nxWis supported in {2h1−τ<|ξ|<3h1−τ}\nwhich is contained in the elliptic set. The proof of Lemma 5.2 will be imitated. Conjugate\ntheχψ(k)∂k\nxWterms in the sum in (27) by Zto take advantage of the location of their\nsupport. Once again care is taken with the regularity of ∂kWwhen applying Lemma A.7.\nSetbk(x,ξ) =χ(x)ψ(k)(ξh1−τ)∂k\nxW(x). RecallZfrom Lemma 3.2. Since z≡1 on\nsupp (bk) and∂k\nxW∈Wk0−k,∞,Lemma A.7 can be applied with N=k0−k\nOp(bk) =ZOp(bk)Z+OL2→L2(h(k0−k)τ−5).\nSo conjugating Op( bk) byZis Op(bk) modulo an error term. Once again terms with larger\nkhave less regularity and have larger error terms. However, a s before, reintroducing the\nhτkfrom the sum improves the error terms\nhτkOp(bk) =hτkZOp(bk)Z+hτkOL2→L2(h(k0−k)τ−5).\nIn particular, and as in Lemma 5.2, the error term is o(h2) becauseτ≥τmin>7\nk0−1and\nτk+(k0−k)τ−5 =k0τ−5>/parenleftbigg7\nk0−1/parenrightbigg\nk0−5>2.\nSo /vextendsingle/vextendsingle/vextendsingle/angbracketleftig\nhτkOp(bk)u,u/angbracketrightig/vextendsingle/vextendsingle/vextendsingle≤hτk|/a\\}b∇ack⌉tl⌉{tZOp(bk)Zu,u/a\\}b∇ack⌉t∇i}ht|+o(h2)||u||2\nL2.\n30Continuing to follow the proof of Lemma 5.2, use the self adjo intness ofZand the H¨ older\ninequality to write\nhτk|/a\\}b∇ack⌉tl⌉{tZOp(bk)Zu,u/a\\}b∇ack⌉t∇i}ht| ≤hτk||Op(bk)Zu||L2||Zu||L2.\nNow by Lemma A.2, Op( bk) is bounded on L2\nhτk||Op(bk)Zu||L2||Zu||L2≤hτk||Zu||2\nL2.\nThen apply the elliptic estimate, Lemma 3.2, to Zu\nhτk||Zu||2\nL2≤Ch5τ+kτ−1||f||2\nL2+o(h2)||u||2\nL2.\nCombining this chain of inequalities and multiplying both s ides byh2−γgives\nh2−γ/vextendsingle/vextendsingle/vextendsingle/angbracketleftig\nhτkOp(bk)u,u/angbracketrightig/vextendsingle/vextendsingle/vextendsingle≤Ch1−γ+(5+k)τ||f||2\nL2+o(h2)||u||2\nL2≤Ch1−γ+6τ||f||2\nL2+o(h2)||u||2\nL2.\n(30)\nWhere the second inequality follows since k≥1.\nTherefore using (28), (29) and (30) to estimate terms in (27)\n|/a\\}b∇ack⌉tl⌉{tOp(χψp)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch2||f||L2||u||L2+Ch2−τ+γ||f||L2||u||L2+Ch1−γ+6τ||f||2\nL2+o(h2)||u||2\nL2.\nMultiplying both sides by hand using that γ−τ≥0 (sinceτ∈(1/2,1] andγ∈ {1,2})\ngives the desired inequality\nh|/a\\}b∇ack⌉tl⌉{tOp(χψp)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3||f||L2||u||L2+Ch2+6τ−γ||f||2\nL2+o(h3)||u||2\nL2.\n5.2.4 Estimate of Op (χψW)\nTo estimate Op( χψW) I write it as vjOp(χψ)vjplus error terms. The vjOp(χψ)vjterms\nare controlled by the damped region estimate (9). The error t erms are either small or\nare supported on the elliptic set of Pand can be further estimated using Lemma 3.2. In\nparticular the following inequality will be shown\nh3−γ|/a\\}b∇ack⌉tl⌉{tOp(χψW)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3||f||L2||u||L2+Ch2+7τ−γ||f||2\nL2+o(h3)||u||2\nL2.(31)\nNote this term appears in (23) as h2−γhOp(χψW), but to simplify notation these extra fac-\ntors ofhare not carried through the intermediate calculations. Bec ause of this remainders\nof sizeo(h2) are acceptably small, instead of the o(h3) of other calculations.\nTo begin recall that W=/summationtextv2\njand so\nOp(χψW) = Op/parenleftig\nχψ/summationdisplay\nv2\nj/parenrightig\n=/summationdisplay\nOp(χψv2\nj).\nThis is exactly the principal symbol of/summationtextvjOp(χψ)vj, an expansion of which will now be\ncomputed.\n31First, since vj∈Wk0,∞apply part 2 of Lemma A.6 with /tildewideN=k0−5, to obtain\nOp(χψ)vj=k0−6/summationdisplay\nk=0(ih)k\n2kk!Op(χ∂k\nξψ(ξhτ−1)∂k\nxvj)+OL2→L2(h(k0−5)τ−5(1−τ)).\nAs in subsection 5.2.3, since τ≥τmin>7\nk0−1the remainder term is o(h2). In particular\n(k0−5)τ−5(1−τ) =k0τ−5>/parenleftbigg7\nk0−1/parenrightbigg\nk0−5>2.\nReplacinghk∂k\nξψ(ξhτ−1) =hkτψ(k)(ξhτ−1) gives\nOp(χψ)vj=k0−6/summationdisplay\nk=0ikhτk\n2kk!Op(χψ(k)∂k\nxvj)+oL2→L2(h2).\nNow compute the following composition of vjand Op(χψ(k)∂k\nxvj), adjusting the number\nof terms taken in the expansion based on how many derivatives fallen on∂k\nxvj.\nvjOp(χψ)vj=vjk0−6/summationdisplay\nk=0ik\n2kk!hτkOp(χψ(k)∂k\nxvj)+oL2→L2(h2). (32)\nIn particular, since vj∈Wk0,∞andχ∂k\nxvjψ(k)∈Wk0−kS1−τ, apply part 1 of Lemma\nA.6 with/tildewideN=k0−5−kandk0−k≥5.\nik\n2kk!vjhτkOp(∂k\nxvjψ(k)χ) =k0−k−6/summationdisplay\nl=0(ih)l\n2ll!ik\n2kk!hτk(−1)lOp(χ∂l\nxvj∂k\nxvj∂l\nξψ(k))+hτkOL2→L2(h(k0−k−5)τ−5(1−τ)).\nAlthough there are fewer terms taken in the expansion for lar ger values of k, the\nadditionalhτkensures that the remainder term is o(h2). In particular\nτk+(k0−k−5)τ−5(1−τ) =k0τ−5>k0/parenleftbigg7\nk0−1/parenrightbigg\n−5>2.\nTherefore\nik\n2kk!vjhτkOp(∂k\nxvjψ(k)χ) =k0−k−6/summationdisplay\nl=0hτ(l+k)\n2l+kl!k!ik+l(−1)lOp(∂l\nxvj∂k\nxvjψ(k+l)χ)+oL2→L2(h2).\nNow plug this into (32) to obtain\nvjOp(χψ)vj=k0−6/summationdisplay\nk=0/parenleftiggk0−k−6/summationdisplay\nl=0i(k+l)\n2l+kl!k!hτ(l+k)(−1)lOp(∂l\nxvj∂k\nxvjψ(k+l)χ)/parenrightigg\n+oL2→L2(h2).\n32Note that the k= 0,l= 1 term and k= 1,l= 0 term are identical except for a minus sign\nand cancel. There are more cancellations which occur in the s um, but only this first one is\nnecessary for the proof. Note also that k+l≤k+k0−k−6 =k0−6,\nvjOp(χψ)vj= Op(v2\njχψ)+k+l≤k0−6/summationdisplay\nk,l=1i(k+l)\n2l+kl!k!hτ(l+k)(−1)lOp(∂l\nxvj∂k\nxvjψ(k+l)χ)+oL2→L2(h2).\n(33)\nIn order to further control the size of the terms in this sum th e technique from Lemma 5.2\nis used. Let/tildewiderbk,l=∂k\nxvj∂l\nxvjψ(k+l)χ. Fork,l≥1,/tildewiderbk,lhas support contained in the elliptic\nset which can be made use of byy conjugating by Zas in Lemma 5.2 and then applying\nthe elliptic estimate to Zu. The proof is almost identical to Lemma 5.2, but is written he re\nfor exactness.\nAs in Lemma 5.2, recall Zfrom Lemma 3.2. Note z≡1 on supp/tildewiderbk,land/tildewiderbk,l∈\nWk0−max(k,l)S1−τ(T∗S1), so Lemma A.7 with N=k0−max(k,l) gives\nOp(/tildewiderbk,l) =ZOp(/tildewiderbk,l)Z+OL2→L2(hτ(k0−max(k,l))−5).\nNote that there is less regularity for larger l,kand so the error term is larger. However\nafter reintroducing the h(k+l)τfrom the sum the error terms are improved\nh(k+l)τOp(bk,l) =h(k+l)τZOp(bk,l)Z+OL2→L2(h(k0−max(k,l))τ−5hτ(k+l)).\nIn particular since τ≥τmin>7\nk0−1andk,l≥1\n(k0−max(k,l))τ−5+τ(k+l) = (k0+min(k,l))τ−5>k0/parenleftbigg7\nk0−1/parenrightbigg\n−5>2.\nTherefore\nh(k+l)τOp(bk,l) =h(k+l)τZOp(bk,l)Z+oL2→L2(h2). (34)\nNow, apply (34) term by term to /tildewideb=/summationtextk+l≤k0−6\nk,l=1Ck,lhτ(l+k)/tildewiderbk,l, the sum in (33)\n|/angbracketleftig\nOp(/tildewideb)u,u/angbracketrightig\n| ≤ |/angbracketleftig\nZOp(/tildewideb)Zu,u/angbracketrightig\n|+o(h2)||u||2\nL2.\nThen use the self-adjointness of Zand the H¨ older inequality to write\n|/angbracketleftig\nOp(/tildewideb)u,u/angbracketrightig\n| ≤ |/angbracketleftig\nZOp(/tildewide(b))Zu,u/angbracketrightig\n| ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleOp(/tildewide(b))Zu/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2||Zu||L2.\nNow note, since /tildewiderbk,lis bounded on L2by Lemma A.2, and k,l≥1,\nh(k+l)τOp(/tildewiderbk,l) =OL2→L2(h2τ),\n33so Op(/tildewideb) =OL2→L2(h2τ) and\n|/angbracketleftig\nOp(/tildewideb)u,u/angbracketrightig\n| ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleOp(/tildewide(b))Zu/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL2||Zu||L2≤Ch2τ||Zu||2\nL2.\nNow apply the elliptic estimate Lemma 3.2 to Zuto see\n|/angbracketleftig\nOp(/tildewideb)u,u/angbracketrightig\n| ≤Ch2τ||Zu||2\nL2≤Ch7τ−1||f||2\nL2+Ch2+2τ||u||2\nL2. (35)\nNow these pieces will be combined to give the final estimate of Op(χψW). Recall that\nW=/summationtext\njv2\njso\n|/a\\}b∇ack⌉tl⌉{tOp(χψW)u,u/a\\}b∇ack⌉t∇i}ht| ≤/summationdisplay\nj|/angbracketleftbig\nOp(χψv2\nj)u,u/angbracketrightbig\n|.\nThe composition computation (33) gives\n/summationdisplay\nj|/angbracketleftbig\nOp(χψv2\nj)u,u/angbracketrightbig\n| ≤/summationdisplay\nj|/a\\}b∇ack⌉tl⌉{tvjOp(χψ)vju,u/a\\}b∇ack⌉t∇i}ht|+|/angbracketleftig\nOp(/tildewideb)u,u/angbracketrightig\n|+o(h2)||u||2\nL2.\nThen (35) gives\n/summationdisplay\nj|/angbracketleftbig\nOp(χψv2\nj)u,u/angbracketrightbig\n| ≤/summationdisplay\nj|/a\\}b∇ack⌉tl⌉{tvjOp(χψ)vju,u/a\\}b∇ack⌉t∇i}ht|+Ch7τ−1||f||2\nL2+o(h2)||u||2\nL2.(36)\nIt remains to control the vjOp(χψ)vjterms with the damping region estimate. Using that\nvjis a multiplier and thus is self-adjoint, as well as the H¨ old er inequality\n|/a\\}b∇ack⌉tl⌉{tvjOp(χψ)vju,u/a\\}b∇ack⌉t∇i}ht| ≤ ||Op(χψ)vju||L2||vju||L2.\nNow note Op( χψ) is bounded on L2by Lemma A.2 so\n||Op(χψ)vju||L2||vju||L2≤C||vju||2\nL2.\nAgain using that W=/summationtextv2\njsov2\nj≤W≤CW1/2and (9)\n||vju||2\nL2≤C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nL2≤Chγ||f||L2||u||L2.\nCombining this chain of inequalities and (36) gives\n|/a\\}b∇ack⌉tl⌉{tOp(χψW)/a\\}b∇ack⌉t∇i}ht| ≤Chγ||f||L2||u||L2+h7τ−1||f||2\nL2+o(h2)||u||2\nL2.\nFinally multiply both sides by h3−γto obtain the desired inequality\nh3−γ|/a\\}b∇ack⌉tl⌉{tOp(χψW)u,u/a\\}b∇ack⌉t∇i}ht| ≤Ch3||f||L2||u||L2+Ch2+7τ−γ||f||2\nL2+o(h3)||u||2\nL2.\n345.2.5 Combining Estimates\nNow use (24), (25), (26) and (31) to estimate terms in (23)\n2h3β||Ju||2\nL2≤h1−τ/vextendsingle/vextendsingle/angbracketleftbig\n[h2∂2\nx,A]u,u/angbracketrightbig/vextendsingle/vextendsingle+C(h3+h3−2τ+γ)||f||L2||u||L2\n+Ch2+6τ−γ||f||2\nL2+(Ch3+2τβ+o(h3))||u||2\nL2.\nUse thatγ−τ >0 to group the ||f||L2||u||L2terms to obtain the desired estimate (22)\n2h3β||Ju||2\nL2≤h1−τ/vextendsingle/vextendsingle/angbracketleftbig\n[h2∂2\nx,A]u,u/angbracketrightbig/vextendsingle/vextendsingle+Ch3−τ||f||L2||u||L2+Ch2+6τ−γ||f||2\nL2\n+(Ch3+2τβ+o(h3))||u||2\nL2.\n5.3 End of Proof of Lemma 3.4\nRecall (16) is\n2h3−τIm/a\\}b∇ack⌉tl⌉{tf,Au/a\\}b∇ack⌉t∇i}ht=h1−τ/angbracketleftbig\n[h2∂2\nx,A]u,u/angbracketrightbig\n+h3−γ−τ/a\\}b∇ack⌉tl⌉{t(AW+WA)u,u/a\\}b∇ack⌉t∇i}ht.\nNow apply (17), (22), to estimate the terms on the right hand s ide, and Lemma A.2 (to\nsee that ||Au||L2≤C||u||L2)\n2h3β||Ju||2\nL2≤Ch3−τ||f||L2||u||L2+Ch2+6τ−γ||f||2\nL2+C/parenleftbig\no(h3)+βh3+2τ/parenrightbig\n||u||2\nL2.\nDivide through by 2 h3βto obtain the desired estimate, which can be done since βis\nbounded away from 0.\nA Pseudodifferential Operators\nThis appendix contains the necessary background informati on on pseudodifferential oper-\nators, as well as a lemma calculating the size of errors from i ntroducing cutoff operators\nand a careful calculation of the regularity required to have remainder terms in composition\nexpansions bounded on L2.\nThis paper uses the semiclassical Weyl quantization, which takes in a function on T∗R\nand produces an operator Op( a) defined by\nOp(a)u(x) =1\n2πh/integraldisplay\nR/integraldisplay\nRei(x−y)ξ\nha/parenleftbiggx+y\n2,ξ/parenrightbigg\nu(y)dydξ. (37)\nOn the torus this formula still makes sense. A function a∈C∞(T∗S1) is equivalent to\na∈C∞(Rx×Rξ) periodic in the xvariable. It is straightforward to see that for such\na,Op(a) preserves the space of 2 πZperiodic distributions on Rand thus preserves D′(S1).\n35Definition 1. a(x,ξ;h)∈Sm\nρ(T∗S1) ifa∈C∞(T∗S1) and satisfies\nsup\nx,ξ|∂α\nx∂θ\nξa(x,ξ;h)| ≤Cαθh−ρ|θ|/a\\}b∇ack⌉tl⌉{tξ/a\\}b∇ack⌉t∇i}htm−|θ|. (38)\nNote that this definition is not the typical one for hdependent symbols. In particular\nonly derivatives in ξproduce unfavorable powers of h, derivatives in xdo not produce\nany. This structure would allow ρ≥1/2 (see [DZ16] section 3) corresponding to τ <1/2\nwhich would give an improved decay rate, however requiremen ts of the elliptic estimate\n(Proposition 3.2) prevent ρfrom being taken this large.\nThe following lemma gives the standard composition and adjo int formula for Sm\nρ(T∗S1)\nsymbols. It follows from Theorems 4.17 and 4.18 of [Zwo12].\nLemma A.1. Leta∈Sm\nρ(T∗S1),b∈Sm′\nρ(T∗S1)then\n1. Op(a)Op(b) =Op(a#b)wherea#b∈Sm+m′\nρ(T∗S1)and for each N\na#b(x,y;h) =N−1/summationdisplay\nk=0(ih)k\n2kk!(∂y∂ξ−∂x∂η)k(a(x,ξ;h)b(y,η;h))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ+OSm+m′\nρ(T∗S1)(h(N(1−ρ)).\n(39)\n2. Op(a)∗=Op(¯a),in particular real symbols have self-adjoint Weyl quantiza tion.\nThe following two definitions are finite regularity analogs o f Definition 1. In particular\nthey define two different symbol classes with a finite number of d erivatives in xand an\ninfinite number of derivatives in ξ. The first only produces unfavorable powers of hwhen\ndifferentiated in ξwhile the second producesunfavorable powers of hwhen differentiated in\nξandx. The notation is again somewhat unusual but is made this way t o mirror Definition\n1.\nDefinition 2. A distribution a∈WkSρ(T∗S1) if forα≤k,θ∈N\nsup\nx,ξ|∂α\nx∂θ\nξa| ≤Ch−ρθ/a\\}b∇ack⌉tl⌉{tξ/a\\}b∇ack⌉t∇i}ht−θ.\nA distribution a∈WkSρ,ρ(T∗S1) if forα≤k,θ∈N\nsup\nx,ξ|∂α\nx∂θ\nξa| ≤Ch−ρ(α+θ)/a\\}b∇ack⌉tl⌉{tξ/a\\}b∇ack⌉t∇i}ht−θ.\nThe following theorem gives a sufficient condition for a pseud odifferential operator to\nbe bounded on L2. It follows immediately from Theorem 1.2 of [Bou99].\n36Lemma A.2. There exists C >0such that for all b(x,ξ)∈ S′(T∗S1)\n||Op(b)||L(L2(S1))≤C/summationdisplay\nα,θ∈{0,1}hθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α\nx∂θ\nξb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞.\nIn particular if b∈W1Sρ(T∗S1)then Op(b)is bounded on L2.\nProof.So\nOp(b) = (2πh)−1/integraldisplay\nR×Rei(x−y)ξ\nhb/parenleftbiggx+y\n2,ξ/parenrightbigg\nv(y)dydξ\n= (2π)−1/integraldisplay\nR×Rei(x−y)ηb/parenleftbiggx+y\n2,ηh/parenrightbigg\nv(y)dydη= Op(b(·,hη))\nwhich by [Bou99] Theorem 1.2 has\n||Op(b)||L(L2(S1))≤C/summationdisplay\nα,θ∈{0,1}/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α\nx∂θ\nηb(x,hη)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞≤C/summationdisplay\nα,θ∈{0,1}hθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α\nx∂θ\nξb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞.\nIn order to prove composition results for finite regularity s ymbols I will make use of\nthe notation and results of [Sj¨ o95]\nDefinition 3. Lete1,...,embe a basis in Rnand Γ =/circleplustextn\n1Zej. Then letχ0∈ S(Rn) be\nsuch that 1 =/summationtext\nj∈Γχj(x) whereχj(x) =χ0(x−j) forj∈Γ. DefineSwas the space of\nu∈ S′(Rn) such that\nU(ξ) = sup\nj∈Γ|Fχju(ξ)| ∈L1(Rn).\nThenSwis a Banach space with the norm\n/ba∇⌈blu/ba∇⌈blΓ,χ0=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesup\nj∈Γ|Fχju|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL1.\nThe following L2boundedness result is from page 8 of [Sj¨ o95] .\nLemma A.3. Ifa∈Swthen Op(a)is bounded on L2and\n/ba∇⌈blOp(a)/ba∇⌈blL2→L2≤ ||a||Sw.\nIfkis taken large enough then WkSm(Rn) is contained in Sw(Rn). Note that this\nresult is stated on a more general space than T∗S1. This is because in the computation\nof an expansion of the composition of symbols a,bthere is an intermediate step where\nc(x,ξ,y,η) =a(x,ξ)b(y,η) is considered as a symbol on T∗S1×T∗S1, which can be thought\nof asR4.\n37Lemma A.4. Ifa∈WkSm\nρ(Rn)fork≥n+1thena∈Sw(Rn)and\n||a||Sw≤Csup\n|γ|≤n+1||∂γa||L∞.\nProof.Starting with the definition of ||·||Sw\n||a||Sw=/integraldisplay\nRnsup\nj∈Γ|F(χju)(ξ)|dξ\n=/integraldisplay\nRn/a\\}b∇ack⌉tl⌉{tξ/a\\}b∇ack⌉t∇i}htn+1/a\\}b∇ack⌉tl⌉{tξ/a\\}b∇ack⌉t∇i}ht−(n+1)sup\nj∈Γ|F(χju)(ξ)|dξ\n≤C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/a\\}b∇ack⌉tl⌉{tξ/a\\}b∇ack⌉t∇i}htn+1sup\nj∈Γ|F(χu)(ξ)|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞≤sup\n|α|≤n+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξαsup\nj∈Γ|F(χju)(ξ)|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞.\nWhere the integrability of /a\\}b∇ack⌉tl⌉{tξ/a\\}b∇ack⌉t∇i}ht−(n+1)onRngives the first inequality. Then\nsup\n|α|≤n+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξαsup\nj∈Γ|F(χju)(ξ)|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞≤sup\n|α|≤n+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesup\nj∈Γ|ξαF(χju)(ξ)|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞\n= sup\n|α|≤n+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesup\nj∈Γ|F(∂α(χju))(ξ)|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞.\nand\n|F(∂α(χju))(ξ)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\ne−ixξ∂α(χju)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\n|∂α(χju)|dx≤C||∂αu||L∞/integraldisplay\nχjdx.\nThe following lemma gives an exact calculation of the regula rity required to show the\nremainder term in a composition is bounded on L2. TheSwsymbol class is used here as\nit allows a more straightforward proof than proceeding dire ctly withWkSρsymbols.\nLemma A.5. Ifa,b∈ S′(R2n)with(∂y∂ξ−∂x∂η)Na(x,ξ)b(y,η)∈Sw(R4n)for some\nN∈NandQis a symmetric nonsingular matrix define\nRN(a,b)(x,ξ) =/integraldisplay1\n0(1−t)N−1eith/an}bracketle{tQD,D/an}bracketri}ht(∂y∂ξ−∂x∂η)N(a(x,ξ;h)b(y,η;h))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ.\nthen forhchosen small enough\n||RN(a,b)||Sw≤C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂y∂ξ−∂x∂η)N(a(x,ξ)b(y,η))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nSw. (40)\n38Therefore Op (RN)is bounded as an operator on L2(Rn)with\n||Op(RN)||L2→L2≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂y∂ξ−∂x∂η)N(a,b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nSw≤sup\n|γ|≤4n+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂γ(∂y∂ξ−∂x∂η)N(a,b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nL∞.\nProof.By [Sj¨ o95] Theorem 1.4 and equation (1.21) (pg. 7), for any ε >0 there exists\nh0>0 such that for h<h0\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0eith/an}bracketle{tQD,D/an}bracketri}ht(∂y∂ξ−∂x∂η)N(a,b)dt−C(∂y∂ξ−∂x∂η)N(a,b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nSw≤ε.\nThen (40) follows by choosing ε </vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂y∂ξ−∂x∂η)N(a,b)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nSwand using the fact that\nrestriction to a linear subspace (i.e. setting y=x,η=ξ) is bounded on Sw(bottom of\npage 2 in [Sj¨ o95]). The L2bound then follows by Lemmas A.3 and A.4, where |γ| ≤4n+1\nbecausea(x,ξ)b(y,η) is a function on R4n.\nThe following lemma is a composition expansion result for lo w regularity symbols. The\nexpansion is of the same form as Lemma A.1, but the remainder t erm is larger by a factor\nofh−5ρ.\nLemma A.6. SupposeN≥6∈Nis some fixed constant and a,bare distributions\n1. Ifb∈WN,∞(S1), let/tildewideN≤N−5and assume for some ρ∈[0,1/2)thata∈\nW5Sρ(T∗S1). Then\nOp(a)Op(b) =/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!Op(∂k\nξa(x,ξ)∂k\nxb(x))+OL2→L2(h/tildewideN(1−ρ)−5ρ).\nOp(b)Op(a) =/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!(−1)kOp(∂k\nξa(x,ξ)∂k\nxb(x))+OL2→L2(h/tildewideN(1−ρ)−5ρ).\n2. If for some ρ∈[0,1/2),b∈WNSρ(T∗S1)anda∈Sρ(T∗S1)let/tildewideN≤N−5then\nOp(a)Op(b) =/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!Op/parenleftigg\n(∂y∂ξ−∂x∂η)ka(x,ξ)b(y,η)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ/parenrightigg\n+OL2→L2(h/tildewideN(1−ρ)−5ρ).\nOp(b)Op(a) =/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!Op/parenleftigg\n(∂y∂ξ−∂x∂η)kb(x,ξ)a(y,η)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ/parenrightigg\n+OL2→L2(h/tildewideN(1−ρ)−5ρ).\n393. If for some ρ∈[0,1/2),b∈WNS0(T∗S1)anda∈WNSρ,ρ(T∗S1)let/tildewideN≤N−5then\nOp(a)Op(b) =/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!Op/parenleftigg\n(∂y∂ξ−∂x∂η)ka(x,ξ)b(y,η)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ/parenrightigg\n+OL2→L2(h/tildewideN(1−ρ)−5ρ).\nCases 1 and 2 are stated separately to emphasize that when one symbol depends only\nonx, less regularity is required of the other symbol. Also note t hat in Case 3 one of the\nsymbols does not produce unfavorable powers of hunder differentiation.\nProof.The proof relies on [Sj¨ o95], although special attention is paid to the minimal regu-\nlarity necessary. Only the proof of the first part of 1) will be show, the other parts follow\nby analogous arguments.\nSetc(x,ξ,y,η) =a(x,ξ)b(y,η) and letQbe the symmetric nonsingular matrix given\nby/a\\}b∇ack⌉tl⌉{tQD,D/a\\}b∇ack⌉t∇i}htR4=/a\\}b∇ack⌉tl⌉{tDξ,Dy/a\\}b∇ack⌉t∇i}htR2− /a\\}b∇ack⌉tl⌉{tDη,Dx/a\\}b∇ack⌉t∇i}htR2whereD= (Dx,Dξ,Dy,Dη). I will first show\nthat\nOp(a)Op(b) = Op((eih\n2/an}bracketle{tQD,D/an}bracketri}htc)(x,ξ,x,ξ)), (41)\nand then provide an expansion of the right hand side of the des ired form.\nTo begin since a,b∈WkSρ(T∗S1) fork≥= 2+1 = 3, then a,b∈Sw(T∗S1) by Lemma\nA.4. Therefore by [Sj¨ o95] [Theorem 2.2, and the discussion on pages 7-8] (41) holds, where\neih\n2/an}bracketle{tQD,D/an}bracketri}htis defined as the unique extension from S.\nSo it remains to provide an expansion of ( eih\n2/an}bracketle{tQD,D/an}bracketri}htc)(x,ξ,x,ξ). Well using a standard\nTaylor expansion of eih\n2/an}bracketle{tQD,D/an}bracketri}htas in [Sj¨ o95] equation (1.20)\neih\n2/an}bracketle{tQD,D/an}bracketri}htc(x,ξ,y,η) =/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!/a\\}b∇ack⌉tl⌉{tQD,D/a\\}b∇ack⌉t∇i}htkc(x,ξ,y,η)\n+(ih)/tildewideN\n2/tildewideN/tildewideN!/integraldisplay1\n0(1−t)/tildewideN−1eih\n2/an}bracketle{tQD,D/an}bracketri}ht/a\\}b∇ack⌉tl⌉{tQD,D/a\\}b∇ack⌉t∇i}ht/tildewideNc(x,ξ,y,η)dt\n=/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!(∂y∂ξ−∂x∂η)ka(x,ξ)b(y)\n+(ih)/tildewideN\n2/tildewideN/tildewideN!/integraldisplay1\n0(1−t)/tildewideN−1eih\n2/an}bracketle{tQD,D/an}bracketri}ht(∂y∂ξ−∂x∂η)/tildewideNa(x,ξ)b(y)dt\n=/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!∂k\nξa(x,ξ)∂k\nyb(y)+(ih)/tildewideN\n2/tildewideN/tildewideN!/integraldisplay1\n0(1−t)/tildewideN−1eih\n2/an}bracketle{tQD,D/an}bracketri}ht∂/tildewideN\nξa(x,ξ)∂/tildewideN\nyb(y)dt.\n40Therefore\neih\n2/an}bracketle{tQD,D/an}bracketri}htc(x,ξ,x,ξ) =/tildewideN−1/summationdisplay\nk=0(ih)k\n2kk!∂k\nξa(x,ξ)∂k\nxb(x)+(ih)/tildewideN\n2/tildewideN/tildewideN!/integraldisplay1\n0(1−t)/tildewideN−1eih\n2/an}bracketle{tQD,D/an}bracketri}ht∂/tildewideN\nξa(x,ξ)∂/tildewideN\nyb(y)dt|y=x,η=ξ.\nFirst consider the terms in the sum. Since ais smooth in ξ,b∈WN,∞andk≤/tildewideN−1≤\nN−6, forα≤1 andθ∈N\n|∂k+θ\nξa(x,ξ)∂k+α\nxb(x)|<∞.\nIn particular each term in the sum is in W1Sρ(T∗S1) and so quantizing it produces an\noperator bounded on L2by Lemma A.2.\nNow consider the integral term. By Lemma A.5 it is in Swand its quantization is\nbounded on L2by\nCh/tildewideNsup\nx,y,ξ,η,|γ|≤5/vextendsingle/vextendsingle/vextendsingle∂γ(∂y∂ξ)/tildewideN(a(x,ξ)b(y))/vextendsingle/vextendsingle/vextendsingle\n≤Ch/tildewideNsup\nx,y,ξ,η/summationdisplay\nγ1+γ2+γ3≤5/vextendsingle/vextendsingle/vextendsingle∂/tildewideN+γ1yb(y)∂γ2x∂/tildewideN+γ3\nξa(x,ξ)/vextendsingle/vextendsingle/vextendsingle\n≤Ch/tildewideNh−ρ(/tildewideN+5),\nwhere the final inequality holds because a∈W5Sρ(T∗S1) andb∈WN,∞(S1) with/tildewideN+5≤\nN.\nThe following lemma calculates the size of errors from intro ducing cutoff operators. It\nis a key tool used to take advantage of symbols with support co ntained in regions of phase\nspace where good estimates hold, namely the elliptic set of Pand the support of W.\nLemma A.7. FixN∈N,N≥6andρ∈[0,1/2). Suppose b∈WNSρ(T∗S1)and\nt∈S0\nρ(T∗S1), such that t≡1on suppb, then\n1.\nOp(t)Op(b) =Op(b)+OL2→L2(hN(1−ρ)−5)\nOp(b)Op(t) =Op(b)+OL2→L2(hN(1−ρ)−5).\n2.\nOp(t)Op(b)Op(t) =Op(b)+OL2→L2(hN(1−ρ)−5).\n41Proof.Well by part 2 of Lemma A.6, setting /tildewideN=N−5\nOp(t)Op(b) =N−6/summationdisplay\nk=0(ih)k\n2kk!Op/parenleftigg\n(∂y∂ξ−∂x∂η)kt(x,ξ)b(y,η)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=x,η=ξ/parenrightigg\n+OL2→L2(h(N−5)(1−ρ)−5ρ)\n= Op(tb)+OL2→L2(hN(1−ρ)−5)\n= Op(b)+OL2→L2(h(N(1−ρ)−5),\nwhere the terms with 1 ≤k≤N−6 all vanish, since ∂k\nξt(x,ξ) =∂k\nxt(x,ξ) = 0 on supp b,\nand Op(tb) = Op(b) ast≡1 on suppb. The second equation of part 1 follows by an\nanalogous proof.\nTo see part 2 use the first half of part 1 of this Lemma\nOp(t)Op(b)Op(t) = (Op(t)Op(b))Op(t)\n=/parenleftig\nOp(b)+OL2→L2(hN(1−ρ)−5)/parenrightig\nOp(t)\n= Op(b)Op(t)+OL2→L2(hN(1−ρ)−5) (42)\nwhere Op(t) is bounded on L2by Lemma A.2. Now apply the second half of part 1 of this\nLemma to Op( b)Op(t) to obtain the desired conclusion.\nReferences\n[AL14] N. Anantharaman and M. L´ eautaud. Sharp polynomial d ecay rates for the\ndamped wave equation on the torus. Anal. PDE , 7(1):159–214, 2014. With an\nappendix by S. Nonnenmacher.\n[BG97] N. Burq and P. G´ erard. Condition n´ ecessaire et suffis ante pour la contrˆ olabilit´ e\nexacte des ondes. , C. R. Math. Acad. Sci. Paris , 325(7):749–752, 1997.\n[BG18] N. Burq and P. G´ erard. Stabilisation of wave equatio ns on the torus with rough\ndampings. arXiv:1801.00983 , 2018.\n[BH07] N. Burq and M. Hitrik. Energy decay for damped wave equ ations on partially\nrectangular domains. 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American Mathematical Soc.,\n2012.\n44"    },    {        "title": "1802.10273v1.Nonexistence_of_global_solutions_of_wave_equations_with_weak_time_dependent_damping_and_combined_nonlinearity.pdf",        "content": "arXiv:1802.10273v1  [math.AP]  28 Feb 2018Nonexistence of global solutions of wave\nequations with weak time-dependent damping\nand combined nonlinearity\nNing-An Lai∗†Hiroyuki Takamura‡\nKeywords: scattering damping, combined nonlinearty, blow-up, lifespan\nMSC2010: primary 35L71, secondary 35B44\nAbstract\nIn our previous two works, we studied the blow-up and lifespa n\nestimates for damped wave equations with a power nonlineari ty of the\nsolution or its derivative, with scattering damping indepe ndently. In\nthis work, we are devoted to establishing a similar result fo r a com-\nbined nonlinearity. Comparing to the result of wave equatio n without\ndamping, one can say that the scattering damping has no influe nce.\n1 Introduction\nRecently, the small data Cauchy problem of damped semilinear wave e qua-\ntions with time dependent variable coefficients attracts more and mo re at-\ntention. The works of Wirth [18, 19, 20] showed that the behavior o f the\nsolution of the following linear problem\n/braceleftBigg\nu0\ntt−∆u0+µ\n(1+t)βu0\nt= 0,inRn×[0,∞),\nu0(x,0) =u1(x), u0\nt(x,0) =u2(x), x∈Rn,\n∗Institute of Nonlinear Analysis and Department of Mathematics, Lis hui University,\nLishui 323000, China.\n†School of Mathematical Sciences, Fudan University, Shanghai 20 0433, China. e-mail:\nhyayue@gmail.com.\n‡Department of Complex and Intelligent Systems, Faculty of System s Information Sci-\nence, Future University Hakodate, 116-2 Kamedanakano-cho, H akodate, Hokkaido 041-\n8655, Japan. e-mail: takamura@fun.ac.jp.\n1heavily relies on the decay rate βand the size of the positive constant µ.\nThen people get interested in the corresponding nonlinear problem, i.e. the\nfollowing small data Cauchy problem\n/braceleftBigg\nutt−∆u+µ\n(1+t)βut=|u|pinRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =εg(x), x∈Rn,(1.1)\nwhereµ >0, n∈Nandβ∈R, andεmeasures the smallness of the\ndata. Beforegoingon, it isnecessary to mention two correspondin g nonlinear\nproblems without damping\n/braceleftbiggut−∆u=|u|pinRn×[0,∞),\nu(x,0) =εf(x), x∈Rn,(1.2)\nand /braceleftbiggutt−∆u=|u|pinRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =εg(x), x∈Rn.(1.3)\nFor Cauchy problem (1.2) we know that it admits the critical value of pby\npF(n) := 1+2\nn,\nwhich isso-calledFujitaexponent, while theoneforproblem(1.3)is so -called\nStrauss exponent pS(n), which is the positive root of the quadratic equation,\nγ(p,n) := 2+(n+1)p−(n−1)p2= 0.\nRemark 1.1 “critical” here means the borderline which divides the doma in\nofpinto the blow-up part and the global existence part of the sol ution.\nRemark 1.2 It is easy to prove that\npF(n)<pS(n)forn≥2.\nNow we come back to Cauchy problem (1.1). It is interesting to study\nthe relation of the critical exponents among (1.1), (1.2) and (1.3). Forβ∈\n[−1,1), due to the works [3, 13, 17, 8, 4, 7], we know that it admits the\nsame critical exponent as that of problem (1.2). For β >1, since the authors\nshowed blow-up result for 1 <p<p S(n) in [11], we may believe that it has\nthe same critical exponent as that of (1.3).\nIf we consider the case β= 1 for Cauchy problem (1.1), the size of the\npositive constant µshould also be taken into account. Generally speaking,\nifµis relatively large, the term {µ/(1+t)}utin the equation will have the\n2main influence on the behavior of the solution, which means that this c ase\nhas the same critical exponent as that of problem (1.2). See the wo rks [1, 2].\nBut, ifµis relatively small, we may conjecture that the influence of utt\nwill dominate over {µ/(1+t)}ut, which means that the critical exponent is\nrelated topS(n). See the work [10] by the authors and Wakasa for 0 <µ<\n(n2+n+2)/{2(n+2)}, which was extended to 0 <µ<(n2+n+2)/(n+2) by\nIkeda and Sobajima [9] and Tu and Lin [15, 16]. Unfortunately, till now we\nare not clear of the boardline of µ, which determines that the critical power\nof Cauchy problem (1.1) with β= 1 will be Fujita or Strauss. We refer the\nreader to a very recent work by Palmieri and Reissig [14].\nIn a recent work [12] by the authors, we study the blow-up for the small\ndata Cauchy problem\n/braceleftBigg\nutt−∆u+µ\n(1+t)βut=|ut|pinRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =εg(x), x∈Rn.(1.4)\nIfβ >1, then we showed that the problem has no global solution for 1 <\np≤pG(n), where\npG(n) :=n+1\nn−1,\nwhich denotes the critical exponent for Glassey conjecture. In t his work,\nwe are devoted to studying the small data Cauchy problem with comb ined\nnonlinear terms, that is:\n/braceleftBigg\nutt−∆u+µ\n(1+t)βut=|ut|p+|u|qinRn×[0,∞),\nu(x,0) =εf(x), ut(x,0) =εg(x), x∈Rn,(1.5)\nwhereβ >1. Inspired by the work [5], in which Han and Zhou studied the\nCauchy problem (1.5) without damping and obtained the blow-up resu lt for\nmax/parenleftbigg\n1,2\nn−1/parenrightbigg\n<p≤2n\nn−1(1.6)\nand\n1<q <min/braceleftbigg\n1+4\n(n−1)p−2,2n\nn−2/bracerightbigg\n,\nwe want to show that whether we have the same blow-up result for C auchy\nproblem (1.5). The difficulty comes from the damping term, which prev ents\nus from getting the lower bound of some functional by using the tes t function\nmethod, and we overcome this by using a multiplier which was first intro -\nduced in the authors [11]. Also, due to the damping term, we can’t get the\nblow-up result and lifespan estimate by using Kato’s Lemma, and we do it\nby using an iteration argument similar to that in [11].\n3Remark 1.3 Hidano, Wang and Yokoyama [6] established global existence\nresult for Cauchy problem (1.5)without damping for n= 2,3and\np>pG(n),q >q S(n)and(q−1)((n−1)p−2)≥4.\nIn the following we are going to find out that whether the globa l existence\nresult holds for Cauchy problem (1.5).\n2 Main Result\nFirst we introduce the definition of the solution as follows.\nDefinition 2.1 As in [11], we say that uis an energy solution of (1.5)on\n[0,T)if\nu∈1/intersectiondisplay\ni=0Ci([0,T),H1−i(Rn))∩C1((0,T),Lp(Rn))∩Lq\nloc(Rn×(0,T))\nsatisfiesu(x,0) =εf(x)inH1(Rn)and\n/integraldisplay\nRnut(x,t)φ(x,t)dx−/integraldisplay\nRnεg(x)φ(x,0)dx\n+/integraldisplayt\n0ds/integraldisplay\nRn{−ut(x,s)φt(x,s)+∇u(x,s)·∇φ(x,s)}dx\n+/integraldisplayt\n0ds/integraldisplay\nRnµut(x,s)\n(1+s)βφ(x,s)dx\n=/integraldisplayt\n0ds/integraldisplay\nRn|ut(x,s)|pφ(x,s)dx+/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|qφ(x,s)dx(2.1)\nwith anyφ∈C∞\n0(Rn×[0,T))and anyt∈[0,T).\nEmploying the integration by parts in (2.1) and letting t→T, we get the\nweak solution of (1.5)\n/integraldisplay\nRn×[0,T)u(x,s)/braceleftbigg\nφtt(x,s)−∆φ(x,s)−/parenleftbiggµφ(x,s)\n(1+s)β/parenrightbigg\ns/bracerightbigg\ndxds\n=/integraldisplay\nRnµεf(x)φ(x,0)dx−/integraldisplay\nRnεf(x)φt(x,0)dx\n+/integraldisplay\nRnεg(x)φ(x,0)dx+/integraldisplay\nRn×[0,T)|ut(x,s)|pφ(x,s)dxds\n+/integraldisplay\nRn×[0,T)|u(x,s)|qφ(x,s)dxds.\nOur main theorem is the following.\n4Theorem 2.1 Letµ>0,β >1andn≥1. Assume that both f∈H1(Rn)\nandg∈L2(Rn)are non-negative, compactly supported, and gdoes not vanish\nidentically. Suppose that an energy solution uof(1.5)on[0,T)satisfies\nsuppu⊂ {(x,t)∈Rn×[0,∞) :|x| ≤t+R} (2.2)\nwith someR≥1. If\np>1 (2.3)\nand\n\n1<q <min/braceleftbigg\n1+4\n(n−1)p−2,2n\nn−2/bracerightbigg\nforn≥2,\n1<q forn= 1,(2.4)\nthen there exists a constant ε0=ε0(f,g,n,p,µ,β,R )>0such thatThas to\nsatisfy\nT≤Cε−2p(q−1)/{2q+2−(n−1)p(q−1)}(2.5)\nfor0<ε≤ε0, whereCis a positive constant independent of ε.\nRemark 2.1 We have less restriction for p, by comparing the conditions\n(2.3)and(1.6), since we use an iteration argument instead of Kato’s type\nlemma. Which means that we may get blow-up result even for lar gepbut\nsmallq. What is more, for relatively large pand smallq, we can establish\nan improved lifespan estimate. See Theorem 2.2 below.\nRemark 2.2 The restriction q <2n/(n−2)forn≥2is necessary to\nguarantee the integrability of the nonlinear term |u|q.\nRemark 2.3 As in [5], we should point out that there exist pairs of (p,q)\nsatisfying\np>pG(n), q>q S(n),\nbut still blow-up will occur. For example, since\nγ/parenleftbigg\nn,1+4\nn−1/parenrightbigg\n=−8\nn−1<0,\nwe may choose such an appropriate pair (p0,q0)by setting small constants ,\nδ1andδ2, such that\np0:=n+1\nn−1+δ1>pG(n)\nand\nqS(n)<q0:= 1+4\nn−1+(n+1)δ1<1−δ2+4\nn−1\n<1+4\nn−1+(n−1)δ1= 1+4\n(n−1)p0−2.\n5Wealso have animprovement onthe estimate of the lifespan forrelat ively\nlargepand smallqas follows.\nTheorem 2.2 Letµ>0,β >1andn≥2. Assume that both f∈H1(Rn)\nandg∈L2(Rn)are non-negative, compactly supported, and gdoes not vanish\nidentically. Suppose that an energy solution uof(1.5)on[0,T)satisfies\nsuppu⊂ {(x,t)∈Rn×[0,∞) :|x| ≤t+R} (2.6)\nwith someR≥1. If\np>2n\nn−1and1<q<n+1\nn−1, (2.7)\nthen there exists a constant ε0=ε0(f,g,n,p,µ,β,R )>0such thatThas to\nsatisfy\nT(ε)≤Cε−(q−1)/{q+1−n(q−1)}(2.8)\nfor0<ε≤ε0, whereCis a positive constant independent of ε.\nRemark 2.4 Under the assumption (2.7), the lifespan estimate (2.8)is bet-\nter than (2.5). For this, we should have\nq−1\nq+1−n(q−1)<2p(q−1)\n2q+2−(n−1)p(q−1)(2.9)\nwhich is equivalent to\np>2(q+1)\n2(q+1)−(n+1)(q−1). (2.10)\nOn the other hand, q <(n+1)/(n−1)is equivalent to\n2(q+1)\n2(q+1)−(n+1)(q−1)<2n\nn−1,\nwhich means that assumption (2.7)guarantees the inequality (2.9). In section\n6 we will give the reason why we have to pose the restriction on pin the form\np>2n\nn−1\ninstead of (2.10).\n63 Lower bound of the first functional\nOne of the key ingredients to the blow-up result is to get the lower bo und\nof\nF1(t) :=/integraldisplay\nRnu(x,t)ψ(x,t)dx,\nwhere\nψ(x,t) :=e−tφ1(x), φ1(x) :=\n\n/integraldisplay\nSn−1ex·ωdSωforn≥2,\nex+e−xforn= 1,(3.1)\nwhich was first introduced in Yordanov and Zhang [21]. Another key p oint\nis a multiplier,\nm(t) := exp/parenleftbigg\nµ(1+t)1−β\n1−β/parenrightbigg\n, (3.2)\nwhich is crucial for our proof and was first introduced in [11]. We note that\nm(t) is bounded as\n0<m(0)≤m(t)≤1.\nThen we have the following lemma.\nLemma 3.1 Letube an energy solution of (1.5)on[0,T). Under the same\nassumption of Theorem 2.1, it holds that\nF1(t)≥m(0)ε\n2/integraldisplay\nRnf(x)φ1(x)dx≥0fort≥0. (3.3)\nProof.The proof of Lemma 3.1 is almost the same as that of Lemma 3.1 in\n[12], which is established by neglecting the spatial integral of the non linear\nterm /integraldisplay\nRn|u(x,t)|pdx\ndue to its positivity. Replacing this quantity by\n/integraldisplay\nRn{|ut(x,t)|p+|u(x,t)|}qdx,\nwe get the desired proof immediately.\n74 Lower bound of the second functional\nWith Lemma 3.1 in hand, we may prove a key inequality for\nF2(t) :=/integraldisplay\nRnut(x,t)ψ(x,t)dx.\nLemma 4.1 Letu(x,t)andψ(x,t)be as in section 3. Under the same\nassumption of Theorem 2.1, it holds that\nF2(t)≥m(0)ε\n2/integraldisplay\nRng(x)φ1(x)dx≥0fort≥0. (4.1)\nProof.Actually Lemma 4.1 is a partial result of the proof of Theorem 2.1 in\n[12]. For convenience we rewrite the detail. By direction calculation we have\nd\ndt/bracketleftbigg\nm(t)/integraldisplay\nRn{ut(x,t)+u(x,t)}ψ(x,t)dx/bracketrightbigg\n=µ\n(1+t)βm(t)/integraldisplay\nRn{ut(x,t)+u(x,t)}ψ(x,t)dx\n+m(t)d\ndt/integraldisplay\nRn{ut(x,t)+u(x,t)}ψ(x,t)dx.(4.2)\nReplacing the test function φinthe definition (2.1) with ψand taking deriva-\ntive to both sides with respect to t, we have that\nd\ndt/integraldisplay\nRnut(x,t)ψ(x,t)dx−/integraldisplay\nRnut(x,t)ψt(x,t)dx\n+/integraldisplay\nRn∇u(x,t)·∇ψ(x,t)dx+µ\n(1+t)β/integraldisplay\nRnut(x,t)ψ(x,t)dx\n=/integraldisplay\nRn|ut(x,t)|pψ(x,t)dx+/integraldisplay\nRn|u(x,t)|qψ(x,t)dx.(4.3)\nSince forψ(x,t) we have\nψt=−ψ, ψ tt= ∆ψ=ψ,\nthen by integration by parts in the first term in the second line of the last\nequality yields that\nd\ndt/integraldisplay\nRn{ut(x,t)+u(x,t)}ψ(x,t)dx\n+µ\n(1+t)β/integraldisplay\nRnut(x,t)ψ(x,t)dx\n=/integraldisplay\nRn|ut(x,t)|pψ(x,t)dx+/integraldisplay\nRn|u(x,t)|qψ(x,t)dx.(4.4)\n8By combining (4.2) and (4.4) we have\nd\ndt/bracketleftbigg\nm(t)/integraldisplay\nRn{ut(x,t)+u(x,t)}ψ(x,t)dx/bracketrightbigg\n=m(t)/integraldisplay\nRn|ut(x,t)|pψ(x,t)dx+m(t)/integraldisplay\nRn|u(x,t)|qψ(x,t)dx\n+µ\n(1+t)βm(t)F1(t)(4.5)\nfort≥0. Then (4.5) and the positivity of F1by Lemma 3.1 yield\nm(t)/integraldisplay\nRn{ut(x,t)+u(x,t)}ψ(x,t)dx\n≥m(0)ε/integraldisplay\nRn{f(x)+g(x)}φ1(x)dx\n+/integraldisplayt\n0ds/integraldisplay\nRnm(s)|ut(x,s)|pψ(x,s)dx\n+/integraldisplayt\n0ds/integraldisplay\nRnm(s)|u(x,s)|qψ(x,s)dx.(4.6)\nOn the other hand, noting that\nm′(t)\nm(t)=µ\n(1+t)β,\nthen (4.3) implies that\nd\ndt/integraldisplay\nRnut(x,t)ψ(x,t)dx+m′(t)\nm(t)/integraldisplay\nRnut(x,t)ψ(x,t)dx\n+/integraldisplay\nRn{ut(x,t)−u(x,t)}ψ(x,t)dx\n=/integraldisplay\nRn|ut(x,t)|pψ(x,t)dx+/integraldisplay\nRn|u(x,t)|qψ(x,t)dx.\nMultiplying the above equality by m(t), we get\nd\ndt/bracketleftbigg\nm(t)/integraldisplay\nRnut(x,t)ψ(x,t)dx/bracketrightbigg\n+m(t)/integraldisplay\nRn{ut(x,t)−u(x,t)}ψ(x,t)dx\n=m(t)/integraldisplay\nRn|ut(x,t)|pψ(x,t)dx+m(t)/integraldisplay\nRn|u(x,t)|qψ(x,t)dx.(4.7)\n9Adding (4.6) and (4.7) together, we obtain that\nd\ndt/bracketleftbigg\nm(t)/integraldisplay\nRnut(x,t)ψ(x,t)dx/bracketrightbigg\n+2m(t)/integraldisplay\nRnut(x,t)ψ(x,t)dx\n≥m(0)ε/integraldisplay\nRn{f(x)+g(x)}φ1(x)dx+m(t)/integraldisplay\nRn|ut(x,t)|pψ(x,t)dx\n+m(t)/integraldisplay\nRn|u(x,t)|qψ(x,t)dx\n+/integraldisplayt\n0m(s)ds/integraldisplay\nRn|ut(x,s)|pψ(x,s)dx\n+/integraldisplayt\n0ds/integraldisplay\nRnm(s)|u(x,s)|qψ(x,s)dx.(4.8)\nSetting\nG(t) :=m(t)/integraldisplay\nRnut(x,t)ψ(x,t)dx−m(0)ε\n2/integraldisplay\nRng(x)φ1(x)dx\n−1\n2/integraldisplayt\n0m(s)ds/integraldisplay\nRn|ut(x,s)|pψ(x,s)dx,(4.9)\nthen we have\nG(0) =m(0)ε\n2/integraldisplay\nRng(x)φ1(x)dx>0.\nIt is easy to get from (4.8) that\nG′(t)+2G(t)\n≥m(t)\n2/integraldisplay\nRn|ut(x,t)|pψ(x,t)dx+m(0)ε/integraldisplay\nRnφ1(x)f(x)dx\n≥0\nwhich implies\nG(t)≥e−2tG(0)>0 fort≥0.\nHence, by the definition (4.9), it holds that\nm(t)/integraldisplay\nRnut(x,t)ψ(x,t)dx\n≥1\n2/integraldisplayt\n0m(s)ds/integraldisplay\nRn|ut(x,s)|pψ(x,s)dx\n+m(0)ε\n2/integraldisplay\nRng(x)φ1(x)dx,(4.10)\n10which implies that\n/integraldisplay\nRnut(x,t)ψ(x,t)dx≥m(0)ε\n2m(t)/integraldisplay\nRng(x)φ1(x)dx\n≥m(0)ε\n2/integraldisplay\nRng(x)φ1(x)dx,\nwhich is exactly the desired inequality in Lemma 4.1.\n5 Iteration argument\nAs mentioned in the introduction, we can’t establish the blow-up resu lt\nand lifespan estimate by using Kato’s lemma, instead of which we will use\nan iteration argument, following the idea in [11]. Set\nF0(t) :=/integraldisplay\nRnu(x,t)dx.\nChoosing the test function φ=φ(x,s) in (2.1) to satisfy φ≡1 in{(x,s)∈\nRn×[0,t] :|x| ≤s+R}, we get\n/integraldisplay\nRnut(x,t)dx−/integraldisplay\nRnut(x,0)dx+/integraldisplayt\n0ds/integraldisplay\nRnµut(x,s)\n(1+s)βdx\n=/integraldisplayt\n0ds/integraldisplay\nRn|ut(x,s)|pdx+/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|qdx,\nwhich implies that by taking derivative with respect to ton the both sides\nF′′\n0(t)+µ\n(1+t)βF′\n0(t) =/integraldisplay\nRn|ut(x,t)|pdx+/integraldisplay\nRn|u(x,s)|qdx.\nMultiplying with m(t) on the both sides yields\n{m(t)F′\n0(t)}′=m(t)/integraldisplay\nRn|ut(x,t)|pdx+m(t)/integraldisplay\nRn|u(x,t)|qdx,(5.1)\nwhich means that\nF′\n0(t)≥m(0)/integraldisplayt\n0ds/integraldisplay\nRn|ut(x,s)|pdx+m(0)/integraldisplayt\n0ds/integraldisplay\nRn|u(x,s)|qdx.(5.2)\nLemma 5.1 (Inequality (2.5) of Yordanov and Zhang [21]) Thereex-\nists a constant C1=C1(n,p,R)>0such that\n/integraldisplay\n|x|≤t+R[ψ(x,t)]p/(p−1)dx≤C1(1+t)(n−1){1−p/(2(p−1))}fort≥0.(5.3)\n11By H¨ older’s inequality, (5.3) and (4.1), we may estimate the nonlinear term\n/integraldisplay\nRn|ut(x,t)|pdx≥Fp\n2(t)/parenleftbigg/integraldisplay\n|x|≤t+R[ψ(x,t)]p/(p−1)dx/parenrightbigg−(p−1)\n≥C2εp(1+t)−(n−1)(p−2)/2,\nwhere\nC2:=C1−p\n1/parenleftbiggm(0)\n2/integraldisplay\nRng(x)φ1(x)dx/parenrightbiggp\n.\nPlugging which into (5.2) we have\nF0(t)≥m(0)C2εp/integraldisplayt\n0/integraldisplays\n0(1+r)n−1−(n−1)p/2drds\n≥m(0)C2εp(1+t)−(n−1)p/2/integraldisplayt\n0/integraldisplays\n0rn−1drds\n≥C3εp(1+t)−(n−1)p/2tn+1,(5.4)\nwhere\nC3:=m(0)C2\nn(n+1).\nBy H¨ older’s inequality again, it follows from (5.2) that\nF0(t)≥C4m(0)/integraldisplayt\n0/integraldisplays\n0(1+r)−n(q−1)Fq\n0(r)drds (5.5)\nwith some positive constant C4independent of ε. In this way, we find two\nkey ingredients for our iteration argument.\nAssuming that\nF0(t)≥Aj(1+t)−ajtbjfort≥0 (j= 1,2,3···)(5.6)\nwith\nA1=C3εp, a1=(n−1)p\n2, b1=n+1. (5.7)\nPlugging (5.6) into (5.5) we have\nF0(t)≥Aj+1(1+t)−qaj−n(q−1)tqbj+2,\nwhere\nAj+1≥C4m(0)Aq\nj\n(qbj+2)2, aj+1=qaj+n(q−1), bj+1=qbj+2.(5.8)\n12By combining (5.7) and (5.8) we come to\naj=qj−1/parenleftbigg(n−1)p\n2+n/parenrightbigg\n−n,\nbj=qj−1/parenleftbigg\nn+1+2\nq−1/parenrightbigg\n−2\nq−1,\nAj≥C5Aq\nj−1\nq2(j−1)\nwith\nC5:=C4m(0)/parenleftBig\nn+1+2\nq−1/parenrightBig2.\nHence we have\nlogAj\n≥qlogAj−1−2(j−1)logq+logC5\n≥q2logAj−2−2/parenleftbig\nq(j−2)+(j−1)/parenrightbig\nlogq+(q+1)logC5.\nRepeating this procedure, we have\nlogAj≥qj−1logA1−j−1/summationdisplay\nk=12klogq−logC5\nqk,\nwhich yields that\nAj≥exp/braceleftbig\nqj−1(logA1−Sq(j))/bracerightbig\n,\nwhere\nSq(j) :=j−1/summationdisplay\nk=12klogq−logC5\nqk.\nBy d’Alembert’s criterion we know that Sq(j) converges for q >1 asj→ ∞.\nAnd therefore we obtain that\nAj≥exp/braceleftbig\nqj−1(logA1−Sq(∞))/bracerightbig\n.\nSo if we come back to (5.6) we have\nF0(t)≥Aj(1+t)−ajtbj\n≥(1+t)nt−2/(q−1)exp/parenleftbig\nqj−1J(t)/parenrightbig\n, t>0,(5.9)\n13where\nJ(t) =−/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog(1+t)+/parenleftbigg\nn+1+2\nq−1/parenrightbigg\nlogt\n+logA1−Sq(∞).\nThen fort≥1,J(t) can be estimated as\nJ(t)≥−/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog(2t)+/parenleftbig\nn+1+2\nq−1/parenrightbig\nlogt\n+logA1−Sq(∞)\n=/parenleftbigg\nn+1+2\nq−1−(n−1)p\n2−n/parenrightbigg\nlogt+logA1\n−/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog2−Sq(∞)\n=log/parenleftbig\nt1+2/(q−1)−(n−1)p/2A1/parenrightbig\n−C6,(5.10)\nwhere\nC6:=/parenleftBig\n(n−1)p\n2+n/parenrightBig\nlog2+Sq(∞).\nRecall the definition of A1in (5.7), we have that J(t)>1 if\nt≥C7ε−2p(q−1)/{2q+2−(n−1)p(q−1)}\nwith\nC7:=/parenleftbig\nC−1\n3e1+C6/parenrightbig2(q−1)/{2q+2−(n−1)p(q−1)}.\nBy (5.9), it is easy to get\nF0(t)→ ∞asj→ ∞.\nHence we get the lifespan estimate in Theorem 2.1.\nRemark 5.1 In the last line of (5.10), we should require that\n1+2\nq−1−(n−1)p\n2>0,\nwhich leads to the restriction (2.4)forqin the case n≥2.\n146 Proof of Theorem 2.2\nDue to (5.4), we roughly get an estimate of the form,\nF0(t)≥Cεptn+1−(n−1)p/2\nfor largetwith some positive constant Cindependent of ε. So if\np>2n\nn−1,\nthen we have\nn+1−(n−1)p/2<1,\nwhich means that (5.4) is weaker than the linear growth. And hence it is\nnatural to get a better result if we have linear growth in the first st ep in the\niteration argument. Actually, due to the assumption of the initial da ta, we\nget from (5.1) that\nF′\n0(t)≥m(0)\nm(t)F′\n0(0)≥/parenleftbigg\nm(0)/integraldisplay\nRng(x)dx/parenrightbigg\nε,\nwhich implies that\nF0(t)≥C8εt, t≥0, (6.1)\nwhere\nC8:=m(0)/integraldisplay\nRng(x)dx.\nPlugging (6.1) into (5.5) we obtain\nF0(t)≥C9εq/integraldisplayt\n0/integraldisplays\n0(1+r)−n(q−1)rqdrds\n≥C10εq(1+t)−n(q−1)tq+2,(6.2)\nwhere\nC9:=C4m(0)Cq\n8andC10:=C9\n(q+1)(q+2).\nThen as in section 5, we may assume that\nF0(t)≥/tildewideAj(1+t)−/tildewideajt/tildewidebjfort≥0 (j= 1,2,3···)(6.3)\nwith\n/tildewideA1=C10εq,/tildewidea1=n(q−1),/tildewideb1=q+2. (6.4)\n15Plugging (6.3) into (5.5) we have\nF0(t)≥/tildewideAj+1(1+t)−q/tildewideaj−n(q−1)tq/tildewidebj+2,\nwhere\n/tildewideAj+1≥C4m(0)/tildewideAq\nj\n(q/tildewidebj+2)2,/tildewideaj+1=q/tildewideaj+n(q−1),/tildewidebj+1=q/tildewidebj+2,\nfrom which we get that\n\n\n/tildewideaj=nqj−n,\n/tildewidebj=qj−1{q+2+2/(q−1)}−2/(q−1),\n/tildewideAj≥C11/tildewideAq\nj−1/q2(j−1)(6.5)\nwith\nC11:=C4m(0)\n{q+2+2/(q−1)}2.\nIn the same way as in section 5, we conclude that\n/tildewideAj≥exp/braceleftBig\nqj−1/parenleftBig\nlog/tildewideA1−/tildewideSq(∞)/parenrightBig/bracerightBig\nwith\n/tildewideSq(∞) := lim\nj→∞/tildewideSq(j) := lim\nj→∞j−1/summationdisplay\nk=12klogq−logC11\nqk,\nand\nF0(t)≥(1+t)nt−2/(q−1)exp/parenleftBig\nqj−1/tildewideJ(t)/parenrightBig\n(6.6)\nwith\n/tildewideJ(t) =−nqlog(1+t)+/parenleftbigg\nq+2+2\nq−1/parenrightbigg\nlogt+log/tildewideA1−/tildewideSq(∞).\nTherefore, if t≥1, we come to\n/tildewideJ(t)≥ −nqlog(2t)+/parenleftbigg\nq+2+2\nq−1/parenrightbigg\nlogt+log/tildewideA1−/tildewideSq(∞)\n=/parenleftbigg\nq+2+2\nq−1−nq/parenrightbigg\nlogt+log/tildewideA1−/tildewideSq(∞)−nqlog2\n= log/parenleftBig\ntq+2+2/(q−1)−nq/tildewideA1/parenrightBig\n−C12\nwith\nC12:=/tildewideSq(∞)+nqlog2.\n16If\nt≥C13ε−(q−1)/{q+1−n(q−1)},\nwhere\nC13:=/parenleftbiggeC12+1\nC10/parenrightbigg1/{q+2+2/(q−1)−nq}\n,\nthen we have /tildewideJ(t)≥1, which will lead to by (6.6)\nF0(t)→ ∞asj→ ∞,\nand we finish the proof of Theorem 2.2.\nAcknowledgment\nThe first author is partially supported by Zhejiang Province Science Foun-\ndation(LY18A010008),NSFC(11501273,11726612,11771359, 11771194),Chi-\nnese Postdoctoral Science Foundation(2017M620128),the Scie ntific Research\nFoundationoftheFirst-ClassDisciplineofZhejiangProvince(B)(20 1601).The\nsecond author is partially supported by the Grant-in-Aid for Scient ific Re-\nsearch(C) (No.15K04964), Japan Society for the Promotion of Sc ience, and\nSpecial Research Expenses in FY2017, General Topics(No.B21), F uture Uni-\nversity Hakodate.\nReferences\n[1] M.D’Abbicco, The threshold of effective damping for semilinear wave\nequations , Mathematical Methods in Applied Sciences, 38(2015), 1032-\n1045.\n[2] M.D’AbbiccoandS.Lucente, A modifiedtest function methodfor damped\nwave equations , Adv. 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Differential Equations, 232(2007), 74-103.\n[21] B.Yordanov and Q.S.Zhang, Finite time blow up for critical wave equa-\ntions in high dimensions , J. Funct. Anal., 231(2006), 361-374.\n[22] Q.S.Zhang, A blow-up result for a nonlinear wave equation with damp-\ning: the critical case , C. R. Math. Acad. Sci. Paris, S´ er. I, 333(2001),\n109-114.\n19"    },    {        "title": "2003.12967v1.Stability_results_for_an_elastic_viscoelastic_waves_interaction_systems_with_localized_Kelvin_Voigt_damping_and_with_an_internal_or_boundary_time_delay.pdf",        "content": "arXiv:2003.12967v1  [math.AP]  29 Mar 2020STABILITY RESULTS FOR AN ELASTIC-VISCOELASTIC WAVES INTER ACTION\nSYSTEMS WITH LOCALIZED KELVIN-VOIGT DAMPING AND WITH AN INT ERNAL\nOR BOUNDARY TIME DELAY\nMOUHAMMAD GHADER1, RAYAN NASSER1,2, AND ALI WEHBE1\nAbstract. We investigate the stability of a one-dimensional wave equa tion with non smooth localized internal\nviscoelastic damping of Kelvin-Voigt type and with boundar y or localized internal delay feedback. The main\nnovelty in this paper is that the Kelvin-Voigt and the delay d amping are both localized via non smooth\ncoefficients. In the case that the Kelvin-Voigt damping is loc alized faraway from the tip and the wave is\nsubjected to a locally distributed internal or boundary del ay feedback, we prove that the energy of the system\ndecays polynomially of type t−4. However, an exponential decay of the energy of the system is established\nprovided that the Kelvin-Voigt damping is localized near a p art of the boundary and a time delay damping\nacts on the second boundary. While, when the Kelvin-Voigt an d the internal delay damping are both localized\nvia non smooth coefficients near the tip, the energy of the syst em decays polynomially of type t−4. Frequency\ndomain arguments combined with piecewise multiplier techn iques are employed.\nContents\n1. Introduction 1\n2. Wave equation with local Kelvin-Voigt damping and with bo undary delay feedback 7\n2.1. Wave equation with local Kelvin-Voigt damping far from the boundary and with boundary\ndelay feedback 8\n2.1.1. Well-posedness of the problem 9\n2.1.2. Strong Stability 12\n2.1.3. Polynomial Stability 16\n2.2. Wave equation with local Kelvin-Voigt damping near the boundary and boundary delay\nfeedback 21\n3. Wave equation with local internal Kelvin-Voigt damping a nd local internal delay feedback 24\n3.1. Well-posedness of the problem 25\n3.2. Polynomial Stability 28\nAppendix A. Notions of stability and theorems used 37\nReferences 38\n1Lebanese University, Faculty of sciences 1, Khawarizmi Lab oratory of Mathematics and Applications-KALMA,\nHadath-Beirut, Lebanon.\n2Université de Bretagne-Occidentale, France.\nE-mail addresses :mhammadghader@hotmail.com, rayan.nasser94@hotmail.co m, ali.wehbe@ul.edu.lb .\n1991Mathematics Subject Classification. 35L05; 35B35; 93D15; 93D20.\nKey words and phrases. Wave equation; Kelvin-Voigt damping; Time delay; Semigrou p; Stability.\niWAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n1.Introduction\nViscoelastic materials feature intermediate characteris tics between purely elastic and purely viscous behav-\niors,i.e.they display both behaviors when undergoing deformation. I n wave equations, when the viscoelastic\ncontrolling parameter is null, the viscous property vanish es and the wave equation becomes a pure elastic wave\nequation. However, time delays arise in many applications a nd practical problems like physical, chemical, bi-\nological, thermal and economic phenomena, where an arbitra ry small delay may destroy the well-posedness of\nthe problem and destabilize it. Actually, it is well-known t hat simplest delay equations of parabolic type,\nut(x,t) = ∆u(x,t−τ),\nor hyperbolic type\nutt(x,t) = ∆u(x,t−τ),\nwith a delay parameter τ >0,are not well-posed. Their instability is due to the existenc e of a sequence of\ninitial data remaining bounded, while the corresponding so lutions go to infinity in an exponential manner at a\nfixed time (see [ 16,23]).\nThe stabilization of a wave equation with Kelvin-Voigt type damping and internal or boundary time delay\nhas attracted the attention of many authors in the last five ye ars. Indeed, in 2016 Messaoudi et al. studied\nthe stabilization of a wave equation with global Kelvin-Voi gt damping and internal time delay in the multidi-\nmensional case (see [ 30]), and they obtained an exponential stability result. In th e same year, Nicaise et al. in\n[33] considered the multidimensional wave equation with local ized Kelvin-Voigt damping and mixed boundary\ncondition with time delay. They obtained an exponential dec ay of the energy regarding that the damping is\nacting on a neighborhood of part of the boundary via a smooth c oefficient. Also, in 2018, Anikushyn et al. in\n[15] considered the stabilization of a wave equation with globa l viscoelastic material subjected to an internal\nstrong time delay where a global exponential decay rate was o btained. Thus, it seems to us that there are\nno previous results concerning the case of wave equations wi th internal localized Kelvin-Voigt type damping\nand boundary or internal time delay, especially in the absen ce of smoothness of the damping coefficient even\nin the one dimensional case. So, we are interested in studyin g the stability of elastic wave equation with local\nKelvin–Voigt damping and with boundary or internal time del ay (see Systems ( 1.1) and ( 1.2)).\nThis paper investigates the study of the stability of a strin g with Kelvin-Voigt type damping localized via\nnon-smooth coefficient and subjected to a localized internal or boundary time delay. Indeed, in the first part of\nthis paper, we study the stability of elastic wave equation w ith local Kelvin–Voigt damping, boundary feedback\nand time delay term at the boundary, i.e.we consider the following system\n(1.1)\n\nUtt(x,t)−/bracketleftbig\nκUx(x,t)+δ1χ(α,β)Uxt(x,t)/bracketrightbig\nx= 0,(x,t)∈(0,L)×(0,+∞),\nU(0,t) = 0, t∈(0,+∞),\nUx(L,t) =−δ3Ut(L,t)−δ2Ut(L,t−τ), t ∈(0,+∞),\n(U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L),\nUt(L,t) =f0(L,t), t ∈(−τ,0),\nwhereL, τ, δ 1andδ3are strictly positive constant numbers, δ2is a non zero real number and the initial data\n(U0,U1,f0)belongs to a suitable space. Here 0≤α<β <L andU=uχ(0,α)+vχ(α,β)+wχ(β,L), withχ(a,b)is\nthe characteristic function of the interval (a,b).We assume that there exist strictly positive constant numbe rs\nκ1, κ2, κ3, such that κ=κ1χ(0,α)+κ2χ(α,β)+κ3χ(β,L).In fact, here we will consider two cases. In the first\ncase, we divide the bar into 3 pieces; the first piece is an elas tic part, the second piece is the viscoelastic part\nand in the third piece, the time delay feedback is effective at the ending point of the piece, i.e.we consider\nthe caseα>0(see Figure 1). While, in the second case, we divide the bar into 2 pieces; t he first piece is the\nviscoelastic part and in the second piece the time delay feed back is effective at the ending point of the piece,\ni.e.we consider the case α= 0(see Figure 2). Remark, here, in both cases, the Kelvin–Voigt damping is\neffective on a part of the piece and the time delay is effective a tL.\n1WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n• •α βv(x) u(x) w(x)\n0 LElastic part Viscoelastic part Boundary delay feedback\nFigure 1. K-V damping is acting localized in the internal of the body an d time delay feedback\nis effective at L\n•\n0 βv(x) w(x)\nLViscoelastic part Boundary delay feedback\nFigure 2. K-V damping is acting localized near the boundary of the body and time delay\nfeedback is effective at L\nIn the second part of this paper, we study the stability of ela stic wave equation with local Kelvin–Voigt damping\nand local internal time delay. This system takes the followi ng form\n(1.2)\n\nUtt(x,t)−/bracketleftbig\nκUx(x,t)+χ(α,β)(δ1Uxt(x,t)+δ2Uxt(x,t−τ))/bracketrightbig\nx= 0,(x,t)∈(0,L)×(0,+∞),\nU(0,t) =U(L,t) = 0, t∈(0,+∞),\n(U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L),\nUt(x,t) =f0(x,t), (x,t)∈(0,L)×(−τ,0),\nwhereL, τandδ1are strictly positive constant numbers, δ2is a non zero real number and the initial data\n(U0,U1,f0)belongs to a suitable space. Here 0<α<β <L andU=uχ(0,α)+vχ(α,β)+wχ(β,L), withχ(a,b)is\nthe characteristic function of the interval (a,b).We assume that there exist strictly positive constant numbe rs\nκ1, κ2, κ3, such that κ=κ1χ(0,α)+κ2χ(α,β)+κ3χ(β,L).In fact, here we will divide the bar into 3 pieces; the\nfirst piece is an elastic part, in the second piece the Kelvin– Voigt damping and the time delay are effective and\nthe third piece is an elastic part (see Figure 3). So, the Kelvin–Voigt damping and the time delay are effecti ve\non(α,β).\n• •α βv(x) u(x) w(x)\n0 LElastic part Viscoelastic part &delay feedback Elastic part\nFigure 3. Local internal K-V damping and Local internal delay feedbac k\nIn the literature, Datko et al. studied in 1985 the one dimensional wave equation which mode ls the vibrations\n2WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nof a string fixed at one end and free at the other one (see [ 14]). The system is given by the following:\n(1.3)\n\nutt(x,t)−uxx(x,t)+2aut(x,t)+a2u(x,t) = 0,(x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t∈(0,+∞),\nux(1,t) =−κut(1,t−τ), t ∈(0,+∞),\nwhere the delay parameter τis strictly positive, a>0andκ>0. So, the above system models a string having\na boundary feedback with delay at the free end. They showed th at ifκ/parenleftbig\ne2a+1/parenrightbig\n<e2a−1,then System ( 1.3)\nis strongly stable for all small enough delays. However, if κ/parenleftbig\ne2a+1/parenrightbig\n>e2a−1,then there exists an open set D\ndense in (0,+∞), such that for all τinD, System ( 1.3) admits exponentially unstable solutions. Moreover, in\nthe absence of delay in System ( 1.3) (i.eτ= 0) anda≥0,κ≥0, its energy decays exponentially to zero under\nthe condition a2+κ2>0(see [12]). In 1990, Datko in [ 13] considered the boundary feedback stabilization of a\none-dimensional wave equation with time delay (see Example 3.5 in [ 13]). The system is given by the following:\n(1.4)\n\nutt(x,t)−uxx(x,t)−δuxxt(x,t) = 0,(x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =−κut(1,t−τ), t ∈(0,+∞),\nwhereτ >0, κ >0andδ >0. He proved that System ( 1.4) is unstable for an arbitrary small value of τ.\nIn 2006, Xu et al. in [44] investigated the following closed loop system with homoge neous Dirichlet boundary\ncondition at x= 0and delayed Neumann boundary feedback at x= 1:\n(1.5)\n\nutt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =−κut(1,t)−κ(1−µ)ut(1,t−τ), t∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈(0,1),\nut(1,t) =f0(1,t), t ∈(−τ,0).\nThe above system represents a wave equation that is fixed at on e end and subjected to a boundary control\ninput possessing a partial time delay of weight (1−µ)at the other end. They proved the following stability\nresults:\n1. Ifµ>2−1,then System ( 1.5) is uniformly stable.\n2. Ifµ= 2−1andτ∈Q∩(0,1), then System ( 1.5) is unstable.\n3. Ifµ= 2−1andτ∈(R\\Q)∩(0,1), then System ( 1.5) is asymptotically stable.\n4. Ifµ<2−1,then System ( 1.5) is always unstable.\nLater on, in 2008, Guo and Xu in [ 18] studied the stabilization of a wave equation in the 1-D case where it is\neffected by a boundary control and output observation sufferi ng from time delay. The system is given by the\nfollowing:\n\n\nutt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =w(t), t ∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1),\ny(t) =ut(1,t−τ), t ∈(0,+∞),\nwherewis the control and yis the output observation. Using the separation principle, the authors proved\nthat the above delayed system is exponentially stable. In 20 10, Gugat in [ 17] studied the wave equation which\nmodels a string of length Lthat is rigidly fixed at one end and stabilized with a boundary feedback and constant\n3WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\ndelay at the other end. The problem is described by the follow ing system\n\n\nutt(x,t)−c2uxx(x,t) = 0, (x,t)∈(0,L)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(L,t) = 0, t ∈(0,2Lc−1),\nux(L,t) =c−1λut/parenleftbig\nL,t−2Lc−1/parenrightbig\n, t ∈(2Lc−1,+∞),\n(u(x,0),ut(x,0),u(0,0)) = (u0(x),u1(x),0), x∈(0,L),\nwhereλis a real number and c >0. Gugat proved that the above system is exponentially stable . In 2011,\nJ. Wang et al. in [41] studied the stabilization of a wave equation under boundar y control and collocated\nobservation with time delay. The system is given by the follo wing:\n\n\nutt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =κut(1,t−τ), t ∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1).\nThey showed that if the delay is equal to even multiples of the wave propagation time, then the above closed\nloop system is exponentially stable under sufficient and nece ssary conditions for κ. Else, if the delay is an odd\nmultiple of the wave propagation time, thus the closed loop s ystem is unstable. In 2013, H. Wang et al. in [43],\nstudied System ( 1.5) under the feedback control law ut(1,t) =w(t)provided that the weight of the feedback\nwith delay is a real βand that of the feedback without delay is a real α. They found a feedback control law\nthat stabilizes exponentially the system for any |α| /\\e}atio\\slash=|β|, by modifying the velocity feedback into the form\nu(t) =βwt(1,t)+αf(w(.,t),wt(.,t)), wherefis a linear functional. Finally, in 2017, Xu et al. in [42], studied\nthe stability problem of a one dimensional wave equation wit h internal control and boundary delay term\n\n\nutt(x,t)−uxx(x,t)+2αut(x,t) = 0,(x,t)∈(0,1)×(0,+∞),\nu(0,t) = 0, t ∈(0,+∞),\nux(1,t) =κut(1,t−τ), t ∈(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1),\nut(1,t) =f0(1,t), t ∈(−τ,0),\nwhereτ >0,α>0andκis real. Based on the idea of Lyapunov functional, they prove d exponential stability\nof the above system under a certain relationship between αandκ.\nGoing to the multidimensional case, the stability of wave eq uation with time delay has been studied in\n[32,6,37,30,33,15,3,4]. In 2006, Nicaise and Pignotti in [ 32] studied the multidimensional wave equation\nconsidering two cases. The first case concerns a wave equatio n with boundary feedback and a delay term at\nthe boundary\n(1.6)\n\nutt(x,t)−∆u(x,t) = 0, (x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈ΓD×(0,+∞),\n∂u\n∂ν(x,t) =−µ1ut(x,t)−µ2ut(x,t−τ),(x,t)∈ΓN×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈ΓN×(−τ,0).\n4WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nThe second case concerns a wave equation with an internal fee dback and a delayed velocity term ( i.ean internal\ndelay) and a mixed Dirichlet-Neumann boundary condition\n(1.7)\n\nutt(x,t)−∆u(x,t)+µ1ut(x,t)+µ2ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈ΓD×(0,+∞),\n∂u\n∂ν(x,t) = 0, (x,t)∈ΓN×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0).\nIn both systems, τ, µ1, µ2are strictly positive constants, ∂u/∂ν is the partial derivative, Ωis an open bounded\ndomain of RNwith a boundary Γof classC2andΓ = ΓD∪ΓN, such that ΓD∩ΓN=∅. Under the assumption\nthat the weight of the feedback with delay is smaller than tha t without delay (µ2< µ1), they obtained an\nexponential decay of the energy of both Systems ( 1.6) and ( 1.7). On the contrary, if the previous assumption\ndoes not hold (i.eµ2≥µ1), they found a sequence of delays for which the energy of some s olutions does not\ntend to zero (see [ 10] for the treatment of Problem ( 1.7) in more general abstract form). In 2009, Nicaise et\nal.in [34] studied System ( 1.6) in the one dimensional case where the delay time τis a function depending on\ntime and they established an exponential stability result u nder the condition that the derivative of the decay\nfunction is upper bounded by a constant d<1and assuming that µ2<√\n1−d µ1. In 2010, Ammari et al. in\n[6] studied the wave equation with interior delay damping and d issipative undelayed boundary condition in an\nopen domain ΩofRN, N≥2.The system is given by the following:\n(1.8)\n\nutt(x,t)−∆u(x,t)+aut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ0×(0,+∞),\n∂u\n∂ν(x,t) =−κut(x,t), (x,t)∈Γ1×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0),\nwhereτ >0,a>0andκ >0. Under the condition that Γ1satisfies the Γ-condition introduced in [ 25], they\nproved that System ( 1.8) is uniformly asymptotically stable whenever the delay coe fficient is sufficiently small.\nIn 2012, Pignotti in [ 37] considered the wave equation with internal distributed ti me delay and local damping\nin a bounded and smooth domain Ω⊂RN,N≥1. The system is given by the following:\n(1.9)\n\nutt(x,t)−∆u(x,t)+aχωut(x,t)+κut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f(x,t), (x,t)∈Ω×(−τ,0),\nwhereκreal,τ >0anda >0. System ( 1.9) shows that the damping is localized, indeed, it acts on a\nneighborhood of a part of the boundary of Ω. Under the assumption that |κ|<κ0<a,the author established\nan exponential decay rate. Later, in 2016, Messaoudi et al. in [30] considered the stabilization of the following\nwave equation with strong time delay\n\n\nutt(x,t)−∆u(x,t)−µ1∆ut(x,t)−µ2∆ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0),\nwhereµ1>0andµ2is a non zero real number. Under the assumption that |µ2|< µ1, they obtained an\nexponential stability result. In addition, in the same year , Nicaise et al. in [33] studied the multidimensional\n5WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwave equation with localized Kelvin-Voigt damping and mixe d boundary condition with time delay\n(1.10)\n\nutt(x,t)−∆u(x,t)−div(a(x)∇ut) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ0×(0,+∞),\n∂u\n∂ν(x,t) =−a(x)∂ut\n∂ν(x,t)−κut(x,t−τ),(x,t)∈Γ1×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω,\nut(x,t) =f0(x,t), (x,t)∈Γ1×(−τ,0),\nwhereτ >0,κis real,a(x)∈L∞(Ω)anda(x)≥a0>0onωsuch thatω⊂Ωis an open neighborhood of Γ1.\nUnder an appropriate geometric condition on Γ1and assuming that a∈C1,1(Ω),∆a∈L∞(Ω), they proved\nan exponential decay of the energy of System ( 1.10). Finally, in 2018, Anikushyn et al. in [15] considered an\ninitial boundary value problem for a viscoelastic wave equa tion subjected to a strong time localized delay in a\nKelvin-Voigt type. The system is given by the following:\n\n\nutt(x,t)−c1∆u(x,t)−c2∆u(x,t−τ)−d1∆ut(x,t)−d2∆ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞),\nu(x,t) = 0, (x,t)∈Γ0×(0,+∞),\n∂u\n∂ν(x,t) = 0, (x,t)∈Γ1×(0,+∞),\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈Ω,\nu(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0).\nUnder appropriate conditions on the coefficients, a global ex ponential decay rate is obtained. We can also\nmention that Ammari et al. in [7] considered the stabilization problem for an abstract equa tion with delay and\na Kelvin-Voigt damping in 2015. The system is given by the fol lowing:\n\n\nutt(t)+aBB∗ut(t)+BB∗u(t−τ), t∈(0,+∞),\n(u(0),ut(0)) = (u0,u1),\nB∗u(t) =f0(t), t ∈(−τ,0),\nfor an appropriate class of operator Banda >0.Using the frequency domain approach, they obtained an\nexponential stability result. Finally, the transmission p roblem of a wave equation with global or local Kelvin-\nVoigt damping and without any time delay was studied by many a uthors in the one dimensional case (see\n[26,2,21,1,20,19,35,39,28]) and in the multidimensional case (see [ 31,45,40,27]) and polynomial and\nexponential stability results were obtained. In addition, the stability of wave equations on tree with local\nKelvin-Voigt damping has been studied in [ 5].\nThus, as we confirmed in the beginning, the case of wave equati ons with localized Kelvin-Voigt type damping\nand boundary or internal time delay; as in our Systems ( 1.1) and ( 1.2), where the damping is acting in a non-\nsmooth region is still an open problem. The aim of the present paper consists in studying the stability of the\nSystems ( 1.1) and ( 1.2). For System ( 1.1), we consider two cases. Case one, if α >0(see Figure 1), then\nusing the semigroup theory of linear operators and a result o btained by Borichev and Tomilov, we show that\nthe energy of the System ( 1.1) has a polynomial decay rate of type t−4. Case two, if α= 0(see Figure 2),\nthen using the semigroup theory of linear operators and a res ult obtained by Huang and Prüss, we prove an\nexponential decay of the energy of System ( 1.1). For System ( 1.2), by using the semigroup theory of linear\noperators and a result obtained by Borichev and Tomilov, we s how that the energy of the System ( 1.2) has a\npolynomial decay rate of type t−4.\nThis paper is organized as follows: In Section 2, we study the stability of System ( 1.1). Indeed, in Subsection\n2.1, we consider the case α>0. First, we prove the well-posedness of System ( 1.1). Next, we prove the strong\nstability of the system in the lack of the compactness of the r esolvent of the generator. Then, we establish\na polynomial energy decay rate of type t−4(see Theorem 2.7). In addition, in Subsection 2.2, we consider\nthe caseα= 0and we prove the exponential stability of system ( 1.1) (see Theorem 2.14). In Section 3, we\nstudy the stability of System ( 1.2). First, we prove the well-posedness of System ( 1.2). Next, we establish a\npolynomial energy decay rate of type t−4(see Theorem 3.2).\n6WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n2.Wave equation with local Kelvin-Voigt damping and with boun dary delay feedback\nThis section is devoted to our first aim, that is to study the st ability of a wave equation with localized Kelvin-\nVoigt damping and boundary delay feedback (see System ( 1.1)). For this aim, let us introduce the auxiliary\nunknown\nη(L,ρ,t) =Ut(L,t−ρτ), ρ∈(0,1), t>0.\nThus, Problem ( 1.1) is equivalent to\n(2.1)\n\nUtt(x,t)−/bracketleftbig\nκUx(x,t)+δ1χ(α,β)Uxt(x,t)/bracketrightbig\nx= 0,(x,t)∈(0,L)×(0,+∞),\nτηt(L,ρ,t)+ηρ(L,ρ,t) = 0, (ρ,t)∈(0,1)×(0,+∞),\nU(0,t) = 0, t∈(0,+∞),\nUx(L,t) =−δ3Ut(L,t)−δ2η(L,1,t), t ∈(0,+∞),\n(U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L),\nη(L,ρ,0) =f0(L,−ρτ), ρ ∈(0,1).\nLetUbe a smooth solution of System ( 2.1), we associate its energy defined by\n(2.2) E(t) =1\n2/integraldisplayL\n0/parenleftbig\n|Ut|2+κ|Ux|2/parenrightbig\ndx+τ\n2/integraldisplay1\n0|η|2dρ.\nMultiplying the first equation of ( 2.1) byUt, integrating over (0,L)with respect to x, then using by parts\nintegration and the boundary conditions in ( 2.1) atx= 0and atx=L, we get\n(2.3)1\n2d\ndt/integraldisplayL\n0/parenleftbig\n|Ut|2+κ|Ux|2/parenrightbig\ndx=−δ1/integraldisplayβ\nα|Uxt|2dx−κ3δ3|Ut(L,t)|2−κ3δ2η(L,1,t)Ut(L,t).\nMultiplying the second equation of ( 2.1) byη, integrating over (0,1)with respect to ρ, then using the fact that\nη(L,0,t) =Ut(L,t), we get\n(2.4)τ\n2d\ndt/integraldisplay1\n0|η|2dρ=−1\n2|η(L,1,t)|2+1\n2|Ut(L,t)|2.\nAdding ( 2.3) and ( 2.4), we get\n(2.5)dE(t)\ndt=−δ1/integraldisplayβ\nα|Uxt|2dx−/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n|Ut(L,t)|2−κ3δ2η(L,1,t)Ut(L,t)−1\n2|η(L,1,t)|2.\nFor allp>0, we have\n(2.6) −κ3δ2η(L,1,t)Ut(L,t)≤κ3|δ2||η(L,1,t)|2\n2p+κ3|δ2|p|Ut(L,t)|2\n2.\nInserting ( 2.6) in (2.5), we get\n(2.7)dE(t)\ndt≤ −δ1/integraldisplayβ\nα|Uxt|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1,t)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|Ut(L,t)|2.\nIn the sequel, the assumption on κ3, δ1, δ2andδ3will ensure that\n(H) κ3>0, δ1>0, δ3>0, δ2∈R∗, δ3>1\n2κ3,|δ2|<1\nκ3/radicalbig\n2κ3δ3−1.\nIn this case, we easily check that there exists a strictly pos itive number psatisfying\n(2.8) κ3|δ2|<p<2\nκ3|δ2|/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n,\nsuch that\n1\n2−κ3|δ2|\n2p>0andκ3δ3−1\n2−κ3|δ2|p\n2>0,\nso that the energies of the strong solutions satisfy E′(t)≤0.Hence, System ( 2.1) is dissipative in the sense\nthat its energy is non increasing with respect to the time t.\nFor studying the stability of System ( 2.1), we consider two cases. In Subsection 2.1, we consider the first case,\n7WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwhen the Kelvin-Voigt damping is localized in the internal o f the body, i.e.α >0. While in Subsection 2.2,\nwe consider the second, when the Kelvin-Voigt damping is loc alized near the boundary of the body, i.e.α= 0.\n2.1.Wave equation with local Kelvin-Voigt damping far from the b oundary and with boundary\ndelay feedback. In this subsection, we assume that there exist αandβsuch that 0< α < β < L , in this\ncase, the Kelvin-Voigt damping is localized in the internal of the body (see Figure 1). For this aim, we denote\nthe longitudinal displacement by Uand this displacement is divided into three parts\nU(x,t) =\n\nu(x,t),(x,t)∈(0,α)×(0,+∞),\nv(x,t),(x,t)∈(α,β)×(0,+∞),\nw(x,t),(x,t)∈(β,L)×(0,+∞).\nIn this case, System ( 2.1) is equivalent to the following system\nutt−κ1uxx= 0,(x,t)∈(0,α)×(0,+∞), (2.9)\nvtt−(κ2vx+δ1vxt)x= 0,(x,t)∈(α,β)×(0,+∞), (2.10)\nwtt−κ3wxx= 0,(x,t)∈(β,L)×(0,+∞), (2.11)\nτηt(L,ρ,t)+ηρ(L,ρ,t) = 0,(ρ,t)∈(0,1)×(0,+∞), (2.12)\nwith the following boundary and transmission conditions\nu(0,t) = 0, t∈(0,+∞), (2.13)\nwx(L,t) =−δ3wt(L,t)−δ2η(L,1,t), t∈(0,+∞), (2.14)\nu(α,t) =v(α,t), t∈(0,+∞), (2.15)\nv(β,t) =w(β,t), t∈(0,+∞), (2.16)\nκ2vx(α,t)+δ1vxt(α,t) =κ1ux(α,t), t∈(0,+∞), (2.17)\nκ2vx(β,t)+δ1vxt(β,t) =κ3wx(β,t), t∈(0,+∞), (2.18)\nand with the following initial conditions\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,α), (2.19)\n(v(x,0),vt(x,0)) = (v0(x),v1(x)), x∈(α,β), (2.20)\n(w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L), (2.21)\nη(L,ρ,0) =f0(L,−ρτ), ρ∈(0,1), (2.22)\nwhere the initial data (u0,u1,v0,v1,w0,w1,f0)belongs to a suitable Hilbert space. So, using ( 2.2), the energy\nof System ( 2.9)-(2.22) is given by\nE(t) =1\n2/integraldisplayα\n0/parenleftbig\n|ut|2+κ1|ux|2/parenrightbig\ndx+1\n2/integraldisplayβ\nα/parenleftbig\n|vt|2+κ2|vx|2/parenrightbig\ndx+1\n2/integraldisplayL\nβ/parenleftbig\n|wt|2+κ3|wx|2/parenrightbig\ndx+τ\n2/integraldisplay1\n0|η|2dρ.\nSimilar to ( 2.5) and ( 2.7), we get\ndE(t)\ndt=−δ1/integraldisplayβ\nα|vxt|2dx−1\n2|η(L,1,t)|2−κ3δ2η(L,1,t)wt(L,t)−/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n|wt(L,t)|2,\n≤ −δ1/integraldisplayβ\nα|vxt|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1,t)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|wt(L,t)|2,\nwherepis defined in ( 2.8). Thus, under hypothesis (H), the System ( 2.9)-(2.22) is dissipative in the sense that\nits energy is non increasing with respect to the time t.Now, we are in position to prove the existence and\nuniqueness of the solution of our system.\n8WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n2.1.1. Well-posedness of the problem. We start this part by formulating System ( 2.9)-(2.22) as an abstract\nCauchy problem. For this aim, let us define\nHm=Hm(0,α)×Hm(α,β)×Hm(β,L), m= 1,2,\nL2=L2(0,α)×L2(α,β)×L2(β,L),\nH1\nL={(u,v,w)∈H1|u(0) = 0, u(α) =v(α), v(β) =w(β)}.\nRemark 2.1. The Hilbert space L2is equipped with the norm:\n/ba∇dbl(u,v,w)/ba∇dbl2\nL2=/integraldisplayα\n0|u|2dx+/integraldisplayβ\nα|v|2dx+/integraldisplayL\nβ|w|2dx.\nAlso, it is easy to check that the space H1\nLis Hilbert space over Cequipped with the norm:\n/ba∇dbl(u,v,w)/ba∇dbl2\nH1\nL=κ1/integraldisplayα\n0|ux|2dx+κ2/integraldisplayβ\nα|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx.\nMoreover, by Poincaré inequality we can easily verify that the re existsC >0, such that\n/ba∇dbl(u,v,w)/ba∇dblL2≤C/ba∇dbl(u,v,w)/ba∇dblH1\nL.\n/square\nWe now define the Hilbert energy space Hby\nH=H1\nL×L2×L2(0,1)\nequipped with the following inner product\n/a\\}b∇acketle{tU,˜U/a\\}b∇acket∇i}htH=κ1/integraldisplayα\n0ux˜uxdx+κ2/integraldisplayβ\nαvx˜vxdx+κ3/integraldisplayL\nβwx˜wxdx+/integraldisplayα\n0y˜ydx+/integraldisplayβ\nαz˜zdx+/integraldisplayL\nβφ˜φdx+τ/integraldisplay1\n0η(L,ρ)˜η(L,ρ)dρ,\nwhereU= (u,v,w,y,z,φ,η (L,·))∈ Hand˜U= (˜u,˜v,˜w,˜y,˜z,˜φ,˜η(L,·))∈ H. We use /ba∇dblU/ba∇dblHto denote the\ncorresponding norm. We define the linear unbounded operator A:D(A)⊂ H −→ H by:\nD(A) =/braceleftbigg\n(u,v,w,y,z,φ,η (L,·))∈H1\nL×H1\nL×H1(0,1)|(u,κ2v+δ1z,w)∈H2,\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β),\nwx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0)/bracerightbigg\nand for all U= (u,v,w,y,z,φ,η (L,·))∈D(A)\nAU=/parenleftbig\ny,z,φ,κ 1uxx,(κ2vx+δ1zx)x,κ3wxx,−τ−1ηρ(L,·)/parenrightbig\n.\nIfU= (u,v,w,u t,vt,wt,η(L,·))is a regular solution of System ( 2.9)-(2.22), then we transform this system into\nthe following initial value problem\n(2.23)/braceleftBigg\nUt=AU,\nU(0) =U0,\nwhereU0= (u0,v0,w0,u1,v1,w1,f0(L,−·τ))∈ H.We now use semigroup approach to establish well-posedness\nresult for the System ( 2.9)-(2.22). According to Lumer-Phillips theorem (see [ 36]), we need to prove that the\noperator Ais m-dissipative in H. Therefore, we prove the following proposition.\nProposition 2.2. Under hypothesis (H), the unbounded linear operator Ais m-dissipative in the energy space\nH.\nProof. For allU= (u,v,w,y,z,φ,η (L,·))∈D(A),we have\nRe/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=κ1Re/integraldisplayα\n0(yxux+uxxy)dx+Re/integraldisplayβ\nα(κ2zxvx+(κ2vx+δ1zx)xz)dx\n+κ3Re/integraldisplayL\nβ/parenleftbig\nφxwx+wxxφ/parenrightbig\ndx−Re/integraldisplay1\n0ηρ(L,ρ)η(L,ρ)dρ.\n9WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nHere Re is used to denote the real part of a complex number. Usi ng by parts integration in the above equation,\nwe get\n(2.24)Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−δ1/integraldisplayβ\nα|zx|2dx−1\n2|η(L,1)|2+1\n2|η(L,0)|2+κ3Re/parenleftbig\nwx(L)φ(L)/parenrightbig\n−κ1Re(ux(0)y(0))+ Re(κ1ux(α)y(α)−κ2vx(α)z(α)−δ1zx(α)z(α))\n+Re/parenleftbig\nκ2vx(β)z(β)+δ1zx(β)z(β)−κ3wx(β)φ(β)/parenrightbig\n.\nOn the other hand, since U∈D(A), we have\n(2.25)\n\ny(0) = 0, y(α) =z(α), z(β) =φ(β),\nκ1ux(α)−κ2vx(α)−δ1zx(α) = 0, κ2vx(β)+δ1zx(β)−κ3wx(β) = 0,\nwx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0).\nInserting ( 2.25) in (2.24), we get\n(2.26) Re /a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=−δ1/integraldisplayβ\nα|zx|2dx−1\n2|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n|η(L,0)|2−κ3δ2Re(η(L,0)η(L,1)).\nUnder hypothesis (H), we easily check that there exists p>0such that\nκ3|δ2|<p<2\nκ3|δ2|/parenleftbigg\nκ3δ3−1\n2/parenrightbigg\n.\nBy Young’s inequality, we get\n−κ3δ2Re(η(L,0)η(L,1))≤κ3|δ2||η(L,1)|2\n2p+κ3|δ2|p|η(L,0)|2\n2.\nInserting the above inequality in ( 2.26), we get\n(2.27) Re /a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH≤ −δ1/integraldisplayβ\nα|zx|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|η(L,0)|2.\nFrom the construction of p, we have\n1\n2−κ3|δ2|\n2p>0andκ3δ3−1\n2−κ3|δ2|p\n2>0.\nTherefore, from ( 2.27), we get\nRe/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH≤0,\nwhich implies that Ais dissipative. Now, let us go on with maximality. Let F= (f1,f2,f3,f4,f5,f6,f7(L,·))∈\nHwe look for U= (u,v,w,y,z,φ,η (L,·))∈D(A)solution of the equation\n(2.28) −AU=F.\nEquivalently, we consider the following system\n−y=f1, (2.29)\n−z=f2, (2.30)\n−φ=f3, (2.31)\n−κ1uxx=f4, (2.32)\n−(κ2vx+δ1zx)x=f5, (2.33)\n−κ3wxx=f6, (2.34)\nηρ(L,ρ) =τf7(ρ). (2.35)\nIn addition, we consider the following boundary conditions\nu(0) = 0, u(α) =v(α), v(β) =w(β), (2.36)\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.37)\nwx(L) =−δ3η(L,0)−δ2η(L,1), (2.38)\nη(L,0) =φ(L). (2.39)\n10WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFrom ( 2.29)-(2.31) and the fact that F∈ H, it is clear that (y,z,φ)∈H1\nL. Next, from ( 2.31), (2.39) and the\nfact thatf3∈H1(β,L), we get\nη(L,0) =φ(L) =−f3(L).\nFrom the above equation and Equation ( 2.35), we can determine\nη(L,ρ) =τ/integraldisplayρ\n0f7(ξ)dξ−f3(L).\nIt is clear that η(L,·)∈H1(0,1)andη(L,0) =φ(L) =−f3(L). Inserting the above equation in ( 2.38), then\nSystem ( 2.29)-(2.39) is equivalent to\ny=−f1, z=−f2, φ=−f3, η(L,ρ) =τ/integraldisplayρ\n0f7(ξ)dξ−f3(L), (2.40)\n−κ1uxx=f4, (2.41)\n−(κ2vx+δ1zx)x=f5, (2.42)\n−κ3wxx=f6, (2.43)\nu(0) = 0, u(α) =v(α), v(β) =w(β), (2.44)\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.45)\nwx(L) = (δ3+δ2)f3(L)−τδ2/integraldisplay1\n0f7(ξ)dξ. (2.46)\nLet(ϕ,ψ,θ)∈H1\nL. Multiplying Equations ( 2.41), (2.42), (2.43) byϕ,ψ,θ, integrating over (0,α),(α,β)and\n(β,L)respectively, taking the sum, then using by parts integrati on, we get\n(2.47)κ1/integraldisplayα\n0uxϕxdx+/integraldisplayβ\nα(κ2vx+δ1zx)ψxdx+κ3/integraldisplayL\nβwxθxdx+κ1ux(0)ϕ(0)\n−κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α))ψ(α)−(κ2vx(β)+δ1zx(β))ψ(β)+κ3wx(β)θ(β)\n=/integraldisplayα\n0f4ϕdx+/integraldisplayβ\nαf5ψdx+/integraldisplayL\nβf6θdx+κ3wx(L)θ(L).\nFrom the fact that (ϕ,ψ,θ)∈H1\nL,we have\nϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β).\nInserting the above equation in ( 2.47), then using ( 2.40) and ( 2.44)-(2.46), we get\n(2.48)κ1/integraldisplayα\n0uxϕxdx+κ2/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx=/integraldisplayα\n0f4ϕdx\n+/integraldisplayβ\nα/parenleftbig\nδ1(f2)xψx+f5ψ/parenrightbig\ndx+/integraldisplayL\nβf6θdx+κ3/parenleftbigg\n(δ3+δ2)f3(L)−τδ2/integraldisplay1\n0f7(ξ)dξ/parenrightbigg\nθ(L).\nWe can easily verify that the left hand side of ( 2.48) is a bilinear continuous coercive form on H1\nL×H1\nL,\nand the right hand side of ( 2.48) is a linear continuous form on H1\nL. Then, using Lax-Milgram theorem,\nwe deduce that there exists (u,v,w)∈H1\nLunique solution of the variational Problem ( 2.48). Using stan-\ndard arguments, we can show that (u,κ2v+δ1z,w)∈H2. Finally, by seting y=−f1, z=−f2, φ=−f3\nandη(L,ρ) =τ/integraldisplayρ\n0f7(ξ)dξ−f3(L)and by applying the classical elliptic regularity we deduce thatU=\n(u,v,w,y,z,φ,η (L,·))∈D(A)is solution of Equation ( 2.28). To conclude, we need to show the uniqueness of\nU. So, letU= (u,v,w,y,z,φ,η (L,·))∈D(A)be a solution of ( 2.28) withF= 0, then we directly deduce that\ny=z=φ=η(L,ρ) = 0 and that (u,v,w)∈H1\nLsatisfies Problem ( 2.48) with zero in the right hand side. This\nimplies that u=v=w= 0, in other words, ker A={0}and0belongs to the resolvent set ρ(A)ofA. Then,\nby contraction principale, we easily deduce that R(λI− A) =Hfor sufficiently small λ >0. This, together\nwith the dissipativeness of A, imply that D(A)is dense in Hand that Ais m-dissipative in H(see Theorems\n4.5, 4.6 in [ 36]). The proof is thus complete. /square\nThanks to Lumer-Philips theorem (see [ 36]), we deduce that Agenerates a C0−semigroup of contractions etA\ninHand therefore Problem ( 2.9)-(2.22) is well-posed. Then we have the following result:\n11WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nTheorem 2.3. Under hypothesis (H), for any U0∈ H,Problem (2.23)admits a unique weak solution,\nU(x,ρ,t) =etAU0(x,ρ), such that\nU∈C0(R+,H).\nMoreover, if U0∈D(A),then\nU∈C1(R+,H)∩C0(R+,D(A)).\n2.1.2. Strong Stability. Our main result in this part is the following theorem.\nTheorem 2.4. Under hypothesis (H), theC0−semigroup of contractions etAis strongly stable on the Hilbert\nspaceHin the sense that\nlim\nt→+∞||etAU0||H= 0, ∀U0∈ H.\nFor the proof of Theorem 2.4, according to Theorem A.2, we need to prove that the operator Ahas no pure\nimaginary eigenvalues and σ(A)∩iRcontains only a countable number of continuous spectrum of A. The\nargument for Theorem 2.4relies on the subsequent lemmas.\nLemma 2.5. Under hypothesis (H), forλ∈R,we haveiλI−Ais injective, i.e.\nker(iλI−A) ={0},∀λ∈R.\nProof. From Proposition 2.2, we have 0∈ρ(A).We still need to show the result for λ∈R∗.Suppose that there\nexists a real number λ/\\e}atio\\slash= 0andU= (u,v,w,y,z,φ,η (L,·))∈D(A)such that\n(2.49) AU=iλU.\nFirst, similar to Equation ( 2.27), we have\n0 =Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH≤ −δ1/integraldisplayβ\nα|zx|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|η(L,0)|2≤0.\nThus,\n(2.50) zx= 0 in(α,β)andη(L,1) =η(L,0) = 0.\nNext, writing ( 2.49) in a detailed form gives\ny=iλu, x∈(0,α), (2.51)\nz=iλv, x∈(α,β), (2.52)\nw=iλφ, x∈(β,L), (2.53)\nκ1uxx=iλy, x∈(0,α), (2.54)\n(κ2vx+δ1zx)x=iλz, x∈(α,β), (2.55)\nκ3wxx=iλφ, x∈(β,L), (2.56)\nηρ(L,ρ) =−iλτη(L,ρ), ρ∈(0,1). (2.57)\nFrom ( 2.57) and ( 2.50), we get\n(2.58) η(L,·) =η(L,0)e−iλτ·= 0 in(0,1).\nCombining ( 2.50) with ( 2.52), we get that\n(2.59) vx=zx= 0 in(α,β).\nThus,\nvxx=zxx= 0 in(α,β).\nInserting the above result in ( 2.55), then taking into consideration ( 2.52), we obtain\n(2.60) v=z= 0 in(α,β).\nFrom the definition of D(A)and using ( 2.58)-(2.60), we get\n\n\nu(α) =v(α) = 0, w(β) =v(β) = 0, y(α) =z(α) = 0, φ(β) =z(β) = 0,\nκ1ux(α) =κ2vx(α)+δ1zx(α) = 0, κ3wx(β) =k2vx(β)+δ1zx(β) = 0,\nw(L) =iλφ(L) =iλη(L,0) = 0, wx(L) =−δ3η(L,0)−δ2η(L,1) = 0.\n12WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nCombining ( 2.51) with ( 2.54) and ( 2.53) with ( 2.56) and using the above equation as boundary conditions, we\nget\n\nuxx+λ2\nκ1u= 0, x∈(0,α),\nu(0) =u(α) =ux(α) = 0,\n\nwxx+λ2\nκ3w= 0, x∈(β,L),\nw(β) =w(L) =wx(β) =wx(L) = 0.\nThus,\n(2.61) u(x) = 0∀x∈(0,α)andw(x) = 0∀x∈(β,L).\nCombining ( 2.61) with ( 2.51) and ( 2.53), we obtain\ny(x) = 0∀x∈(0,α)andφ(x) = 0∀x∈(β,L).\nFinally, from the above result, ( 2.58), (2.60) and ( 2.61), we get that U= 0.The proof is thus complete. /square\nLemma 2.6. Under hypothesis (H), forλ∈R,we haveiλI−Ais surjective, i.e.\nR(iλI−A) =H,∀λ∈R.\nProof. Since0∈ρ(A),we still need to show the result for λ∈R∗.For anyF= (f1,f2,f3,f4,f5,f6,f7(L,·))∈ H\nandλ∈R∗,we prove the existence of U= (u,v,w,y,z,φ,η (L,·))∈D(A)solution for the following equation\n(iλI−A)U=F.\nEquivalently, we consider the following problem\ny=iλu−f1inH1(0,α), (2.62)\nz=iλv−f2inH1(α,β), (2.63)\nφ=iλw−f3inH1(β,L), (2.64)\niλy−κ1uxx=f4inL2(0,α), (2.65)\niλz−(κ2vx+δ1zx)x=f5inL2(α,β), (2.66)\niλφ−κ3wxx=f6inL2(β,L), (2.67)\nηρ(L,·)+iτλη(L,·) =τf7(L,·)inL2(0,1), (2.68)\nwith the following boundary conditions\nu(0) = 0, u(α) =v(α), v(β) =w(β), (2.69)\nκ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.70)\nwx(L) =−δ3η(L,0)−δ2η(L,1), (2.71)\nη(L,0) =φ(L). (2.72)\nIt follows from ( 2.68), (2.72) and ( 2.64) that\n(2.73) η(L,ρ) = (iλw(L)−f3(L))e−iτλρ+τ/integraldisplayρ\n0eiτλ(ξ−ρ)f7(L,ξ)dξ.\nInserting ( 2.62)-(2.64) and ( 2.73) in (2.65)-(2.72) and deriving ( 2.63) with respect to x, we get\n−λ2u−κ1uxx=iλf1+f4, (2.74)\n−λ2v−(κ2vx+δ1zx)x=iλf2+f5, (2.75)\n−λ2w−κ3wxx=iλf3+f6, (2.76)\nzx=iλvx−(f2)x, (2.77)\nu(0) = 0, u(α) =v(α), κ1ux(α) =κ2vx(α)+δ1zx(α), (2.78)\nw(β) =v(β), κ3wx(β) =κ2vx(β)+δ1zx(β), (2.79)\nwx(L) =−iλ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nw(L)+/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nf3(L)−τδ2/integraldisplay1\n0eiτλ(ξ−1)f7(L,ξ)dξ. (2.80)\n13WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nLet(ϕ,ψ,θ)∈H1\nL. Multiplying Equations ( 2.74), (2.75), (2.76) byϕ,ψ,θ, integrating over (0,α),(α,β)and\n(β,L)respectively, taking the sum, then using by parts integrati on, we get\n(2.81)κ1/integraldisplayα\n0uxϕxdx+/integraldisplayβ\nα(κ2vx+δ1zx)ψxdx+κ3/integraldisplayL\nβwxθxdx+κ1ux(0)ϕ(0)\n−κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α))ψ(α)−(κ2vx(β)+δ1zx(β))ψ(β)+κ3wx(β)θ(β)\n−λ2/integraldisplayα\n0uϕdx−λ2/integraldisplayβ\nαvψdx−λ2/integraldisplayL\nβwθdx−κ3wx(L)θ(L)\n=/integraldisplayα\n0(iλf1+f4)ϕdx+/integraldisplayβ\nα(iλf2+f5)ψdx+/integraldisplayL\nβ(iλf3+f6)θdx.\nFrom the fact that (ϕ,ψ,θ)∈H1\nL,we have\nϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β).\nInserting the above equation in ( 2.81), then using ( 2.77)-(2.80), we get\n(2.82) a((u,v,w),(ϕ,ψ,θ)) = F(ϕ,ψ,θ),∀(ϕ,ψ,θ)∈H1\nL,\nwhere\nF(ϕ,ψ,θ) =/integraldisplayα\n0(iλf1+f4)ϕdx+/integraldisplayβ\nα(iλf2+f5)ψdx+δ1/integraldisplayβ\nα(f2)xψxdx\n+/integraldisplayL\nβ(iλf3+f6)θdx+κ3/parenleftbigg/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nf3(L)−τδ2/integraldisplay1\n0eiτλ(ξ−1)f7(L,ξ)dξ/parenrightbigg\nθ(L)\nand\na((u,v,w),(ϕ,ψ,θ)) =a1((u,v,w),(ϕ,ψ,θ))+a2((u,v,w),(ϕ,ψ,θ)),\nsuch that\n\na1((u,v,w),(ϕ,ψ,θ)) =κ1/integraldisplayα\n0uxϕxdx+(κ2+iδ1λ)/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx,\na2((u,v,w),(ϕ,ψ,θ)) =−λ2/integraldisplayα\n0uϕdx−λ2/integraldisplayβ\nαvψdx−λ2/integraldisplayL\nβwθdx+iκ3λ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nw(L)θ(L).\nLet/parenleftbig\nH1\nL/parenrightbig′be the dual space of H1\nL. We define the operators A,A1andA2by\n/braceleftBigg\nA :H1\nL→/parenleftbig\nH1\nL/parenrightbig′\n(u,v,w)→A(u,v,w)/braceleftBigg\nA1:H1\nL→/parenleftbig\nH1\nL/parenrightbig′\n(u,v,w)→A1(u,v,w)/braceleftBigg\nA2:H1\nL→/parenleftbig\nH1\nL/parenrightbig′\n(u,v,w)→A2(u,v,w)\nsuch that\n(2.83)\n\n(A(u,v,w))(ϕ,ψ,θ) =a((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1\nL,\n(A1(u,v,w))(ϕ,ψ,θ) =a1((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1\nL,\n(A2(u,v,w))(ϕ,ψ,θ) =a2((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1\nL.\nOur aim is to prove that the operator Ais an isomorphism. For this aim, we proceed the proof in three steps.\nStep 1. In this step we proof that the operator A1is an isomorphism. For this aim, according to ( 2.83), we\nhave\na1((u,v,w),(ϕ,ψ,θ)) =κ1/integraldisplayα\n0uxϕxdx+(κ2+iδ1λ)/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx\nWe can easily verify that a1is a bilinear continuous coercive form on H1\nL×H1\nL. Then, by Lax-Milgram lemma,\nthe operator A1is an isomorphism.\nStep 2. In this step we proof that the operator A2is compact. First, for1\n2<r<1, we introduce the Hilbert\nspaceHr\nLby\nHr\nL={(ϕ,ψ,θ)∈Hr(0,α)×Hr(α,β)×Hr(β,L)|ϕ(0) = 0, ϕ(α) =ψ(α), ψ(β) =θ(β)}.\n14WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nThus by trace theorem, there exists C >0, such that\n(2.84) |θ(L)| ≤C/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr\nL.\nNow, according to ( 2.83), we have\na2((u,v,w),(ϕ,ψ,θ)) =−λ2/integraldisplayα\n0uϕdx−λ2/integraldisplayβ\nαvψdx−λ2/integraldisplayL\nβwθdx+iκ3λ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\nw(L)θ(L).\nThen, by using ( 2.84), we get\n|a2((u,v,w),(ϕ,ψ,θ))| ≤C1/ba∇dbl(u,v,w)/ba∇dblH1\nL/ba∇dbl(ϕ,ψ,θ)/ba∇dblL2+C1/ba∇dbl(u,v,w)/ba∇dblH1\nL/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr\nL,\nwhereC1>0. Therefore, for all r∈(1\n2,1)there exists C2>0, such that\n|a2((u,v,w),(ϕ,ψ,θ))| ≤C2/ba∇dbl(u,v,w)/ba∇dblH1\nL/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr\nL,\nwhich implies that\nA2∈ L/parenleftbig\nH1\nL,(Hr\nL)′/parenrightbig\n.\nFinally, using the compactness embedding from (Hr\nL)′into/parenleftbig\nH1\nL/parenrightbig′we deduce that A2is compact.\nFrom steps 1 and 2, we get that the operator A = A 1+A2is a Fredholm operator of index zero 0. Consequently,\nby Fredholm alternative, proving the operator Ais an isomorphism reduces to proving ker(A) = {0}.\nStep 3. In this step we proof that the ker(A) = {0}. For this aim, let (˜u,˜v,˜w)∈ker(A) ,i.e.\na((˜u,˜v,˜w),(ϕ,ψ,θ)) = 0,∀(ϕ,ψ,θ)∈H1\nL.\nEquivalently,\n/integraldisplayα\n0/parenleftbig\nκ1˜uxϕx−λ2˜uϕ/parenrightbig\ndx+/integraldisplayβ\nα/parenleftbig\n(κ2+iδ1λ)˜vxψx−λ2˜vψ/parenrightbig\ndx+/integraldisplayL\nβ/parenleftbig\nκ3˜wxθx−λ2˜wθ/parenrightbig\ndx\n+iκ3λ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\n˜w(L)θ(L) = 0,∀(ϕ,ψ,θ)∈H1\nL.\nThen, we find that\n\n−λ2˜u−κ1˜uxx= 0,\n−λ2˜v−(κ2+iδ1λ)˜vxx= 0,\n−λ2˜w−κ3˜wxx= 0,\n˜u(0) = 0,˜u(α) = ˜v(α), κ1˜ux(α) = (κ2+iδ1λ)˜vx(α),\n˜w(β) = ˜v(β), κ3˜wx(β) = (κ2+iδ1λ)˜vx(β),\n˜wx(L) =−iλ/parenleftbig\nδ3+δ2e−iτλ/parenrightbig\n˜w(L).\nTherefore, the vector ˜Vdefine by\n˜V=/parenleftbig\n˜u,˜v,˜w,iλ˜u,iλ˜v,iλ˜w,iλ˜w(L)e−iτλ·/parenrightbig\nbelongs toD(A)and we have\niλ˜V−A˜V= 0.\nThus,˜V∈ker(iλI−A), therefore by Lemma 2.5, we get ˜V= 0, this implies that ˜u= 0,˜v= 0and˜w= 0, so\nker(A) = {0}.\nTherefore, from step 3 and Fredholm alternative, we get that the operator Ais an isomorphism. It easy to\nsee that the operator Fis continuous form on H1\nL. Consequently, Equation ( 2.82) admits a unique solution\n(u,v,w)∈H1\nL. Thus, using ( 2.62)-(2.64), (2.73) and a classical regularity arguments, we conclude that (iλI−\nA)U=Fadmits a unique solution U∈D(A). The proof is thus complete. /square\nProof of Theorem 2.4.Form Lemma 2.5, we have that the operator Ahas no pure imaginary eigenvalues and\nby Lemma 2.6,R(iλI−A) =Hfor allλ∈R.Therefore, the closed graph theorem implies that σ(A)∩iR=∅.\nThus, we get the conclusion by applying Theorem A.2of Arendt and Batty. The proof is thus complete. /square\n15WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\n2.1.3. Polynomial Stability. In this part, we will prove the polynomial stability of Syste m (2.9)-(2.22). Our\nmain result in this part is the following theorem.\nTheorem 2.7. Under hypothesis (H), for all initial data U0∈D(A),there exists a constant C >0independent\nofU0such that the energy of System (2.9)-(2.22)satisfies the following estimation\n(2.85) E(t)≤C\nt4/ba∇dblU0/ba∇dbl2\nD(A),∀t>0.\nFrom Lemma 2.5and Lemma 2.6, we have seen that iR⊂ρ(A),then for the proof of Theorem 2.7, according\nto Theorem A.5(part (ii)), we need to prove that\n(2.86) sup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble\nL(H)=O/parenleftBig\n|λ|1\n2/parenrightBig\n.\nWe will argue by contradiction. Indeed, suppose there exist s\n{(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(L,·)))}n≥1⊂R∗\n+×D(A),\nsuch that\n(2.87) λn→+∞,/ba∇dblUn/ba∇dblH= 1\nand there exists sequence Fn:= (f1,n,f2,n,f3,n,f4,n,f5,n,f6,n,f7,n(L,·))∈ H, such that\n(2.88) λℓ\nn(iλnI−A)Un=Fn→0inH.\nIn case that ℓ=1\n2, we will check condition ( 2.86) by finding a contradiction with /ba∇dblUn/ba∇dblH= 1such as/ba∇dblUn/ba∇dblH=\no(1).From now on, for simplicity, we drop the index n. By detailing Equation ( 2.88), we get the following\nsystem\niλu−y=λ−ℓf1inH1(0,α), (2.89)\niλv−z=λ−ℓf2inH1(α,β), (2.90)\niλw−φ=λ−ℓf3inH1(β,L), (2.91)\niλy−κ1uxx=λ−ℓf4inL2(0,α), (2.92)\niλz−(κ2vx+δ1zx)x=λ−ℓf5inL2(α,β), (2.93)\niλφ−κ3wxx=λ−ℓf6inL2(β,L), (2.94)\nηρ(L,·)+iτλη(L,·) =τλ−ℓf7(L,·)inL2(0,1). (2.95)\nRemark that, since U= (u,v,w,y,z,φ,η (L,·))∈D(A), we have the following boundary conditions\n(2.96)/braceleftBigg\n|ux(α)|=κ−1\n1|κ2vx(α)+δ1zx(α)|,|y(α)|=|z(α)|,\n|wx(β)|=κ−1\n3|κ2vx(β)+δ1zx(β)|,|z(β)|=|φ(β)|\nand\n(2.97) wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0).\nThe proof of Theorem 2.7is divided into several lemmas.\nLemma 2.8. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n/integraldisplayβ\nα|zx|2=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.98)\n|φ(L)|2=|η(L,0)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n,|η(L,1)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.99)\n/integraldisplayβ\nα|vx|2dx=o/parenleftbig\nλ−ℓ−2/parenrightbig\n, (2.100)\n|wx(L)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n. (2.101)\n16WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nProof. Taking the inner product of ( 2.88) withUinH, then using the fact that Uis uniformly bounded in H,\nwe get\n−Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=Re/a\\}b∇acketle{t(iλI−A)U,U/a\\}b∇acket∇i}htH=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nNow, under hypothesis (H), similar to Equation ( 2.27), we get\n(2.102) 0≤δ1/integraldisplayβ\nα|zx|2dx+C1|η(L,1)|2+C2|η(L,0)|2≤ −Re/a\\}b∇acketle{tAU,U/a\\}b∇acket∇i}htH=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nwhere\nC1=1\n2−κ3|δ2|\n2p>0andC2=κ3δ3−1\n2−κ3|δ2|p\n2>0.\nTherefore, from ( 2.102), we get ( 2.98) and ( 2.99). Next, from ( 2.90), (2.98) and the fact that (f2)x→0in\nL2(α,β), we get ( 2.100). Finally, from ( 2.97) and ( 2.99), we obtain ( 2.101). Thus, the proof of the lemma is\ncomplete. /square\nLemma 2.9. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimation\n(2.103)/integraldisplay1\n0|η(L,ρ)|2dρ=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nProof. It follows from ( 2.95) that\nη(L,ρ) =η(L,0)e−iτλρ+τλ−ℓ/integraldisplayρ\n0eiτλ(ξ−ρ)f7(L,ξ)dξ∀ρ∈(0,1).\nBy using Cauchy Schwarz inequality, we get\n|η(L,ρ)|2≤2|η(L,0)|2+2τ2λ−2ℓ/parenleftbigg/integraldisplay1\n0|f7(L,ξ)|dξ/parenrightbigg2\n≤2|η(L,0)|2+2τ2λ−2ℓ/integraldisplay1\n0|f7(L,ξ)|2dξ∀ρ∈(0,1).\nIntegrating over (0,1)with respect to ρ, then using ( 2.99) and the fact that f7(L,·)→0inL2(0,1), we get\n/integraldisplay1\n0|η(L,ρ)|2dρ≤2|η(L,0)|2+2τ2λ−2ℓ/integraldisplay1\n0|f7(L,ξ)|2dξ=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nhence, we get ( 2.103). Thus, the proof of the lemma is complete. /square\nLemma 2.10. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n/integraldisplayL\nβ|φ|2dx=o/parenleftbig\nλ−ℓ/parenrightbig\n,/integraldisplayL\nβ|wx|2dx=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.104)\n|wx(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n,|φ(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n, (2.105)\n|κ2vx(β)+δ1zx(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n,|z(β)|2=o/parenleftbig\nλ−ℓ/parenrightbig\n. (2.106)\nProof. Multiplying Equation ( 2.94) byxwxand integrating over (β,L),we get\n(2.107) iλ/integraldisplayL\nβxφwxdx−κ3/integraldisplayL\nβxwxxwxdx=λ−ℓ/integraldisplayL\nβxf6wxdx.\nFrom ( 2.91), we deduce that\niλwx=−φx−λ−ℓ(f3)x.\nInserting the above result in ( 2.107), then using the fact that φ, wxare uniformly bounded in L2(β,L)and\n(f3)x, f6converge to zero in L2(β,L)gives\n−/integraldisplayL\nβxφφxdx−κ3/integraldisplayL\nβxwxxwxdx=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nTaking the real part in the above equation, then using by part s integration, we get\n1\n2/integraldisplayL\nβ|φ|2dx+κ3\n2/integraldisplayL\nβ|wx|2dx+β\n2/parenleftbig\nκ3|wx(β)|2+|φ(β)|2/parenrightbig\n=L\n2/parenleftbig\nκ3|wx(L)|2+|φ(L)|2/parenrightbig\n+o/parenleftbig\nλ−ℓ/parenrightbig\n.\n17WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nInserting ( 2.99) and ( 2.101) in the above equation, we get\n1\n2/integraldisplayL\nβ|φ|2dx+κ3\n2/integraldisplayL\nβ|wx|2dx+β\n2/parenleftbig\nκ3|wx(β)|2+|φ(β)|2/parenrightbig\n=o/parenleftbig\nλ−ℓ/parenrightbig\n,\nhence, we get ( 2.104) and ( 2.105). Finally, from ( 2.96) and ( 2.105), we obtain ( 2.106). The proof is thus\ncomplete. /square\nLemma 2.11. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n/integraldisplayβ\nα|z|2dx=o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n, (2.108)\n|z(α)|2=o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n,|z(β)|2=o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n, (2.109)\n|κ2vx(α)+δ1zx(α)|2=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\n. (2.110)\nProof. Letg∈C1([α,β])such that\ng(β) =−g(α) = 1,max\nx∈[α,β]|g(x)|=cgandmax\nx∈[α,β]|g′(x)|=cg′,\nwherecgandcg′are strictly positive constant numbers independent from λ. The proof is divided into three\nsteps.\nStep 1. In this step, we prove the following asymptotic behavior est imate\n(2.111) |z(β)|2+|z(α)|2≤/parenleftBigg\nλ1\n2\n2+2cg′/parenrightBigg/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n.\nFirst, from ( 2.90), we have\n(2.112) zx=iλvx−λ−ℓ(f2)xinL2(α,β).\nMultiplying ( 2.112) by2gzand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nαg(x)(|z|2)xdx=Re/braceleftBigg\n2iλ/integraldisplayβ\nαg(x)vxzdx/bracerightBigg\n−Re/braceleftBigg\n2λ−ℓ/integraldisplayβ\nαg(x)(f2)xzdx/bracerightBigg\n,\nusing by parts integration in the left hand side of above equa tion, we get\n/bracketleftbig\ng(x)|z|2/bracketrightbigβ\nα=/integraldisplayβ\nαg′(x)|z|2dx+Re/braceleftBigg\n2iλ/integraldisplayβ\nαg(x)vxzdx/bracerightBigg\n−Re/braceleftBigg\n2λ−ℓ/integraldisplayβ\nαg(x)(f2)xzdx/bracerightBigg\n,\nconsequently,\n(2.113) |z(β)|2+|z(α)|2≤cg′/integraldisplayβ\nα|z|2dx+2λcg/integraldisplayβ\nα|vx||z|dx+2λ−ℓcg/integraldisplayβ\nα|(f2)x||z|dx.\nOn the other hand, we have\n2λcg|vx||z| ≤λ1\n2|z|2\n2+2λ3\n2c2\ng|vx|2and2λ−ℓcg|(f2)x||z| ≤cg′|z|2+c2\ngλ−2ℓ\ncg′|(f2)x|2.\nInserting the above equation in ( 2.113), then using ( 2.100) and the fact that (f2)x→0inL2(α,β), we get\n|z(β)|2+|z(α)|2≤/parenleftBigg\nλ1\n2\n2+2cg′/parenrightBigg/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−min(2ℓ,ℓ+1\n2)/parenrightBig\n,\nhence, we get ( 2.111).\nStep 2. In this step, we prove the following asymptotic behavior est imate\n(2.114) |κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2≤λ3\n2\n2/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−ℓ+1\n2/parenrightBig\n.\n18WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFirst, multiplying ( 2.93) by−2g(κ2vx+δ1zx)and integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nαg(x)/parenleftBig\n|κ2vx+δ1zx|2/parenrightBig\nxdx= 2Re/braceleftBigg\niλ/integraldisplayβ\nαg(x)z(κ2vx+δ1zx)dx/bracerightBigg\n−2λ−ℓRe/braceleftBigg/integraldisplayβ\nαg(x)f5(κ2vx+δ1zx)dx/bracerightBigg\n,\nusing by parts integration in the left hand side of above equa tion, we get\n/bracketleftBig\ng(x)|κ2vx+δ1zx|2/bracketrightBigβ\nα=/integraldisplayβ\nαg′(x)|κ2vx+δ1zx|2dx+2Re/braceleftBigg\niλ/integraldisplayβ\nαg(x)z(κ2vx+δ1zx)dx/bracerightBigg\n−2λ−ℓRe/braceleftBigg/integraldisplayβ\nαg(x)f5(κ2vx+δ1zx)dx/bracerightBigg\n,\nconsequently,\n|κ2vx(β)+δ1zx(β)|2+|κ2vx(α)+δ1zx(α)|2≤cg′/integraldisplayβ\nα|κ2vx+δ1zx|2dx+2λcg/integraldisplayβ\nα|z||κ2vx+δ1zx|dx\n+2λ−ℓcg/integraldisplayβ\nα|f5||κ2vx+δ1zx|dx.\nNow, using Cauchy Schwarz inequality, Equations ( 2.98), (2.100) and the fact that f5→0inL2(α,β)in the\nright hand side of above equation, we get\n(2.115) |κ2vx(β)+δ1zx(β)|2+|κ2vx(α)+δ1zx(α)|2≤2λcg/integraldisplayβ\nα|z||κ2vx+δ1zx|dx+o/parenleftbig\nλ−ℓ/parenrightbig\n.\nOn the other hand, we have\n2λcg|z||κ2vx+δ1zx| ≤λ3\n2\n2|z|2+2λ1\n2c2\ng|κ2vx+δ1zx|2.\nInserting the above equation in ( 2.115), then using Equations ( 2.98) and ( 2.100), we get\n|κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2≤λ3\n2\n2/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−ℓ+1\n2/parenrightBig\n,\nhence, we get ( 2.114).\nStep 3. In this step, we prove the asymptotic behavior estimations o f (2.108)-(2.110). First, multiplying ( 2.93)\nby−iλ−1zand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nα|z|2dx=−Re/braceleftBigg\niλ−1/integraldisplayβ\nα(κ2vx+δ1zx)xzdx/bracerightBigg\n−Re/braceleftBigg\niλ−ℓ−1/integraldisplayβ\nαf5zdx/bracerightBigg\n,\nconsequently,\n(2.116)/integraldisplayβ\nα|z|2dx≤λ−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\nα(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+λ−ℓ−1/integraldisplayβ\nα|f5||z|dx.\nFrom the fact that zis uniformly bounded in L2(α,β)andf5→0inL2(α,β), we get\n(2.117) λ−ℓ−1/integraldisplayβ\nα|f5||z|dx=o/parenleftbig\nλ−ℓ−1/parenrightbig\n.\n19WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nOn the other hand, using by parts integration and ( 2.98), (2.100), we get\n(2.118)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\nα(κ2v+δ1z)xxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle[(κ2vx+δ1zx)z]β\nα−/integraldisplayβ\nα(κ2vx+δ1zx)zxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ |κ2vx(β)+δ1zx(β)||z(β)|+|κ2vx(α)+δ1zx(α)||z(α)|+/integraldisplayβ\nα|κ2vx+δ1zx||zx|dx\n≤ |κ2vx(β)+δ1zx(β)||z(β)|+|κ2vx(α)+δ1z(α)||z(α)|+o/parenleftbig\nλ−ℓ/parenrightbig\n.\nInserting ( 2.117) and ( 2.118) in (2.116), we get\n(2.119)/integraldisplayβ\nα|z|2dx≤λ−1|κ2vx(β)+δ1zx(β)||z(β)|+λ−1|κ2vx(α)+δ1zx(α)||z(α)|+o/parenleftbig\nλ−ℓ−1/parenrightbig\n.\nNow, forζ=βorζ=α, we have\nλ−1|κ2vx(ζ)+δ1zx(ζ)||z(ζ)| ≤λ−1\n2\n2|z(ζ)|2+λ−3\n2\n2|κ2vx(ζ)+δ1zx(ζ)|2.\nInserting the above equation in ( 2.119), we get\n/integraldisplayβ\nα|z|2dx≤λ−1\n2\n2/parenleftbig\n|z(α)|2+|z(β)|2/parenrightbig\n+λ−3\n2\n2/parenleftBig\n|κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2/parenrightBig\n+o/parenleftbig\nλ−ℓ−1/parenrightbig\n.\nNext, inserting Equations ( 2.111) and ( 2.114) in the above inequality, we obtain\n/integraldisplayβ\nα|z|2dx≤/parenleftbigg1\n2+cg′\nλ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx+o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n,\nconsequently,/parenleftbigg1\n2−cg′\nλ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx≤o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n.\nSinceλ→+∞, by choosing λ>4c2\ng′, we get\n0</parenleftbigg1\n2−cg′\nλ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx≤o/parenleftBig\nλ−min(2ℓ+1\n2,ℓ+1)/parenrightBig\n,\nhence, we get ( 2.108). Finally, inserting ( 2.108) in (2.111) and ( 2.114) and using the first asymptotic estimates\nof (2.106), we get ( 2.109) and ( 2.110). Thus, the proof of the lemma is complete. /square\nRemark 2.12. An example about g, we can take g(x) = cos/parenleftbigg(β−x)π\nβ−α/parenrightbigg\nto get\ng(β) =−g(α) = 1, g∈C1([α,β]),max\nx∈[α,β]|g(x)|= 1,max\nx∈[α,β]|g′(x)|=π\nβ−α.\nAlso, we can take\ng(x) =/parenleftbiggβ−x\nβ−α/parenrightbigg2\n−3/parenleftbiggβ−x\nβ−α/parenrightbigg\n+1.\n/square\nLemma 2.13. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations\n(2.89)-(2.95)satisfies the following asymptotic behavior estimations\n(2.120)/integraldisplayα\n0|y|2dx=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\nand/integraldisplayα\n0|ux|2dx=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\n.\nProof. Multiply Equation ( 2.92) byxuxand integrating over (0,α),we get\n(2.121) iλ/integraldisplayα\n0xyuxdx−κ1/integraldisplayα\n0xuxxuxdx=λ−ℓ/integraldisplayα\n0xf4uxdx.\nFrom ( 2.89), we deduce that\niλux=−yx−λ−ℓ(f1)x.\n20WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nInserting the above result in ( 2.121), then using the fact that ux, yare uniformly bounded in L2(0,α)and\n(f1)x, f4converge to zero in L2(0,α)gives\n−/integraldisplayα\n0xyyxdx−κ1/integraldisplayα\n0xuxxuxdx=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nTaking the real part in the above equation, then using by part s integration, we get\n(2.122)1\n2/integraldisplayα\n0|y|2dx+κ1\n2/integraldisplayα\n0|ux|2dx−α\n2/parenleftbig\nκ1|ux(α)|2+|y(α)|2/parenrightbig\n=o/parenleftbig\nλ−ℓ/parenrightbig\n.\nInserting the boundary conditions ( 2.96) atx=αin (2.122) gives\n1\n2/integraldisplayα\n0|y|2dx+κ1\n2/integraldisplayα\n0|ux|2dx=α\n2/parenleftbig\nκ−1\n1|κ2vx(α)+δ1zx(α)|2+|z(α)|2/parenrightbig\n+o/parenleftbig\nλ−ℓ/parenrightbig\n.\nInserting ( 2.109)-(2.110) in the above equation, we obtain the first and the second asym ptotic estimates of\n(2.120). The proof is thus complete. /square\nProof of Theorem 2.7.From Lemma 2.8, Lemma 2.9, Lemma 2.10, Lemma 2.11and Lemma 2.13, we get\n/ba∇dblU/ba∇dbl2\nH=κ1/integraldisplayα\n0|ux|2dx+κ2/integraldisplayβ\nα|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx+/integraldisplayα\n0|y|2dx\n+/integraldisplayβ\nα|z|2dx+/integraldisplayL\nβ|φ|2dx+τ/integraldisplay1\n0|η(L,ρ)|2dρ=o/parenleftBig\nλ−min(2ℓ−1,ℓ−1\n2)/parenrightBig\n.\nTo obtain /ba∇dblU/ba∇dblH=o(1), we need min/parenleftbigg\n2ℓ−1,ℓ−1\n2/parenrightbigg\n≥0, so we choose ℓ=1\n2as the optimal value. Hence,\nwe obtain that /ba∇dblU/ba∇dblH=o(1)which contradicts ( 2.87). Therefore, the energy of System ( 2.9)-(2.22) satisfies\nestimation ( 2.85) for all initial data U0∈D(A). /square\n2.2.Wave equation with local Kelvin-Voigt damping near the boun dary and boundary delay\nfeedback. In this subsection, we study the stability of System ( 2.1), but in the case that the Kelvin-Voigt\ndamping is near the boundary, i.e.α= 0and0<β <L (see Figure 2). For this aim, we denote the longitudinal\ndisplacement by Uand this displacement is divided into two parts\nU(x,t) =/braceleftBigg\nv(x,t),(x,t)∈(α,β)×(0,+∞),\nw(x,t),(x,t)∈(β,L)×(0,+∞).\nIn this case, System ( 2.1) is equivalent to the following system\n(2.123)\n\nvtt−(κ2vx+δ1vxt)x= 0, (x,t)∈(0,β)×(0,+∞),\nwtt−κ3wxx= 0, (x,t)∈(β,L)×(0,+∞),\nτηt(L,ρ,t)+ηρ(L,ρ,t) = 0, (ρ,t)∈(0,1)×(0,+∞),\nv(0,t) = 0, t ∈(0,+∞),\nwx(L,t) =−δ3wt(L,t)−δ2η(L,1,t), t∈(0,+∞),\nv(β,t) =w(β,t), t ∈(0,+∞),\nκ2vx(β,t)+δ1vxt(β,t) =κ3wx(β,t), t∈(0,+∞),\n(v(x,0),vt(x,0)) = (v0(x),v1(x)), x ∈(α,β),\n(w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L),\nη(L,ρ,0) =f0(L,−ρτ), ρ ∈(0,1),\nwhere the initial data (v0,v1,w0,w1,f0)belongs to a suitable space. Similar to Section 2.1, we define\nXm=Hm(0,β)×Hm(β,L), m= 1,2,\nX0=L2(0,β)×L2(β,L),X1\nL={(v,w)∈X1|v(0) = 0, v(β) =w(β)},\n21WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwhere the Hilbert space X0is equipped with the norm:\n/ba∇dbl(v,w)/ba∇dbl2\nX0=/integraldisplayβ\n0|v|2dx+/integraldisplayL\nβ|w|2dx.\nMoreover, it is easy to check that the space X1\nLis Hilbert space over Cequipped with the norm:\n/ba∇dbl(v,w)/ba∇dbl2\nX1\nL=κ2/integraldisplayβ\n0|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx.\nIn addition, by Poincaré inequality we can easily verify tha t there exists C >0, such that\n/ba∇dbl(v,w)/ba∇dblX0≤C/ba∇dbl(v,w)/ba∇dblX1\nL.\nWe now define the Hilbert energy space by\nH1=X1\nL×X0×L2(0,1)\nequipped with the following inner product\n/a\\}b∇acketle{tU,˜U/a\\}b∇acket∇i}htH1=κ2/integraldisplayβ\n0vx˜vxdx+κ3/integraldisplayL\nβwx˜wxdx+/integraldisplayβ\n0z˜zdx+/integraldisplayL\nβφ˜φdx+τ/integraldisplay1\n0η(L,ρ)˜η(L,ρ)dρ,\nwhereU= (v,w,z,φ,η (L,·))∈ H1and˜U= (˜v,˜w,˜z,˜φ,˜η(L,·))∈ H1. We use /ba∇dblU/ba∇dblH1to denote the corresponding\nnorm. We define the linear unbounded operator A1:D(A1)⊂ H1−→ H 1by:\nD(A1) =/braceleftbigg\nU= (v,w,z,φ,η (L,·))∈X1\nL×X1\nL×H1(0,1)|(κ2v+δ1z,w)∈X2,\nκ2vx(β)+δ1zx(β) =κ3wx(β), wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0)/bracerightbigg\nand for all U= (v,w,z,φ,η (L,·))∈D(A1)\nA1U=/parenleftbig\nz,φ,(κ2vx+δ1zx)x,κ3wxx,−τ−1ηρ(L,·)/parenrightbig\n.\nIfU= (v,w,v t,wt,η(L,·))is a regular solution of System ( 2.123), then we transform this system into the\nfollowing initial value problem\n(2.124)/braceleftBigg\nUt=A1U,\nU(0) =U0,\nwhereU0= (v0,w0,v1,w1,f0(L,−·τ))∈ H1.Note thatD(A1)is dense in H1and that for all U∈D(A1), we\nhave\n(2.125) Re /a\\}b∇acketle{tA1U,U/a\\}b∇acket∇i}htH1≤ −δ1/integraldisplayβ\n0|zx|2dx−/parenleftbigg1\n2−κ3|δ2|\n2p/parenrightbigg\n|η(L,1)|2−/parenleftbigg\nκ3δ3−1\n2−κ3|δ2|p\n2/parenrightbigg\n|η(L,0)|2,\nwherepis defined in ( 2.8). Consequently, under hypothesis (H), the system becomes d issipative. We can easily\nadapt the proof in Subsection 2.1.1to prove the well-posedness of System ( 2.124).\nTheorem 2.14. Under hypothesis (H), for all initial data U0∈ H1,the System (2.123)is exponentially stable.\nAccording to Theorem A.5(part (i)), we have to check if the following conditions hold ,\n(2.126) iR⊆ρ(A1)\nand\n(2.127) sup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A1)−1/vextenddouble/vextenddouble/vextenddouble\nL(H1)=O(1).\nProof. First, we can easily adapt the proof in Subsection 2.1.2to prove the strong stability (condition ( 2.126))\nof System ( 2.123). Next, we will prove condition ( 2.127) by a contradiction argument. Indeed, suppose there\nexists\n{(λn,Un:= (vn,wn,zn,φn,ηn(L,·)))}n≥1⊂R∗\n+×D(A1),\nsuch that\n(2.128) λn→+∞,/ba∇dblUn/ba∇dblH1= 1\n22WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nand there exists sequence Gn:= (g1,n,g2,n,g3,n,g4,n,g5,n(L,·))∈ H1, such that\n(2.129) (iλnI−A1)Un=Gn→0inH1.\nWe will check condition ( 2.127) by finding a contradiction with /ba∇dblUn/ba∇dblH1= 1such as/ba∇dblUn/ba∇dblH1=o(1).From now\non, for simplicity, we drop the index n. By detailing Equation ( 2.129), we get the following system\niλv−z=g1inH1(0,β), (2.130)\niλw−φ=g2inH1(β,L), (2.131)\niλz−(κ2vx+δ1zx)x=g3inL2(0,β), (2.132)\niλφ−κ3wxx=g4inL2(β,L), (2.133)\nηρ(L,·)+iτλη(L,·) =τg5(L,·)inL2(0,1). (2.134)\nRemark that, since U= (v,w,z,φ,η (L,·))∈D(A1), we have the following boundary conditions\n|wx(β)|=κ−1\n3|κ2vx(β)+δ1zx(β)|,|z(β)|=|φ(β)|, (2.135)\nwx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0). (2.136)\nTaking the inner product of ( 2.129) withUinH1, then using ( 2.125), hypothesis (H) and the fact that Uis\nuniformly bounded in H1, we obtain\n(2.137)/integraldisplayβ\n0|zx|2=o(1),|φ(L)|2=|η(L,0)|2=o(1),|η(L,1)|2=o(1).\nFrom ( 2.130), then using the first asymptotic estimate of ( 2.137) and the fact that (g1)x→0inL2(0,β), we\nget\n(2.138)/integraldisplayβ\n0|vx|2dx=o/parenleftbig\nλ−2/parenrightbig\n.\nFrom the first asymptotic estimate of ( 2.136), then using the second and the third asymptotic estimates o f\n(2.137), we obtain\n(2.139) |wx(L)|2=o(1).\nSimilar to Lemma 2.9, withℓ= 0, from ( 2.134), then using the second and the third asymptotic estimates o f\n(2.137), we obtain\n(2.140)/integraldisplay1\n0|η(L,ρ)|2dρ=o(1).\nSimilar to Lemma 2.10, withℓ= 0, multiplying Equation ( 2.133) byxwxand integrating over (β,L),after that\nusing the fact that iλwx=−φx−(g2)x, then using the fact that φ, wxare uniformly bounded in L2(β,L)and\n(g2)x, g4converge to zero in L2(β,L)gives\n−/integraldisplayL\nβxφφxdx−κ3/integraldisplayL\nβxwxxwxdx=o(1).\nTaking the real part in the above equation, then using by part s integration, Equation ( 2.139) and the second\nasymptotic estimate of ( 2.137), we obtain\n1\n2/integraldisplayL\nβ|φ|2dx+κ3\n2/integraldisplayL\nβ|wx|2dx+β\n2/parenleftbig\nκ3|wx(β)|2+|φ(β)|2/parenrightbig\n=o(1),\nhence, we get\n(2.141)/integraldisplayL\nβ|φ|2dx=o(1),/integraldisplayL\nβ|wx|2dx=o(1),|wx(β)|2=o(1),|φ(β)|2=o(1).\nInserting the third and the fourth asymptotic estimates of ( 2.141) in (2.135), we get\n(2.142) |κ2vx(β)+δ1zx(β)|=o(1),|z(β)|=o(1).\n23WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nSimilar to step 3 of Lemma 2.11, withα= 0andℓ= 0, multiplying ( 2.132) by−iλ−1zand integrating over\n(0,β),taking the real part, then using the fact that zis uniformly bounded in L2(0,β)andg3→0inL2(0,β),\nwe get\n(2.143)/integraldisplayβ\n0|z|2dx≤λ−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\n0(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+o/parenleftbig\nλ−1/parenrightbig\n.\nOn the other hand, using by parts integration, the fact that z(0) = 0 , and Equations ( 2.137)-(2.138), (2.142),\nwe get\n(2.144)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\n0(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle[(κ2vx+δ1zx)z]β\n0−/integraldisplayβ\n0(κ2vx+δ1zx)zxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ |κ2vx(β)+δ1zx(β)||z(β)|+/integraldisplayβ\n0|κ2vx+δ1zx||zx|dx=o(1).\nInserting ( 2.144) in (2.143), we get\n(2.145)/integraldisplayβ\n0|z|2dx=o/parenleftbig\nλ−1/parenrightbig\n.\nFinally, from ( 2.138), (2.140), (2.141) and ( 2.145), we get\n/ba∇dblU/ba∇dblH1=o(1),\nwhich contradicts ( 2.128). Therefore, ( 2.127) holds and the result follows from Theorem A.5(part (i)). /square\n3.Wave equation with local internal Kelvin-Voigt damping and local internal delay\nfeedback\nIn this section, we study the stability of System ( 1.2). We assume that there exists αandβsuch that\n0< α < β < L , in this case, the Kelvin-Voigt damping and the time delay fe edback are locally internal (see\nFigure 3). For this aim, we denote the longitudinal displacement by Uand this displacement is divided into\nthree parts\nU(x,t) =\n\nu(x,t),(x,t)∈(0,α)×(0,+∞),\nv(x,t),(x,t)∈(α,β)×(0,+∞),\nw(x,t),(x,t)∈(β,L)×(0,+∞).\nFurthermore, let us introduce the auxiliary unknown\nη(x,ρ,t) =vt(x,t−ρτ), x∈(α,β), ρ∈(0,1), t>0.\nIn this case, System ( 1.2) is equivalent to the following system\nutt−κ1uxx= 0,(x,t)∈(0,α)×(0,+∞), (3.1)\nvtt−(κ2vx+δ1vxt(x,t)+δ2ηx(x,1,t))x= 0,(x,t)∈(α,β)×(0,+∞), (3.2)\nwtt−κ3wxx= 0,(x,t)∈(β,L)×(0,+∞), (3.3)\nτηt(x,ρ,t)+ηρ(x,ρ,t) = 0,(x,ρ,t)∈(α,β)×(0,1)×(0,+∞), (3.4)\nwith the Dirichlet boundary conditions\n(3.5) u(0,t) =w(L,t) = 0, t∈(0,+∞),\nwith the following transmission conditions\n(3.6)\n\nu(α,t) =v(α,t), v(β,t) =w(β,t), t ∈(0,+∞),\nκ1ux(α,t) =κ2vx(α,t)+δ1vxt(α,t)+δ2ηx(α,1,t), t∈(0,+∞),\nκ3wx(β,t) =κ2vx(β,t)+δ1vxt(β,t)+δ2ηx(β,1,t), t∈(0,+∞),\n24WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nand with the following initial conditions\n(3.7)\n\n(u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,α),\n(v(x,0),vt(x,0)) = (v0(x),v1(x)), x ∈(α,β),\n(w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L),\nη(x,ρ,0) =f0(x,−ρτ), (x,ρ)∈(α,β)×(0,1),\nwhere the initial data (u0,u1,v0,v1,w0,w1,f0)belongs to a suitable space. To a strong solution of System\n(3.1)-(3.7), we associate the energy defined by\nE(t) =1\n2/integraldisplayα\n0/parenleftbig\n|ut(x,t)|2+κ1|ux(x,t)|2/parenrightbig\ndx+1\n2/integraldisplayβ\nα/parenleftbig\n|vt(x,t)|2+κ2|vx(x,t)|2/parenrightbig\ndx\n+1\n2/integraldisplayL\nβ/parenleftbig\n|wt(x,t)|2+κ3|wx(x,t)|2/parenrightbig\ndx+τ|δ2|\n2/integraldisplay1\n0/integraldisplayβ\nα|ηx(x,ρ,t)|2dρdx.\nMultiplying ( 3.1), (3.2), (3.3) and ( 3.4)xbyut,yt,wtand|δ2|ηx, integrating over (0,α),(α,β),(β,L)and\n(α,β)×(0,1)respectively, taking the sum, then using by parts integrati on and the boundary conditions in\n(3.5)-(3.6), we get\nE′(t) =/parenleftbigg\n−δ1+|δ2|\n2/parenrightbigg/integraldisplayβ\nα|vxt(x,t)|2dx−|δ2|\n2/integraldisplayβ\nα|ηx(x,1,t)|2dx−δ2/integraldisplayβ\nαvxt(x,t)ηx(x,1,t)dx.\nUsing Young’s inequality for the third term in the right, we g et\nE′(t)≤(−δ1+|δ2|)/integraldisplayβ\nα|vxt(x,t)|2dx.\nIn the sequel, the assumption on δ1andδ2will ensure that\n(H1) δ1>0, δ2∈R∗,|δ2|<δ1.\nIn this case, the energies of the strong solutions satisfy E′(t)≤0.Hence, the System ( 3.1)-(3.7) is dissipative\nin the sense that its energy is non increasing with respect to the timet.\n3.1.Well-posedness of the problem. We start this part by formulating System ( 3.1)-(3.7) as an abstract\nCauchy problem. For this aim, let us define\nL2\n∗=L2(0,α)×L2\n∗(α,β)×L2(β,L),\nH1\n∗={(u,v,w)∈H1(0,α)×H1\n∗(α,β)×H1(β,L)|u(0) = 0, u(α) =v(α), v(β) =w(β), w(L) = 0},\nH2=H2(0,α)×H2(α,β)×H2(β,L).\nHere we consider\nL2\n∗(α,β) =/braceleftBigg\nz∈L2(α,β)|/integraldisplayβ\nαzdx= 0/bracerightBigg\nandH1\n∗(α,β) =H1(α,β)∩L2\n∗(α,β).\nThe spaces L2\n∗andH1\n∗are obviously a Hilbert spaces equipped respectively with t he norms\n/ba∇dbl(u,v,w)/ba∇dbl2\nL2∗=/integraldisplayα\n0|u|2dx+/integraldisplayβ\nα|v|2dx+/integraldisplayL\nβ|w|2dx\nand\n/ba∇dbl(u,v,w)/ba∇dbl2\nH1∗=κ1/integraldisplayα\n0|ux|2dx+κ2/integraldisplayβ\nα|vx|2dx+κ3/integraldisplayL\nβ|wx|2dx.\nIn addition by Poincaré inequality we can easily verify that there exists C >0, such that\n/ba∇dbl(u,v,w)/ba∇dblL2∗≤C/ba∇dbl(u,v,w)/ba∇dblH1∗.\nLet us define the energy Hilbert space H2by\nH2=H1\n∗×L2\n∗×L2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n25WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nequipped with the following inner product\n/a\\}b∇acketle{tU,˜U/a\\}b∇acket∇i}htH2=κ1/integraldisplayα\n0ux˜uxdx+κ2/integraldisplayβ\nαvx˜vxdx+κ3/integraldisplayL\nβwx˜wxdx\n+/integraldisplayα\n0y˜ydx+/integraldisplayβ\nαz˜zdx+/integraldisplayL\nβφ˜φdx+τ|δ2|/integraldisplay1\n0/integraldisplayβ\nαηx(x,ρ)˜ηx(x,ρ)dxdρ,\nwhereU= (u,v,w,y,z,φ,η (·,·))∈ H2and˜U= (˜u,˜v,˜w,˜y,˜z,˜φ,˜η(·,·))∈ H2. We use /ba∇dblU/ba∇dblH2to denote the\ncorresponding norm. We define the linear unbounded operator A2:D(A2)⊂ H2−→ H 2by:\nD(A2) =/braceleftbigg\n(u,v,w,y,z,φ,η (·,·))∈ H2|(y,z,φ)H1\n∗\n(u,κ2v+δ1z+δ2η(·,1),w)∈H2, κ2vx(α)+δ1zx(α)+δ2ηx(α,1) =κ1ux(α),\nκ2vx(β)+δ1zx(β)+δ2ηx(β,1) =κ3wx(β), η, ηρ∈L2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n, z(·) =η(·,0)/bracerightbigg\nand for all U= (u,v,w,y,z,φ,η (·,·))∈D(A2)\nA2U=/parenleftbig\ny,z,φ,κ 1uxx,(κ2vx+δ1zx+δ2ηx(·,1))x,κ3wxx,−τ−1ηρ(·,·)/parenrightbig\n.\nIfU= (u,v,w,u t,vt,wt,η(·,·))is a regular solution of System ( 3.1)-(3.7), then we transform this system into\nthe following initial value problem\n(3.8)/braceleftBigg\nUt=A2U,\nU(0) =U0,\nwhereU0= (u0,v0,w0,u1,v1,w1,f0(·,−·τ))∈ H2.We now use semigroup approach to establish well-posedness\nresult for the System ( 3.1)-(3.7). We prove the following proposition.\nProposition 3.1. Under hypothesis (H1), the unbounded linear operator A2is m-dissipative in the energy\nspaceH2.\nProof. For allU= (u,v,w,y,z,φ,η (·,·))∈D(A2),we have\nRe/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2=κ1Re/integraldisplayα\n0(yxux+uxxy)dx+Re/integraldisplayβ\nα(κ2zxvx+(κ2vx+δ1zx+δ2ηx(·,1))xz)dx\n+κ3Re/integraldisplayL\nβ/parenleftbig\nφxwx+wxxφ/parenrightbig\ndx−|δ2|Re/integraldisplayβ\nα/integraldisplay1\n0ηxρ(x,ρ)ηx(x,ρ)dxdρ.\nUsing by parts integration in the above equation, we get\n(3.9)Re/a\\}b∇acketle{tA2U,U2/a\\}b∇acket∇i}htH2=−δ1/integraldisplayβ\nα|zx|2dx−δ2Re/integraldisplayβ\nαηx(·,1)zxdx+|δ2|\n2/integraldisplayβ\nα|ηx(x,0)|2dx\n−|δ2|\n2/integraldisplayβ\nα|ηx(x,1)|2dx−κ1Re(ux(0)y(0))+κ3Re/parenleftbig\nwx(L)φ(L)/parenrightbig\n+Re(κ1ux(α)y(α)−κ2vx(α)z(α)−δ1zx(α)z(α)−δ2ηx(α,1)z(α))\n+Re/parenleftbig\nκ2vx(β)z(β)+δ1zx(β)z(β)+δ2ηx(β,1)z(β)−κ3wx(β)φ(β)/parenrightbig\n.\nSinceU∈D(A2), we have\n/braceleftBigg\ny(0) =φ(0) = 0, y(α) =z(α), z(β) =φ(β), z(x) =η(x,0),\nκ1ux(α)−κ2vx(α)−δ1zx(α)−δ2ηx(α,1) = 0, κ2vx(β)+δ1zx(β)+δ2ηx(β,1)−κ3wx(β) = 0.\nSubstituting the above boundary conditions in ( 3.9), then using Young’s inequality, we get\n(3.10)Re/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2=/parenleftbigg\n−δ1+|δ2|\n2/parenrightbigg/integraldisplayβ\nα|zx|2dx−|δ2|\n2/integraldisplayβ\nα|ηx(x,1)|2dx−δ2Re/integraldisplayβ\nαηx(·,1)zxdx\n≤(−δ1+|δ2|)/integraldisplayβ\nα|zx|2dx,\n26WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nhence under hypothesis (H1), we get\nRe/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2≤0,\nwhich implies that A2is dissipative. To prove that A2is m-dissipative, it is enough to prove that 0∈ρ(A2)\nsinceA2is a closed operator and D(A2) =H2. LetF= (f1,f2,f3,f4,f5,f6,f7(·,·))∈ H2.We should prove\nthat there exists a unique solution U= (u,v,w,y,z,φ,η (·,·))∈D(A2)of the equation\n−A2U=F.\nEquivalently, we consider the following system\n−y=f1, (3.11)\n−z=f2, (3.12)\n−φ=f3, (3.13)\n−κ1uxx=f4, (3.14)\n−(κ2vx+δ1zx+δ2ηx(·,1))x=f5, (3.15)\n−κ3wxx=f6, (3.16)\nηρ(x,ρ) =τf7(x,ρ). (3.17)\nIn addition, we consider the following boundary conditions\nu(0) = 0, u(α) =v(α), v(β) =w(β), w(L) = 0, (3.18)\nκ2vx(α)+δ1zx(α)+δ2ηx(α,1) =κ1ux(α), κ2vx(β)+δ1zx(β)+δ2ηx(β,1) =κ3wx(β), (3.19)\nη(·,0) =z(·). (3.20)\nFrom ( 3.11)-(3.13) and the fact that F∈ H, we obtain (y,z,φ)∈H1\n∗. Next, from ( 3.12), (3.20) and the fact\nthatf2∈H1\n∗(α,β), we get\nη(·,0) =z(·) =−f2(·)∈H1\n∗(α,β).\nFrom the above equation and Equation ( 3.17), we can determine\n(3.21) η(x,ρ) =τ/integraldisplayρ\n0f7(x,ξ)dξ−f2(x).\nSincef2∈H1\n∗(α,β)andf7∈L2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n, then it is clear that η, ηρ∈L2((0,1),H1\n∗(0,1)). Now, let\n(ϕ,ψ,θ)∈H1\n∗. Multiplying Equations ( 3.14), (3.15), (3.16) byϕ,ψ,θ, integrating over (0,α),(α,β)and(β,L)\nrespectively, taking the sum, then using by parts integrati on, we get\n(3.22)κ1/integraldisplayα\n0uxϕxdx+/integraldisplayβ\nα(κ2vx+δ1zx+δ2ηx(·,1))ψxdx+κ3/integraldisplayL\nβwxθxdx+κ1ux(0)ϕ(0)−κ3wx(L)θ(L)\n−κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α)+δ2ηx(α,1))ψ(α)−(κ2vx(β)+δ1zx(β)+δ2ηx(β,1))ψ(β)\n+κ3wx(β)θ(β) =/integraldisplayα\n0f4ϕdx+/integraldisplayβ\nαf5ψdx+/integraldisplayL\nβf6θdx.\nFrom the fact that (ϕ,ψ,θ)∈H1\n∗,we have\nϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β), θ(L) = 0.\nInserting the above equation in ( 3.22), then using ( 3.12), (3.19) and ( 3.21), we get\n(3.23)κ1/integraldisplayα\n0uxϕxdx+κ2/integraldisplayβ\nαvxψxdx+κ3/integraldisplayL\nβwxθxdx\n=/integraldisplayα\n0f4ϕdx+/integraldisplayβ\nαf5ψdx+/integraldisplayL\nβf6θdx+/integraldisplayβ\nα/parenleftbigg\n(δ1+δ2)(f2)x−δ2τ/integraldisplay1\n0(f7(·,ξ))xdξ/parenrightbigg\nψxdx.\nWe can easily verify that the left hand side of ( 3.23) is a bilinear continuous coercive form on H1\n∗×H1\n∗, and\nthe right hand side of ( 3.23) is a linear continuous form on H1\n∗. Then, using Lax-Milgram theorem, we deduce\nthat there exists (u,v,w)∈H1\n∗unique solution of the variational Problem ( 3.23). Using standard arguments,\nwe can show that (u,κ2v+δ1z+δ2η(·,1),w)∈H2. Thus, from ( 3.11)-(3.13), (3.21) and applying the classical\nelliptic regularity we deduce that U= (u,v,w,y,z,φ,η (·,·))∈D(A2). The proof is thus complete. /square\n27WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nThanks to Lumer-Philips theorem (see [ 36]), we deduce that A2generates a C0−semigroup of contractions etA2\ninH2and therefore Problem ( 3.1)-(3.7) is well-posed.\n3.2.Polynomial Stability. The main result in this subsection is the following theorem.\nTheorem 3.2. Under hypothesis (H1), for all initial data U0∈D(A2),there exists a constant C >0indepen-\ndent ofU0such that the energy of System (3.1)-(3.7)satisfies the following estimation\nE(t)≤C\nt4/ba∇dblU0/ba∇dbl2\nD(A2),∀t>0.\nAccording to Theorem A.5(part (ii)), we have to check if the following conditions hol d,\n(3.24) iR⊆ρ(A2)\nand\n(3.25) sup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A2)−1/vextenddouble/vextenddouble/vextenddouble\nL(H2)=O/parenleftBig\n|λ|1\n2/parenrightBig\n.\nThe next proposition is a technical result to be used in the pr oof of Theorem 3.2given below.\nProposition 3.3. Under hypothesis (H1), let(λ,U:= (u,v,w,y,z,φ,η (·,·)))∈R∗×D(A2),such that\n(3.26) (iλI−A2)U=F:= (f1,f2,f3,f4,f5,f6,g(·,·))∈ H2,\ni.e.\niλu−y=f1 inH1(0,α), (3.27)\niλv−z=f2 inH1\n∗(α,β), (3.28)\niλw−φ=f3 inH1(β,L), (3.29)\niλy−κ1uxx=f4 inL2(0,α), (3.30)\niλz−(κ2vx+δ1zx+δ2ηx(·,1))x=f5 inL2\n∗(α,β), (3.31)\niλφ−κ3wxx=f6 inL2(β,L), (3.32)\nηρ(·,·)+iτλη(·,·) =τg(·,·)inL2/parenleftbig\n(0,1),H1\n∗(α,β)/parenrightbig\n. (3.33)\nThen, we have the following inequality\n(3.34) /ba∇dblU/ba∇dbl2\nH2≤K1λ−4(|λ|+1)6/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nIn addition, if |λ| ≥M >0,then we have\n(3.35) /ba∇dblU/ba∇dbl2\nH2≤K2/parenleftbigg√\nM+1√\nM/parenrightbigg2\n|λ|1\n2/parenleftBig\n1+|λ|−1\n2/parenrightBig8/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nHere and below we denote by Kja positive constant number independent of λ.\nBefore stating the proof of Proposition 3.3, leth∈C1([α,β])such that\nh(α) =−h(β) = 1,max\nx∈[α,β]|h(x)|=Chandmax\nx∈[α,β]|h′(x)|=Ch′,\nwhereChandCh′are strictly positive constant numbers independent of λ. An example about h, we can take\nh(x) =−2(x−α)\nβ−α+1to get\nh(α) =−h(β) = 1, h∈C1([α,β]), Ch= 1, Ch′=2\nβ−α.\nFor the proof of Proposition 3.3, we need the following lemmas.\n28WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nLemma 3.4. Under hypothesis (H1), the solution (u,v,w,y,z,φ,η (·,·))∈D(A2)of Equation (3.26)satisfies\nthe following estimations\n/integraldisplayβ\nα|zx|2dx≤K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2, (3.36)\n/integraldisplayβ\nα|vx|2dx≤K4λ−2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n, (3.37)\n/integraldisplayβ\nα/integraldisplay1\n0|ηx(x,ρ)|2dxdρ≤K5/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n, (3.38)\n/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx≤K6/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n, (3.39)\nwhere /braceleftBigg\nK3= (δ1−|δ2|)−1, K4= 2max/parenleftbig\nK3,κ−1\n2/parenrightbig\n,\nK5= 2max/parenleftbig\nK3,τ|δ2|−1/parenrightbig\n, K6= 3max/parenleftbig\nκ2\n2K4,δ2\n1K3+δ2\n2K5/parenrightbig\n.\nProof. First, taking the inner product of ( 3.26) withUinH2, then using hypothesis (H1), arguing in the same\nway as ( 3.10), we obtain\n/integraldisplayβ\nα|zx|2dx≤ −1\nδ1−|δ2|Re/a\\}b∇acketle{tA2U,U/a\\}b∇acket∇i}htH2=1\nδ1−|δ2|Re/a\\}b∇acketle{tF,U/a\\}b∇acket∇i}htH2≤1\nδ1−|δ2|/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2,\nhence we get ( 3.36). Next, from ( 3.28), (3.36) and the fact that κ2/integraldisplayβ\nα|(f2)x|2dx≤ /ba∇dblF/ba∇dbl2\nH2, we obtain\n/integraldisplayβ\nα|vx|2dx≤2λ−2/integraldisplayβ\nα|zx|2dx+2λ−2/integraldisplayβ\nα|(f2)x|2dx\n≤2λ−2/parenleftBig\nK3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+κ−1\n2/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤2λ−2max/parenleftbig\nK3,κ−1\n2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\ntherefore we get ( 3.37). Now, from ( 3.33) and using the fact that U∈D(A2) (i.e.η(·,0) =z(·)), we obtain\n(3.40) η(x,ρ) =z(x)e−iτλρ+τ/integraldisplayρ\n0eiτλ(ξ−ρ)g(x,ξ)dξ(x,ρ)∈(α,β)×(0,1),\nconsequently, we obtain\n/integraldisplayβ\nα/integraldisplay1\n0|ηx(x,ρ)|2dxdρ≤2/integraldisplayβ\nα|zx|2dx+2τ2/integraldisplayβ\nα/integraldisplay1\n0|gx(x,ξ)|2dξdx.\nInserting ( 3.36) in the above equation, then using the fact that τ|δ2|/integraldisplayβ\nα/integraldisplay1\n0|gx(x,ξ)|2dξdx≤ /ba∇dblF/ba∇dbl2\nH2, we obtain\n/integraldisplayβ\nα/integraldisplay1\n0|ηx(x,ρ)|2dxdρ≤2K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+2τ|δ2|−1/ba∇dblF/ba∇dbl2\nH2\n≤2max/parenleftbig\nK3,τ|δ2|−1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.38). On the other hand, from ( 3.40), we get\nηx(x,1) =zx(x)e−iτλ+τ/integraldisplay1\n0eiτλ(ξ−1)gx(x,ξ)dξ x∈(α,β).\nFrom the above equation and ( 3.36), we obtain\n/integraldisplayβ\nα|ηx(x,1)|2dx≤2/integraldisplayβ\nα|zx|2dx+2τ2/integraldisplayβ\nα/integraldisplay1\n0|gx(x,ξ)|2dξdx\n≤2K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+2τ|δ2|−1/ba∇dblF/ba∇dbl2\nH2\n≤2max/parenleftbig\nK3,τ|δ2|−1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\n29WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFinally, from ( 3.36), (3.37) and the above inequality, we get\n/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx≤3κ2\n2/integraldisplayβ\nα|vx|2dx+3δ2\n1/integraldisplayβ\nα|zx|2dx+3δ2\n2/integraldisplayβ\nα|ηx(·,1)|2dx\n≤3/parenleftbig\nκ2\n2K4λ−2+δ2\n1K3+δ2\n2K5/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤3max/parenleftbig\nκ2\n2K4,δ2\n1K3+δ2\n2K5/parenrightbig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.39). /square\nLemma 3.5. Under hypothesis (H1), for alls1, s2∈Randr1, r2∈R∗\n+, the solution (u,v,w,y,z,φ,η (·,·))∈\nD(A2)of Equation (3.26)satisfies the following estimations\n(3.41) |z(β)|2+|z(α)|2≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+K7r1C2\nh|λ|−1\n2+s1/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.42)|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n≤|λ|3\n2−s2\nr2/integraldisplayβ\nα|z|2dx+K8/parenleftBig\nCh′+Ch+r2C2\nh|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nK7= 2/parenleftbig\nK4+κ−1\n2/parenrightbig\n, K8=K6+1.\nProof. First, from Equation ( 3.28), we have\n−zx= (f2)x−iλvx.\nMultiplying the above equation by 2hz, integrating over (α,β)and taking the real parts, then using by parts\nintegration and the fact that h(α) =−h(β) = 1, we get\n(3.43)|z(β)|2+|z(α)|2=−/integraldisplayβ\nαh′|z|2dx−2Re/braceleftBigg/integraldisplayβ\nαh(iλvx−(f2)x)zdx/bracerightBigg\n≤Ch′/integraldisplayβ\nα|z|2dx+2Ch|λ|/integraldisplayβ\nα|vx||z|dx+2Ch/integraldisplayβ\nα|(f2)x||z|dx.\nOn the other hand, for all s1∈Randr1∈R∗\n+, we have\n2Ch|λ||vx||z| ≤|λ|1\n2−s1|z|2\n2r1+2r1C2\nh|λ|3\n2+s1|vx|2and2Ch|(f2)x||z| ≤|λ|1\n2−s1|z|2\n2r1+2r1C2\nh|λ|−1\n2+s1|(f2)x|2.\nInserting the above equation in ( 3.43), then using ( 3.37) and the fact that/integraldisplayβ\nα|(f2)x|2dx≤κ−1\n2/ba∇dblF/ba∇dbl2\nH2,we get\n|z(β)|2+|z(α)|2\n≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+2r1C2\nh|λ|s1/parenleftBigg\n|λ|3\n2/integraldisplayβ\nα|vx|2dx+|λ|−1\n2/integraldisplayβ\nα|(f2)x|2dx/parenrightBigg\n≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+2r1C2\nh|λ|s1−1\n2/parenleftBig\nK4/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+κ−1\n2/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤/parenleftBigg\nCh′+|λ|1\n2−s1\nr1/parenrightBigg/integraldisplayβ\nα|z|2dx+2r1C2\nh|λ|s1−1\n2/parenleftbig\nK4+κ−1\n2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.41). Next, multiplying Equation ( 3.31) by2h(κ2vx+δ1zx+δ2ηx(·,1)), integrating over (α,β)\nand taking the real parts, then using by parts integration, w e get\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n=−/integraldisplayβ\nαh′|κ2vx+δ1zx+δ2ηx(·,1)|2dx+2Re/integraldisplayβ\nαh(f5−iλz)(κ2vx+δ1zx+δ2ηx(·,1))dx,\n30WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nconsequently, we have\n(3.44)|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n≤Ch′/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx+2Ch/integraldisplayβ\nα|f5||κ2vx+δ1zx+δ2ηx(·,1)|dx\n+2Ch|λ|/integraldisplayβ\nα|z||κ2vx+δ1zx+δ2ηx(·,1)|dx.\nOn the other hand, for all s2∈Randr2∈R∗\n+, we have\n\n\n2Ch|f5|||κ2vx+δ1zx+δ2ηx(·,1)| ≤Ch|f5|2+Ch|κ2vx+δ1zx+δ2ηx(·,1)|2,\n2Ch|λ||z|||κ2vx+δ1zx+δ2ηx(·,1)| ≤|λ|3\n2−s2\nr2|z|2+r2C2\nh|λ|1\n2+s2||κ2vx+δ1zx+δ2ηx(·,1)|2.\nInserting the above equation in ( 3.44), we get\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2≤|λ|3\n2−s2\nr2/integraldisplayβ\nα|z|2dx\n+/parenleftBig\nCh′+Ch+r2C2\nh|λ|1\n2+s2/parenrightBig/bracketleftBigg/integraldisplayβ\nα||κ2vx+δ1zx+δ2ηx(·,1)|2dx+/integraldisplayβ\nα|f5|2dx/bracketrightBigg\n.\nInserting ( 3.39) in the above equation, then using the fact that\n/integraldisplayβ\nα|f5|2dx≤ /ba∇dblF/ba∇dbl2\nH2≤/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwe get\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2\n≤|λ|3\n2−s2\nr2/integraldisplayβ\nα|z|2dx+(K6+1)/parenleftBig\nCh′+Ch+r2C2\nh|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.42). /square\nLemma 3.6. Under hypothesis (H1), for alls1, s2∈R, the solution (u,v,w,y,z,φ,η (·,·))∈D(A2)of Equation\n(3.26)satisfies the following estimation\n(3.45)/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx\n≤K9/parenleftBig\n1+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K10/parenleftBig\n1+|λ|−1\n2+s1+|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nK9= max/bracketleftbig\nmax(α,L−β),max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/bracketrightbig\nmax(Ch′,1)\nand\nK10= 2max/bracketleftBig\nK7C2\nhmax(α,L−β), K8max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig\nmax/parenleftbig\nCh′+Ch,C2\nh/parenrightbig\n,4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig/bracketrightBig\n.\nProof. First, multiplying Equation ( 3.30) by2xux, integrating over (0,α)and taking the real parts, then using\nby parts integration, we get\n(3.46) 2Re/braceleftbigg\niλ/integraldisplayα\n0xyuxdx/bracerightbigg\n+κ1/integraldisplayα\n0|ux|2dx=κ1α|ux(α)|2+2Re/braceleftbigg/integraldisplayα\n0xf4uxdx/bracerightbigg\n.\nFrom ( 3.27), we deduce that\niλux=−yx−(f1)x.\n31WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nInserting the above result in ( 3.46), then using by parts integration, we get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx=κ1α|ux(α)|2+α|y(α)|2+2Re/braceleftbigg/integraldisplayα\n0xf4uxdx/bracerightbigg\n+2Re/braceleftbigg/integraldisplayα\n0xy(f1)xdx/bracerightbigg\n,\nconsequently, we get\n(3.47)/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx≤κ1α|ux(α)|2+α|y(α)|2+2α/parenleftbigg/integraldisplayα\n0|ux||f4|dx+/integraldisplayα\n0|y||(f1)x|dx/parenrightbigg\n.\nUsing Cauchy Schwarz inequality, we get\n(3.48)/integraldisplayα\n0|ux||f4|dx+/integraldisplayα\n0|y||(f1)x|dx≤2κ−1\n2\n1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nOn the other hand, since U∈D(A2), we have\n(3.49)/braceleftBigg\nκ1|ux(α)|=|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|,|y(α)|=|z(α)|,\nκ3|wx(β)|=|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|,|φ(β)|=|z(β)|.\nSubstituting ( 3.48) and the boundary conditions ( 3.49) atx=αin (3.47), we obtain\n(3.50)/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx≤ακ−1\n1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2+α|z(α)|2+4ακ−1\n2\n1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nNext, by the same way, we multiply equation ( 3.32) by2(x−L)wxand integrate over (β,L),then we use ( 3.29).\nArguing in the same way as ( 3.47), we get\n/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx≤κ3(L−β)|wx(β)|2+(L−β)|φ(β)|2+4(L−β)κ−1\n2\n3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nSubstituting the boundary conditions ( 3.49) atx=βin the above equation, we obtain\n/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx≤(L−β)/bracketleftBig\nκ−1\n3|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|z(β)|2+4κ−1\n2\n3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2/bracketrightBig\n.\nNow, adding the above equation and ( 3.50), we get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx≤max(α,L−β)/parenleftbig\n|z(α)|2+|z(β)|2/parenrightbig\n+max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/parenleftbig\n|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2+|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2/parenrightbig\n+4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nInserting ( 3.41) and ( 3.42) withr1=r2= 1in the above estimation, we get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx\n≤max/bracketleftbig\nmax(α,L−β),max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/bracketrightbig/parenleftBig\nCh′+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+max(α,L−β)K7C2\nh|λ|−1\n2+s1/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig\nK8/parenleftBig\nCh′+Ch+C2\nh|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2.\nIn the above equation, using the fact that\n\n\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2≤/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2≤/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n32WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nwe get\n/integraldisplayα\n0|y|2dx+κ1/integraldisplayα\n0|ux|2dx+/integraldisplayL\nβ|φ|2dx+κ3/integraldisplayL\nβ|wx|2dx\n≤max/bracketleftbig\nmax(α,L−β),max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig/bracketrightbig\nmax(Ch′,1)/parenleftBig\n1+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K7C2\nhmax(α,L−β)|λ|−1\n2+s1/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+K8max/parenleftbig\nκ−1\n1α,κ−1\n3(L−β)/parenrightbig\nmax/parenleftbig\nCh′+Ch,C2\nh/parenrightbig/parenleftBig\n1+|λ|1\n2+s2/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+4/parenleftBig\nακ−1\n2\n1+(L−β)κ−1\n2\n3/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence, we get ( 3.45). /square\nLemma 3.7. Under hypothesis (H1), for alls1, s2, s3∈Randr1, r2,r3∈R∗\n+, the solution (u,v,w,y,z,φ,η (·,·))∈\nD(A2)of Equation (3.26)satisfies the following estimations\n(3.51)/ba∇dblU/ba∇dbl2\nH2≤K11/parenleftBig\n1+|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K12/parenleftBig\n1+|λ|1\n2+s2+|λ|−1\n2+s1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.52) R1,λ/integraldisplayβ\nα|z|2dx≤K13R2,λ/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nsuch that\n\nR1,λ= 1−1\n2/parenleftbigg|λ|−s3−s2\nr2r3+r3|λ|−s1+s3\nr1+r3Ch′|λ|s3−1\n2/parenrightbigg\n,\nR2,λ=C2\nh|λ|−1(r1r3|λ|s1+s3+r2r−1\n3|λ|s2−s3)+r−1\n3|λ|−s3−3\n2(Ch′+Ch)+|λ|−1,\nwhere\nK11=K9+1, K12=K10+max(κ2K4,τ|δ2|K5), K13=max(K3+K6+2,max(K7,K8))\n2.\nProof. First, from ( 3.37), (3.38) and ( 3.45), we get\n/ba∇dblU/ba∇dbl2\nH2≤/parenleftBig\nK9+1+K9/parenleftBig\n|λ|1\n2−s1+|λ|3\n2−s2/parenrightBig/parenrightBig/integraldisplayβ\nα|z|2dx\n+K10/parenleftBig\n1+|λ|1\n2+s2+|λ|−1\n2+s1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+max(κ2K4,τ|δ2|K5)/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.51). Next, multiplying ( 3.31) by−iλ−1zand integrating over (α,β),then taking the real part,\nthen using by parts integration, we get\n/integraldisplayβ\nα|z|2dx=−Re/braceleftBigg\niλ−1/integraldisplayβ\nαf5zdx/bracerightBigg\n+Re/braceleftBigg\niλ−1/integraldisplayβ\nα(κ2v+δ1z+δ2η(·,1))xzxdx/bracerightBigg\n−Re/braceleftbig\niλ−1(κ2vx(β)+δ1zx(β)+δ2ηx(β,1))z(β)/bracerightbig\n+Re/braceleftbig\niλ−1(κ2vx(α)+δ1zx(α)+δ2ηx(α,1))z(α)/bracerightbig\n,\nconsequently,\n(3.53)/integraldisplayβ\nα|z|2dx≤ |λ|−1/integraldisplayβ\nα|f5||z|dx+|λ|−1/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)||zx|dx\n+|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|.\nUsing Cauchy Schwarz inequality, we have\n(3.54) |λ|−1/integraldisplayβ\nα|f5||z|dx≤ |λ|−1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2≤ |λ|−1/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\n33WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nFrom ( 3.36) and ( 3.39), we get\n/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)||zx|dx\n≤1\n2/integraldisplayβ\nα|zx|2dx+1\n2/integraldisplayβ\nα|κ2vx+δ1zx+δ2ηx(·,1)|2dx\n≤K3\n2/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+K6\n2/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤K3+K6\n2/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nInserting ( 3.54) and the above estimation in ( 3.53), we get\n(3.55)/integraldisplayβ\nα|z|2dx≤K3+K6+2\n2|λ|−1/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n+|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|.\nNow, for all s3∈R,r3∈R∗\n+and forζ=αorζ=β, we get\n|λ|−1|κ2vx(ζ)+δ1zx(ζ)+δ2ηx(ζ,1)||z(ζ)| ≤r3|λ|s3−1\n2\n2|z(ζ)|2+|λ|−s3−3\n2\n2r3|κ2vx(ζ)+δ1zx(ζ)+δ2ηx(ζ,1)|2.\nFrom the above inequality, we get\n|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|\n≤|λ|−s3−3\n2\n2r3/parenleftBig\n|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2/parenrightBig\n+r3|λ|s3−1\n2\n2/parenleftbig\n|z(α)|2+|z(β)|2/parenrightbig\n.\nInserting ( 3.41) and ( 3.42) in the above estimation, we obtain\n|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|\n≤1\n2/parenleftbigg|λ|−s3−s2\nr2r3+r3|λ|−s1+s3\nr1+r3Ch′|λ|s3−1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx\n+max(K7,K8)\n2R3,λ/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nR3,λ=C2\nh|λ|−1(r1r3|λ|s1+s3+r2r−1\n3|λ|s2−s3)+r−1\n3|λ|−s3−3\n2(Ch′+Ch).\nFinally, inserting the above equation in ( 3.55), we get\n/bracketleftbigg\n1−1\n2/parenleftbigg|λ|−s3−s2\nr2r3+r3|λ|−s1+s3\nr1+r3Ch′|λ|s3−1\n2/parenrightbigg/bracketrightbigg/integraldisplayβ\nα|z|2dx\n≤max(K3+K6+2,max(K7,K8))\n2/parenleftbig\nR3,λ+|λ|−1/parenrightbig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.52). /square\nProof of Proposition 3.3.We now divide the proof in two steps:\nStep 1. In this step, we prove the asymptotic behavior estimate of ( 3.34). Takings3=s1=−s2=1\n2,\nr1=1\nCh′, r2= 9Ch′andr3=1\n3Ch′in Lemma 3.7, we get\n\n\n1\n2/integraldisplayβ\nα|z|2dx≤K13λ−4/parenleftbiggC2\nh\n3C2\nh′λ2+|λ|+3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig/parenrightbigg/parenleftbig\nλ2+1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n/ba∇dblU/ba∇dbl2\nH2≤2K11/parenleftbig\nλ2+1/parenrightbig/integraldisplayβ\nα|z|2dx+3K12λ−2/parenleftbig\nλ2+1/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\n34WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nIn the above equation, using the fact that\nC2\nh\n3C2\nh′λ2+|λ|+3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n≤max/parenleftbiggC2\nh\n3C2\nh′,3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n,1/parenrightbigg/parenleftbig\nλ2+|λ|+1/parenrightbig\n≤max/parenleftbiggC2\nh\n3C2\nh′,3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n,1/parenrightbigg\n(|λ|+1)2\nand\nλ2+1≤(|λ|+1)2,\nwe get\n(3.56)/integraldisplayβ\nα|z|2dx≤K14λ−4(|λ|+1)4/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.57) /ba∇dblU/ba∇dbl2\nH2≤2K11(|λ|+1)2/integraldisplayβ\nα|z|2dx+3K12λ−2(|λ|+1)2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nwhere\nK14= 2K13max/parenleftbiggC2\nh\n3C2\nh′,3Ch′/parenleftbig\n9C2\nhCh′+Ch+Ch′/parenrightbig\n,1/parenrightbigg\n.\nInserting ( 3.56) in (3.57), we get\n/ba∇dblU/ba∇dbl2\nH2≤/parenleftBig\n2K11K14(|λ|+1)4+3K12λ2/parenrightBig\nλ−4(|λ|+1)2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\n≤(2K11K14+3K12)λ−4(|λ|+1)6/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get ( 3.34).\nStep 2. In this step, we prove the asymptotic behavior estimate of ( 3.35). LetM∈R∗such that |λ| ≥M >0.\nIn this case, taking s1=s2=s3= 0,r1=3√\nM\n2Ch′,r2=3Ch′√\nMandr3=√\nM\n2Ch′in Lemma 3.7, we get\n(3.58)/ba∇dblU/ba∇dbl2\nH2≤K11|λ|3\n2/parenleftBig\n1+|λ|−1+|λ|−3\n2/parenrightBig/integraldisplayβ\nα|z|2dx\n+K12|λ|1\n2/parenleftBig\n1+|λ|−1\n2+|λ|−1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\nand\n(3.59)1\n2/parenleftBigg\n1−√\nM\n2|λ|1\n2/parenrightBigg/integraldisplayβ\nα|z|2dx\n≤K13|λ|−1/bracketleftBigg\n1+3C2\nhM\n4C2\nh′+6C2\nhC2\nh′\nM+2Ch′(Ch+Ch′)|λ|−1\n2√\nM/bracketrightBigg\n/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nFrom the fact that |λ| ≥M, we get\n1\n2/parenleftBigg\n1−√\nM\n2|λ|1\n2/parenrightBigg\n≥1\n4>0.\nTherefore, from the above inequality and ( 3.59), we get\n(3.60)/integraldisplayβ\nα|z|2dx≤K15|λ|−1/parenleftBigg\n1+M+1\nM+|λ|−1\n2√\nM/parenrightBigg\n/parenleftbig\n1+λ−2/parenrightbig/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nwhere\nK15= 4K13max/bracketleftbigg\n1,3C2\nh\n4C2\nh′,6C2\nhC2\nh′,2Ch′(Ch+Ch′)/bracketrightbigg\n.\n35WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nIn Estimation ( 3.59), using the fact that\n\n\n1+M+1\nM≤/parenleftbigg√\nM+1√\nM/parenrightbigg2\n,1√\nM≤/parenleftbigg√\nM+1√\nM/parenrightbigg2\n,\n1+λ−2≤/parenleftBig\n1+|λ|−1\n2/parenrightBig4\n.\nwe get\n(3.61)/integraldisplayβ\nα|z|2dx≤K15|λ|−1/parenleftbigg√\nM+1√\nM/parenrightbigg2/parenleftBig\n1+|λ|−1\n2/parenrightBig5/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n.\nInserting ( 3.61) in (3.58), then using the fact that\n1+|λ|−1+|λ|−3\n2≤/parenleftBig\n1+|λ|−1\n2/parenrightBig3\n,/parenleftBig\n1+|λ|−1\n2+|λ|−1/parenrightBig/parenleftbig\n1+λ−2/parenrightbig\n≤/parenleftBig\n1+|λ|−1\n2/parenrightBig6\n≤/parenleftBig\n1+|λ|−1\n2/parenrightBig8\n,\nwe get\n/ba∇dblU/ba∇dbl2\nH2≤max(K11K15,K12)|λ|1\n2/parenleftBig\n1+|λ|−1\n2/parenrightBig8/bracketleftBigg/parenleftbigg√\nM+1√\nM/parenrightbigg2\n+1/bracketrightBigg/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n≤2max(K11K15,K12)|λ|1\n2/parenleftBig\n1+|λ|−1\n2/parenrightBig8/parenleftbigg√\nM+1√\nM/parenrightbigg2/parenleftBig\n/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2\nH2/parenrightBig\n,\nhence we get estimation of ( 3.35). The proof is thus complete. /square\nProof of Theorem 3.2.First, we will prove condition ( 3.24). Remark that it has been proved in Proposition\n3.1that0∈ρ(A2).Now, suppose ( 3.24) is not true, then there exists ω∈R∗such thatiω/\\e}atio\\slash∈ρ(A2). According\nto Lemma A.3and Remark A.4, there exists\n{(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(·,·)))}n≥1⊂R∗×D(A2),\nwithλn→ωasn→ ∞,|λn|<|ω|and/ba∇dblUn/ba∇dblH2= 1, such that\n(iλnI−A2)Un=Fn:= (f1,n,f2,n,f3,n,f4,n,f5,6,f6,n,f7,n(·,·))→0inH2,asn→ ∞.\nWe will check condition ( 3.24) by finding a contradiction with /ba∇dblUn/ba∇dblH2= 1such as/ba∇dblUn/ba∇dblH2→0.According to\nEquation ( 3.34) in Proposition 3.3withU=Un, F=Fnandλ=λn, we obtain\n0≤ /ba∇dblUn/ba∇dbl2\nH2≤K1|λn|−4(|λn|+1)6/parenleftBig\n/ba∇dblFn/ba∇dblH2/ba∇dblUn/ba∇dblH2+/ba∇dblFn/ba∇dbl2\nH2/parenrightBig\n,\nasn→ ∞,we get/ba∇dblUn/ba∇dbl2\nH2→0,which contradicts /ba∇dblUn/ba∇dblH2= 1. Thus, condition ( 3.24) is holds true. Next, we\nwill prove condition ( 3.25) by a contradiction argument. Suppose there exists\n{(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(·,·)))}n≥1⊂R∗×D(A2),\nwith|λn| ≥1without affecting the result, such that |λn| →+∞,and/ba∇dblUn/ba∇dblH2= 1and there exists a sequence\nGn:= (g1,n,g2,n,g3,n,g4,n,g5,6,g6,n,g7,n(·,·))∈ H2, such that\n(iλnI−A2)Un=λ−1\n2nGn→0inH2.\nWe will check condition ( 3.25) by finding a contradiction with /ba∇dblUn/ba∇dblH2= 1such as/ba∇dblUn/ba∇dblH2=o(1).According\nto Equation ( 3.35) in Proposition 3.3withU=Un, F=λ−1\n2Gn, λ=λnandM= 1, we get\n/ba∇dblUn/ba∇dbl2\nH2≤4K2/parenleftBig\n1+|λn|−1\n2/parenrightBig8/parenleftBig\n/ba∇dblGn/ba∇dblH2/ba∇dblUn/ba∇dblH2+|λn|−1\n2/ba∇dblGn/ba∇dbl2\nH2/parenrightBig\n,\nas|λn| → ∞,we get/ba∇dblUn/ba∇dbl2\nH2=o(1),which contradicts /ba∇dblUn/ba∇dblH2= 1. Thus, condition ( 3.25) is holds true. The\nresult follows from Theorem A.5(part (ii)). The proof is thus complete. /square\nRemark 3.8. In the case that α= 0andβ/\\e}atio\\slash=Lorβ=Landα/\\e}atio\\slash= 0, we can proceed similar to the proof of\nTheorem 3.2to check that the energy of System (3.1)-(3.7)decays polynomially of order t−4. /square\n36WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nAppendix A.Notions of stability and theorems used\nWe introduce here the notions of stability that we encounter in this work.\nDefinition A.1. Assume that Ais the generator of a C 0-semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space\nH. TheC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is said to be\n1.strongly stable if\nlim\nt→+∞/ba∇dbletAx0/ba∇dblH= 0,∀x0∈H;\n2.exponentially (or uniformly) stable if there exist two posit ive constants Mandǫsuch that\n/ba∇dbletAx0/ba∇dblH≤Me−ǫt/ba∇dblx0/ba∇dblH,∀t>0,∀x0∈H;\n3.polynomially stable if there exists two positive constants Candαsuch that\n/ba∇dbletAx0/ba∇dblH≤Ct−α/ba∇dblAx0/ba∇dblH,∀t>0,∀x0∈D(A).\n/square\nFor proving the strong stability of the C0-semigroup/parenleftbig\netA/parenrightbig\nt≥0, we will recall two methods, the first result\nobtained by Arendt and Batty in [ 8].\nTheorem A.2 (Arendt and Batty in [ 8]).Assume that Ais the generator of a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space H. IfAhas no pure imaginary eigenvalues and σ(A)∩iRis countable, where\nσ(A)denotes the spectrum of A, then theC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is strongly stable. /square\nThe second one is a classical method based on Arendt and Batty theorem and the contradiction argument\n(see page 25 in [ 29]).\nLemma A.3. Assume that Ais the generator of a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0on a Hilbert space\nH. Furthermore, Assume that 0∈ρ(A).If there exists ω∈R∗, such that iω/\\e}atio\\slash∈ρ(A), then\n(A.1)/braceleftBig\niλsuch thatλ∈R∗and|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|/bracerightBig\n⊂ρ(A)and sup\n|λ|</bardblA−1/bardbl−1≤|ω|/vextenddouble/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble/vextenddouble=∞.\nProof. Since0∈ρ(A), for any real number λwith|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1, we deduce from the contraction mapping\ntheorem that operator iλI−A=−A−1(I−iλA)is invertible. Therefore, we get\n/braceleftBig\niλsuch thatλ∈R∗and|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1/bracerightBig\n⊂ρ(A).\nIn addition, if there exists ω∈R∗, such that iω/\\e}atio\\slash∈ρ(A), then/vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |w|, hence we get the first estimation\nof (A.1). Next, since iω/\\e}atio\\slash∈ρ(A), then for all |λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|the operator\niωI−A= (iλI−A)/parenleftBig\nI+i(ω−λ)(iλI−A)−1/parenrightBig\nis not invertible, hence from the contraction mapping theor em we deduce\n/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble≥1\n|ω−λ|,∀ |λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|,\ntherefore\nsup\n|λ|</bardblA−1/bardbl−1≤|ω|/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble≥sup\n|λ|</bardblA−1/bardbl−1≤|ω|1\n|ω−λ|=∞,\nhence we get second estimation of ( A.1). /square\nRemark A.4. Condition (A.1)turns out that there exists {(λn,Un)}n≥1⊂R∗×D(A),withλn→ωas\nn→ ∞,|λn|<|ω|and/ba∇dblUn/ba∇dblH2= 1, such that\n(iλnI−A)Un=Fn→0inH, asn→ ∞.\nThen, we will check condition iR⊂ρ(A)by finding a contradiction with /ba∇dblUn/ba∇dblH= 1such as/ba∇dblUn/ba∇dblH=o(1)./square\nWe now recall the following standard result which is stated i n a comparable way (see [ 22,38] for part (i) and\n[9,11,9] for part (ii)).\n37WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY\nTheorem A.5. Assume that Ais the generator of a strongly continuous semigroup of contr actions/parenleftbig\netA/parenrightbig\nt≥0\nonH. Assume that iR⊂ρ(A).Then;\n(i)The semigroup etAis exponentially stable if and only if\nlim\nλ→∞/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble<∞.\n(ii)The semigroup etAis polynomially stable of order α>0if and only if\nlim\nλ→∞|λ|−1\nα/vextenddouble/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble/vextenddouble<∞.\n/square\nReferences\n[1] M. Alves, J. M. Rivera, M. Sepúlveda, and O. V. Villagrán. The Lack of Exponential Stability in Certain Transmission\nProblems with Localized Kelvin–Voigt Dissipation .SIAM Journal on Applied Mathematics , 74(2):345–365, Jan. 2014. 6\n[2] M. Alves, J. M. Rivera, M. Sepúlveda, O. V. Villagrán, and M. Z. Garay. The asymptotic behavior of the linear transmission\nproblem in viscoelasticity .Mathematische Nachrichten , 287(5-6):483–497, Oct. 2013. 6\n[3] K. Ammari and B. Chentouf. 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A constructive method for the stabilization of the wave equa tion with localized Kelvin–Voigt damping .Comptes\nRendus Mathematique , 350(11-12):603–608, June 2012. 6\n[41] J.-M. Wang, B.-Z. Guo, and M. Krstic. Wave Equation Stabilization by Delays Equal to Even Multipl es of the Wave Propa-\ngation Time .SIAM Journal on Control and Optimization , 49(2):517–554, Jan. 2011. 4\n[42] Y. Xie and G. Xu. 10(3):557–579, 2017. 4\n[43] Y. Xie and G. Xu. Exponential stability of 1-d wave equation with the boundar y time delay based on the interior control .\nDiscrete &Continuous Dynamical Systems - S , 10(3):557–579, 2017. 4\n[44] G. Q. Xu, S. P. Yung, and L. K. Li. Stabilization of wave systems with input delay in the bounda ry control .ESAIM: Control,\nOptimisation and Calculus of Variations , 12(4):770–785, Oct. 2006. 3\n[45] Q. Zhang. Polynomial decay of an elastic/viscoelastic waves interac tion system .Zeitschrift für angewandte Mathematik und\nPhysik , 69(4), June 2018. 6\n39"    },    {        "title": "0802.1143v1.On_the_scaling_of_the_damping_time_for_resonantly_damped_oscillations_in_coronal_loops.pdf",        "content": "arXiv:0802.1143v1  [astro-ph]  8 Feb 2008On the scaling of the damping time for resonantly damped\noscillations in coronal loops\nI˜ nigo Arregui1, Jos´ e Luis Ballester1, Marcel Goossens2,1\nABSTRACT\nThere is not as yet full agreement on the mechanism that causes th e rapid\ndamping of the oscillations observed by TRACE in coronal loops. It ha s been\nsuggested that the variation of the observed values of the dampin g time as func-\ntion of the corresponding observed values of the period contains in formation on\nthe possible damping mechanism. The aim ofthis Letter is toshow that , for reso-\nnant absorption, this is definitely not the case unless detailed a prior i information\non the individual loops is available.\nSubject headings: MHD — Sun: corona — Sun: magnetic fields — waves\n1. Introduction\nTransverse oscillations in coronal loops have been detected in obse rvations made with\nthe EUV telescope on board of the Transition Region and Coronal Ex plorer (TRACE) in\n1999 by Aschwanden et al. (1999) and Nakariakov et al. (1999). Sin ce then the detection\nof these oscillations has been confirmed and in addition damped oscillat ions have been ob-\nserved in hot coronal loops by the SUMER instrument on board SOHO (Wang et al. 2002;\nKliem et al. 2002). The TRACE oscillations have periods of the order of ≃2−10 min-\nutes and comparatively short damping times of the order of ≃3−20 minutes. There is\ngeneral consensus that these oscillations are fast standing kink m ode oscillations. How-\never, there is still debate about the mechanism that causes the ob served fast damping.\nDifferent mechanisms have been suggested: phase mixing (Ofman & A schwanden 2002), res-\nonant absorption (Hollweg & Yang 1988; Goossens et al. 2002; Rude rman & Roberts 2002),\nlateral and foot-point wave leakage (Smith et al. 1997; De Pontieu e t al. 2001), drag due\n1Departament de F´ ısica, Universitat de les Illes Balears, E-07122 Pa lma de Mallorca, Spain. Email:\ninigo.arregui@uib.es, dfsjlb0@uib.es\n2Centre Plasma Astrophysics, Katholieke Universiteit Leuven, Leuv en, B-3001, Belgium. Email: mar-\ncel.goossens@wis.kuleuven.be– 2 –\nto the ambient plasma (Chen & Schuck 2007). In order to discriminat e between different\ndamping mechanisms Ofman & Aschwanden (2002) suggested to stu dy how the observed\ndamping times vary as a function of the corresponding observed pe riods. In particular\nOfman & Aschwanden (2002) claimed that the observed values are c ompatible with phase\nmixing if the damping time increases with period TasT4/3and with resonant absorption if\nit increases as T. The aim of the present Letter is to show that random observation s of os-\ncillation events in coronal loops are very unlikely to produce any part icular relation between\ndamping times and periods whatever the actual mechanism is that ca uses the damping. In\nparticular we focus on resonant absorption and use synthetic dat a for periods and damping\ntimes to show that various samples of pairs of periods and damping tim es can lead to various\nand widely different scaling laws.\n2. Analytical theory\nThe suggestion that the damping time increases linearly with period, τd∼T, was\ntriggered by analytical asymptotic expressions for the damping of the quasi-mode given by\ne.g. Goossens et al. (1992), Ruderman & Roberts (2002), Goosse ns et al. (2002) (see re-\nviews Goossens et al. 2006 and Goossens 2008). These asymptotic expressions are derived\nin the approximation that the non-uniform layer is thin. This is the so- called thin bound-\nary (TB) approximation. Jump conditions are used to connect the s olution over the ideal\nsingularity and to avoid solving the non-ideal MHD wave equations (se e e.g. Sakurai et al.\n1991; Goossens et al. 1995). The asymptotic expression for the d amping time in the TB\napproximation is\nτd\nT=FR\nlζ+1\nζ−1, (1)\nwhereζ=ρi/ρeisthedensity contrast, ρiandρetheconstant internal andexternal densities,\nlthe thickness of the non-uniform layer defined in [ R−l/2,R+l/2], andRthe mean radius.\nFis a parameter that depends on the variation of density in the trans itional layer. For\na linear variation F= 4/π2(Hollweg & Yang 1988; Goossens et al. 1992), for a sinusoidal\nvariation F= 2/π(Ruderman & Roberts 2002). In what follows we shall use the versio n\nwithF= 2/π.\nEquation (1) might be considered to be an invitation for predicting a lin ear variation\nof damping time with period. However, the crucial point for this pred iction to make sense\nis that the remaining quantities in equation (1) are constants; i.e. all the oscillations events\nunder considerations have to occur in loops with the same density co ntrastζand the same– 3 –\nradial inhomogeneity l/R. When applied to a collection of oscillating loops on which no\nprior information on their structure is available equation (1) does no t allow any prediction.\nA naive use of equation (1) indicates that for a given period Tthe damping times can differ\nby f.e. a factor 50 /3 for the combinations ζ= 1.5,l/R= 0.1 andζ= 5,l/R= 0.5.\nThis is indeed a naive use of equation (1)since Tisa function of ζalso. In order to make\nthat clear we use results from the thin tube (TT) or long wave-lengt h approximation for the\nperiodT. Our starting point is the well-known expression for the square of t he frequency of\nthe kink mode in a cylinder with a uniform and straight magnetic field alon g thez-axis (see\ne.g. Edwin & Roberts 1983)\nω2=ω2\nk=ρiω2\nA,i+ρeω2\nA,e\nρi+ρe, (2)\nwhereωA=kzvAis the local Alfv´ en frequency and vA=B/√µρthe local Alfv´ en velocity\n(the subscripts “i” and “e” refer to internal andexternal, respe ctively). Now we note that for\nthe fundamental mode kz=π/L, withLthe length of the loop, and we convert frequencies\nto periods and rewrite equation (2) as\nT\nτA,i=√\n2A(ζ), (3)\nwhereτA,i=L/vA,iis the internal Alfv´ en travel time and the function A(ζ) is defined as\nA(ζ) =/parenleftbiggζ+1\nζ/parenrightbigg1/2\n. (4)\nIt is convenient to adopt areference loopwith magnetic fieldstreng thB0, lengthL0, and\ninternal density ρi0. This reference loop allows us to consider loops of different dimension s\nand to introduce the normalized period T⋆defined as\nT⋆=T\n(√\n2)τA,i0. (5)\nT⋆can be interpreted as the period measured in the unit τA,i0√\n2. With this normalized\nperiod equation (3) takes the simple form\nT⋆=a A(ζ), (6)– 4 –\nwitha=τA,i/τA,i0. From equation (6) we learn that, for a given value of a,T⋆varies\ncontinuously from√\n2atoawhenζis allowed to vary from 1 (no loop) to ∞(outside\nvacuum).\nLet us now turn back to equation (1) and rewrite it as\nτd\nτA,i=2√\n2\nπB(ζ)1\nl/R, (7)\nwith the function B(ζ) defined as\nB(ζ) =/parenleftbiggζ+1\nζ/parenrightbigg1/2ζ+1\nζ−1=A(ζ)ζ+1\nζ−1. (8)\nIn what follows we shall refer to the combination of the TB approxima tion to compute the\ndamping time and the TT approximation for the period as the TTTB app roximation.\nIn the same way as for the period it is convenient to introduce the no rmalized decay\ntimeτ⋆\ndas\nτ⋆\nd=τd\n(2√\n2/π)τA,i0. (9)\nτ⋆\ndcan be interpreted as the decay time measured in the unit τA,i0(2√\n2/π). With the use\nof this normalized decay time we can write equation (7) as\nτ⋆\nd=a B(ζ)1\nl/R, (10)\nwhich tells us that, for a given value of a,τ⋆\nddepends on ζandl/R. The dependency on\nl/Ris straightforward. The dependency on ζis slightly more complicated. Basically, τ⋆\nd\nis a decreasing function of ζvarying from + ∞for limζ→1 toa\nl/Rfor limζ→+∞. In\nsummary, τ⋆\ndis a decreasing function of both l/Randζ.\nWe now combine equations (6) and (10). On Figure 1 we have plotted τ⋆\ndversusT⋆for\nvalues of ζfrom 1.5 up to 10 and of l/Rfrom 0.01 to 2. For every value of athe pairs ( T⋆,τ⋆\nd)\ndefine a vertical strip in the ( T⋆,τ⋆\nd)-plane with a maximum horizontal extent of (√\n2−1)a.\nAs a matter of fact acan take on any positive real value, but for clarity we have limited the\nvalues to a= 1,2,3,4,5. The inclusion of intermediate values of a, such as a= 4/3,5/3,7/3,\nproduces vertical strips overlapping those shown on Figure 1. The iso-lines corresponding to– 5 –\na constant value of ζare vertical lines since the period is independent of the inhomogeneit y\nlength scale, in the thin tube approximation. The iso-lines correspon ding to a constant value\nofl/Rare not straight and defined by the equation\nτ⋆\nd=(T⋆)3\n2a2−(T⋆)21\nl/RforT⋆∈[a,a√\n2[. (11)\nIt is clear from Figure 1 that the model of resonant absorption in its most simple mathemat-\nical formulation using the TTTB approximation produces a wide variet y of combinations of\n(T⋆,τ⋆\nd).Now let us see what happens when a collection of pairs ( T⋆,τ⋆\nd) is drawn from this\nreservoir.\n3. Scaling laws: how many do we want?\nThe aim of this section is to show that different collections of data can be produced by\nthe simple TTTB mathematical model of resonant absorption with ea ch of them leading to\nscaling laws, τ⋆\nd/(T⋆)n=C(withCa function of the equilibrium parameters), with different\nindices,n. We use equations (6) and (10), together with the particular relat ion between\nτ⋆\ndandT⋆for each index n, and derive those collections of data. As a first example we\nhave plotted on Figure 2 a collection of ( T⋆,τ⋆\nd) combinations drawn from the big reservoir\ndepicted on Figure 1. These combinations define a scaling law with index n=−1. It is clear\nthat we have engineered the data presented on Figure 2 so as to fit the scaling law with such\nindex . Actually, the engineering is relatively straightforward. We re quireT⋆τ⋆\nd=Cs. The\nleft-hand side is the value of the function Cfor a starting configuration with a=aS,ζ=ζS,\nandl/R= (l/R)S. We fix a loop to start with by prescribing the values aS= 1,ζS= 1.5,\nand (l/R)S= 0.1 and compute values of a,ζ, andl/Rthat satisfy the equation\nl\nR=/parenleftbiggl\nR/parenrightbigg\nS/parenleftbigga\naS/parenrightbigg2f(ζ)\nf(ζS);f(ζ) =A(ζ)B(ζ). (12)\nLikewise on Figure 2 we have also plotted a second collection of data th at now define\na scaling law with index n= 2. To obtain these data, we start from a loop with prescribed\nvaluesaS= 1,ζS= 5, and ( l/R)S= 1 and compute values of a,ζ, andl/Rthat satisfy the\nequation\nl\nR=/parenleftbiggl\nR/parenrightbigg\nSaS\nag(ζ)\ng(ζS);g(ζ) =B(ζ)\n(A(ζ))2. (13)\nAthirdcollection ofdata, that nowdefine a scaling law withindex n= 4/3is also shown\nin Figure 2. According to Ofman & Aschwanden (2002) this value of th e index singles out– 6 –\nphase mixing as damping mechanism. Here it is obtained for a collection o f data that are\nproduced by the theoretical prediction for resonant absorption . We start from a loop with\nprescribed values aS= 1,ζS= 3, and ( l/R)S= 0.5 and compute values of a,ζ, andl/Rthat\nsatisfy the equation\nl\nR=/parenleftbiggl\nR/parenrightbigg\nS/parenleftBigaS\na/parenrightBig1/3h(ζ)\nh(ζS);h(ζ) =B(ζ)\n(A(ζ))4/3. (14)\nFinally, on Figure 2 we have also plotted a collection of data that define a scaling law\nwith the canonical value n=1 for the index. We start from a loop with p rescribed values\naS= 1,ζS= 2.5, and (l/R)S= 0.2 and compute values of a,ζ, andl/Rthat satisfy the\nequation\nl\nR=/parenleftbiggl\nR/parenrightbigg\nSc(ζ)\nc(ζS);c(ζ) =B(ζ)\nA(ζ)=ζ+1\nζ−1. (15)\nNote that ais absent from (15) meaning that a solution of (15) can combined with any value\nofa >0. An obvious solution to (15) is\nζ=ζS,l\nR=/parenleftbiggl\nR/parenrightbigg\nS, a >0. (16)\nEquation (16) defines a straight line in the ( T⋆,τ⋆\nd)-plane. Note that equation (16) is not the\nonly solution in the ( T⋆,τ⋆\nd)-plane. Any combination ( T⋆, τ⋆\nd) that satisfies equation (15)\ncombined with a∈]0,∞[ produces the same straight line in the ( T⋆,τ⋆\nd) plane as solution\n(16). Since the function c(ζ) is a decreasing function of its argument, a value ζ > ζ Sis\ncombined with a value l/R <(l/R)Sand conversely a value ζ < ζSis combined with a value\nl/R >(l/R)S.\nIt is clear fromFigure2 that almost any scaling law can beobtained fro mdata produced\nby the simple TTTB mathematical model of resonant absorption. All the periods and\ndamping times plotted in Figure 2 correspond to combinations of ζandl/R, obtained from\nequations (12), (13), (14), and (15), that are reasonable value s of these quantities in the\nrangesζ∈[1.5,10] andl/R∈[0.016,1.59]. The discrete sets of values in Figure 2 arise due\nto the considered discrete values of a, but loops with different internal travel times, with\nrespect to the reference loop, would give the full set of values for τ⋆\ndandT⋆along the solid\nlines depicted in Figure 2.– 7 –\n4. Beyond the TTTB approximation\nTheTTTB approximationasamathematical model forresonant abs orptionhasitsclear\nlimitations. First, the values of the period defined by equation (6) ar e independent of the\nradiusRand the inhomogeneity length scale l/R. In reality, the period is a function of the\ndensity contrast ζ,l/R, andR. As a consequence the iso-lines corresponding to a constant\nvalue ofζin the (T⋆,τ⋆\nd)-plane are no longer straight vertical lines. For a given value of a\nthis dependency of the period on ζproduces additional spread on the original vertical strips.\nSecond, the TTTB approximation is an accurate approximation as lon g as the damping time\nis sufficiently longer thanthe period, since this isan inherent assumpt ion for carrying out the\nasymptotic analysis leading to equation (1). We can expect this equa tion to be inaccurate\nfor large values of the density contrast ζand large values of the inhomogeneity length scale\nl/R. Figure 3 is the twin version of Figure 1 with the values of T⋆andτ⋆\ndnow computed\nfor fully non-uniform 1-D loops (Van Doorsselaere et al. 2004). Figu re 3 is very similar in\nappearance to Figure 1. The differences occur where they expect ed to occur. The vertical\nstrips of Figure 1 are now replaced with strips slightly oblique to the ve rtical axis and the\nlargest differences appear at the short values of thedamping times corresponding to the large\nvalues of the density contrast ζand large values of the inhomogeneity length scale l/R. The\nbasic message from both figures is the same. The model of resonan t absorption produces a\nwide variety of combinations in the ( T⋆,τ⋆\nd)-plane. If we draw a curve defined by a given\nrelation between T⋆andτ⋆\nd, for instance τ⋆\nd/(T⋆)2=C, in this plane we can graphically\ndetermine as many points on this curve as we want. If we select the d ata determined in this\nway the result is a scaling law with index n= 2 as the one in Figure 2. The only difference\nis that now the procedure is numerical all the way.\n5. Conclusion\nIn this Letter we have explained that scaling laws for a group of obse rvations of oscil-\nlating loops to discriminate between different damping mechanisms are of not much use if\nthere is no a priori information on the properties of the loops. The a nalytical expressions\nobtained by the TTTB approximation enable us to show in a straightfo rward and easy to\nfollow manner that in the framework of resonant absorption collect ions of synthetic data can\nbe produced that follow almost any scaling law. Then, we have backed up our findings by\nnumerical eigenvalue computations that do not suffer from the TTT B limitations but are on\ntheir own rather less instructive.\nItmightbearguedthatnatureisnotascunningastheauthorsoft hispaperanddoesnot\nattempt to engineer data according to prescribed rules as those d efined in these equations.– 8 –\nOn the other hand, there is no reason why nature would want to pro duce coronal loops that\nall have the same values of ζandl/R. So, unless there is accurate a priori information on\nthe coronal loops available, the use of scaling laws to discriminate bet ween different damping\nmechanisms is questionable, to say the least.\nIt is pleasure for MG to acknowledge the warm hospitality of JLB, the friendly at-\nmosphere of the Solar Physics Group at the UIB, and the visiting pos ition from the UIB.\nMG also acknowledges the FWO-Vlaanderen for awarding him a sabbat ical leave. IA and\nJLB acknowledge the funding provided under projects AYA2006-0 7637 (Spanish MEC) and\nPRIB-2004-10145 and PCTIB2005GC3-03 (Government of the Ba learic Islands). The au-\nthors are grateful to R. Oliver, J. Terradas, and T. Van Doorsse laere for comments and\nsuggestions.– 9 –\nREFERENCES\nAschwanden, M. J., Fletcher, L., Schrijver, C. J., & Alexander, D. 1 999, ApJ, 520, 880\nChen, J. & Schuck, P. W. 2007, Sol. Phys., 246, 145\nDe Pontieu, B., Martens, P. C. H., & Hudson, H. S. 2001, ApJ, 558, 8 59\nEdwin, P. 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M., Roberts, B., & Oliver, R. 1997, A&A, 317, 752\nVan Doorsselaere, T., Andries, J., Poedts, S., & Goossens, M. 2004 , ApJ, 606, 1223\nWang, T., Solanki, S. K., Curdt, W., Innes, D. E., & Dammasch, I. E. 20 02, ApJ, 574, L101\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 10 –\nFig. 1.— Normalized damping time as a function of the normalized period f or discrete values\nofζ∈[1.5,10] andl/R∈[0.01,2], and five values of a.– 11 –\nFig. 2.— Four synthetic scaling laws corresponding to n=−1,4/3,1,2, and values of a,\nfrom left to right, a= 1,2,3,4.– 12 –\nFig. 3.— Normalized damping time as a function of the normalized period f or fully non-\nuniform loop models with ζ∈[1.5,20] andl/R∈[0.1,2]. The six vertical strips correspond\nto six values of kz=π/L."    },    {        "title": "0903.0227v3.K_essence_Explains_a_Lorentz_Violation_Experiment.pdf",        "content": "arXiv:0903.0227v3  [gr-qc]  18 Jun 2009CAS-KITPC/ITP-087\nK-essence Explains a Lorentz Violation Experiment\nMiao Li,∗Yi Pang,†and Yi Wang‡\nKavli Institute for Theoretical Physics,\nKey Laboratory of Frontiers in Theoretical Physics,\nInstitute of Theoretical Physics, Chinese Academy of Scien ces,\nBeijing 100190, People’s Republic of China\nAbstract\nRecently, a state of the art experiment shows evidence for Lo rentz violation in the gravitational\nsector. To explain this experiment, we investigate a sponta neous Lorentz violation scenario with a\ngeneralized scalar field. We find that when the scalar field is n onminimally coupled to gravity, the\nLorentz violation induces a deformation in the Newtonian po tential along the direction of Lorentz\nviolation.\n∗Electronic address: mli@itp.ac.cn\n†Electronic address: yipang@itp.ac.cn\n‡Electronic address: wangyi@itp.ac.cn\n1The pursuit of Lorentz violation has attracted increasing attentio n. The local Lorentz\nsymmetry has been examined in many sectors of the standard mode l, including the sectors\nrelating to photons, electrons, protons, and neutrons [1, 2, 3]. N o Lorentz violation has been\nidentified so far in these sectors. The theoretical studies of Lore ntz violation can be found\nin [2, 4, 5].\nRecently, M¨ uller, Chiow, Herrmann, Chu, and Chung [6] performed an experiment to\nprobe the local Lorentz symmetry in the gravitational sector. Th ey measured the phase\nshift of atoms using atom interferometry. Finally, they found a mor e than 2σdeparture of\nLorentz symmetry. This result may be a signal of Lorentz symmetr y violation.\nIn [6], the deviation from Lorentz symmetry is parametrized in the SM E (standard model\nextension) framework [2, 5]. In SME, the Lorentz violation originate s from a Lorentz vio-\nlating coupling in the action\nSLV=M2\np\n2/integraldisplay\nd4x√−g(sµνRT\nµν+tµναβCµναβ), (1)\nwheresµν,tµναβindicate Lorentz violation in gravity, RT\nµνis the traceless part of Rµνand\nCµναβis the conformal Weyl tensor. As a result of this coupling, sµνandtµναβinherit the\nsymmetries of the Ricci tensor and the Riemann curvature tensor respectively.\nThe Lagrangian for a nonrelativistic test particle takes the form\nLp=1\n2mv2+GMm\nr/parenleftbigg\n1+sjkrjrk\n2r2+···/parenrightbigg\n, (2)\nwhere “···” denotes terms that are irrelevant to the measurement.\nIf one assumes standard dispersion relation for photons, the Lor entz violating tensor sµν\nis measured to be\nsXX−sYY=−(5.6±2.1)×10−9. (3)\nIn this article, we explain the anomaly in Eq. (3) in terms of scalar fields with a gener-\nalized kinetic term. We consider the action\nS=/integraldisplay\nd4x√−g/parenleftbigg1\n2M2\npR+P(X)−1\n˜M2∂µφ∂νφRµν/parenrightbigg\n, (4)\nwhereX=−gµν∂µφ∂νφ, with the signature of the metric ( −,+,+,+), and ˜Mis an energy\nscale denoting the coupling strength between φandRµν. This kind of generalized kinetic\nterm in the action has been widely used in cosmology (see, e.g.[7, 8, 9, 10], and references\n2therein). For our purpose of breaking rotational invariance, it is im portant that we have\na negative vacuum expectation value for X; thusP(X) is a function of Xonly. This is\nguaranteed at the effective action level since we assume that ther e is a shift symmetry of φ:\nφ→φ+a, and this symmetry is respected also by the coupling between the Ric ci tensor and\nthe gradient of φas in Eq. (4). This action has another feature that is to retain the s hift\nsymmetry of φ,∂µφ∂νφRµνis the unique nontrivial coupling between φ,Rµν, andCµναβ\nwhen our discussion is just relevant up to the first order derivative ofφ. The reason is that\none cannot construct tµναβfrom the gradient of φ, since∂µφ∂νφis symmetric about indices\nµ,ν, whiletµναβinheriting the symmetry of the Riemann curvature tensor, is antisy mmetric\nabout indices µ,ν, andα,β. Thus we only need to introduce one parameter to describe the\ninteraction strength between φand gravity in the framework of SME. On the contrary, if the\nLorentz violation is induced by the vector field or tensor field [11, 12, 13, 14, 15]; then one\ncan check that up to the first order derivative of these fields, the ir couplings to gravity will\ninclude both the Ricci term and the Weyl term. Therefore to param etrize these couplings,\nat least two parameters are necessary.\nTo have Lorentz violation in spacelike direction, the Hamiltonian derive d from the scalar\nfield Lagrangian P(X) must have a minima at X <0. Note that the X <0 regime is an\nopposite limit compared with the ghost condensation scenario, wher eXhas an expectation\nvalue at X >0. As the expectation value of Xcomes from spontaneously breaking, our\nX <0 model has equal probability to be realized compared with the ghost condensation\nscenario, so our model has to be considered as seriously as ghost c ondensation. We also\nfind that our model has interesting cosmological implications, which a re to be discussed in\na forthcoming work.\nWe use the Lagrangian\nP(X) =X2\n4M4+1\n2X+1\n4M4. (5)\nTo have the vacuum expectation value of X, we solve the equation of motion of φwith\ngµν=ηµν. The equation of motion takes the form\n∂µ/parenleftbigg√−g/parenleftbiggX\nM4+1/parenrightbigg\ngµν∂νφ/parenrightbigg\n= 0. (6)\nNote that the solution ∂νφ= 0 is not a stable solution, since the corresponding energy is\nnot minimal. The stable solution of the above equation is X=−M4, implying a Lorentz\nviolation.\n3Without losing generality, we assume the gradient of the scalar field t o be along the z\ndirection. We have ∂zφ=M2. Inspired by the experiment [6], the interaction strength\nshould take the form\nα≡M4\nM2p˜M2≃10−9. (7)\nIn the remainder of this article, we will show explicitly that our model c an explain the\nproposed Lorentz violation. To do this, we will solve the perturbatio nequations with a point\nmassmδ(x). We find that the gravitational potential induced by mis indeed deformed in\nthezdirection. We also find that the perturbation has positive mass squa red, so the\nperturbation is well defined and stable.\nTo consider perturbations, we let φ(x) =M2z+π(x) and the perturbation of the metric\nds2=−(1+2Φ(x))dt2+(1−2Ψ(x))(dx2+dy2)+(1−2˜Ψ(x))dz2. (8)\nThe Einstein equation contains the following constraint equations\n∂0(M2\np(Ψ+˜Ψ)+2M2\n˜M2χ) = 0,\nM2\np(˜Ψ−Φ)+2M2\n˜M2χ= 0,\nM2\np(Ψ−Φ)+2M2\n˜M2χ+2M4\n˜M2˜Ψ−2M4\n˜M2(Ψ−Φ) = 0. (9)\nIn the above equations, the first one can be rewritten as\nM2\np(Ψ+˜Ψ)+2M2\n˜M2χ=ϕ . (10)\nwhereϕ≡ϕ(x) is a function with no time dependence.\nInserting these constraint equations into the rest of Einstein equ ation, we derive following\ntwo independent equations of motion:\n∇2ϕ−2αϕ,33=mδ(x), (11)\nM2\n˜M2/parenleftBig\n−2Ψ,00+∇2(Ψ−Φ)+/square˜Ψ+(Ψ−˜Ψ),33/parenrightBig\n+χ+M2˜Ψ = 0, (12)\nwhereχ≡π,3,/square≡ −∂2\nt+∂2\nx, and Eq. (12) is consistent with the equation of motion of π.\n4To solve above equations, we first express Φ, Ψ, and ˜Ψ in terms of ϕandχ,\nΦ =1−2α\n1−3αϕ\n2M2p+1−2α\n1−3αM2\n˜M2M2pχ ,\nΨ =1−4α\n1−3αϕ\n2M2p−1−2α\n1−3αM2\n˜M2M2\npχ ,\n˜Ψ =1−2α\n1−3αϕ\n2M2p−1−4α\n1−3αM2\n˜M2M2pχ . (13)\nIn terms of ϕandχ, Eq. (12) can be rewritten as\n−/squareχ−2α\n3−8αχ,33+˜M2\n2M2(3−8α)/parenleftbig\n∇2ϕ−2αϕ,33−4α∇2ϕ/parenrightbig\n+m2\n1χ−m2\n2ϕ= 0,(14)\nwhere\nm2\n1≡(1−2α)2\nα(3−8α)˜M2, m2\n2≡1−2α\n2(3−8α)˜M4\nM2. (15)\nWhen we set m= 0 in Eq. (11), we have ϕ= 0, and the χfield has an oscillating\nsolution with positive mass m1, much larger than ˜M. In other words, the perturbation of χ\nhas positive mass squared. The perturbation is stable.\n˜Mis the mass scale appearing in the Ricci term in Eq. (4), representing the mass scale\nat which this term is generated. If this term is due to quantum gravit y effect, ˜Mis close to\nMp. If for some purpose we want to have a much lower scale, we will have to assume new\nphysics (for instance a large extra dimension) from which this Ricci t erm arises.\nWe further consider gravity with source. When m/negationslash= 0, the solution of Eq. (11) is\nϕ=−m√1−2α4πr′, r′2≡x2+y2+z′2, z′≡z√1−2α. (16)\nInserting Eq. (16) into Eq. (14), we have\nχ=m˜M2(1−4α)\n2M2(3−8α)e−m1r′′\n4πr′′−amm2\n2\nm1√1−2α/integraldisplay1\n01√\nte−√\ntm1r′′′dt\n−4aα2\n(3−8α)√1−2αm˜M2\nm1M2∂2\n3/integraldisplay1\n01√\nte−√\ntm1r′′′dt , (17)\nwhere\nr′′2≡x2+y2+z′′2, z′′≡z/radicalbig\n1+2α/(3−3α),\nr′′′2≡x2+y2+z′′′2, z′′′≡z′′\n/radicalbig\nt+(1−t)a2, a≡/radicalbigg\n(1−2α)(3−3α)\n3−α.(18)\n5To the first order in α, Eq. (17) takes the form\nM2\n˜M2M2pχ=1\n3Gme−m1r′′\nr′′−αGm\nr/parenleftbig\n1−e−m1r/parenrightbig\n. (19)\nNote that the contribution from χto the gravitation potential Φ is either suppressed by the\nYukawa factor e−m1r′′or by the small number α. So at a long distance, the only contribution\nfromχto Φ is a shift in the Newtonian constant G. Insert Eqs. (16) and (19) into Eq. (13),\nand we have\nΦ =−(1+3α)Gm\nr/parenleftbigg\n1−z2\nr2α/parenrightbigg\n, (20)\nwhereG≡1/(8πM2\np), and the factor 1+3 αcan be absorbed into a redefinition of G, so it is\nnot measurable. Meanwhile the term 1 −z2\nr2αgives an explicit Lorentz violation. Comparing\nwith (2), we have\ns33=−2α . (21)\nTo compare with experiments, we can identify the third direction (de noted by zor 3 in\nthe article) with the Xdirection in [6]. Then a value α= 2.8×10−9gives an explanation to\nthe measurement [6]. Alternatively, we can also identify the zdirection with the Ydirection\nin [6], and let α=−2.8×10−9to explain the experiment. In this case, ˜M2<0, while the\nperturbation is still stable.\nTheoretically, we find that physics at string scale may be responsible for the small value\nofα. The reason is below. As mentioned before, it is reasonable to assum e that the scalar-\ngravity coupling term in Eq. (4) originates from the quantum gravity effects, so ˜M≃Mp.\nThen from the expression of αEq. (7), we read\nM=|α|1/4Mp≃1.7×1016GeV, (22)\nwhereMp≃2.4×1018GeV has been used ( Mpis the reduced Planck mass defined as\nMp≡1/√\n8πG). The energy scale of Mcan naturally arise from string theory by requiring\nthe scale of extra dimension approach Planck scale. At this stage, w e cannot guarantee the\nLorentz violation is due to stringy effects, but there is the possibility that Lorentz violation\nis induced by some stringy physics effectively described by the gener alized scalar field.\nFinally, we consider some signatures of our model at small length sca les. At small length\nscales, the first term in the right hand side of Eq. (19) becomes impo rtant. Combining this\nterm and the contribution from ϕ, we will obtain a gravitation potential with a running\n6Newtonian constant,\nΦ =−G(r)m\nr, (23)\nand\nG(r) = (1−1\n3e−m1r)G, (24)\nwhere we have neglected the terms proportional to α. As we remarked before, ˜Mis the\nenergy scale at which the Ricci term is generated, so naturally it is no t small, while m1is a\nfactor 1/αlarger than ˜M, an even larger mass scale, so it is not conceivable to measure the\nrunning of the Newton constant.\nTo conclude, we have considered spontaneous Lorentz violation fr om a generalized scalar\nfield. We show that when Xhas a nonzero vacuum expectation value, the Lorentz symme-\ntry is spontaneously broken. When coupling to gravity, this Lorent z violation affects the\nNewtonian potential. This modification of gravity can explain the curr ent experiment [6],\nand can be tested in future experiments. As stated in [6], future ex periments may reach the\naccuracy of 10−14. It is very interesting to see whether the Lorentz violation is confir med in\nthe future.\nAcknowledgments\nWe thank M. L. Yan for discussion. This work was supported by NSFC Grant No.\n10525060, and a 973 project Grant No. 2007CB815401.\n[1] G. Amelino-Camelia, C. 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Bluhm and V.A. Kosteleck´ y, Phys. Rev. D 71, 065008 (2005)\n[13] R. Bluhm, S. Fung, V.A. Kosteleck´ y, Phys. Rev. D 77, 065020 (2008)\n[14] V.A. Kosteleck´ y and R. Lehnert, Phys. Rev. D 63, 065008 (2001); B. Altschul and V.A.\nKosteleck´ y, Phys. Lett. B 628, 106 (2005); T. Jacobson and D. Mattingly, Phys. Rev. D 64,\n024028 (2001); C. Eling and T. Jacobson, Phys. Rev. D 69, 064005 (2004); Class. Quant.\nGrav.23, 5643 (2006); P. Kraus and E.T. Tomboulis, Phys. Rev. D 66, 045015 (2002); S.M.\nCarroll and E.A. Lim, Phys. Rev. D 70, 123525 (2004); O. Bertolami and J. Paramos, Phys.\nRev. D72, 044001 (2005); M.V. Libanov and V.A. Rubakov, JHEP 0508, 001 (2005); J.W.\nElliottet al., JHEP 0508, 066 (2005); R. Bluhm et al., Phys. Rev. D 77, 125007 (2008);\nS.M. Carroll et al., Phys. Rev. D 79, 065012 (2009), arXiv:0812.1050; 79, 065011 (2009),\narXiv:0812.1049.\n[15] M.D. Seifert, Phys. Rev. D 76, 064002 (2007); arXiv:0903.2279v1.\n8"    },    {        "title": "2111.12418v1.Quantum_corrections_to_the_Lorentz_algebra_due_to_mixed_gravitational__U_1___chiral_anomalies.pdf",        "content": "arXiv:2111.12418v1  [hep-th]  24 Nov 2021Quantum corrections to the Lorentz algebra due to mixed\ngravitational- U(1)-chiral anomalies.\nSandeep S. Cranganore1,2,∗\n1Institute for Theoretical Physics, University of Cologne, 50923 Cologne, Germany\n2TU Wien - Atominstitut, Stadionallee 2, 1020 Vienna, Austri a\n(Dated: November 25, 2021)\nAbstract\nWe calculate the quantum corrections to the Lorentz algebra for chiral Weyl fermions interact-\ning with an external U(1)gauge field in a background Riemann-Cartan (RC) spacetime. T his\nwas achieved by setting up the equal-time commutation relat ions (ETCR) for the canonical spin\ncurrent of chiral Weyl fermions. Furthermore, these quantu m corrections lead to an order sensitive\ncommutator, i.e., swapping the Lorentz generators in the co mmutator doesn’t merely lead to a\nsign change, but rather a completely different correction ter m to the Lorentz algebra. Thus, the\nalgebra of Lorentz is altered due to anomalies associated wi th chiral particles.\n∗sandeep.cranganore@tuwien.ac.at\n1NOTATIONS\nSpacetime coordinates will be labeled with Latin indices i,j,k...=0,1,2,3. Spatial co-\nordinates will be denoted by a, b,....= 1, 2, 3. The frame (tetrad) fields are denoted as\neαwith components eiα, whereα,β,...=0,1,2,3 are the Lorentz indices. The coframe\nfield is denoted as ϑβwith components (vierbeins) eβ\nj.ωα=∗ϑα,ωαβ=∗(ϑα∧ϑβ),\nωαβγ=∗(ϑα∧ϑβ∧ϑγ),e∶=det(eβ\nj)=√−g. Parentheses around the indices denote antisym-\nmetrization[ij]∶=1\n2(ij−ji). The (Fock-Ivanenko) covariant exterior derivative component s\nfor spinors are Dα=ei\nαDiwithDi=∂i+i\n4Γαβ\niΣαβ, where Γαβ\niis the Lorentz (spin) con-\nnection and Σαβare the representations of the Lorentz generators. The metric field will be\ndenoted as gij(x)and the Minkowski metric as ηij.\n2I. INTRODUCTION\nThe Einstein-Cartan (EC) theory is a gauge theory of gravity, obt ained by gauging the\nPoincar´ e group. The Poincar´ e group is a 10 parameter non-Abelia n and non-compact Lie\ngroup which is the semi-direct product of 4 translations T4and 6 Lorentz transformations\nSO(1,3), i.e.,P(1,3)=T4⋊SO(1,3). This approach is built over the concept of Wigner\nclassification, where quantum particles are indexed by its invariants mass and spin which are\nlinked totheset oftranslations andLorentztransformationsres pectively. Thegaugeconnec-\ntion one-forms (potentials) are the spin connection Γαβ(x)linked to the Lorentz group and\nthe orthonormal coframe field (tetrads) ϑα(x)linked to the translations. The translational\nfield strength two-form Tα(x)is called torsionwhile the rotational field strength two-form\nRαβ(x)is called Lorentzcurvature. Thus the underlying arena of the Einstein-Cartan the -\nory is called a Riemann-Cartan (RC) spacetime. The matter current three-forms associated\nwith these gauge potentials are the canonical energy-momentum c urrentTα(x), which are\nthe currents coupled to ϑα(x)and sources the underlying spacetime curvature while the\ncanonical spin current Sαβ(x)are currents coupled to the spin (Lorentz) connection and are\nsources of spacetime torsion [ 1]. In this paper, we concentrate specifically on the canonical\nspin current for spin-1\n2spinorial fields which are surprisingly dualto the chiral (axial) vector\ncurrentsJi\n5(x).\nInspinor electrodynamics, quantum anomalous non-conservation of the chiral (axial) vec-\ntor current even for massless fermions leads to a topological term . This was firstly explained\nin the context of PCAC problem of a neutral meson decaying into two photonsπ0−>2γ.\nThis is the celebrated Adler-Bell–Jackiw (ABJ) anomaly or chiral anom aly problem [ 2].1\nIn particle physics, the intrinsic angular momentum or spin plays a pivo tal role. In [ 3] it\nwas already claimed that the conservation law for spin current of ha drons would satisfy a\nPCAC (partially conserved axial vector current) like equation. It w as also shown that the\ntensorial part of the spin current could source massive spin- 2mesons. Hence, we take this\nstudy further to set up an analogous PCAC and chiral anomaly like ex pression for the spin\ntensor.\n1Definitely, there exists chiral anomalies for non-Abelian gauge fields too.\n3Recently, a major breakthrough happened in condensed matter s ystems, where the chi-\nral anomaly analogue was experimentally confirmed in new topological phases of matter,\nnamely the Weyl-semimetal . Electrons in these material behave like a Weyl fermion which\nare massless relativistic particles. A Weyl semimetal carries non-tr ivial topological propeties\nand the Weyl fermions with opposite chiralities are separated in mome ntum space and host\na monopole and an antimonopole of Berry flux in momentum space resp ectively. In this\nsituation, parallel magnetic and electric fields ( ⃗E⋅⃗B) can pump electrons between Weyl\ncones of opposite chirality that are separated in momentum space. This process violates the\nchiral charge conservation and the number of particles of left and right chirality are not sep-\narately conserved. This arises because ofthe topological term(1\n2ǫijklFij(x)Fkl(x)≈⃗E⋅⃗B)[4].\nRecently, there have been attempts in the direction of mixed gravit ational-axial anoma-\nlies in Weyl-semimetals [ 5]. But what is important to note is that mixed gravitational-axial\nanomalies are purely based on General relativity (GR). GR is built over the concept of a\nsymmetric metric field gij(x)whose second-derivatives lead to the Riemann curvature tensor\nRijkl(x). This doesn’t offer a way to rather probe the interaction of spin of c hiral particles\nwith gravitational fields, which is only possible in the context of EC gra vity.\nIdeally, there have not been enough concrete experimental proo fs of spin current aris-\ning from a gauge perspective. Although, one should not forget tha t there have been\nsome experimental signatures related to the anomalous phases of3Hein the A phase\nat low temperatures having a net macroscopic spin. This also points s trongly towards\nthe existence of an asymmetric energy-momentum tensor. Also, t he EC length scale\nlEC≈(λComptonl2\nPlanck)1\n3≈10−29mis seven orders of magnitude larger than the reduced\nPlanck scale lPlanck≈10−36m. PLANCK data indicate that General Relativity can be ver-\nified to scales of≈10−28m[6]. Thus, signatures of EC gravity may well be detected in the\nnear future.\nSince, we have some hopes of spacetime being RC and also have acces s to such topological\nmaterialstolookintoanomaliesassociatedwithchiralWeylfermionse xperimentally, wetake\nup an interesting problem which deals with finding quantum non-conservation laws for the\ncanonical spin current of fermions .\n4A. Motivation\nThe question that we want to address in this paper is if the conserva tion of spin angular\nmomentum holds in the quantum level. If not, then the “anomalies” ar ising in QFT, would\nlead to the breakdown of Lorentz symmetry in the quantum regime. This would imply that\nthealgbra of Lorentz suffers from anomalies due to quantum effect s.\nIt is quite surprising that the canonical spin current (sources of t orsion) of spinorial\nfields are related to the chiral (axial) currents. Hence, it is motivat ing enough to look for\ngravitational analogues of the chiral anomaly. This phenomena is no t merely calculating\nthe chiral anomaly in a curved background [ 7,8], but rather a completely different anomaly\nwhere we compute the divergence of the spin current in a backgrou nd Riemann-Cartan (RC)\nspacetime (curved + contorted) or interacting with a combination o f external Abelian U(1)\ngauge fields in a RC background.2\nInfact, such an anomaly has already been calculated in [ 9,10] and is also famously known\nas the “Lorentz-Anomaly”. Thus computing these anomalies could a lways be useful since\nexperimental tests of these anomalies could be possible using the to pologically non-trivial\n“Weyl semimetal”.\nIt was shown in [ 9] that, purely gravitational anomalies do not exist in 4 ndimensions\nas a consequence of charge conjugation properties of the gravit ational interactions. On the\ncontrary they are present only in 4 n+2 dimensions [ 9,11,12]. Infact, the existence of such\nchiral Lorentz anomalies leads to a breakdown of Lorentz symmetr y in 4n+ 2 dimensions\ndue to quantum effects.\nThus our main motivation is to calculate using the results obtained in [ 10], the quantum\ncorrections to the algebra of Lorentz present for chiral particles interacting with a combined\ngravitational and U(1)gauge field. Thus, by studying chiral gauge theories in curved spac e-\ntime, one can understand the interactions between intrinsic angular momentum of fermions\ncoupled to the underlying geometry of spacetime .\n2It would be further clarified why non-Abelian fields were not mentione d here.\n5II. PLAN OF THE PAPER\nWe firstly describe the canonical spin current of Dirac fields and its ir reducible decom-\nposition into different components. It turns out to be that only the totallyantisymmetric\naxial vector part survives, i.e.,AXSi\nαβ(x)=i̵hc\n4eeiγǫαβγδJδ\n5(x).Then in subsection ( IIIC),\nwe revisit the ETCR/ spin current algebra. Later, we briefly go thro ugh the similarities\nbetween the ABJ anomaly and the Lorentz anomaly by considering, ( i) a purely gravita-\ntional (RC) background, (ii) spin of chiral Weyl fermions interactin g with a combination of\nan external U(1)gauge field in a background RC spacetime.\nOur main work can be found in section ( IV), wherein, we compute the quantum mod-\nifications to the spin current algebra which shows up as “anomalies” d ue to U(1) gauge\nfields and Riemann-Cartan spacetime combined. Furthermore, usin g these results, we re-\nduce these spin current commutators to the Lorentz algebra for chiral Weyl fermions. This\nleads to a very interesting consequence that, one obtains differen t quantum correction terms\neither proportional to α(“Fine-structure constant”) or rather a Dirac quantized product of\nthe electric and magnetic charges , if we include magnetic monopoles in Maxwell’s theory. A\nvery interesting result that we show here is that, merely swapping t he Lorentz generators in\nthe commutation relation not only leads to a sign change but rather a completely different\ncorrection terms. Thus, we show that “quantum anomalies” leads t o the breakdown of local\nLorentz symmetry and ultimately, alterations to the Lorentz algeb ra.\nIII. THE CANONICAL SPIN CURRENT OF A SPINORIAL FIELD.\nA. Spin currents irreducible decomposition\nThe irreducible decomposition of the spin tensor (6 ×4=24 components) consists of a\ntensor part, a vector part and an axial vector part [ 13],\nSk\nij=TENSk\nij+VECSk\nij+AXSk\nij. (1)\nThe tensor piece,TENSk\nijconsists of 16 components. The vector pieceVECSk\nij∶=2\n3Sl\n[i/divides.alt0lδk\n/divides.alt0j]\ncontains 4 components and the axial vector piece being completely a ntisymmetricAXSijk=\nS[ijk]also contains 4 components.\n6The Dirac Lagrangian in a Riemann-Cartan spacetime reads,\nLD=̵hc\n2e/parenleft.alt1i(¯ψγαDαψ−Dα¯ψγαψ)−2mc\n̵h¯ψψ/parenright.alt1. (2)\nWhereγαare the constant Dirac matrices and are the components of the Cliff ord-algebra\nvalued exterior forms, i.e.,\nγ∶=γαϑα,\nand the Fock-Ivanenko type covariant derivative acts on the spin ors,\nDiψ(x)=/parenleft.alt1∂i+i\n4Γαβ\ni[γα,γβ]/parenright.alt1ψ(x)\nWhere, Σαβ=i\n4[γα,γβ]is the spinor representation of the Lorentz generators. The Dira c\nequation for the spinor field ψand the conjugate field ¯ψare,\niek\nαγα∂kψ=mψ−iek\nαγαΓkψ, (3a)\n−i∂k¯ψγαek\nα=¯ψm−i¯ψΓkγαek\nα, (3b)\nwhereDαψ=ekα(∂k+Γk)ψ,Dα¯ψ=¯ψ(∂k−Γk)ekα.\nB. Relation between spinorial spin current and chiral (axial) vector currents\nThe canonical spin current of the Dirac field are totally antisymmetr ic (axial part with\n4-components) and is dual to the axial vector currents ,\nAXSi\nαβ(x)∶=−2δLD\nδΓαβ\ni=i̵hc\n4eeiγǫαβγδ¯ψ(x)γ5γδψ(x)=i̵hc\n4eeiγǫαβγδJδ\n5(x).(4)\nThe above Eq. ( 4), obeys the classical conservation equation for spin angular momentum\n,i.e., [13],\nDiSi\nαβ(x)−2T[αβ]=0. (5)\nWhereT[αβ]is the antisymmetric part of the canonical energy-momentum curr ent. Our\ngoal is to study whether the above Eq. ( 5) is subjected to “quantum non-conservation” due\nto anomalies. Hence, we derive an analogue of the chiral anomaly. Fr om this point on, we\nwork with chiral Weyl fermions which are massless and carry a hande dness (chirality).\n7C. Spin current algebra\nThe ETCRs for the spin current can be derived using the Schwinger q uantum action\nprinciple by varying the matrix elements of Eq. ( 5), w.r.t to the spin (Lorentz) connection\nΓαβ\ni(x). Thus, the canonical spin current Lie algebra reads,3\n[S0\nαβ(x),Sj\nγδ(x′)]x0=x′\n0=2i/parenleft.alt1ηγ[αSj\nβ]δ(x)−ηδ[αSj\nβ]γ(x)/parenright.alt1δ3(x−x′)\n+i/parenleft.alt4∂iδSi\nαβ(x)\nδΓγδ\nj(x′)−2δT[αβ](x)\nδΓγδ\nj(x′)/parenright.alt4. (6)\nThe second bracket contains the Schwinger terms. The first term in the second bracket of\nEq. (6) is always zero for fermions sinceδSi\nαβ(x)\nδΓγδ\nj(x′)=0, i.e the current is independent of the\nconnection. For the sake of ease, one could avoid the other term in the second bracket.\nD. Anomalous divergence of spin currents\nUsing theLagrangianforchiral particles (chiral Lagrangian)andlo cal Lorentzinvariance,\nwe find that the chiral Weyl spinors also satisfy the classical conse rvation law,\nDiSi\nαβ(x)χ−2T[αβ](x)χ=0, (7)\nwhere subscript χdenotes chiral Weyl fermions. The spin tensor and the antisymmet ric\ncanonical energy-momentum tensors are,\nSi\nαβ(x)=ie̵hc\n4eiγǫαβγδ¯χ(x)γ5(x)γδ(x)χ(x)=i̵hc\n2ǫαβγδeeiγJδ\n5(x), (8a)\nT[αβ](x)=i̵hc\n2e/parenleft.alt1¯χγ[αDβ]χ−D[β¯χγα]χ/parenright.alt1. (8b)\nThese two currents Eqs. ( 8a,8b) are related to Lorentz invariance anddiffeomorphism\ninvariance respectively. Hence, fermions interacting with purely gravitationa l field should\npossess both Lorentz invariance anddiffeomorphism invariance .\nIn order to study the anomalies associated with the divergence of s pin currents, we\nconcentrate on the local Lorentz symmetry. In order to achieve this we construct a gauge\n3The spin current Lie-algebra corresponds to so(1,3)⊗so(1,3).\n8invariant axial vector current interacting with a background Riemann-Cartan spacetime,\nJl\n5(x/divides.alt0ǫ)=lim\nǫ→0i\n2¯ψ(x+ǫ\n2)γ5γlexp/braceleft.alt4iλ/integral.dispx+ǫ\n2\nx−ǫ\n2dyjΓαβ\nj(y)Σαβ/braceright.alt4ψ(x−ǫ\n2). (9)\nHere, the Fujikawa point splitting on the fields in terms of the spin connection has been\nused since Jδ\n5(x)is singular. Here, λcorresponds to a coupling associated with gravity.\nSubstituting Eqs. ( 9,8a) into Eq. ( 7), we obtain the anomalous divergence of the spin\ntensor,4\nDiSi\nαβ(x)−2T[αβ]≈/parenleft.alt1̃Rγij\nαRβγij−̃Rγij\nβRαγij/parenright.alt1=0. (10)\nWherẽRij\nαβ(x)=1\n2ǫαβγδ̃Rγδij(x)is the dual Lorentz curvature. The anomaly term ( ̃RR) in\nthe r.h.s. of Eq. ( 10) is always zero for 4 ndimensions because of charge conjugation prop-\nerties of the gravitational interactions. Thus the quantum corrections to the spin current\nalgebra is non-existent in 4-dimensional spacetime . Hence, the quantum conservation equa-\ntion holds for the spin in a background RC spacetime . This is a marked difference between\nthe above Eq. ( 10) and the ABJ anomaly, where the later contains anomalous topologic al\nterms in 4-dimensional spacetime even for the massless case.\nIV. ANOMALY DUE TO GRAVITATIONAL +U(1) GAUGE FIELDS AND\nBREAKDOWN OF LORENTZ INVARIANCE\nIn order to better understand the interaction between spin and t he fundamental interac-\ntions, consider the chiral Weyl spinors interacting with an externa l AbelianU(1)gauge field\nAi(x) and other non-Abelian SU(N)gauge fields AA\ni(x) in a background Riemann-Cartan\nspacetime.\nIt is pretty surprising that anomalies are solely contributed by the AbelianU(1)gauge\nfieldson a curved background. Surprisingly, the non-Abelian gauge fields don’t contribute\nto anomalies since there is no breakdown of Poincar´ e invariance, Th is leads to the break-\ndown of Lorentz invariance in physical spacetime. Thus correspon ding expression for the\nLorentz-chiral anomaly is, [ 10],5\nDiSi\nαβχ−2T[αβ]χ=−iqe\n96π2/parenleft.alt1R̃Fαβ+̃RαβklFkl+2Fi\nαβ;i/parenright.alt1. (11)\n4We avoid the torsion tensor coupling to the chiral currents in Eq. ( 10). This term potentially leads to\nanomalies and would be investigated more in the future.\n5The covariant derivative ; in Eq. ( 11) contains both Lorentz+affine connections. we only vary w.r.t the\nspin connection and don’t bother about the affine part.\n9Here,̃Fαβ(x)=1\n2ǫαβγδFγδ(x)is the dual of the electromagnetic field strength tensor. A\nmajor difference between Eq. ( 11) and the chiral anomaly is that the presence of a Laplacian\nof the electromagnetic field strength tensor,Fi\nαβ;i. Also, the first two terms in the r.h.s of\nEq.(11) shows coupling between the electromagnetic field strength and th e rotational field\nstrength (curvature) of the gravitational fields. On the contra ry the chiral anomaly contains\nthe topological termq2\ne\n8π2̃FijFij(x)[2]. Thus, the anomaly due to spin leads to several new\nterms which are absent in chiral anomalies.\nThe spin current commutators are obtained by varying Eq. ( 11) w.r.t. the spin connection\nΓγδ\nj(x′). We record directly the spin current commutators with anomalies inc luded,\n[S0\nαβ(x),Sj\nγδ(x′)]=2i/parenleft.alt1ηγ[αSj\nβ]δ(x)−ηδ[αSj\nβ]γ(x)/parenright.alt1δ3(x−x′) (12)\n−W0,j\nαβ,γδ(x,x′)\n−iqe\n24π2/parenleft.alt1̃Fαβej\n[γek\nδ](x)∂kδ3(x−x′)+1\n2ǫαβγδFjk(x)∂kδ3(x−x′)/parenright.alt1.\nThe third term in r.h.s of Eq. ( 12) is the quantum modification or the anomaly term. By\nthe same argument as above, the Wterms for spinorial fields completely vanish. Here, we\nexplicitly write down the different components,\n[S0\nab(x),Se\ncd(x′)]=2i/parenleft.alt1ηc[aSe\nb]d(x)−ηd[aSe\nb]c(x)/parenright.alt1δ3(x−x′) (13)\n−iqe\n24π2/parenleft.alt1ǫabgEg(x)ee\n[cef\nd](x)/parenright.alt1∂fδ3(x−x′),\n[S0\na0(x),Se\nb0(x′)]=iSe\nab(x)δ3(x−x′) (14)\n−iqec\n24π2/parenleft.alt1Ba(x)ee\n[bef\n0](x)/parenright.alt1∂fδ3(x−x′),\n[S0\nab(x),Se\nc0(x′)]=2iηc[aSe\nb]0δ3(x−x′) (15)\n−iqe\n24π2/parenleft.alt1̃Fab(x)ee\n[cef\n0](x)∂fδ3(x−x′)−1\n2Fef∂fδ3(x−x′)/parenright.alt1,\n[S0\na0(x),Se\nbc(x′)]=2iηa[bSe\nc]0δ3(x−x′) (16)\n−iqec\n24π2/parenleft.alt1̃Fa0(x)ee\n[bef\nc](x)∂fδ3(x−x′)+1\n2Fef∂fδ3(x−x′)/parenright.alt1\n10Writing down the equations explicitly in terms of the electric and magne tic components,\nyields,\n[S0\na0(x),Se\nbc(x′)]=2iηa[bSe\nc]0(x)δ3(x−x′) (17)\n−iqec\n24π2/parenleft.alt11\n2(B×∇)eδ3(x−x′)+Baee\n[bef\nc](x)∂fδ3(x−x′)/parenright.alt1,\n[S0\nab(x),S0\nc0(x′)]=2iηc[aS0\nb]0δ3(x−x′) (18)\n−iqe\n24π2/parenleft.alt11\n2(E⋅∇)δ3(x−x′)+ǫabgEg(x)ef\nc(x)∂fδ3(x−x′)/parenright.alt1,\n[S0\na0(x),S0\na0(x′)]≈−iqec\n24π2(B⋅∇)δ3(x−x′), (19)\n[S0\nab(x),Se\nab(x′)]≈−iqe\n24π2(E×∇)eδ3(x−x′). (20)\nThe symbol≈means there are some other terms which are not of our interest. I nfact the\nLorentz generators are nothing but the integral of the time comp onent of the spin current\nover a space-like 3 dimensional hypersurface, i.e.,\nΣbc∶=/integral.dispd3xS0\nbc(x) (21a)\nΣa0=/integral.dispd3xS0\na0(x). (21b)\nWhere Σabare the Lorentz rotations while Σa0are the Lorentz boosts.\nIntegrating Eq. ( 18) w.r.t. x & x’ and using the Gauss theorem yields,\n[Σab,Σc0]=/parenleft.alt1ηacΣb0−ηbcΣ0a/parenright.alt1−i̵hc\n12πα+.... (22)\nWhereα=q2\ne̵hcis thefine structure constant .\nThis is a remarkable result, since the second term in Eq. ( 22) is the alteration to the\nLorentz algebra due to quantum corrections. Hence, the the exis tencequantum anomalies\nleads to the breakdown of local Lorentz symmetry.\nInfact, integrating Eq. ( 19) w.r.t. x and x’ and using the Gauss theorem yields a very\nsurprising result,\n[Σ0a,Σ0a]=−i̵hc\n12πn. (23)\nWheren=2qeqm̵hc∈Zis theDirac quantization .qmis the magnetic charge. Ideally, the\nalgebra of two same boost generators is zero. If we don’t consider magnetic monopoles, this\nyields the standard result, which is zero.\n11V. DISCUSSION AND CONCLUSION\nFirstly we infer that the purely gravitational anomaly term proport ional tõRγij\nαRβγij−\ñRγij\nβRαγij, vanishes identically in four dimensions. This is related to the charge c onjuga-\ntion properties of the gravitational field. Hence a quantum conservation law holds for spin\ncurrents in four dimensions and the spin current algebra contains no anomalous terms.\nWe see that[Sr,Sr,b]6always contains correction terms related to the ⃗Efields. On the\ncontrary,[Sb,Sr,b], contains terms proportional to ⃗Bfields as closure failure. Thus local\nactive transformations performed on Weyl fermions in the presen ce ofU(1)gauge fields ,\nproduces, apart from a mere sign change, completely different fieldsandcharges(electric\nor magnetic). Hence, anomalies are sensitive to the order of perfo rming symmetry transfor-\nmations.\nA very interesting result is Eq. ( 23), where the rotation-boost commutation contains a\ncorrection term proportional toq2\ne̵hcor the “fine-structure constant” .\nThe commutation between the rotation and boosts generators co ntains a quantum cor-\nrection term proportional to the electric charge squared. While th e same boost generators\nclose on a non-trivial topological number n∈Zwhich is the Dirac quantized product of\nelectric and magnetic charges.\nIf magnetic charges are not taken into consideration, Eqs. ( 17,18) (rotation-boost or\nboost-rotationcommutators)containtheMaxwell source equations . Thesource free Maxwell\nequations show up in - Eqs. ( 19,20) (rotation-rotation or boost-boost commutators). This\ncould have some important implications and would be investigated in fut ure works.\nSimilar anomalous current commutation relations were found for the vector and axial\nvector currents (cf. [ 14]). Here too, the additional anomaly terms corresponded to electr ic\nand magnetic fields or charges and the current commutation were o rder sensitive.\n6Here,Sr,bimplies either SrorSb\n12Our further goal would be to rather come up with viable experimenta l approaches to look\nfor chiral-Lorentz anomaly signatures using Weyl semimetals. This s hall be investigated in\nthe near future.\nACKNOWLEDGMENTS\nI am indebted to my supervisor Prof. Dr. Claus Kiefer who constant ly guided and sup-\nported me to work on this problem accompanied by many helpful and in sightful discussions.\nI am also indebted to Prof. Dr. Friedrich W. Hehl for for consistent ly guiding me with nu-\nmerous insightful discussions and helpful remarks. This project w as partly supported by the\nBonn–Cologne Graduate School of Physics and Astronomy scholar ship.\n[1] M. Blagojevi´ c and F. W. Hehl, Gauge Theories of Gravitation. A Reader with Commentaries\n(Imperial College Press, London, 2013).\n[2] S. B. Treiman, R. Jackiw, and D. J. Gross, Lectures on Current Algebra and Its Applications\n(Princeton University Press, 1972).\n[3] K. Hayashi and R. Stuller, On the dynamical role of spin in particle physics, Lett. Nuovo Cim.\n9S2, 211 (1974) .\n[4] C.-L. Zhang, S.-Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. T ong, G. Bian, N. Alidoust, C.-C.\nLee, S.-M. Huang, T.-R. Chang, G. Chang, C.-H. Hsu, H.-T. Jen g, M. Neupane, D. S. Sanchez,\nH. Zheng, J. Wang, H. Lin, C. Zhang, H.-Z. Lu, S.-Q. Shen, T. Ne upert, M. Zahid Hasan,\nand S. Jia, Signatures of the Adler–Bell–Jackiw chiral anom aly in a weyl fermion semimetal,\nNature Communications 7, 10735 (2016).\n[5] J. Gooth, A. C. Niemann, T. Meng, A. G. Grushin, K. Landste iner, B. Gotsmann, F. Menges,\nM. Schmidt, C. Shekhar, V. S¨ uß, R. H¨ uhne, B. Rellinghaus, C . Felser, B. Yan, and K. Nielsch,\nExperimental signatures of the mixed axial–gravitational anomaly in the weyl semimetal NbP,\nNature547, 324 (2017).\n[6] J.BoosandF.W.Hehl,Gravity-inducedfour-fermioncon tact interaction impliesgravitational\nintermediate W and Z type gauge bosons, International Journal of Theoretical Physics 56,\n751–756 (2016) .\n13[7] T. Kimura, Divergence of axial-vector current in the gra vitational field. 2. anomalous commu-\ntators and model theory, Prog. Theor. Phys. 44, 1353 (1970) .\n[8] S. Treiman, R. Jackiw, B. Zumino, and E. Witten, Current Algebra and Anomalies (WORLD\nSCIENTIFIC, 1985).\n[9] L. N. Chang and H. T. Nieh, Lorentz anomalies, Phys. Rev. Lett. 53, 21 (1984) .\n[10] H. T. Nieh, Quantum effects on four-dimensional space-ti me symmetries, Phys. Rev. Lett. 53,\n2219 (1984) .\n[11] C. Kiefer, Quantum Gravity: Third Edition , International Series of Monographs on Physics\n(OUP Oxford, 2012).\n[12] R. A. Bertlmann, Anomalies in Quantum Field Theory , International Series of Monographs\non Physics (Oxford University Press, Oxford, 2000).\n[13] F. W. Hehl, On energy-momentum and spin/helicity of qua rk and gluon fields, in Proceedings,\n15th Workshop on High Energy Spin Physics (DSPIN-13): Dubna, Russia, Oct 8-12, 2013\n(2014)arXiv:1402.0261 .\n[14] S. Deser and J. Rawls, Generalizations of the sugawara m odel,Phys. Rev. 187, 1935 (1969) .\n14"    },    {        "title": "1205.2545v1.Quantum_dynamics_of_the_damped_harmonic_oscillator.pdf",        "content": "Quantum dynamics of the damped harmonic\noscillator\nT G Philbin\nSchool of Physics and Astronomy, University of St Andrews, North Haugh, St\nAndrews, Fife KY16 9SS, Scotland, UK.\nE-mail: tgp3@st-andrews.ac.uk\nAbstract. The quantum theory of the damped harmonic oscillator has been a subject\nof continual investigation since the 1930s. The obstacle to quantization created by the\ndissipation of energy is usually dealt with by including a discrete set of additional\nharmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield\ndynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest.\nBy using a continuum of oscillators as a reservoir, we canonically quantize the harmonic\noscillator with Ohmic damping and also with general damping behaviour. The\ndynamics of a damped oscillator is determined by an arbitrary e\u000bective susceptibility\nthat obeys Kramers-Kronig relations. This approach o\u000bers an alternative description\nof nano-mechanical oscillators and opto-mechanical systems.\nPACS numbers: 03.65.-w, 03.65.Yz, 03.70.+k\n1. Introduction\nFew classical dynamical systems are as simple or important as the one-dimensional\ndamped harmonic oscillator:\nq+\r_q+!2\n0q=f(t): (1)\nBut the simplicity of this dynamical system does not survive the transition to quantum\nmechanics. The presence of dissipation in (1) (or ampli\fcation with t!\u0000t) leads\nto severe di\u000eculties with its quantization, a problem that has attracted repeated\ninvestigation from the 1930s to the present day (some historical information is given\nin [1, 2, 3]). A brief account of the di\u000berent approaches to quantizing (1) will explain\nthe rather straightforward, but crucial, respect in which the starting point of this paper\ndi\u000bers from previous work.\nThe central obstacle in quantizing the dynamics (1) is that a Hamiltonian is required\nthat will generate the quantum time evolution (a Lagrangian will su\u000ece for the path-\nintegral approach). A Lagrangian and Hamiltonian exist that give the equation of\nmotion (1), but they are time dependent because of the dissipation (or ampli\fcation)\nand this leads to di\u000eculties in implementing the canonical commutation relation [1, 2].\nIf the dissipated energy is included through extra dynamical degrees of freedom, thenarXiv:1205.2545v1  [quant-ph]  11 May 2012Quantum dynamics of the damped harmonic oscillator 2\nthe problem of a time-dependent Hamiltonian can be avoided. The simplest approach\nis to write a two-body Hamiltonian describing one damped and one ampli\fed oscillator,\nwith conserved total energy [4]. But the canonical variables of this system [4] are not the\npositions and momenta of the two oscillators, so again there is no straightforward way of\nimposing canonical commutation relations on each oscillator [1, 2, 5, 6, 7, 8]. In recent\ntimes the most popular approach is to treat a damped harmonic oscillator as a free\noscillator coupled to a reservoir of oscillators of di\u000berent frequencies. The apparently\nuniversal practice for investigations of the damped harmonic oscillator has been to use\na discrete set of oscillators for the reservoir. zThe resulting form of the Hamiltonian\nis attributed to Magalinskii [9], and it is also the most popular starting point for\nattempts to describe quantum Brownian motion (with a free particle coupled to the\nreservoir) [3, 10, 11, 12, 13, 14, 15, 16, 17, 18]. This approach does not give the dynamics\n(1) for the oscillator of interest, with a damping proportional to velocity; instead, the\noscillator equation of motion can be written with a termRt\nt0dsg(t\u0000s) _q(s), whereg(t)\nis an integral kernel dependent on the coupling to the reservoir [3]. In order to describe\ndamping proportional to velocity, often called Ohmic damping in view of the electrical\napplication of (1), a limiting procedure g(t)!2\r\u000e(t) must be employed at some\npoint [3]. The limit that produces Ohmic damping (or ampli\fcation) involves arbitrarily\ndecreasing the frequency spacing between oscillators in the reservoir while maintaining\nreservoir oscillators of arbitrarily high frequencies; in other words, the reservoir is\ne\u000bectively regarded as a continuum, but only after the dynamical equations have been\nsolved under the assumption that the reservoir is a discrete set. The impossibility of\nachieving Ohmic damping with a discrete reservoir and the emergence of Ohmic damping\nthrough a delicate continuum limit, after the dynamics has been solved, is discussed in\ngreat detail by Tatarskii [19]. In addition to the approaches just described, which seek to\nemploy the standard quantization rules, there are phenomenological approaches to the\ndamped harmonic oscillator where no rigorous quantization is attempted (reference [20]\nis just one example of such approaches).\nThe starting point of the results presented here is a harmonic oscillator coupled\nto a reservoir, where the latter is a continuum of oscillators of all positive frequencies.\nUse of a continuum reservoir from the outset gives a much richer dynamical system\ncompared to the use of a discrete reservoir. As the system we will be analyzing has an\nuncountable number of degrees of freedom, it acquires many of the properties of a \feld\ntheory, and is qualitatively di\u000berent from a countable set of coupled oscillators, even if\nthe latter set is in\fnite. We will describe the general method for solving the dynamical\nsystem with a continuum reservoir and solve exactly the case of Ohmic damping and\nampli\fcation, which will emerge from a particular choice of coupling to the reservoir.\nThe quantization for general damping and Ohmic damping is treated in detail, including\nthe diagonalization of the Hamiltonian and the case of thermal equilibrium.\nzThere are of course countless other contexts in which use is made of a discrete reservoir.Quantum dynamics of the damped harmonic oscillator 3\nThe continuum reservoir appears to originate with Huttner and Barnett [21], a\npaper that is very well known and yet whose technical innovation and importance\nhave been under-appreciated. An example of what can be achieved with a continuum\nreservoir is the canonical quantization of the macroscopic Maxwell equations for\narbitrary media obeying the Kramers-Kronig relations [22, 23], including bi-anisotropic\nand moving media [24, 25]. In the present paper the continuum reservoir will again\nshow its power by allowing an exact treatment of the dynamics (1) and its canonical\nquantization. More importantly, the continuum reservoir will naturally lead to the\ncharacterization of a quantum damped harmonic oscillator by an arbitrary e\u000bective\nsusceptibility obeying Kramers-Kronig relations. Because of this last property, the\nquantum damped harmonic oscillator will be found to have much in common with the\nquantum theory of light in macroscopic media. This o\u000bers a alternative framework for\ndescribing macroscopic quantum oscillators, which are now a subject of some remarkable\nexperiments [26, 27, 28, 29, 30]. Rather than attempting to capture the immensely\ncomplicated microscopic physics, the results in this paper suggest that macroscopic\nquantum oscillators may be describable by e\u000bective susceptibilities that are to be\nexperimentally measured, just as the electromagnetic properties of macroscopic media\nare captured by measured permittivities and permeabilities.\nThe price to be paid for employing the continuum reservoir is the extra\nmathematical complexity compared to the discrete case. Many aspects of this\nmathematical apparatus, which provides an important and unusual addition to standard\nquantum \feld theory, have still not been fully explored. As well as treating the speci\fc\nproblem of the damped harmonic oscillator, the results presented here give further\ninsight into the remarkably rich classical and quantum physics of a continuum reservoir.\nSection 2 gives the Lagrangian and equations of motion of an oscillator coupled to\na continuum reservoir. In sections 3 and 4 the dynamics for a particular coupling that\ngives damping proportional to velocity is solved in detail. The system is quantized in\nsection 5 and the diagonalization of the Hamiltonian is addressed. In sections 6 and 7\ncoherent-state solutions and thermal equilibrium are treated. Possible applications and\nextensions of the results are discussed in section 8.\n2. Lagrangian and dynamical equations\nWe consider the dynamical system with Lagrangian\nL=1\n2_q2\u00001\n2!2\n0q2+1\n2Z1\n0d!\u0010\n_X2\n!\u0000!2X2\n!\u0011\n+Z1\n0d!\u000b(!)qX!: (2)\nThis describes a harmonic oscillator with displacement qand frequency !0and a\nreservoir of oscillators with displacements X!and frequencies !2[0;1), with the\nq-oscillator linearly coupled to the reservoir by an arbitrary coupling function \u000b(!). We\ncouple only displacements in (2), not velocities, and the reservoir oscillators are not\ndirectly coupled to each other. The variables q(t) andX!(t) are functions only of time,\nso the entire system may be viewed as located at one point in space. The dynamicalQuantum dynamics of the damped harmonic oscillator 4\nvariables in (2) are thus not \felds in the conventional sense, but the dependence of\nX!(t) on the continuous quantity !will give this system many of the properties of a\n\feld theory, in sharp contrast to the case of a discrete reservoir. The total energy of the\nsystem described by (2) is\nE=1\n2_q2+1\n2!2\n0q2+1\n2Z1\n0d!\u0010\n_X2\n!+!2X2\n!\u0011\n\u0000Z1\n0d!\u000b(!)qX!: (3)\nThe system (2) is close to being a drastic simpli\fcation of the Huttner-Barnett\nmodel [21] of a dielectric coupled to the electromagnetic \feld: if the electromagnetic\n\feld is removed along with the spatial dependence of the medium and the reservoir,\nthen the Huttner-Barnett model almost reduces to the system (2), the di\u000berence being\nthat theq-oscillator would be coupled to _X!rather than X!. The relationship of\nour system to the Huttner-Barnett model will be commented on at several points, as\nsome of the technical achievements of Huttner-Barnett will be closely related to results\nhere. The main di\u000berences from Huttner-Barnett, in addition to the simpli\fcations just\ndescribed, are that (i) we do not rely solely on a retarded or advanced solution of the\nreservoir dynamics, (ii) much of our time will be spent in obtaining the exact solution\nfor a speci\fc coupling function, whereas Huttner and Barnett consider only a general\ncoupling function, and (iii) our coupling term is of a di\u000berent form.\nThe Euler-Lagrange equations of (2) are\nq+!2\n0q\u0000Z1\n0d!\u000b(!)X!= 0; (4)\nX!+!2X!\u0000\u000b(!)q= 0: (5)\nThe general solution of the reservoir equation (5) can be written\nX!(t) =A0(!) cos!t+B0(!) sin!t+\u000b(!)\n!Zt\n0dt0q(t0) sin [!(t\u0000t0)]e\u00000+jt\u0000t0j;\nA0(!) =X!(0); B 0(!) =1\n!_X!(0);(6)\nwhere 0+is a positive in\fnitesimal quantity. By means of (6) we can impose arbitrary\ndisplacements X!(0) and velocities _X!(0) on the reservoir at t= 0 (these are not \\initial\"\nconditions because the solution (6) is valid for all t). The choice of t= 0 is of course\narbitrary, but no generality is lost by the form (6) if we wish to impose conditions on the\nreservoir at some \fnite time (the imposition of conditions in the in\fnite past or future\nis dealt with later in this section). The presence of the exponential in (6) is important\nfor taking the Fourier transform of this general solution, and it can be understood as\nfollows. Solutions of (5) can be constructed using a Green function G(t) de\fned by\nG+!2G=\u000e(t): (7)\nFor example, the retarded Gr(t) and advanced Ga(t) Green functions are\nGr(t) =1\n!\u0012(t) sin!te\u00000+t; (8)\nGa(t) =\u00001\n!\u0012(\u0000t) sin!te0+t; (9)Quantum dynamics of the damped harmonic oscillator 5\nwhere\u0012(t) is the step function. The exponential factors are necessary in the Green\nfunctions (8) and (9) for their Fourier transforms to exist, and the in\fnitesimal number\n0+gives the familiar pole prescriptions in the frequency domain, with Gr(!) analytic in\nthe upper-half complex !-plane andGa(!) analytic in the lower-half plane. The general\nsolution (6) is constructed with the di\u000berence\nGr(t)\u0000Ga(t) =1\n!sin!te\u00000+jtj; (10)\nwhich is a solution of the homogeneous version of (7) (i.e. (7) without the delta function).\nThis gives the exponential factor in (6) that is required to de\fne the Fourier transform\nofX!(t).\nTo complete the solution for the dynamics we must substitute (6) into (4) and solve\nthe resulting equation for q(t):\nq+!2\n0q\u0000Z1\n0d!Zt\n0dt0q(t0)\u000b2(!)\n!sin [!(t\u0000t0)]e\u00000+jt\u0000t0j\n\u0000Z1\n0d!\u000b(!)\u001a1\n2[A0(!) + iB0(!)] exp(\u0000i!t) + c:c:\u001b\n= 0: (11)\nFor most coupling functions \u000b(!) this integro-di\u000berential equation is di\u000ecult to solve\nin the time domain. When written in the frequency domain however, (11) becomes\nan integral equation that can be solved by a systematic procedure once the coupling\nfunction\u000b(!) is chosen. Instead of directly Fourier transforming (11), it is a little\nsimpler to Fourier transform (6) and substitute the result into the Fourier transform of\n(4). De\fning the relation\nf(t) =1\n2\u0019Z1\n\u00001d!f(!) exp(\u0000i!t) (12)\nbetween the time and frequency domains, we obtain from (6)\nX!(!0) =\u0019[A0(!) + iB0(!)]\u000e(!\u0000!0) +\u0019[A0(!)\u0000iB0(!)]\u000e(!+!0)\n+P\u000b(!)q(!0)\n!2\u0000!02+\u000b(!)\n2![\u000e(!\u0000!0)\u0000\u000e(!+!0)] PZ1\n\u00001d\u0018q(\u0018)\n\u0018\u0000!0; (13)\nwhere the in\fnitesimal quantity 0+in (6) has given rise to principal-value terms denoted\nwith a P (delta-function terms arising from 0+are found to cancel). The Fourier\ntransforms of the basic equations (4) and (5) are\n(\u0000!02+!2\n0)q(!0)\u0000Z1\n0d!\u000b(!)X!(!0) = 0; (14)\n(\u0000!02+!2)X!(!0)\u0000\u000b(!)q(!0) = 0: (15)\nOne veri\fes that (13) solves (15), and substitution of (13) in (14) yields an integral\nequation for q(!0):\n\u0014\n!02\u0000!2\n0+ PZ1\n0d!\u000b2(!)\n!2\u0000!02\u0015\nq(!0) +\u000b2(j!0j)\n2!0PZ1\n\u00001d\u0018q(\u0018)\n\u0018\u0000!0\n=\u0000\u0019\u000b(j!0j) [A0(j!0j) + i sgn(!0)B0(j!0j)] (16)Quantum dynamics of the damped harmonic oscillator 6\nEquation (16) is consistent with the relation q\u0003(!0) =q(\u0000!0), which holds because q(t)\nis real. Once q(!0) is found from (16), the reservoir solution X!(t) is most easily found\nby substituting q(t) into (6).\nIn deriving the frequency-domain result (16) we have indi\u000berently commuted time\nand frequency integrations; if this commutation is not valid then the solution q(!0) of\n(16) will not give solutions q(t) andX!(t) of the dynamical equations (4) and (5). It\nturns out that for a constant (i.e. frequency-independent) coupling \u000b(!) =a, the time\nand frequency integrations do notcommute, but this case of constant coupling is easily\nsolved in the time domain. For (16) to be valid the coupling function must also be such\nthat the integral containing \u000b2(!) converges. Needless to say, it must always be directly\nveri\fed, regardless of the solution method employed, that the dynamical equations (4)\nand (5) are satis\fed.\nOnce a coupling function \u000b(!) and thet= 0 state of the reservoir are chosen,\n(16) presents a principal-value integral equation for q(!0), also known as a singular\nintegral equation. In the standard classi\fcation, (16) is an inhomogeneous singular\nintegral equation of the third kind. There is a systematic and very elegant method of\nsolving singular integral equations, which exploits some complex analysis of Riemann\nand Hilbert [31, 32]. Integral equations such as (16) are solved by relating the equation\nto a boundary problem of the theory of analytic functions, usually called a Riemann\nproblem but sometimes called a Hilbert problem [31, 32]. If the integral equation has a\nsolution it can be found using any solution of the associated Riemann problem [31, 32].\nWe refer the reader to Pipkin's text [31] for the details, which are too lengthy to be\ndescribed here (the classic text of Muskhelishvili [32] does not appear to treat directly\nequations on an in\fnite interval).\nThe following additional remarks on the dynamical equations are worthwhile. Note\nthat for any coupling function for which (16) is valid, we can insert any function\nq(!0)(=q\u0003(\u0000!0)) and solve for A0(!0) andB0(!0), provided the integral involving q(!0)\nconverges. In other words, we can choose the dynamics of the q-oscillator from an\nenormous class of functions q(t) whose Fourier transforms exist, and use (16) to \fnd the\nvalues of the displacements and velocities of the reservoir at t= 0 that will produce this\ndynamicsq(t). It must be borne in mind that there may well be solutions q(t),X!(t)\nwhose Fourier transforms do not exist; for example, there is no mathematical reason why\nthe function q(t) should be bounded. Even if q(!) exists for a solution q(t), it can only\nbe found by solving (16) if the integral containing q(!) is well de\fned. Transformation\nto the frequency domain is an essential part of diagonalizing the Hamiltonian, which in\nturn gives the most convenient description of quantum states of the system. But there is\nno guarantee that the diagonalized Hamiltonian will give all the dynamical solutions of\nthe original Hamiltonian, and moreover there are couplings for which the Hamiltonian\ncannot be diagonalized (see section 5).\nIt is already clear that our dynamical system is completely di\u000berent from that\nin which the continuum reservoir is replaced by a discrete one. The analysis of\ndynamical systems consisting of a discrete number of coupled harmonic oscillators isQuantum dynamics of the damped harmonic oscillator 7\nlargely a matter of algebra [19]. In our case the mathematics is more challenging and\nthis is because the dynamics is much richer. The possible couplings \u000b(!) constitute\nthe entire space of real functions, rather than a discrete set of numbers. Moreover,\nfor some coupling functions \u000b(!), the existence of a solution of (16) requires severe\nconstraints on the functions A0(!0) andB0(!0), whereas for other coupling functions\nthere will be no such constraints [31]. The general mathematical theory also shows that\nthere is a subspace of coupling functions for which (16) gives only the trivial solution\nq(t) =X!(t) = 0 when X!(0) = _X!(0) = 0 [31], although it is by no means clear what\nthis subspace is. It would be very interesting to know how the space of solutions behaves\nas a functional of coupling \u000b(!), but that is far beyond the scope of this paper. We\nwill however \fnd a constraint on the coupling function that must be satis\fed for the\nHamiltonian to be diagonalizable.\nIn addition to the form (6), we also require the general solution of the reservoir\nequation (5) in a form that allows the imposition of conditions in the in\fnite past or\nfuture. We choose the in\fnite past t!\u00001 , and the required form of the general\nsolution of (5) is obtained by using the retarded Green function (8), yielding\nX!(t) =AR(!) cos!t+BR(!) sin!t+\u000b(!)\n!Zt\n\u00001dt0q(t0) sin [!(t\u0000t0)]e\u00000+(t\u0000t0):(17)\nHere the functions AR(!) andBR(!) can be viewed as determining X!(t) in the limit\nt!\u00001 , provided the last term vanishes as t!\u00001 (this last property, however, may\nnot hold for all solutions). The form (17) of the solution for the reservoir gives, in place\nof (11), the q-oscillator equation\nq+!2\n0q\u0000Z1\n0d!Zt\n\u00001dt0q(t0)\u000b2(!)\n!sin [!(t\u0000t0)]e\u00000+jt\u0000t0j\n\u0000Z1\n0d!\u000b(!)\u001a1\n2[AR(!) + iBR(!)] exp(\u0000i!t) + c:c:\u001b\n= 0: (18)\nIn the frequency domain, (17) is\nX!(!0) =\u0019[AR(!) + iBR(!)]\u000e(!\u0000!0) +\u0019[AR(!)\u0000iBR(!)]\u000e(!+!0)\n+P\u000b(!)q(!0)\n!2\u0000!02+i\u0019\u000b(!)q(!0)\n2![\u000e(!\u0000!0)\u0000\u000e(!+!0)]; (19)\nwhich inserted into (14) gives the following equation for q(!0):\n\u0014\n!02\u0000!2\n0+ PZ1\n0d!\u000b2(!)\n!2\u0000!02+i\u0019\u000b2(j!0j)\n2!0\u0015\nq(!0)\n=\u0000\u0019\u000b(j!0j) [AR(j!0j) + i sgn(!0)BR(j!0j)]: (20)\nThis is the frequency-domain version of (18). The solution q(!0) of (20) gives the\nreservoir solution X!(t) by substitution of q(t) into (17). This second formulation of the\ndynamical equations presents a much easier path to solutions, as we avoid the integral\nequation (16). Moreover, as every solution can in principle be obtained by solving either\n(16) or (20), it may seem that (16) should be avoided entirely. But the ability in (16)\nto impose directly conditions on the reservoir at a \fnite time allows the discovery ofQuantum dynamics of the damped harmonic oscillator 8\ninteresting particular solutions that in practice would not be found from (20). This will\nbe clearly demonstrated in sections 3 and 4.\nBefore proceeding to quantize the system for a general coupling function \u000b(!),\nwe show classically that a particular coupling function gives damping proportional\nto velocity. This coupling function will therefore allow an exact quantization of the\ndynamics (1).\n3. Coupling for damping proportional to velocity I\nWe consider the coupling function\n\u000b(!) =!0!\u00142\r\n\u0019(!2+\r2)\u00151=2\n; \r > 0; (21)\nwhich increases monotonically from 0 at != 0 to an asymptotic value of !0p\n2\r=\u0019as\n!!1 . The function (21) satis\fes the integral relation\nPZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2=\r2!2\n0\n!2+\r2; (22)\nand for this choice of coupling the integral equation (16) reduces to\nPZ1\n\u00001d\u0018q(\u0018)\n\u0018\u0000!=\u0000\u0019!\n\r!2\n0\u0002\n!2+\r2\u0000!2\n0\u0003\nq(!)\n\u0000\u0019\n!0\u00142\u0019\n\r(!2+\r2)\u00151=2\n[sgn(!)A0(j!j) + iB0(j!j)]: (23)\nThe general solution of (23) will contain the solution of the homogeneous equation\nPZ1\n\u00001d\u0018q(\u0018)\n\u0018\u0000!=\u0000\u0019!\n\r!2\n0\u0002\n!2+\r2\u0000!2\n0\u0003\nq(!): (24)\nWe \frst solve the homogeneous equation (24) and then solve the inhomogeneous\nequation (23) for arbitrary A0(!) andB0(!) by solving an associated integral equation\nfor a type of Green function.\nFollowing the standard procedure [31], the general solution of the homogeneous\nequation (24) is found to be\nq(!) =a\n(!2\u0000!2\n0)2+\r2!2; (25)\nwhereais an arbitrary constant. Transforming (25) to the time domain, we obtain\nthree di\u000berent behaviours depending on the size of the constant \r >0:\nq(t) =8\n>>>>><\n>>>>>:bexp\u0012\n\u0000\rjtj\n2\u0013\u0014\ncos\u0012!1t\n2\u0013\n+\r\n!1sin\u0012!1jtj\n2\u0013\u0015\n; for\r <2!0, (26 a)\nbexp (\u0000!0jtj) (1 +!0jtj); for\r= 2!0, (26 b)\ncexp\u0012\n\u0000\rjtj\n2\u0013\u0014\nexp\u0012\r1jtj\n2\u0013\n\u0000\r\u0000\r1\n\r+\r1exp\u0012\n\u0000\r1jtj\n2\u0013\u0015\n;for\r >2!0, (26 c)Quantum dynamics of the damped harmonic oscillator 9\nwherebandcare arbitrary constants, and !1and\r1are de\fned by\n!1=q\n4!2\n0\u0000\r2; \r 1=q\n\r2\u00004!2\n0: (27)\nNote that!1in (26 a) is real, as is \r1in (26 c). The presence of the absolute value of t\nin the solution (26 a){(26 c) may at a glance appear to give a discontinuity in _ q(t) and a\ndelta function in  q(t), but in fact _ q(t) and q(t) are continuous for all t.\nTheq-oscillator dynamics (26 a){(26 c) is exactly the solution of\nq+\r_q+!2\n0q= 0; t\u00150;\nq\u0000\r_q+!2\n0q= 0; t\u00140;(28)\nwith the condition _ q(0) = 0. We have thus obtained Ohmic damping and ampli\fcation\nof theq-oscillator in a closed system that can be canonically quantized. Recall that\nthe timet= 0, which divides the damping and ampli\fcation epochs in (28), was an\narbitrary choice. If we wish to impose arbitrary values of qand _qat some time t1, with\nOhmic damping for all later times, we need only replace t= 0 in our derivations by a\ntimet0\u0014t1, such that t0and the arbitrary constant in q(t) give the required values of\nqand _qatt1.\nThe dynamics (26 a){(26 c) is the complete solution for the q-oscillator when (23) is\nvalid andA0(!) =B0(!) = 0. This last condition sets the displacements and velocities\nof all the reservoir oscillators equal to zero at t= 0 (see (6)). The solution for the\nreservoir is obtained from (6) by substituting (26 a){(26 c) andA0(!) =B0(!) = 0; the\nresult is\n\r <2!0:\nX!(t) =b!! 0\n(!2\u0000!2\n0)2+\r2!2\u00142\r\n\u0019(!2+\r2)\u00151=2\u001a\n(!2\n0\u0000\r2\u0000!2) cos(!t) +\r!2\n0\n!sin(!jtj)\n+ exp\u0012\n\u0000\rjtj\n2\u0013\u0014\n(!2+\r2\u0000!2\n0) cos\u0012!1t\n2\u0013\n+\r\n!1(!2+\r2\u00003!2\n0) sin\u0012!1jtj\n2\u0013\u0015\u001b\n(29a)\n\r= 2!0:\nX!(t) =2b!! 0\n(!2+!2\n0)2\u0014!0\n\u0019(!2+ 4!2\n0)\u00151=2\u001a\nexp(\u0000!0jtj)\u0002\n!2+ 3!2\n0+ (!2+!2\n0)!0jtj\u0003\n\u0000(!2+ 3!2\n0) cos(!t) + 2!3\n0\n!sin(!jtj)\u001b\n(29b)\n\r >2!0:\nX!(t) =c!! 0\n(!2\u0000!2\n0)2+\r2!2\u0014\r\n8\u0019(!2+\r2)\u00151=2\n\u0002\u001a8\r1\n\r+\r1\u0014\n(!2\n0\u0000\r2\u0000!2) cos(!t) +\r!2\n0\n!sin(!jtj)\u0015\n+ exp\u0012\n\u0000\rjtj\n2\u0013\n\u0002\u0014\n(4!2+ (\r+\r1)2) exp\u0012\r1jtj\n2\u0013\n\u0000\r\u0000\r1\n\r+\r1(4!2+ (\r\u0000\r1)2) exp\u0012\n\u0000\r1jtj\n2\u0013\u0015\u001b\n:(29c)\nThe under-damped case (26 a) and (29 a) is plotted in Figure 1 for a choice of\nparameters. Viewed from the in\fnite past to the in\fnite future, the q-oscillator isQuantum dynamics of the damped harmonic oscillator 10\n-10-50510-0.50.00.51.0\ntqHtL\nFigure 1. The solution (26 a) and (29 a) with!0= 3,\r= 1 andb= 1. This solution\nis for coupling (21) with X!(0) = _X!(0) = 0. The q-oscillator is damped into the past\nand future from t= 0, with damping proportional to velocity as shown in (28). As\njtj!1 , most of the q-oscillator energy goes into reservoir oscillators with frequencies\naround!0(= 3).\ninitially (t!\u00001 ) at rest with all of the energy in the reservoir; the q-oscillator is then\nampli\fed by the reservoir until at t= 0 it has extracted all of the reservoir energy;\nthen the energy is returned to the reservoir as the q-oscillator is damped. Alternatively,\nviewed as a consequence of the imposition of a t= 0 condition, the q-oscillator is given\na displacement, with _ q(0) = 0 and X!(0) = _X!(0) = 0, and is then damped into the\npast and future, transferring its energy to the reservoir. Note in Figure 1 that as the\nreservoir is ampli\fed ( jtjincreasing from 0) and removes energy from the q-oscillator,\nthe reservoir oscillators that attain the largest amplitudes as jtj!1 are those withQuantum dynamics of the damped harmonic oscillator 11\nfrequencies close to !0, the free oscillation frequency of the q-oscillator. Compared\nto Figure 1, the over-damped case \r\u00152!0has the following qualitative di\u000berences:\ntheq-oscillator is exponentially damped into the past and future from t= 0 without\noscillating, and the amplitudes of the reservoir oscillators for jtj!1 decrease as !\nincreases from 0 (so there is no peak in the reservoir amplitudes at !=!0).\nThe homogeneous version of equation (11) (i.e. with X!(0) = _X!(0) = 0) has in\nfact a slightly more general solution q(t) than (26 a){(26 c) for coupling (21). We did not\nobtain this more general solution in the frequency domain by solving the homogeneous\nintegral equation (24), because its Fourier transform q(!) is not well enough behaved to\nmake the derivation of (24) valid. For coupling (21), the time-domain equation (11) is\nq+!2\n0q\u0000sgn(t)\r!2\n0Zt\n0dt0q(t0) exp(\u0000\rjt\u0000t0j) =f(t);\nf(t) =Z1\n0d!!0!\u0014\r\n2\u0019(!2+\r2)\u00151=2\nf[A0(!) + iB0(!)] exp(\u0000i!t) + c:c:g;(30)\nwhich is (23) in the time domain. The homogeneous version of (30) ( f(t) = 0) has a\nsolution (26 a){(26 c), but the more general solution is\nq(t) =1\n2(b2\u0000b1)\n+ exp\u0012\n\u0000\rt\n2\u0013\u0014\nb1cos\u0012!1t\n2\u0013\n+b1\r2+ (b2\u0000b1)!2\n0\n\r!1sin\u0012!1t\n2\u0013\u0015\n; t\u00150;(31a)\nq(t) =\u00001\n2(b2\u0000b1)\n+ exp\u0012\rt\n2\u0013\u0014\nb2cos\u0012!1t\n2\u0013\n\u0000b2\r2\u0000(b2\u0000b1)!2\n0\n\r!1sin\u0012!1t\n2\u0013\u0015\n; t\u00140;(31b)\nwhereb1andb2are arbitrary constants and !1is again given by (27). For simplicity we\nhave written (31 a){(31 b) in the form appropriate for the underdamped case \r < 2!0;\nthis solution is also valid for \r >2!0and the solution for \r= 2!0can be obtained by\ntaking the limit !1!0. The solution (26 a){(26 c) corresponds to the choice b2=b1\nin (31 a){(31 b). The more general solution (31 a){(31 b) di\u000bers from (26 a){(26 c) by not\nhaving _q(0) = 0 and by having a constant displacement of the q-oscillator asjtj!1 .\nNote that the displacement q(t!1 ) in (31 a) is minus the displacement q(t!\u00001 ) in\n(31b). In the solution (31 a){(31 b) the reservoir extracts all the kinetic energy from the\nq-oscillator asjtj!1 but does not bring it to its uncoupled equilibrium displacement\nq= 0. By inserting (31 a){(31 b) in (6) with A0(!) =B0(!) = 0, the more general\nversion of the solution (29 a){(29 c) forX!(t) is obtained, but we refrain from writing\nthe result.\nTo complete the solution for the dynamics with coupling (21), we must solve the\nequations of motion for non-zero X!(0) and _X!(0). We can write the solution in closed\nform for any A0(!) andB0(!) by \fnding a Green function for (30). If we \fnd a solution\nG(t;t0) of\nG(t;t0) +!2\n0G(t;t0)\u0000sgn(t)\r!2\n0Zt\n0dt0G(t0;t0) exp(\u0000\rjt\u0000t0j) =\u000e(t\u0000t0); (32)Quantum dynamics of the damped harmonic oscillator 12\nthen the general solution of (30) is\nq(t) =Z1\n\u00001dt0G(t;t0)f(t0) (33)\nplus the solution (31 a){(31 b) of the homogeneous version of (30) (with f(t) = 0). The\nsolution for X!(t) is then obtained from (6). A Green function G(t;t0) can be found by\nFourier transforming (32) in tand solving the resulting integral equation for G(!;t0).\nAs (23) is the Fourier transform in tof (30), we see from (32) that the integral equation\nforG(!;t0) is\nPZ1\n\u00001d\u0018G(\u0018;t0)\n\u0018\u0000!=\u0000\u0019!\n\r!2\n0\u0002\n!2+\r2\u0000!2\n0\u0003\nG(!;t0)\u0000\u0019(!2+\r2)\n\r!2\n0(!\u0000i\u000f)exp(i!t0); (34)\nwhere a pole at != 0 in the last term has been moved o\u000b the real line by inserting\n\u000f6= 0. The two choices of sign of \u000fin (34) give two Green functions G(t;t0) in the\ntime domain that di\u000ber by a solution of the homogeneous version of (32) (with 0 on\nthe right-hand side); this homogeneous solution is of course given by (31 a){(31 b) with\na particular b1andb2. The Green functions have rather lengthy expressions and the\nresults forG(!;t0) andG(t;t0) are given in Appendix A for the choice \u000f>0.\nAlthough the classical dynamics is now completely solved for the coupling (21), the\ndiagonalization of the Hamiltonian (in both the classical and quantum cases) will make\nuse of an alternative formulation based on (17).\n4. Coupling for damping proportional to velocity II\nThe forms (17) and (18) of the dynamical equations can be obtained by pushing back\nthe (arbitrary) integration limit t= 0 in (6) and (11) to t!\u00001 . The part of X!(t)\nin (17) that is the solution of the homogeneous version of (5) (with q(t) = 0) describes\nthe state of the reservoir at t!\u00001 if the last term in (17) vanishes as t!\u00001 .\nFor the coupling (21), which satis\fes (22), the q-oscillator equation (20) reduces to\n!\n!+ i\r(!2+ i\r!\u0000!2\n0)q(!) =g(!);\ng(!) =\u0000!!0\u00142\u0019\r\n!2+\r2\u00151=2\n[sgn(!)AR(j!j) + iBR(j!j)]:(35)\nIn the time domain this is (18) with coupling (21):\nq+!2\n0q\u0000\r!2\n0Zt\n\u00001dt0q(t0) exp[\u0000\r(t\u0000t0)] =g(t); (36)\ng(t) =Z1\n0d!!0!\u0014\r\n2\u0019(!2+\r2)\u00151=2\nf[AR(!) + iBR(!)] exp(\u0000i!t) + c:c:g: (37)\nBecause of the factor of !on the left-hand side of (35), the homogeneous equation\n(g(!) = 0) has the solution\nq(!) = 2\u0019a\u000e(!) =)q(t) =a; (38)Quantum dynamics of the damped harmonic oscillator 13\nwhereais an arbitrary constant. The corresponding solution for the reservoir is obtained\nfrom (17) with q(t) =aandAR(!) =BR(!) = 0; the reservoir is also independent of\ntime and the full solution is\nq(t) =a; X !(t) =a!0\n!\u00142\r\n\u0019(!2+\r2)\u00151=2\n: (39)\nThis \\zero-mode\" solution has zero energy, as can be veri\fed using (3). It is interesting\nto check when a zero mode occurs in the general dynamical equations (4) and (5).\nInsertingq(t) =ainto (4) and (5) one \fnds that for a6= 0 there exists a consistent\nsolution\nq(t) =a; X !(t) =a\u000b(!)\n!2(40)\nif and only if the coupling function satis\fes\n!2\n0=Z1\n0d!\u000b2(!)\n!2: (41)\nThe coupling (21) is one example of a function that satis\fes (41), and this is why there\nis a zero mode (39). Zero modes are always a possibility for oscillators coupled as in (2);\neven for just two oscillators coupled in this fashion there exists a zero mode for one value\nof the square of the coupling constant. In the case of the system (2) there is an in\fnite\nclass of coupling functions that give a zero mode, and an in\fnite class that do not. It\nis interesting that the coupling function that gives Ohmic damping of the q-oscillator\nalso allows a zero mode, but the wider signi\fcance of this fact is not immediately clear.\nNote that the zero mode (39) is not related to the constant displacement as t!\u00061\nof theq-oscillator in the solution (31 a){(31 b); that solution has X!(0) = _X!(0) = 0\nand so has no relation to (39). The formulation of the dynamical equations used in this\nsection has shown that the zero mode (39) is completely determined by the condition\nAR(!) =BR(!) = 0 on the reservoir, and we see from (39) that this is not equivalent\nto the condition X!(t)jt!\u00001 = 0.\nThe general solution of (35) is the solution (38) of the homogeneous equation plus\nthe solution\nq(!) =\u0000GR(!)g(!); (42)\nGR(!) =\u0000!+ i\r\n(!+ i0+)(!2+ i\r!\u0000!2\n0): (43)\nIn (43) we have de\fned a Green function and given a prescription for dealing with its\npole at!= 0. We have moved the != 0 pole into the lower-half complex !-plane, where\nthe other two poles of GR(!) lie, and have therefore chosen the retarded Green function\n(analytic in the upper-half !-plane). It is easy to see that a di\u000berent prescription for\ndealing with the != 0 pole will give a Green function that di\u000bers from (43) by a solution\n(38) of the homogeneous equation. The Green function (43) in the time domain is\nGR(t) =\u0012(t)\r\n!2\n0\u001a\ne\u00000+t\u0000exp\u0012\n\u0000\rt\n2\u0013\u0014\ncos\u0012!1t\n2\u0013\n+\r2\u00002!2\n0\n\r!1sin\u0012!1t\n2\u0013\u0015\u001b\n; (44)Quantum dynamics of the damped harmonic oscillator 14\nwith!1given by (27). The result (44) is valid for all values of \rif the case \r= 2!0\nis understood as the limit !1!0. As is expected from (36), the Green function (44)\nsatis\fes\nGR+!2\n0GR\u0000\r!2\n0Zt\n\u00001dt0GR(t0) exp[\u0000\r(t\u0000t0)] =\u000e(t): (45)\nIn the time domain the general solution for the q-oscillator is\nq(t) =a+Z1\n\u00001dt0GR(t\u0000t0)g(t0); (46)\nas can be con\frmed by showing that the second term satis\fes (36) because of (45). The\nsolution for X!(t) is given by (17).\nBy removing the step-function factor \u0012(t) in the retarded Green function (44), we\nobtain a solution of the homogeneous version of (45) (with zero on the right-hand side).\nThis is of course also a solution of the homogeneous version of (36), so there seems\nto be a more general solution for q(t) withAR(!) =BR(!) = 0 than the zero-mode\nsolution (38). In fact the general solution of the homogeneous version of (36) is given\nby (31 a), with the expression valid for all trather than t\u00150. (The Green function\n(44) with\u0012(t) removed corresponds to (31 a) withb1=\u0000\r=!2\n0andb2=\r=!2\n0.) But the\nexpression in (31 a), taken to be valid for all t, is not a solution for the dynamical system\nfor the simple reason that the corresponding reservoir solution, given by (17), diverges.\nThis behaviour of the dynamics can be described with reference to Fig. 1, which shows\na particular case of the solution (31 a){(31 b) and the corresponding reservoir solution.\nThe timet= 0 in Fig. 1 is arbitrary and the solution holds with the t= 0 peak in q(t),\nand zero in X!(t), moved to any other \fnite value of t. If this peak in q(t) is pushed\nback tot!\u00001 , however, the amplitude of the peak diverges and this causes the entire\nsolutionX!(t) to diverge. The general solution for the dynamics with the condition\nAR(!) =BR(!) = 0 is therefore the zero mode (39).\nThe solution shown in Fig. 1 must be a particular case of the general solution as\nformulated in this section: it must correspond to some choice of AR(!) andBR(!). As\nthe solution in Fig. 1 has q(t)!0 asjtj!1 , the corresponding functions AR(!) and\nBR(!) turn out to specify the state of the reservoir at t!\u00001 (see (17)). By taking\nthe limitt!\u00001 in (29 a) we see that X!(t) takes the form of the \frst two terms in\n(17), with\nAR(!) =\u0000b\u00142\r\n\u0019(!2+\r2)\u00151=2!!0(!2+\r2\u0000!2\n0)\n(!2\u0000!2\n0)2+\r2!2; (47)\nBR(!) =\u0000b\u00142\r\n\u0019(!2+\r2)\u00151=2\r!3\n0\n(!2\u0000!2\n0)2+\r2!2: (48)\nOne can verify that with AR(!) andBR(!) given by (47) and (48), and with a= 0,\nequations (46), (44), (37) and (17) reproduce the solution (26 a){(26 c) and (29 a){(29 c).\nNeedless to say, it is by far from obvious that the speci\fcation (47) and (48) of the\nreservoir at t!\u00001 implies the interesting behaviour depicted in Fig. 1 (modulo a\nzero-mode solution (39)).Quantum dynamics of the damped harmonic oscillator 15\nWe have formulated the general solution for the dynamics with coupling (21) in two\nways. Every particular solution is completely speci\fed by either the functions A0(!)\nandB0(!) and constants b1andb2in section 3, or by the functions AR(!) andBR(!)\nand constant ain this section. The relationship between the pairs fA0(!);B0(!)gand\nfAR(!);BR(!)gfor the same solutionfq(t);X!(t)gis the one that transforms (13) into\n(19) and (16) into (20); it is easy to show that this relationship is\nA0(j!j) =AR(j!j)\u0000\u000b(j!j)\n2!\u001a\nIm[q(!)] +2!\n\u0019PZ1\n0d\u0018Re[q(\u0018)]\n\u00182\u0000!2\u001b\n; (49)\nB0(j!j) =BR(j!j) + sgn(!)\u000b(j!j)\n2!\u001a\nRe[q(!)]\u00002\n\u0019PZ1\n0d\u0018\u0018Im[q(\u0018)]\n\u00182\u0000!2\u001b\n: (50)\nThe results (47) and (48) for the particular solution (26 a){(26 c) and (29 a){(29 c) can\nalso be derived from (49) and (50) by inserting A0(!) =B0(!) = 0, (21) and (25).\n5. Quantization and diagonalization of the Hamiltonian\nThe canonical momenta for the Lagrangian (2) are\n\u0005q(t) = _q(t); \u0005X!(t) = _X!(t): (51)\nWe quantize the system in the Heisenberg picture by imposing the equal-time canonical\ncommutation relations\n[^q(t);^\u0005q(t)] = i ~; [^X!(t);^\u0005X!0(t)] = i ~\u000e(!\u0000!0); (52)\n[^X!(t);^X!0(t)] = 0;[^\u0005X!(t);^\u0005X!0(t)] = 0;[^q(t);^X!(t)] = 0;[^q(t);^\u0005X!(t)] = 0:(53)\nThe commutation relations for the reservoir are similar to that of a \feld theory because\nof the continuum of frequencies !. The Hamiltonian is\n^H=1\n2^\u00052\nq+1\n2!2\n0^q2+1\n2Z1\n0d!\u0010\n^\u00052\nX!+!2^X2\n!\u0011\n\u00001\n2Z1\n0d!\u000b(!)h\n^q^X!+^X!^qi\n; (54)\nwhere a Hermitian combination of the operators has been taken in the last term;\nclassically, (54) gives the total energy (3).\nThe solution of Hamilton's equations for the canonical operators can be immediately\nwritten down using the classical results in sections 3 and 4. But in order to describe\ngeneral quantum states we need some basis in the Hilbert space, and a solution for\nthe canonical operators does not by itself reveal such a basis. Diagonalization of the\nHamiltonian solves Hamilton's equations for the canonical operators in terms of the\nenergy eigenstates of the system and thus solves the dynamics of general quantum states.\nThe diagonalization process can also be performed on a purely classical level, where the\nclassical normal modes of the system correspond to the quantum energy eigenstates.\nWhat is required is a new set of dynamical variables for which the system reduces to\na set of uncoupled harmonic oscillators; each of these free oscillations (normal modes)\nhas a conserved energy and gives an energy eigenstate in the quantum theory.\nFor a discrete set of oscillators with coupling terms of the same form as in (2), the\nnormal-mode displacements are linear combinations of the displacements of the coupledQuantum dynamics of the damped harmonic oscillator 16\noscillators [19]. The continuum reservoir in (2), however, has the remarkable e\u000bect that\nthe normal modes are given by a canonical transformation of the original variables, in\nwhich displacements and canonical momenta are mixed. It is well known that in the\ndiscrete case the frequencies of the normal modes become complex when the coupling\nof the form (2) is too large [19]. We shall also \fnd that there is a restriction on the\ncoupling function \u000b(!) in order for the Hamiltonian (54) to be diagonalizable with real-\nfrequency eigenmodes. This restriction on \u000b(!) is met by the particular coupling (21).\nIn contrast, Huttner and Barnett [21] showed, as part of their model of a dielectric,\nthat the system (2) with X!in the coupling term replaced by _X!has real-frequency\neigenmodes for essentially any coupling function \u000b(!).\nWe attempt to show that the Hamiltonian (54) can be written\n^H=1\n2Z1\n0d!\u0010\n^\u00052\n\b!+!2^\b2\n!\u0011\n(55)\n=1\n2Z1\n0d!~!h\n^Cy(!;t)^C(!;t) +^C(!;t)^Cy(!;t)i\n; (56)\nwhere ^\b!(t),!2[0;1) are displacement operators for a set of uncoupled harmonic\noscillators, the normal modes or energy eigenmodes of the system. The normal-mode\ncreation and annihilation operators ^Cy(!;t) and ^C(!;t) are given by the usual harmonic-\noscillator expressions\n^C(!;t) =r!\n2~\u0014\n^\b!(t) +i\n!^\u0005\b!(t)\u0015\n; ^Cy(!;t) =r!\n2~\u0014\n^\b!(t)\u0000i\n!^\u0005\b!(t)\u0015\n; (57)\nand they obey the commutation relations\n[^C(!;t);^Cy(!0;t)] =\u000e(!\u0000!0); [^C(!;t);^C(!0;t)] = 0: (58)\nFrom (56) and (58) we obtain\n[^C(!;t);^H] =~!^C(!;t); (59)\nso that the eigenmode creation and annihilation operators have a stationary time-\ndependence given by\n^C(!;t) = exp(\u0000i!t)^C(!): (60)\nThe diagonalized form (55) and (56) of the Hamiltonian (54) will be achieved with a\nrestriction on the coupling function \u000b(!). The following derivation can be performed\nclassically if commutators are replaced with Poisson brackets.\nIf the diagonalization is possible then there exists a transformation from the original\ndynamical degrees of freedom in (54) to eigenmode degrees of freedom for which the\nHamiltonian takes the form (55). In the quantum theory it is convenient to use the\ncreation and annihilation operators (57) as the eigenmode degrees of freedom, rather\nthan the canonical operators ^\b!and ^\u0005\b!(the quantities (57) can also be used in the\nclassical theory). We thus seek a transformation\n^q(t) =Z1\n0d!h\nfq(!)^C(!;t) + h:c:i\n; ^\u0005q(t) =Z1\n0d!h\nf\u0005q(!)^C(!;t) + h:c:i\n;(61)Quantum dynamics of the damped harmonic oscillator 17\n^X!(t) =Z1\n0d!0h\nfX(!;!0)^C(!0;t) + h:c:i\n; (62)\n^\u0005X!(t) =Z1\n0d!0h\nf\u0005X(!;!0)^C(!0;t) + h:c:i\n; (63)\nfor which (54) becomes (56). The unknown coe\u000ecients fq(!), etc., in (61){(63) can be\nwritten as commutators by utilizing (58):\nfq(!) = [^q(t);^Cy(!;t)]; f \u0005q(!) = [ ^\u0005q(t);^Cy(!;t)]; (64)\nfX(!;!0) = [ ^X!(t);^Cy(!0;t)]; f \u0005X(!;!0) = [ ^\u0005X!(t);^Cy(!0;t)]:(65)\nThe eigenmodes variables must be expressible in terms of the original variables, i.e.\n(61){(63) must be invertible, and (64), (65), (52) and (53) imply\n^C(!;t) =\u0000i\n~\u001a\nf\u0003\n\u0005q(!)^q(t)\u0000f\u0003\nq(!)^\u0005q(t)\n+Z1\n0d!0h\nf\u0003\n\u0005X(!0;!)^X!0(t)\u0000f\u0003\nX(!0;!)^\u0005X!0(t)i\u001b\n: (66)\nIt is clear from (61){(63) and (60) that the f-coe\u000ecients are closely related to\nFourier transforms of the original canonical operators. This suggests that the f-\ncoe\u000ecients must satisfy the original dynamical equations written in the frequency\ndomain. We derive these equations for the f-coe\u000ecients as follows. By inserting (66)\nand (54) in (59), and using the canonical commutation relations (52) and (53), we obtain\n~!^C(!;t) =f\u0003\n\u0005q(!)^\u0005q(t) +!2\n0f\u0003\nq(!)^q(t) +Z1\n0d!0n\nf\u0003\n\u0005X(!0;!)^\u0005X!0(t)\n+!02f\u0003\nX(!0;!)^X!0(t)\u0000\u000b(!0)h\nf\u0003\nq(!)^X!0(t) +f\u0003\nX(!0;!)^q(t)io\n: (67)\nComparing coe\u000ecients of the canonical operators in (66) and (67) we \fnd\nf\u0005q(!) =\u0000i!fq(!); i!f\u0005q(!) =!2\n0fq(!)\u0000Z1\n0d!0\u000b(!0)fX(!0;!); (68)\nf\u0005X(!0;!) =\u0000i!fX(!0;!); i!f\u0005X(!0;!) =!02fX(!0;!)\u0000\u000b(!0)fq(!); (69)\nwhich give\n!2fq(!) =!2\n0fq(!)\u0000Z1\n0d!0\u000b(!0)fX(!0;!); (70)\n!2fX(!0;!) =!02fX(!0;!)\u0000\u000b(!0)fq(!): (71)\nThese two equations for fq(!) andfX(!0;!) are indeed identical to the frequency-domain\nequations (14) and (15) for q(!) andX!0(!). We can therefore write the general solution\nof (70) and (71) using the results (19) and (20), in which a retarded Green function was\nused to solve the reservoir equation. As only positive frequency arguments are used in\nthef-coe\u000ecients, we can drop delta functions containing sums of frequencies and so\n(19) gives the following general solution for fX(!0;!):\nfX(!0;!) =hX(!)\u000e(!\u0000!0) + P\u000b(!0)\n!02\u0000!2fq(!) +i\u0019\u000b(!0)\n2!0fq(!)\u000e(!\u0000!0) (72)\n=hX(!)\u000e(!\u0000!0) +\u000b(!0)\n2!0\u00121\n!0\u0000!\u0000i0++1\n!0+!\u0013\nfq(!); (73)Quantum dynamics of the damped harmonic oscillator 18\nwherehX(!) is an arbitrary complex function. The corresponding general solution for\nfq(!) is found from (70) and (72), which yields (20) with positive frequency arguments:\n\u0014\n!2\u0000!2\n0+ PZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2+i\u0019\u000b2(!)\n2!\u0015\nfq(!) =\u0000\u000b(!)hX(!): (74)\nThe solution of (74) can be written\nfq(!) =hq(!) +G(!)\u000b(!)hX(!);\nG(!) =\u0000\u0014\n!2\u0000!2\n0+ PZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2+i\u0019\u000b2(!)\n2!\u0015\u00001\n;(75)\nwhere we have de\fned a Green function G(!), andhq(!) is the solution of\n\u0014\n!2\u0000!2\n0+ PZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2+i\u0019\u000b2(!)\n2!\u0015\nhq(!) = 0: (76)\nThe existence of a non-zero solution of (76) requires at least one real root of the function\nin brackets; the resulting hq(!) will contain a delta-function factor and an example is the\nzero mode (38) for the coupling (21). The f-coe\u000ecients are now completely determined\nby the two functions hX(!) andhq(!); it remains to show that values for these two\nfunctions exist that give the required diagonalization. As follows from section 2, there\nare alternative ways of writing the general solution for the f-coe\u000ecients, but it turns out\nthat the form (72){(76), based on the retarded solution of the reservoir equation, is very\nconvenient for solving the diagonalization problem. An equally convenient approach,\nwhich is in fact adopted by Huttner and Barnett [21], is based on the advanced solution\nof the reservoir equation.\nA value for the function hX(!) is obtained by demanding that the commutation\nrelations (58) hold for the operator (66) and its hermitian conjugate. The \frst of (58),\nafter elimination of f\u0005q(!) andf\u0005X(!0;!) using (68) and (69), yields\n(!+!0)f\u0003\nq(!)fq(!0) + (!+!0)Z1\n0d!00f\u0003\nX(!00;!)fX(!00;!0) =~\u000e(!\u0000!0): (77)\nThe key part of the diagonalization is the insertion of (73) into (77). A product of\nterms containing the in\fnitesimal number 0+is encountered, whose value depends\nquadratically on 0+. This means that the form (73) of fX(!0;!) must be used in (77)\nrather than the form (72). Any diagonalization involving the continuum reservoir will\nencounter such products, and their treatment is described in detail in [22], where the\nHamiltonian of macroscopic QED is diagonalized. The result of substituting (73) into\n(77) and applying (74) is\n2!h\u0003\nX(!)hX(!) =~; (78)\nwhich has a simple solution\nhX(!) =\u0012~\n2!\u00131=2\n: (79)\nOther solutions of (78) di\u000ber from (79) by a phase factor. This freedom in the choice of\nthe diagonalizing transformation is clear from the outset, as the diagonalizing operatorsQuantum dynamics of the damped harmonic oscillator 19\n(57) are only de\fned by (56) up to a phase factor. A similar analysis shows that the\nsecond commutation relation in (58) is identically satis\fed by (66) because of (73) and\n(74).\nWe must also check that the commutation relations (52) and (53) of the original\ncanonical operators are satis\fed by the representations (61){(63). This is a consistency\ncheck on the f-coe\u000ecients, which have already been largely determined by (78). It is here\nthat we \fnd a restriction on the coupling function \u000b(!). The full set of coupling functions\nthat satis\fes the restriction is determined by rather complicated integral relations, but\nfor a subclass of coupling functions that includes (21) the restriction is much simpler.\nConsider the set of functions \u000b(!) for which\n\u000b2(!) is an even function of ! and\u000b2(!)\n!>0 except possibly at != 0:(80)\nFor coupling functions satisfying (80) the diagonalization can be achieved if and only if\neither!2\n0>Z1\n0d\u0018\u000b2(\u0018)\n\u00182;\nor!2\n0=Z1\n0d\u0018\u000b2(\u0018)\n\u00182and!2\n0+\u00142\u0000Z1\n0d\u0018\u000b2(\u0018)\n\u00182+\u00142\u0018\u0014n; n\u00141;as\u0014!0:(81)\nThe proof of this is given in Appendix B, which furthermore shows that the diagonalizing\ntransformation requires the choice\nhq(!) = 0 (82)\nas the solution of (76). It is also shown in Appendix B that for the class of coupling\nfunctions (80), there is only one possible real root of the function in brackets in (76),\nat!= 0. Moreover this != 0 root occurs if and only if !2\n0=R1\n0d\u0018\u000b2(\u0018)=\u00182, which\nwe recognize as the condition (41) for the existence of a zero mode. Thus, the second\ncondition in (81) is a restriction on coupling functions that have a zero mode.\nAssuming the conditions (80) and (81) are met, a set of f-coe\u000ecients for the\ndiagonalizing transformation is speci\fed by (73), (75), (79) and (82). The resulting\nrelations (61){(63) for the canonical operators in terms of the eigenmode creation and\nannihilation operators are most simply written in the frequency domain. With the\nFourier de\fnition\n^q(t) =1\n2\u0019Z1\n0d![^q(!) exp(\u0000i!t) + h:c:]; (83)\nfor ^q(!), etc., the frequency-domain canonical operators are\n^q(!) = 2\u0019\u0012~\n2!\u00131=2\n\u000b(!)G(!)^C(!) =i\n!^\u0005q(!); (84)\n^X!(!0) = 2\u0019\u0012~\n2!\u00131=2\n\u000e(!\u0000!0)^C(!0) +\u000b(!)\n2!\u00121\n!\u0000!0\u0000i0++1\n!+!0\u0013\n^q(!0) (85)\n=i\n!0^\u0005X!(!0); (86)Quantum dynamics of the damped harmonic oscillator 20\nwhere the Green function G(!) is de\fned in (75). The Green function can be written\nG(!) =\u00001\n!2\u0000!2\n0[1\u0000\u001f(!)]; (87)\nin terms of a dimensionless quantity \u001f(!) that gives the dependence on the coupling\nfunction\u000b(!):\n!2\n0\u001f(!) = PZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2+i\u0019\u000b2(!)\n2!: (88)\nAs noted in Appendix B, if \u000b2(!) is an even function of !then\u001f(!) exhibits a Kramers-\nKronig relation between its real and imaginary parts and is therefore analytic in the\nupper-half complex !-plane. We can view \u001f(!) as an e\u000bective \\susceptibility\" for the\ndampedq-oscillator; remarkably, for couplings such that \u000b2(!) is even, a single damped\noscillator exhibits dispersion and dissipation connected by Kramers-Kronig relations.\nThere are in fact many similarities between the quantum damped harmonic oscillator\nand macroscopic QED [22], where the incorporation of Kramers-Kronig relations for\nelectromagnetic susceptibilities is an essential part of the theory.\nIn sections 3 and 4 we solved the classical dynamics for the coupling function (21).\nWe note that this coupling function is of the class (80). In addition, the function (21)\nhas the properties\nZ1\n0d\u0018\u000b2(\u0018)\n\u00182=Z1\n0d\u00182\r!2\n0\n\u0019(\u00182+\r2)=!2\n0; (89)\n!2\n0+\u00142\u0000Z1\n0d\u0018\u000b2(\u0018)\n\u00182+\u00142=!2\n0+\u00142\u0000\r!2\n0\n\u0014+\r=!2\n0\n\r\u0014+O(\u00142): (90)\nThe second of the restrictions (81) is thus satis\fed, and therefore the Hamiltonian is\ndiagonalizable in this case. The e\u000bective susceptibility for the coupling (21) is, from\n(88),\n\u001f(!) =\r\n\r\u0000i!: (91)\nThe Green function (87) is then\nGR(!) =\u0000!+ i\r\n(!+ i0+)(!2+ i\r!\u0000!2\n0); (92)\nwhere we have moved a pole at != 0 into the lower-half complex plane; this gives the\nretarded Green function GR(!), analytic in the upper-half plane. The retarded Green\nfunction (92) is identical to (43), which featured in the classical solution (as discussed\nabove, thef-coe\u000ecients are a particular solution of the classical dynamical equations).\nThe choice of the retarded Green function is not required for the diagonalization; other\nchoices will give f-coe\u000ecients that di\u000ber by a zero-mode solution, described in section 4.\n6. Coherent states\nHaving diagonalizaed the quantum (and classical) Hamiltonian for coupling functions\nthat include (21), we can now construct the quantum state of the system that is closestQuantum dynamics of the damped harmonic oscillator 21\nto the interesting classical solution derived in section 3 and plotted in Fig. 1. We expect\nthe quantum states that are closest to classical solutions to be coherent states, and one\nbene\ft of the diagonalization results (84) and (85) is that they allow us to construct\ncoherent states of the eigenmodes of the system. Classically, what we are about to do is\nto relate the particular solution in section 3 to the classical normal modes of the system.\nThe eigenmode annihilation operator ^C(!) de\fnes continuous-mode coherent\nstates [34] by\n^C(!)jC(!)i=C(!)jC(!)i; (93)\nwhereC(!) is an arbitrary complex function. The classical solution for coupling (21)\ngives the relations (35), (42) and (43) for q(!), while the quantum solution for ^ q(!) with\ncoupling (21) is given by (84) with Green function (92). Comparing these classical and\nquantum results we see that every classical solution given by the real functions AR(!)\nandBR(!) has a corresponding quantum coherent state with complex amplitude\nC(!) =\u0000\u0010!\n2~\u00111=2\n[AR(!) + iBR(!)]: (94)\nThe expectation values of ^ q(t) and ^X!(t) in the coherent states are equal to the\ncorresponding classical solutions q(t) andX!(t). It follows from the results at the end\nof section 4 that the coherent state corresponding to the classical solution (26 a){(26 c)\nand (29 a){(29 c) has amplitude (94) with AR(!) andBR(!) given by (47) and (48).\nThe expectation value of the normal-mode displacement operator ^\b!(t) in a\ncoherent state is, from (57), (93) and (94),\nhC(!)j^\b!(t)jC(!)i=\u0012~\n2!\u00131=2\n[C(!) exp(\u0000i!t) + c:c:] (95)\n=\u0000AR(!) cos(!t)\u0000BR(!) sin(!t): (96)\nThis is just the general solution for the classical normal modes \b !(t) of the system.\nThere are many other results that can be derived for the coherent states, but in this\npaper the quantum state we will consider in detail is the thermal state.\n7. Thermal equilibrium\nAnother bene\ft of diagonalizing the Hamiltonian is that it allows a straightforward\ncalculation of the thermodynamic quantities for the q-oscillator. In this section we\ntreat the case of thermal equilibrium for all couplings for which the diagonalization in\nsection 5 is valid; the results for the particular coupling (21), corresponding to damping\nproportional to velocity, will also be described. Given the close relationship between the\ndamped harmonic oscillator and macroscopic QED, noted in the last section, it is not\nsurprising that the following treatment of the thermal state of the damped oscillator\nhas similarities to the Casimir e\u000bect, which is simply macroscopic QED in thermal\nequilibrium [23].Quantum dynamics of the damped harmonic oscillator 22\nIn thermal equilibrium the excitation level of each eigenmode oscillator of the\nsystem will be given by the Planck distribution, so the thermal mixed state is de\fned\nby\nh^Cy(!)^C(!0)i=N(!)\u000e(!\u0000!0) =h^C(!)^Cy(!0)i\u0000\u000e(!\u0000!0); (97)\nN(!) =\u0014\nexp\u0012~!\nkBT\u0013\n\u00001\u0015\n; (98)\nh^C(!)^C(!0)i= 0: (99)\nThe continuum of eigenmodes necessitates the use of delta functions in (97). These\ndelta functions lead to some awkwardness in the thermal calculations, but as in [23]\nsuch di\u000eculties can be negotiated by regarding such delta functions as limits to be\nstrictly imposed only at the end of calculations. Using (97){(99) it is straightforward to\ncompute the thermal position and momentum correlation functions for the q-oscillator,\nas well as its thermal (including zero-point) energy and other thermodynamic quantities.\nOur procedure will be to derive the results for a general coupling function for which the\ndiagonalization in section 5 is valid, and then to examine those results for the particular\ncoupling (21).\n7.1. Thermal correlation functions for the q-oscillator\nIt follows from (83), (84) and (99) that the thermal correlation function of ^ q(t) can be\nwritten\nh^q(t)^q(t0)i=1\n4\u00192Z1\n0d!Z1\n0d!0\b\nexp [\u0000i(!t\u0000!0t0)]h^q(!)^qy(!0)i\n+ exp [i(!t\u0000!0t0)]h^qy(!)^q(!0)i\t\n; (100)\nwith a similar result for the other canonical operators of the system. The frequency-\ndomain correlation functions h^qy(!)^q(!0)iandh^q(!)^qy(!0)iare, from (84) and (97),\nh^qy(!)^q(!0)i=2\u00192~\u000b2(!)\n!G\u0003(!)G(!0)N(!)\u000e(!\u0000!0)\n=N(!)\nN(!) + 1h^q(!)^qy(!0)i:(101)\nMultiplying the Green function expression (87) by G\u0003(!), we obtain\n\b\n!2\u0000!2\n0[1\u0000\u001f(!)]\t\nG\u0003(!)G(!) =\u0000G\u0003(!); (102)\nand the imaginary part of this equation is found using (88):\n\u0019\u000b2(!)\n2!G\u0003(!)G(!) = ImG(!): (103)\nInserting (103) in (101) we \fnd\nh^qy(!)^q(!0)i= 4\u0019~\u000e(!\u0000!0)N(!)ImG(!)\n=N(!)\nN(!) + 1h^q(!)^qy(!0)i;(104)Quantum dynamics of the damped harmonic oscillator 23\nwhich shows that the real part of the temporal correlation function (100) is\n1\n2h^q(t)^q(t0) + ^q(t0)^q(t)i=~\n\u0019Z1\n0d!cos [!(t\u0000t0)] coth\u0012~!\n2kBT\u0013\nImG(!); (105)\nwhere 2N(!) + 1 has been rewritten as a hyperbolic tangent. The momentum operator\nfor theq-oscillator is given by (84) and is just the time derivative of ^ q(t), so we obtain\nfrom (105) the momentum correlation function\n1\n2D\n^\u0005q(t)^\u0005q(t0) +^\u0005q(t0)^\u0005q(t)E\n=~\n\u0019Z1\n0d!!2cos [!(t\u0000t0)] coth\u0012~!\n2kBT\u0013\nImG(!):(106)\nWe now consider the general results (105) and (106) in the case of the coupling\nfunction (21), corresponding to damping of the q-oscillator proportional to velocity. We\n\frst note a slight modi\fcation of the result (103) in the case of (21). Equation (102)\ndoes not include a possible pole prescription at != 0 for a zero mode, such as occurs in\n(92). Evaluating the left-hand side of (103) for the retarded Green function (92), and\nreplacing 0+by\u0011, we \fnd\n\u0019\u000b2(!)\n2!G\u0003\nR(!)GR(!) =\r!!2\n0\n(!2+\u00112)[(!2\u0000!2\n0)2+\r2!2]: (107)\nAs\u0011!0, the right-hand side of (107) has a pole at != 0 and\u0011prescribes the treatment\nof this pole by a principal value; apart from the pole prescriptions, (107) is the result\n(103). With use of (107) the correlation function (105) gives the following expectation\nvalueh^q2(t)ifor coupling (21):\n\n^q2(t)\u000b\n=~\n2\u0019Z1\n\u00001d!\r!!2\n0\n(!2+\u00112)[(!2\u0000!2\n0)2+\r2!2]coth\u0012~!\n2kBT\u0013\n:(108)\nIn (108) the integration has been extended over negative !(the integrand is an even\nfunction of !); this allows evaluation of the integral by closing the integration contour\nin the upper (or lower) half complex plane. It is clear that the integral in (108) does\nnot converge when \u0011= 0, and evaluation by contour integration for \u0011 >0 shows that\nthis divergence is given by a term kBT\r=(!2\n0\u0011). The form of the divergent term shows\nthat (108) is \fnite at T= 0, so that the ground state value of h^q2(t)iis well de\fned.\nThe divergence at T >0 can be attributed to the zero mode, as it is associated with the\n!= 0 pole in the Green function. As the zero mode has zero energy, it is perhaps not\nsurprising that the zero-mode squared displacement diverges at non-zero temperature.\nAlthough it is tempting to drop the divergent zero-mode term kBT\r=(!2\n0\u0011) from (108)\nwhenT > 0, this would be a violation of quantum mechanics because the canonical\ncommutation relations of the system only hold for the full displacement operator ^ q(t).\nThere is thus an inherent pathology in the thermal state for the coupling (21), although\nthe ground state is well-de\fned as are other quantum states such as the in\fnite class of\ncoherent states described in section 6. Note that the divergent term kBT\r=(!2\n0\u0011) also\nvanishes if the limit \r!0 in the integration result is taken before the limit \u0011!0; in\nthis case the result for h^q2(t)ireduces to the free-oscillator value~\n2!0coth\u0010\n~!0\n2kBT\u0011\n. As\ndiscussed in section 5, coupling functions in the class (80) give rise to a zero-mode only ifQuantum dynamics of the damped harmonic oscillator 24\nthey obey the special condition (41); for general coupling functions in the class (80) the\ncorrelation function (105) will be well behaved. The momentum-squared expectation\nvalueh^\u00052\nq(t)ifor the coupling (21) is found from (106) and (107) to be\nD\n^\u00052\nq(t)E\n=~\n2\u0019Z1\n\u00001d!\r!!2\n0\n(!2\u0000!2\n0)2+\r2!2coth\u0012~!\n2kBT\u0013\n; (109)\nwhere there is no longer any contribution of the != 0 pole in the Green function.\nThe integral in (109) is well de\fned and can be evaluated by contour methods but,\ngiven the pathology in the corresponding position quantity h^q2(t)iatT >0, the rather\ncomplicated details are omitted here. We note however that when the the limit \r!0\nis taken in the \fnal result, (109) gives the free-oscillator value1\n2~!0coth\u0010\n~!0\n2kBT\u0011\n.\n7.2. Reservoir contributions to thermal correlation functions\nThe thermal correlation functions of the reservoir are of interest because of their\ncontribution to the thermal energy of the q-oscillator. As discussed in section 7.3,\nfor this purpose we must calculate correlation functions containing the reservoir using\n(85) and (86), but only retain terms that contain ^ q(!). This is the prescription used in\nmacroscopic QED to obtain the Casimir stress-energy of the electromagnetic \feld in a\nmedium, where the electric and magnetic \felds play the role of the q-oscillator [23]. In\nfact the results in this subsection are largely a simpler version of the reservoir correlation\nfunctions in [23].\nThe expectation value of the Hamiltonian (54) in thermal equilibrium contains\nthe termR1\n0d!h(@t^X!)2+!2^X2\n!i=2. We compute this expectation value using (85),\nretaining only terms in which ^ q(!) occurs. Essentially the same calculation is described\nin detail in [23] for macroscopic QED. We therefore refer the reader to [23] for the\nvarious steps in the derivation and here state the result:\n1\n2Z1\n0d!D\n(@t^X!)2+!2^X2\n!E\n=~!2\n0\n2\u0019ImZ1\n0d!coth\u0012~!\n2kBT\u0013d [!\u001f(!)]\nd!G(!): (110)\nThe thermal expectation value of the Hamiltonian (54) also containsR1\n0d!\u000b(!)h^qX!i.\nThis expectation value is also found using (85), but here ^ q(!) occurs in all contributions\nto the expectation value, so no terms are dropped. There is again an essentially identical\ncalculation in [23] for macroscopic QED, and the same derivation gives the result\nZ1\n0d!\u000b(!)D\n^q(t)^X!(t)E\n=~!2\n0\n\u0019ImZ1\n0d!coth\u0012~!\n2kBT\u0013\n\u001f(!)G(!): (111)\nThis quantity is real so the term containing (^ q^X!+^X!^q)=2 in the Hamiltonian (54) gives\nthe thermal expectation value on the right-hand side of (111). The thermal (including\nzero-point) energy of the q-oscillator can now be obtained.\n7.3. Thermal energy of the q-oscillator\nThe energy of the q-oscillator in any state is given by the expectation value of the\nHamiltonian (54) with the reservoir X!traced out. Due to the coupling of qtoX!,Quantum dynamics of the damped harmonic oscillator 25\nthis tracing operation is in general not a trivial matter. The same issue arises in\nmacroscopic QED, where the value of the electromagnetic energy-momentum tensor\nin a dispersive, dissipative medium requires the tracing out of the reservoir. In the\ncase of thermal equilibrium the electromagnetic stress-energy was found in [23] to\nbe given by a straightforward recipe: after the reservoir operators are expressed in\nterms of electromagnetic-\feld operators, all contributions containing electromagnetic-\n\feld operators are to be retained, the other contributions are to be dropped. We follow\nthe same recipe here to trace out the reservoir from the total energy of the system in\nthermal equilibrium and thereby obtain the thermal energy of the q-oscillator. The\nreservoir operators are expressed in terms of the q-operator using (85) and (86), and the\nthermal expectation value of the Hamiltonian (54) is evaluated with only contributions\nthat contain ^ q(!) included. The result is the thermal energy of the q-oscillator, which\nwe denote byh^Hiq. The required expectation values involving the reservoir are given\nby (110) and (111), and the thermal expectation values of the \frst two terms in the\nHamiltonian (54) are found from (105) and (106). This gives the following expression\nforh^Hiq:\nh^Hiq=~\n4\u0019ImZ1\n\u00001d!coth\u0012~!\n2kBT\u0013\u001a\n!2\n0\u0014\n!d\u001f(!)\nd!\u0000\u001f(!) + 1\u0015\n+!2\u001b\nG(!): (112)\nDue to the parity properties of the susceptibility and the Green function, the real part\nof the integrand in (112) does not contribute and the integral is imaginary. The q-\noscillator thermal energy (112) is the analogue for the damped oscillator of the Casimir\nenergy density of electromagnetic \felds in a medium [23]. A dispersive contribution to\nthe energy from the derivative of the susceptibility is familiar in the electromagnetic\ncase [23], and we \fnd a similar contribution from the damped-oscillator \\susceptibility\"\nin (112). As is familiar in Casimir theory and elsewhere, the integral in (112) can be\nconverted to a sum over imaginary Matsubara frequencies.\nIn the case of the coupling (21) with susceptibility (91), the real part of the integrand\nin (112) does not give a convergent integral so that it is not correct to take the Im outside\nthe integration as has been done in the derivation of (112). For the coupling (21) we\nmust keep the Im inside the integration and take the imaginary part of the integrand;\nthe resulting expression for h^Hiqis\nh^Hiq=~\n4\u0019Z1\n\u00001d!\r!!2\n0(\r2+ 3!2\u0000!2\n0)\n(!2+\r2) [(!2\u0000!2\n0)2+\r2!2]coth\u0012~!\n2kBT\u0013\n:(113)\nThere is no zero-mode contribution in (113), as is expected from the fact that the\nzero mode has zero energy. The thermal energy (113) is \fnite and can be evaluated\nanalytically by closing the integration contour in the upper (or lower) half complex\nplane. When the zero-damping limit \r!0 is taken, however, the result for the thermal\nenergy is found to reduce to\n1\n2~!0coth\u0012~!0\n2kBT\u0013\n\u00001\n2kBT; (114)Quantum dynamics of the damped harmonic oscillator 26\ni.e. in addition to the free-oscillator thermal energy there is an additional term \u0000kBT=2.\nThis additional term can be traced back to the dispersive contribution d \u001f(!)=d!in\n(112); although \u001f(!) vanishes as \r!0, the contribution from this dispersive term to\nthe energy becomes \u0000kBT=2 as\r!0. The non-zero contribution of the d \u001f(!)=d!-term\nwhen\r!0 relies on the assumption that the susceptibility (91) is valid up to in\fnite\nfrequencies. Although it is mathematically convenient to employ functions like (91) over\nthe entire frequency range, this is an unphysical assumption and therefore may produce\nsome unphysical results. As will be discussed in section 8, application of the general\nresults presented here to real physical systems would probably require a measurement\nof the susceptibility characterizing the particular damped oscillator; although these\nmeasured susceptibilities could be \ftted to mathematical functions, non-zero values of\nthese functions at very large frequencies beyond the measured range would be physically\nmeaningless. We note that the di\u000eculty with the zero-coupling limit of the thermal\nenergy for damping (21) does not occur for the ground state T= 0, as we see from\n(114) that the correct free-oscillator limit ~!0=2 is obtained. These thermal-energy\nresults support the comment in section 7.1 that the coupling (21) gives an inherent\npathology in the non-zero temperature state.\n8. Conclusions\nThe use of a discrete set of oscillators as a reservoir is almost universal in studying\ndissipation in quantum systems using exact quantization rules. Yet the power of a\ncontinuum reservoir in the study of dissipation was demonstrated twenty years ago by\nHuttner and Barnett [21], and it has recently been shown that a continuum reservoir\nallows an exact canonical quantization of the macroscopic Maxwell equations in arbitrary\nmedia obeying Kramers-Kronig relations [22, 23, 24, 25]. The important results that\nhave been obtained with a continuum reservoir would not have been achieved if a limiting\nprocedure had been applied to results featuring a discrete reservoir, whereas the discrete\nreservoir almost never captures the desired physics without a subsequent, and delicate,\nlimit to the continuum case [19]. In view of this, it is highly advisable to employ a\ncontinuum reservoir from the outset. We have considered a single damped quantum\nharmonic oscillator and shown that a continuum reservoir allows the exact treatment of\ndamping proportional to velocity. In addition we have found that for general damping\nthe quantum oscillator is remarkably similar to light in macroscopic media. For functions\n\u000b(!) coupling the oscillator to the reservoir, with \u000b2(!) an even function of !, the\ndynamics of the damped oscillator is governed by an e\u000bective susceptibility obeying\nKramers-Kronig relations.\nThe experimental investigation of macroscopic oscillators that exhibit quantum\nbehaviour has made remarkable progress in recent years; oscillators have been cooled to\ntheir quantum ground states and even placed in a superposition of energy states [26, 27,\n28, 29, 30]. A theoretical description of nano-oscillators and opto-mechanical systems\ncan be approached in various ways, depending among other things on how much detailQuantum dynamics of the damped harmonic oscillator 27\nof the microscopic structure of the oscillator is included. The approach to the damped\noscillator taken in this paper naturally leads to a description of the oscillator by means\nof an e\u000bective susceptibility. As in macroscopic electromagnetism, the strength of this\napproach is that the complicated microscopic substructure is not included, rather the\nsusceptibility is a quantity to be measured for real physical systems. We have found\nthat diagonalization of the Hamiltonian requires restrictions on the coupling function\nthat determines the susceptibility, but this may not be signi\fcant for real macroscopic\noscillators. The condition (81) constrains the relationship between the coupling function\n\u000b(!) and the free-oscillation frequency !0. But most mechanical oscillators do not\nexhibit a \\free\" oscillation state; the oscillation degree of freedom is produced by the\nsame material geometry that causes damping. This means that the parameter !0cannot\nbe separated from the susceptibility, and they must together be \ftted to a given system\nthrough experimental measurement. A choice of !0for which!2\n0>R1\n0d!\u000b2(!)=!2may\ntherefore not be restrictive. This also suggests that the there is freedom in the form of\nthe coupling term between the q-oscillator and the reservoir. We chose a coupling of the\nformqX!, whereas Huttner and Barnett [21] use q_X!, and the latter does not give the\nrestriction (81) on the coupling function. But the Huttner-Barnett coupling leads to\na renormalization of the free-oscillation frequency !0[21], which is why the restriction\n(81) is avoided. If !0has no direct physical meaning one can e\u000bectively make the same\nrede\fnition with the coupling term chosen here, through the joint characterization of\n!0and the susceptibility \u001f(!).\nThe considerations in this paper are of course only a \frst step towards describing\nphysical quantum oscillators, and other ingredients would have to be included such as\ncoupling to light in the case of opto-mechanical systems. There is also the crucial\nquestion of how the e\u000bective susceptibility is to be measured for real oscillators,\nor how physically realistic susceptibilities could be deduced from simple microscopic\nconsiderations, as is done in electromagnetism. In regard to this last point, damping\nproportional to velocity would not be expected to be physically relevant. While the\ndynamics (1) may be a reasonable description, for small _ q, of a free oscillator in a\n\ruid, nano-oscillators with mechanical damping determined by the particular material\ngeometry would not experience a simple damping proportional to velocity. Real\nmacroscopic oscillators may however be describable by an e\u000bective susceptibility and\nby results such as the thermal energy (112).\nAcknowledgments\nI am indebted to Simon Horsley for many helpful disscussions. I also thank Gabriel\nBarton for useful literature and information on several aspects of this work. This\nresearch is supported by the Royal Society of Edinburgh and the Scottish Government.Quantum dynamics of the damped harmonic oscillator 28\nAppendix A. Green function\nHere we present a Green function G(t;t0) that satis\fes (32). We obtain the solution by\nsolving (34) for G(!;t0) with\u000f>0. The Green function corresponds to an in\fnitesimal\nvalue of\u000f, but (34) is \frst solved for a non-in\fnitesmal number \u000f >0 for which the\nstandard solution method [31] can be applied without di\u000eculty; after the solution is\nobtained the limit of in\fnitesimal \u000fis taken and then G(!;t0) is (inverse) Fourier\ntransformed to \fnd G(t;t0). The solution of (34) with \u000f>0 di\u000bers depending on whether\nt0is positive or negative, as this a\u000bects the analytic properties of the inhomogeneous\nterm as a function of !. The general solution of (34) will also contain the solution (25)\nof the homogeneous integral equation (24), which we do not require here and therefore\ndrop. We \fnd the solution G(!;t0) by the standard technique [31] and then take \u000f!0+,\nwhereupon the in\fnitesimal number 0+serves only to regulate a pole at != 0:\nG(!;t0) =\u0000(!+ i\r) exp(i!t0)\n(!\u0000i0+)(!2\u0000!2\n0+ i\r!)\u00002i\r!2\n0\n(!\u0000i0+)[(!2\u0000!2\n0)2+\r2!2]\n+exp\u0000\n\u0000\rt0\n2\u0001\u0002\n(!2\n0\u0000\r2)!1cos\u0000!1t0\n2\u0001\n\u0000\r(\r2\u00003!2\n0) sin\u0000!1t0\n2\u0001\u0003\n!1[(!2\u0000!2\n0)2+\r2!2]; t 0\u00150;(A.1)\nG(!;t0) =\u0000(!\u0000i\r) exp(i!t0)\n(!\u0000i0+)(!2\u0000!2\n0\u0000i\r!)\n+exp\u0000\rt0\n2\u0001\u0002\n(!2\n0\u0000\r2)!1cos\u0000!1t0\n2\u0001\n+\r(\r2\u00003!2\n0) sin\u0000!1t0\n2\u0001\u0003\n!1[(!2\u0000!2\n0)2+\r2!2]; t 0\u00140:(A.2)\nThe solution (A.1){(A.2) is continuous at t0= 0, and!1is again given by (27). Fourier\ntransformation of (A.1){(A.2) to the time domain gives the Green function\nG(t;t0) = [2\u0012(\u0000t)\u0000\u0012(t0\u0000t)]\r\n!2\n0e0+t\n+exp\u0010\n\u0000\rjtj\n2\u0011\n!1!2\n0\u0014\n\r!1sgn(t) cos\u0012!1t\n2\u0013\n+ (\r2\u00002!2\n0) sin\u0012!1t\n2\u0013\u0015\n\u0000\u0012(t\u0000t0)exp\u0002\n\u0000\r\n2(t\u0000t0)\u0003\n!1!2\n0n\n\r!1cosh!1\n2(t\u0000t0)i\n+ (\r2\u00002!2\n0) sinh!1\n2(t\u0000t0)io\n\u0000exp\u0002\n\u0000\r\n2(jtj+t0)\u0003\n2\r!2\n1!2\n0\u0014\n!1cos\u0012!1t\n2\u0013\n+\rsin\u0012!1jtj\n2\u0013\u0015\n\u0002\u0014\n(\r2\u0000!2\n0)!1cos\u0012!1t0\n2\u0013\n+ (\r2\u00003!2\n0)\rsin\u0012!1t0\n2\u0013\u0015\n; t 0\u00150; (A.3)\nG(t;t0) =\u0012(t0\u0000t)\r\n!2\n0e0+t\n\u0000\u0012(t0\u0000t)exp\u0002\r\n2(t\u0000t0)\u0003\n!1!2\n0n\n\r!1cosh!1\n2(t\u0000t0)i\n\u0000(\r2\u00002!2\n0) sinh!1\n2(t\u0000t0)io\n\u0000exp\u0002\n\u0000\r\n2(jtj\u0000t0)\u0003\n2\r!2\n1!2\n0\u0014\n!1cos\u0012!1t\n2\u0013\n+\rsin\u0012!1jtj\n2\u0013\u0015Quantum dynamics of the damped harmonic oscillator 29\n\u0002\u0014\n(\r2\u0000!2\n0)!1cos\u0012!1t0\n2\u0013\n\u0000(\r2\u00003!2\n0)\rsin\u0012!1t0\n2\u0013\u0015\n; t 0\u00140; (A.4)\nwhere\u0012(x) is the step function. The values of G(t;t0) at zeros of the step functions in\n(A.3){(A.4) are to be taken as limits; G(t;t0) is in fact continuous across t= 0,t0= 0\nandt=t0. It is straightforward to verify that (A.3){(A.4) satis\fes (32); there is a\ndiscontinuity in d G(t;t0)=dtatt=t0which gives a delta function in d2G(t;t0)=dt2.\nThe in\fnitesimal number 0+in (A.3){(A.4) regularizes its Fourier transform in t, giving\ntheG(!;t0) of (A.1){(A.2).\nAppendix B. Coupling functions for a diagonalizable Hamiltonian\nHere we complete the \fnal steps in diagonalizing the Hamiltonian. In section 5\nwe constructed a transformation, given by (61){(63), (73), (75) and (79), that will\ndiagonalize the Hamiltonian provided the commutation relations (52) and (53) are\nsatis\fed by (61){(63). Inserting (61){(63) into (52) and (53), and using (58), we obtain\nthe \fnal non-trivial conditions on the f-coe\u000ecients for the diagonalizing transformation:\nZ1\n0d!fq(!)f\u0003\n\u0005q(!)\u0000Z1\n0d!f\u0003\nq(!)f\u0005q(!) = i~; (B.1)\nZ1\n0d!00fX(!;!00)f\u0003\n\u0005X(!0;!00)\u0000Z1\n0d!00f\u0003\nX(!;!00)f\u0005X(!0;!00) = i~\u000e(!0\u0000!00); (B.2)\nZ1\n0d!00fX(!;!00)f\u0003\nX(!0;!00)\u0000Z1\n0d!00f\u0003\nX(!;!00)fX(!0;!00) = 0; (B.3)\nZ1\n0d!00f\u0005X(!;!00)f\u0003\n\u0005X(!0;!00)\u0000Z1\n0d!00f\u0003\n\u0005X(!;!00)f\u0005X(!0;!00) = 0: (B.4)\nThe last two of these conditions are easily seen to hold because of (73) and the \frst of\n(69). We proceed to show that the \frst two conditions, i.e. (B.1) and (B.2), hold for\ncoupling functions satisfying (80) and (81), a class that includes the coupling function\n(21) corresponding to damping proportional to velocity. The following analysis has an\nexact counterpart in the Huttner-Barnett model [21], although the results are di\u000berent\ndue to a di\u000berent type of coupling between the q-oscillator and the reservoir.\nConsider \frst the condition (B.1). If we insert the \frst of (68) together with (75)\ninto (B.1), we encounter a term quadratic in hq(!). As noted after (76), a non-zero\nhq(!) will contain a delta function, so to avoid a square of a delta function in (B.1) we\nare forced to choose the solution\nhq(!) = 0 (B.5)\nof (76). The condition (B.1) is then, with use of (79),\n1 =Z1\n0d!\u000b2(!)\f\f\f\f!2\u0000!2\n0+ PZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2+i\u0019\u000b2(!)\n2!\f\f\f\f\u00002\n: (B.6)\nThis complicated restriction on the coupling function \u000b(!) is a requirement for the\nHamiltonian to be diagonalizable. For a very broad class of coupling functions however,Quantum dynamics of the damped harmonic oscillator 30\n(B.6) reduces to a much simpler requirement. If \u000b2(!) is an even function of !then\n(B.6) can be written\n1 =1\ni\u0019Z1\n\u00001d!!\u0012\n!2\n0\u0000!2\u0000PZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2\u0000i\u0019\u000b2(!)\n2!\u0013\u00001\n: (B.7)\nThe assumption that \u000b2(!) is an even function also implies that the combination\nPZ1\n0d\u0018\u000b2(\u0018)\n\u00182\u0000!2+i\u0019\u000b2(!)\n2!; (B.8)\nwhich appears in the denominator in (B.7), is analytic on the upper-half complex !-\nplane. This follows from the familiar analysis of susceptibilities that are analytic on\nthe upper-half frequency plane and that therefore obey Kramers-Kronig relations [33];\nthe real and imaginary parts of (B.8) exhibit one of the Kramers-Kronig relations for a\ncomplex susceptibility, provided \u000b2(!) is an even function of !. If we further assume that\n\u000b2(!)=!> 0 except possibly at != 0, then it is shown in [33] that the \\susceptibility\"\n(B.8) does not take real values at any \fnite point in the upper-half complex !-plane\nexcept on the imaginary axis, where it varies monotonically from its value at i0 to its\nvalue at i1. Moreover, the (real) value of (B.8) on the positive imaginary axis != i\u0014,\n\u0014\u00150, is [33]\nZ1\n0d\u0018\u000b2(\u0018)\n\u00182+\u00142: (B.9)\nWe now consider the analytic properties of the integrand in (B.7) in the upper-half\nfrequency plane. As (B.8) is real in the upper-half plane only on the positive imaginary\naxis, the denominator in (B.7) can vanish only at positive imaginary frequencies != i\u0014,\nwhere it has the value (using (B.9))\n!2\n0+\u00142\u0000Z1\n0d\u0018\u000b2(\u0018)\n\u00182+\u00142: (B.10)\nIt is clear that (B.10) has a zero if and only if\n!2\n0\u0014Z1\n0d\u0018\u000b2(\u0018)\n\u00182: (B.11)\nIf the equality holds in (B.11) then (B.10) has a zero at != i\u0014= 0, but this does not\ngive a pole in the integrand in (B.7) unless (B.10) goes as \u0014n,n>1, as\u0014!0. There\nare therefore no poles in the integrand in (B.7) in the upper-half plane if and only if\neither!2\n0>Z1\n0d\u0018\u000b2(\u0018)\n\u00182;\nor!2\n0=Z1\n0d\u0018\u000b2(\u0018)\n\u00182and (B:10)\u0018\u0014n; n\u00141;as\u0014!0:(B.12)\nIf (B.12) holds then the integral in (B.7) is equal to minus the integral over an in\fnite\nsemi-circle in the upper-half plane. Taking !=Rei\u001e,R!1 , 0\u0014\u001e\u0014\u0019as the\nintegration parameter along the in\fnite semi-circle, we \fnd that the integral in (B.7) is\ni\u0019, so that (B.7) is satis\fed.Quantum dynamics of the damped harmonic oscillator 31\nThere remains the condition (B.2). We must substitute the \frst of (69), (73),\n(75), (B.5) and (79). This gives an integral relation that is again easy to evaluate for\ncoupling functions for which \u000b2(!) is an even function of !and\u000b2(!)=! > 0 except\npossibly at != 0. Evaluation of the integral by closing the countour in the upper-half\nplane shows that the integral relation holds without restrictions on the coupling function\nbeyond those already derived from (B.1). This analysis completes the proof that the\nHamiltonian is diagonalizable for the class of coupling functions (80) if and only if (81)\nholds.\nReferences\n[1] Dekker H 1981 Phys. Rep. 801\n[2] Um C I, Yeon K H and George T F 2002 Phys. 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Usp. 30134\n[20] Grabert H and Weiss U 1984 Z. Phys. B5587\n[21] Huttner B and Barnett S M 1992 Phys. Rev. A464306\n[22] Philbin T G 2010 New J. Phys. 12123008\n[23] Philbin T G 2011 New J. Phys. 13063026\n[24] Horsley S A R 2011 Phys. Rev. A84063822\n[25] Horsley S A R 2012 arXiv:1205.0486\n[26] O'Connell A D et al. 2010 Nature 464697\n[27] Teufel J D, Donner T, Li D, Harlow J W, Allman M S, Cicak K, Sirois A J, Whittaker J D,\nLehnert K W and Simmonds R W 2011 Nature 475359\n[28] Chan J, Alegre T P M, Safavi-Naeini A H, Hill J T, Krause A, Gr oblacher S, Aspelmeyer M and\nPainter O 2011 Nature 47889\n[29] Aspelmeyer M, Gr oblacher S, Hammerer K and Kiesel N 2010 J. Opt. Soc. Am. B27A189\n[30] Poot M and Zant H S J 2012 Phys. Rep. 511273\n[31] Pipkin A C 1991 A Course On Integral Equations (New York: Springer)\n[32] Muskhelishvili N I 2008 Singular Integral Equations (New York: Dover)\n[33] Landau L D and Lifshitz E M 1980 Statistical Physics, Part 1 3rd ed (Oxford: Butterworth-\nHeinemann)\n[34] Mandel L and Wolf E 1995 Optical Coherence and Quantum Optics (Cambridge: Cambridge\nUniversity Press)"    },    {        "title": "0706.3982v3.Lorentz_anomaly_in_the_semi_light_cone_gauge_superstrings.pdf",        "content": "arXiv:0706.3982v3  [hep-th]  20 Sep 2007YITP-07-38\nKEK-TH-1161\nJune, 2007\nLorentz Anomaly in Semi-Light-Cone Gauge\nSuperstrings\nHiroshiKunitomo1and Shun’ya Mizoguchi2,3\n1Yukawa Institute for Theoretical Physics, Kyoto Universit y,\nKyoto 606-8502, Japan\n2High Energy Accelerator Research Organization (KEK),\nTsukuba 305-0801, Japan\n3Department of Particle and Nuclear Physics,\nThe Graduate University for Advanced Studies,\nTsukuba 305-0801, Japan\nAbstract\nWe study the Lorentz invariance of D= 4 and 6 superstrings in the double-spinor\nformalism, which are equivalent to the D= 4 and 6 superstrings in the pure-spinor\nformalisminthesenseoftheBRSTcohomology. Wefirstre-examineh owtheconformal\nand Lorentz anomalies appear in the D= 4 and 6 Green-Schwarz superstrings in the\nsemi-light-cone gauge in the framework of BRST quantization. We co nstruct a set of\nBRST invariant Lorentz generators and show that they do not for m a closed algebra,\neven cohomologically. We then turn to the construction of Lorentz generators in the\nD= 4 and 6 double-spinor superstrings, and show that the Lorentz in variance is again\nanomalous. We also discuss the relation between the anomaly-free L orentz generators\nin the lower-dimensional pure-spinor formalisms and that obtained in this paper.\n1§1. Introduction\nRecently, it has been recognized that the covariant quantization o f superstrings using\npure spinors1)can be naturally understood in terms of a Green-Schwarz-like supe rstring\nwith twice as many fermionic degrees of freedom, the double-spinor (DS) formalism.2)The\nsuperstring in the DS formalism possesses an additional local symme try, and is classically\ngauge equivalent to the ordinary Green-Schwarz (GS) superstrin g. Imposing the semi-light-\ncone gauge condition on one half of the fermionic variables, Aisaka an d Kazama completed a\nDirac/BRST quantization of the D= 10 DS superstring, finding that the resulting system is\ncohomologically equivalent to the PS superstring.∗)In this way, they uncovered the “origin”\nof the formalism, and, in particular, they derived the previously mys terious seventeen first-\nclass constraints4)assumed to clarify the relation between GS and PS superstrings.\nIn a previous paper, Ref. 5), we applied this idea to lower-dimensiona l (D= 4 and\n6) cases.6),7)The primary motivation of that work was to understand how the con cept of\nthe critical dimension emerges in the PS formalism. We have shown tha t, starting from\nsimilar Lagrangians, D= 4 andD= 6 DS superstrings can be BRST quantized to yield free\nCFTs similar to the semi-light-cone gauge GS superstrings, along with additional conjugate\npair systems and extra constraints. The BRST charges again redu ce to those of the lower-\ndimensional PS superstrings through similarity transformations.\nThus, the DS superstrings “interpolate” between the GS and PS su perstrings, but this\nraises some questions. The GS superstring theories have a Lorent z anomaly in lower dimen-\nsional cases, while the PS superstring theories have anomaly-free Lorentz generators.6),7)\nWhere does this difference come from? Then, as a related question, what do “quantum\nmechanically consistent D= 4 and 6 superstrings” describe?\nThe DS superstrings are closely related to the GS superstrings in th e semi-light-cone\ngauge.2)The presence or absence of Lorentz and conformal anomalies for theD= 10 semi-\nlight-cone gauge GS superstring was a subject of great debate in t he late 1980s and early\n1990s. In Ref. 8), it was revealed that, contrary to the prevailing belief at that time,9)the\nD= 10 GS superstring in the semi-light-cone gauge has a non-vanishing conformal anomaly.\nLater, it was shown that this conformal anomaly is canceled by intro ducing a certain local\ncounterterm, and the Lorentz algebras become closed with a suita ble modification of the\nLorentz generators.10),11)This local counterterm can be viewed as a coupling to a certain\ndilaton background. More recently, the Lorentz invariance of the D= 10 GS superstring in\nthe semi-light-cone gauge has been re-examined and proved using t he BRST method.4)\nIn this paper, we first examine the conformal and Lorentz anomalie s of theD= 4 and\n∗)See 3) for a different formulation which also relates GS and PS supers trings.\n26 GS superstrings in the semi-light-cone gauge. We BRST quantize th ese lower-dimensional\nGS superstrings in a manner similar to that for the DS superstring in R ef. 5). The key\nstep in this procedure is the modification of the quantum constraint s, and we argue that it\neffectively changes the background from a flat space-time to a linea r-dilaton-like one. We\nthenconstruct aset ofBRST invariant Lorentz generators ands how thatthey arenot closed,\nas expected.\nNext, we turn to an examination of the Lorentz invariance of the D= 4 and 6 DS super-\nstrings studied in Ref. 5). We present a complete set of BRST-invar iant Lorentz generators\nin both cases. We then show that they form the correct Lorentz a lgebra, except for the\ncommutators between the “ i−” generators, which, again, are not BRST exact. Finally, we\ninvestigate the relation between these charges and the anomaly-f ree Lorentz generators in\ntheD= 4 PS formalism described in Refs. 6) and 7).\nThe organization of this paper is as follows. In §2, we study the conformal and Lorentz\nanomalies of the D= 4 and 6 GS superstrings in the semi-light-cone gauge using the BRST\nmethod. We derive the BRST-invariant Lorentz generators of the semi-light-cone gauge DS\nsuperstring and compute their algebras in §3. In the final section, we discuss the difference\nbetween the anomaly-free Lorentz generators of Refs. 6) and 7 ) and those obtained in this\npaper.\n§2. The Lorentz invariance of lower-dimensional GS superstr ings in the\nsemi-light-cone gauge\n2.1.TheD= 4GS superstring in the semi-light-cone gauge\nThe Lagrangian of the D= 4 Green-Schwarz (GS) superstring is an obvious generaliza-\ntionofthe D= 10GSLagrangian,2)withanappropriatespinor structure infourdimensions:\nL=LK+LWZ, (2.1a)\nLK=−1\n2√−ggabΠµ\naΠµb, (2.1b)\nLWZ=ǫabΠµ\na(Wµb−ˆWµb)−ǫabWµ\naˆWµb (2.1c)\nwith\nΠµ\na=∂aXµ−2/summationdisplay\nA=1WAµ\na, (2.2)\nWAµ\na=iθAσµ∂a¯θA−i∂aθAσµ¯θA. (2.3)\n3Here, we employ the notation used in Ref. 5): µ,ν= 0,1,2,3 are the flat space-time indices\nwith the metric ηµν= diag[+1 ,−1,−1,−1],a,b= 0,1 are the worldsheet indices, σµare the\ntwo-by-two hermitian off-diagonal blocks of the gamma matrices in t he chiral representation,\nθAarecomplexWeylspinors, with A= 1,2labelingtheleftandrightdegreesoffreedomafter\nthe semi-light-cone gauge fixing. We also adopt the notation Wµ\na=WA=1,µ\na,ˆWµ\na=WA=2,µ\na,\netc.\nThe fermionic constraints are simply\nDA\nα=kA\nα−i(kµ+ηA(Πµ\n1+Wµ¯A\n1))(σµ¯θA)α≈0, (2.4a)\n¯DA\nα=¯kA\n˙α−i(kµ+ηA(Πµ\n1+Wµ¯A\n1))(θAσµ)˙α≈0, (2.4b)\nwhere¯A= 1(2) if A= 2(1). Parameterizing the worldsheet metric as\ngab=/parenleftBigg\n−N2+γ(N1)2γN1\nγN1γ/parenrightBigg\n, (2.5)\nin the ADM form, we obtain the Hamiltonian\nH=N√γT0+N1T1+˙θAαDA\nα+˙¯θA˙α¯DA\n˙α, (2.6)\nwhere\nT+=1\n2(T0+T1) =1\n4ΠµΠµ, (2.7a)\nT−=1\n2(T0−T1) =1\n4ˆΠµˆΠµ, (2.7b)\nΠµ=kµ+X′µ−2Wµ\n1, (2.8a)\nˆΠµ=kµ−X′µ+2ˆWµ\n1. (2.8b)\nIn fact, all the above formulas can be derived from the correspon ding ones in the D= 4 DS\nformalism5)by setting all the variables with tildes to zero. Assuming the Poisson b rackets\n{Xµ(σ),kν(σ′)}P=ηµνδ(σ−σ′), (2.9a)\n{θAα(σ),kB\nβ(σ′)}P=−δABδα\nβδ(σ−σ′), (2.9b)\n{¯θA˙α(σ),¯kB\n˙β(σ′)}P=−δABδ˙α\n˙βδ(σ−σ′), (2.9c)\nwe find that two of the four fermionic constraints are first class, g enerating the kappa sym-\nmetry, and the other two are second class. Imposing the semi-light -cone gauge condition\nθ2≈¯θ˙2≈0, (2.10)\n4the kappa symmetry is fixed, and all the fermionic constraints beco me second class. Then,\nthe only first-class constraints are the left and right Virasoro con straints generated by (2 .7).\nThe Dirac bracket can be computed straightforwardly, and the re sult is identical to the\nDS superstring given in Ref. 5), with the variables with tildes replaced by variables without\ntildes, and TandΠ+(≡Π0+Π3) replaced by variables appropriate for the GS superstring.\nIn this case, unlike in the case of the DS superstrings, the Dirac bra ckets among Xµandkν\nremain canonical; the only necessary modifications are the familiar re scalings\nS≡√\n2Π+θ1,¯S≡√\n2Π+¯θ˙1, (2.11)\nwhich satisfy the relations\n{S(σ),¯S(σ′)}D=iδ(σ−σ′), (2.12a)\n{Xµ(σ),S(σ′)}D= 0, (2.12b)\n{Xµ(σ),¯S(σ′)}D= 0. (2.12c)\nWe now turn to the quantization of the D= 4 GS superstring. As in Ref. 5), we replace\nthe Dirac brackets obtained above with appropriate OPEs. With som e rescalings, the left\nconstraint, T0+T1, becomes the energy-momentum tensor Tmatter(z) composed of free fields:\nT(z) =1\n2∂Xµ∂Xµ−1\n2(S∂¯S−∂S¯S). (2.13)\nThe OPEs for the basic holomorphic fields are\nXµ(z)Xν(w)∼ηµνlog(z−w), (2.14a)\nS(z)¯S(w)∼1\nz−w. (2.14b)\nBecause the central charge of T(z) is 5, the ghost contribution −26 cannot be cancelled in\nfour dimensions. To compensate for the shortage, we modify the e nergy-momentum tensor\nT(z) similarly to that in Ref. 5), as\nT(z)→ˇT(z) =1\n2∂Xµ∂Xµ−1\n2(S∂¯S−∂S¯S)+7\n8∂2log∂X+, (2.15)\nwithη+−= 2,ηij=−δij. In general, a family of energy-momentum tensors\nTX+X−(z) =1\n2∂X+∂X−+ξ ∂2log∂X+(2.16)\nwith a parameter ξhas central charge\nc(ξ) = 1+24 ξ (2.17)\n5ifX+(z)X−(w)∼+2log(z−w). Therefore, the logarithm term correctly shifts the central\ncharge to 26. Using this modified energy-momentum tensor, we can construct a standard\nnilpotent BRST charge:\nQGS=/contintegraldisplaydz\n2πi/parenleftbig\ncˇT+bc∂c/parenrightbig\n. (2.18)\nNote that although this modification of the energy-momentum tens or may seem ad hoc,\nitisrequired even in the D= 10 GS superstring in the semi-light-cone gauge. Indeed, a\none-loop analysis reveals the existence of a conformal anomaly of c=−12, including the\nbcghosts, which can only be canceled with a special dilaton coupling intro duced as a local\ncounterterm.10),11)This causes a change of theenergy-momentum tensor asin(2 .15), though\nwith a coefficient of 1 /2 instead of 7 /8. The inclusion of the counterterm also results in a\nmodification of the spacetime Lorentz transformation rules, which have been shown to have\nno anomaly.10),11),4)Similarly, we can add a local counterterm to the D= 4 GS action so\nthat the total conformal anomaly vanishes, and this gives rise to a change of the energy-\nmomentum tensor (2 .15). The question is whether, with that counterterm, the rigid Lor entz\nsymmetry is preserved in the theory. Below we examine this point.\nALorentzgeneratorfortheGSsuperstringsinthesemi-light-con egaugebasicallyconsists\nof a Noether current and, if it does not preserve the semi-light-co ne gauge condition (2 .10),\nan additional, compensating kappa-symmetry current. In addition , we need some extra\nterms for the BRST invariance of the generators. For the D= 4 case, we find\nNij=1\n4/parenleftbig\n−Xi∂Xj+Xj∂Xi+iǫijS¯S/parenrightbig\n, (2.19a)\nN+−=1\n4/parenleftbig\n−X+∂X−+X−∂X+/parenrightbig\n, (2.19b)\nNi+=1\n4/parenleftbig\n−Xi∂X++X+∂Xi/parenrightbig\n, (2.19c)\nNi−=1\n4/parenleftbigg\n−Xi∂X−+X−∂Xi+2iǫij∂Xj\n∂X+S¯S−7\n2∂2Xi\n∂X+/parenrightbigg\n, (2.19d)\nwherei,j= 1,2 andǫ12=−ǫ21= 1,ǫ11=ǫ22= 0. The third term in Ni−(2.19d) comes\nfrom the compensating kappa transformation, and the fourth te rm is required for the BRST\ninvariance.∗)Lorentz generators constructed from these currents all comm ute with QGS.\nDefining the charges as\nMµν=/contintegraldisplaydz\n2πiNµν(z), (2.20)\n∗)An analogous term is also needed for the D= 10 GS superstring. In this case, one must add +∂2Xi\n∂X+\ntoNi−in Eq. (3.6) of Ref. 4).\n6it can be verified that they form the D= 4 Lorentz algebra, except for [ M1−, M2−], which\nis given by\n[M1−, M2−] =/contintegraldisplaydz\n2πi/parenleftbigg\niS¯S\n(∂X+)2/parenleftbigg1\n2∂Xµ∂Xµ−7\n8∂3X+\n∂X++7\n4(∂2X+)2\n(∂X+)2/parenrightbigg\n−3\n4∂X1∂2X2−∂2X1∂X2\n(∂X+)2/parenrightbigg\n. (2.21)\nUnlike the D= 10 GS superstring analyzed in Ref. 4), the right hand side cannot b e BRST-\nexact. This can be proven as follows. Suppose that the terms prop ortional to S¯Sin (2.21)\ncould be written as a commutator of QGSand some BRST “parent.” Then, since ˇTdoes not\nhave such a term, the parent itself must contain S¯S. It is not difficult to show that the only\npossible choice isbS¯S\n(∂X+)2multiplied by some constant. However, we have\n/bracketleftbigg\nQGS,/contintegraldisplaydz\n2πiibS¯S\n(∂X+)2(z)/bracketrightbigg\n=/contintegraldisplaydz\n2πi/parenleftbigg\niS¯S\n(∂X+)2/parenleftbigg1\n2∂Xµ∂Xµ−1\n8∂3X+\n∂X+−7\n8(∂2X+)2\n(∂X+)2/parenrightbigg\n+3\n4iS∂2¯S+∂2S¯S\n(∂X+)2/parenrightbigg\n, (2.22)\nwhichisinconsistent. Thus, wehaveshownthat(2 .21)doesnotvanish, evencohomologically,\nand therefore the Lorentz invariance is broken. This is a natural r esult, because we know\nthat the Lorentz algebra is not closed in the light-cone quantization , and this should be\nindependent of the gauge choice.\n2.2.TheD= 6GS superstring in the semi-light-cone gauge\nThe BRST quantization of the D= 6 GS superstring in the semi-light-cone gauge is\ncompletely analogous, and therefore we give only a brief summary. A gain, the Diracbrackets\nfor theD= 6 GS superstring are derived from the D= 6 DS superstring5)by similar\nreplacements. The matter energy-momentum tensor is given by\nT(z) =1\n2∂Xµ∂Xµ−1\n2SI\na∂Sa\nI. (2.23)\nThe relevant OPEs are\nXµ(z)Xν(w)∼ηµνlog(z−w), (2.24a)\nSa\nI(z)Sb\nJ(w)∼ −ǫIJǫab\nz−w. (2.24b)\nAgain, we modify the energy-momentum tensor to\nˇT(z) =1\n2∂Xµ∂Xµ−1\n2SI\na∂Sa\nI+3\n4∂2log∂X+, (2.25)\n7so that the BRST charge\nQGS=/contintegraldisplaydz\n2πi/parenleftbig\ncˇT+bc∂c/parenrightbig\n(2.26)\nbecomes nilpotent. The BRST-invariant Lorentz generators are f ound to be\nNij=1\n4/parenleftbigg\n−Xi∂Xj+Xj∂Xi+i\n2(SIγijSI)/parenrightbigg\n, (2.27a)\nN+−=1\n4/parenleftbig\n−X+∂X−+X−∂X+/parenrightbig\n, (2.27b)\nNi+=1\n4/parenleftbig\n−Xi∂X++X+∂Xi/parenrightbig\n, (2.27c)\nNi−=1\n4/parenleftbigg\n−Xi∂X−+X−∂Xi+i∂Xj\n∂X+(SIγijSI)−3∂2Xi\n∂X+/parenrightbigg\n.(2.27d)\nIt can be verified that they form the correct D= 6 Lorentz algebra, except that\n[Mi−, Mj−]\n=/contintegraldisplaydw\n2πi/parenleftbiggi\n2(SIγijSI)\n(∂X+)2/parenleftbigg1\n2∂Xµ∂Xµ−1\n8SJ\nb∂Sb\nJ−3\n4∂3X+\n∂X++(∂2X+)2\n(∂X+)2/parenrightbigg\n−1\n2∂Xi∂2Xj−∂2Xi∂Xj\n(∂X+)2+i\n4(SIγij∂2SI)\n(∂X+)2\n+i\n8(SIγij∂SJ)\n(∂X+)2SJ\nbSb\nI/parenrightbigg\n. (2.28)\nAgain, the right hand side is not BRST-exact: As in the D= 4 case, the S-bilinear terms\ncan only arise from a product of cˇTand something proportional toSIγijSI\n(∂X+)2, but we have\n/bracketleftbigg\nQGS,/contintegraldisplaydz\n2πiib(SIγijSI)\n2(∂X+)2(z)/bracketrightbigg\n=/contintegraldisplaydw\n2πi/parenleftbiggi\n2(SIγijSI)\n(∂X+)2/parenleftbigg1\n2∂Xµ∂Xµ−1\n2SJ\nb∂Sb\nJ+7\n4∂3X+\n∂X+−3\n4(∂2X+)2\n(∂X+)2/parenrightbigg\n+3\n4i(SIγij∂2SI)\n(∂X+)2/parenrightbigg\n, (2.29)\nwhich does not coincide with (2 .28).\n§3. The Lorentz invariance of the lower-dimensional DS super strings\n3.1.TheD= 4DS superstring\nWe now focus on the issue of the Lorentz invariance of the lower-dim ensional DS super-\nstrings studied in Ref. 5). We first briefly review the relevant result s in theD= 4 case. The\n8Lagrangian of the D= 4 DS superstring is\nL=LK+LWZ, (3.1a)\nLK=−1\n2√−ggabΠµ\naΠµb, (3.1b)\nLWZ=ǫabΠµ\na(Wµb−ˆWµb)−ǫabWµ\naˆWµb, (3.1c)\nwith\nΠµ\na=∂aXµ−2/summationdisplay\nA=1i∂a(θAσµ˜¯θA−˜θAσµ¯θA)−2/summationdisplay\nA=1WAµ\na (3.2)\nand\nWAµ\na=iΘAσµ∂a¯ΘA−i∂aΘAσµ¯ΘA, (3.3a)\nΘA=˜θA−θA,¯ΘA=˜¯θA−¯θA. (3.3b)\nHere,˜θAand¯˜θAare the spinors newly added to the GS superstring, and if they are s et to\nzero, the Lagrangian reduces to that of the GS superstring. Follo wing Ref. 2), we impose\nthe semi-light-cone gauge condition only on the spinors with tildes and compute the Dirac\nbracket. Then, we obtain a new set of canonical variables with resp ect to the Dirac bracket,\nin terms of which the remaining holomorphic first-class constraints r ead as follows:5)\nD1=d1−i√\n2π+¯S, (3.4a)\nD2=d2−i/radicalbigg\n2\nπ+π¯S−2\nπ+S¯S∂¯θ˙2, (3.4b)\n¯D˙1=¯d˙1+i√\n2π+S, (3.4c)\n¯D˙2=¯d˙2+i/radicalbigg\n2\nπ+¯πS+2\nπ+S¯S∂θ2, (3.4d)\nT=−1\n2πµπµ\nπ+−1\n2S∂¯S\nπ++1\n2∂S¯S\nπ++i/radicalbigg\n2\nπ+(S∂¯θ˙1+∂θ1¯S)\n+i/radicalBigg\n2\n(π+)3/parenleftBig\n¯πS∂¯θ˙2+π∂θ2¯S/parenrightBig\n+4S¯S∂θ2∂¯θ˙2\n(π+)2, (3.4e)\nwhere\ndα=pα−i∂Xµ(σµ¯θ)α−1\n2/parenleftbig\n(θσµ∂¯θ)−(∂θσµ¯θ)/parenrightbig\n(σµ¯θ)α, (3.5a)\n¯d˙α= ¯p˙α−i∂Xµ(θσµ)˙α−1\n2/parenleftbig\n(θσµ∂¯θ)−(∂θσµ¯θ)/parenrightbig\n(θσµ)˙α, (3.5b)\nπµ=i∂Xµ+θσµ∂¯θ−∂θσµ¯θ. (3.5c)\n9Here, thesymbol ∂represents∂\n∂z. The quantities π±=π0±π3,π=π1+iπ2and ¯π=π1−iπ2\nare also introduced.\nThe relevant OPEs among the basic fields are all free:\nXµ(z)Xν(w)∼ηµνlog(z−w), (3.6a)\npα(z)θβ(w)∼δβ\nα\nz−w, (3.6b)\n¯p˙α(z)¯θ˙β(w)∼δ˙β\n˙α\nz−w, (3.6c)\nS(z)¯S(w)∼1\nz−w. (3.6d)\nAgain, the algebras of the constraints (3 .4) are not closed, due to the presence of multiple\ncontractions in the OPEs, and this prevents us from constructing a nilpotent BRST charge.\nTo remedy this, as in Ref. 2), we modify the constraints as\nD1→ˇD1≡D1, (3.7a)\n¯D˙1→ˇ¯D˙1≡¯D˙1, (3.7b)\nD2→ˇD2≡D2−∂2¯θ˙2\nπ++1\n2∂π+∂¯θ˙2\n(π+)2, (3.7c)\n¯D˙2→ˇ¯D˙2≡¯D˙2−∂2θ2\nπ++1\n2∂π+∂θ2\n(π+)2, (3.7d)\nT → ˇT ≡ T +∂θ2∂2¯θ˙2\n(π+)2−∂2θ2∂¯θ˙2\n(π+)2−1\n8∂2logπ+\nπ+. (3.7e)\nThe additional terms above can be viewed as arising from the normal- ordering ambiguities\nof the constraints, and the precise values of the coefficients have been determined so that\nthe algebras are closed. One can verify that these modified constr aints have the OPE\nˇD2(z)ˇ¯D˙2(w)∼4ˇT(w)\nz−w, (3.8)\nwithout higher singularities, and is regular otherwise. In this way, we obtain a set of first-\nclass constraints which can be used to construct a nilpotent BRST c harge in a conventional\nmanner as\n˜Q=/contintegraldisplaydz\n2πi/parenleftBig\nλαˇDα+¯λ˙αˇ¯D˙α+cˇT −4λ2¯λ˙2b/parenrightBig\n. (3.9)\nHere,bandcare the usual fermionic ghosts, satisfying\nb(z)c(w)∼1\nz−w, (3.10)\n10whileλαand¯λ˙αareunconstrained bosonic spinor ghosts, a part of which is identified as the\npure spinor ghosts after the similarity transformations described in the next section.∗)\nLet us now consider the Lorentz generators. All of them but Ni−are obtained by adding\ngenerators constructed from p,θ,λandω, the conjugate of λ, with\nλα(z)ωβ(w)∼δα\nβ\nz−w, (3.11a)\n¯λ˙α(z)¯ω˙β(w)∼δ˙α\n˙β\nz−w, (3.11b)\nto those of the GS superstring in the semi-light-cone gauge,\nNij=1\n4/parenleftbig\n−Xi∂Xj+Xj∂Xi+iǫijS¯S\n+iǫij(θσ3p+ ¯pσ3¯θ−λσ3ω+ ¯ωσ3¯λ)/parenrightbig\n, (3.12a)\nN+−=1\n4/parenleftbig\n−X+∂X−+X−∂X+\n+2(θσ3p−¯pσ3¯θ−λσ3ω−¯ωσ3¯λ)+4bc/parenrightbig\n, (3.12b)\nNi+=1\n4/parenleftbig\n−Xi∂X++X+∂Xi\n+2(siθ2p1+ ¯si¯θ˙2¯p˙1−siλ2ω1−¯si¯λ˙2¯ω˙1)/parenrightBig\n, (3.12c)\nwhere\nsi=/braceleftBigg\n1 (i= 1)\ni(i= 2)and ¯si=/braceleftBigg\n1 (i= 1)\n−i(i= 2). (3.13)\nThe generator N+−also contains a contribution from the bc-ghost. This is because that\nthese ghost fields are not Lorentz scalars, which can be seen from the form of the BRST\ncharge (3 .9). On the other hand, Ni−involves extra terms coming from the compensating\nκsymmetry, and also other terms for the BRST invariance. The resu lt is\nNi−=1\n4/parenleftBigg\n−Xi∂X−+X−∂Xi+2(¯siθ1p2+si¯θ˙1¯p˙2−¯siλ1ω2−si¯λ˙1¯ω˙2)\n+4πibc\nπ++2iǫijπjS¯S\nπ++4√\n2ibc(¯siS∂¯θ˙2+si∂θ2¯S)\n(π+)3\n2−3\n2∂πi\nπ+\n−2√\n2i¯si∂S∂¯θ˙2+si∂θ2∂¯S\n(π+)3\n2−√\n2i¯siS∂¯θ˙2+si∂θ2¯S\n(π+)5\n2∂π+\n+12iǫijπj∂θ2∂¯θ˙2\n(π+)2+6¯si∂θ1∂¯θ˙2−si∂θ2∂¯θ˙1\nπ+\n∗)Note that the D= 4 pure spinor condition implies λα= 0 or¯λ˙α= 0, treating them as independent\nquantities (rather than complex conjugates), as usual in the PS f ormalism.\n11+4√\n2ib(¯siS∂¯λ˙2−si∂λ2¯S)\n(π+)1\n2/parenrightBigg\n. (3.14)\nWith the exception of [ Mi−,Mj−], these generators form the correct D= 4 Lorentz algebra:\n[Mµν, Mρσ] =−1\n2(ηνρMµσ−ηµρMνσ−ηνσMµρ+ηµσMνρ), (3.15)\nMµν≡/contintegraldisplaydz\n2πiNµν(z). (3.16)\nThe commutator [ Mi−, Mj−] is given by\n[Mi−, Mj−] =/contintegraldisplaydz\n2πi/bracketleftbigg1\n2iǫijπ+π−−π¯π\n(π+)2S¯S+3\n4−πi∂πj+πj∂πi\n(π+)2\n+√\n2ǫijbc/parenleftBigg\n¯πS∂¯θ˙2−π∂θ2¯S\n(π+)5/2+S∂¯θ˙1−∂θ1¯S\n(π+)3/2/parenrightBigg\n−iǫij∂/parenleftbiggbc\nπ+/parenrightbiggS¯S\nπ++4iǫij/parenleftbigg\n−2∂(bc)\n(π+)3+bc∂π+\n(π+)4/parenrightbigg\n∂θ2∂¯θ˙2\n+√\n2ǫij/parenleftBigg\n3\n2∂¯πS∂¯θ˙2−∂π∂θ2¯S\n(π+)5/2−1\n2¯π∂S∂¯θ˙2−π∂θ2∂¯S\n(π+)5/2\n−7\n4(¯πS∂¯θ˙2−π∂θ2¯S)∂π+\n(π+)7/2−1\n2∂S∂¯θ˙1−∂θ1∂¯S\n(π+)3/2−1\n4(S∂¯θ˙1−∂θ1¯S)∂π+\n(π+)5/2/parenrightBigg\n+3iǫij/parenleftBigg\n(π+π−−2π¯π)∂θ2∂¯θ˙2\n(π+)2−¯π∂θ1∂¯θ˙2+π∂θ2∂¯θ˙1\n(π+)2−∂θ1∂¯θ˙1\nπ+/parenrightBigg\n−iǫij/parenleftBigg\n∂2∂θ2∂2∂¯θ˙2\n(π+)3+∂π+∂(∂θ2∂¯θ˙2)\n(π+)4−4(∂π+)2∂θ2∂¯θ˙2\n(π+)5/parenrightBigg\n+√\n2ǫij/parenleftBigg\nb(¯πS¯λ˙2+πλ2¯S)\n(π+)3/2+b(S¯λ˙1+λ1¯S)√\nπ+/parenrightBigg\n+2iǫij/parenleftBigg\n2∂b∂¯θ˙2λ2−∂θ2¯λ˙2\n(π+)2+b∂2¯θ˙2λ2−∂2θ2¯λ˙2\n(π+)2−b∂¯θ˙2λ2−∂θ2¯λ˙2\n(π+)3∂π+/parenrightBigg\n+2iǫij−∂(S∂¯θ˙2)∂θ2¯S+∂(∂θ2¯S)S∂¯θ˙2\n(π+)3\n+4iǫijbS¯S(λ2∂¯θ˙2+∂θ2¯λ˙2)\n(π+)2/bracketrightBigg\n. (3.17)\nWe can show that the right hand side cannot be written in a BRST exac t form as follows.\nFirst, suppose that all the terms in (3 .17) could be written in the form\n[˜Q,/contintegraldisplaydz\n2πi(parent)] (3 .18)\n12forsome (parent). Then, notethat (parent) cannot contain ωαor ¯ω˙α, because (3 .17) contains\nneitherωαand ¯ω˙αnorpαand ¯p˙α, which necessarily follows from the contraction with λαdα+\n¯λ˙α¯d˙α. In analogy to the previous section, let us focus on terms that do n ot contain any of\nλ,¯λ,∂θand∂¯θ:\n[Mi−, Mj−] =/contintegraldisplaydz\n2πi/parenleftbigg\nǫijiS¯S\nπ+/parenleftbigg1\n2πµπµ\nπ+−∂/parenleftbiggbc\nπ+/parenrightbigg/parenrightbigg\n+3\n4−πi∂πj+πj∂πi\n(π+)2/parenrightbigg\n+O(∂θ)+O(λ). (3.19)\nThis contribution could only arise from contraction with cˇT, and thus (parent) must contain\nb. Taking into account the π+dependence of (3 .19), theS¯Sterms can only arise from the\nOPE between terms of cT(≡cT0), that are independent of both ∂θand∂¯θ, andǫijS¯S\nπ+.\nHowever, we find\n/bracketleftbigg/contintegraldisplaydz\n2πicT0,/contintegraldisplaydw\n2πi/parenleftbigg\n−iǫijbS¯S\nπ+/parenrightbigg/bracketrightbigg\n=/contintegraldisplaydz\n2πi/parenleftbigg\nǫijiS¯S\nπ+/parenleftbigg1\n2πµπµ\nπ+−∂/parenleftbiggbc\nπ+/parenrightbigg/parenrightbigg\n−3\n8∂2π+\n(π+)2+15\n8(∂π+)2\n(π+)3\n−3\n4i∂2S¯S+S∂2¯S\n(π+)2/parenrightbigg\n, (3.20)\nwhich is inconsistent with (3 .19). Therefore, the commutator (3 .17) is not BRST-exact.\nThus we have shown that the D= 4 DS superstring has only partial Lorentz invariance, like\ntheD= 4GSsuperstring in the light-cone or semi-light-cone gauge.\n3.2.TheD= 6DS superstring\nThe Lagrangian of the D= 6 DS superstrings is similarly given by\nLK=−1\n2√−ggmnΠµ\nmΠµn, (3.21a)\nLWZ=ǫmnΠµ\nm(Wµn−ˆWµn)−ǫmnWµ\nmˆWµn, (3.21b)\nwhere\nΠµ\nm=∂mXµ−2/summationdisplay\nA=1i∂m(θIACγµ˜θA\nI)−2/summationdisplay\nA=1WAµ\nm, (3.22)\nWAµ\nm=i(ΘIACγµ∂mΘA\nI), (3.23)\nΘA\nI=˜θA\nI−θA\nI. (3.24)\nHere we use the same convention as in Ref. 5), except that, for lat er convenience, we put\na bar on the lower component in the light-cone decomposition of a SU(2) Majorana-Weyl\n13(MW) spinor:\nθα\nI=/parenleftBigg\nθa\nI\n¯θ˙a\nI/parenrightBigg\n,(a,˙a= 1,2) (3 .25)\nwhereaand ˙aare the spinor indices of the transverse rotation SO(4)∼SU(2)×SU(2).\nTheSU(2) MW condition is given by\n(θa\nI)∗=ǫIJθb\nJǫba≡θI\na, (3.26a)\n(¯θ˙a\nI)∗=ǫIJ¯θ˙b\nJǫ˙b˙a≡¯θI\n˙a. (3.26b)\nAfter some field redefinitions, we find that the constraint generat ors are classically given\nby\nDI\na=dI\na+√\n2π+SI\na, (3.27a)\n¯DI\n˙a=¯dI\n˙a+/radicalbigg\n2\nπ+πi(SI¯γi)˙a+2\nπ+SI\nbSb\nJ∂¯θJ\n˙a, (3.27b)\nT=−1\n2πµπµ\nπ+−1\n2SJ\na∂Sa\nJ\nπ+−/radicalbigg\n2\nπ+∂θJ\naSa\nJ\n−/radicalbigg\n2\nπ+πi(∂¯θJγiSJ)\nπ++2∂¯θI\n˙a∂¯θ˙a\nJSJ\naSa\nI\n(π+)2, (3.27c)\nwhere the super-covariant currents dI\nαandπµare defined by\ndI\nα=pI\nα+i∂Xµ(CγµθI)α+1\n2(θJCγµ∂θJ)(CγµθI)α, (3.28a)\nπµ=i∂Xµ+(θICγµ∂θI). (3.28b)\nThe redefined fields are free and satisfy the relations\nXµ(z)Xν(w)∼ηµνlog(z−w), (3.29a)\npI\nα(z)θβ\nJ(w)∼δI\nJδβ\nα\nz−w, (3.29b)\nSa\nI(z)Sb\nJ(w)∼−ǫIJǫab\nz−w. (3.29c)\nIncluding quantum corrections, we define ˇDI\na,ˇ¯DI\n˙aandˇTas\nˇDI\na=DI\na, (3.30a)\nˇ¯DI\n˙a=¯DI\n˙a−2∂2¯θI\n˙a\nπ++∂π+∂¯θI\n˙a\n(π+)2+8\n3∂¯θI\n˙b∂¯θ˙b\nJ∂¯θJ\n˙a\n(π+)2, (3.30b)\nˇT=T −1\n4∂2logπ+\nπ+−2∂2¯θJ\n˙b∂¯θ˙b\nJ\n(π+)2+8\n3∂¯θI\n˙a∂¯θ˙a\nJ∂¯θJ\n˙b∂¯θ˙b\nI\n(π+)3, (3.30c)\n14then they satisfy\nˇ¯DI\n˙a(z)ˇ¯DJ\n˙b(w)∼−4ǫIJǫ˙a˙bˇT(w)\nz−w, (3.31a)\n[all other combinations] ∼0. (3.31b)\nThe BRST charge can be straightforwardly constructed from this constraint algebra as\n˜Q=/contintegraldisplaydz\n2πi/parenleftbig\nλα\nIˇDI\nα+cˇT −2¯λI\n˙a¯λ˙a\nIb/parenrightbig\n, (3.32)\nwith the unconstrained bosonic ghost pair λα\nIandωI\nαand the fermionic ghost pair bandc,\nwith\nc(z)b(w)∼1\nz−w, (3.33a)\nλα\nI(z)ωJ\nβ(w)∼δα\nβδJ\nI\nz−w. (3.33b)\nThe BRST charge given in (3 .32) is exactly nilpotent.\nUsing the light-cone decomposition, it is convenient to use the rewrit ten forms\nπ+=i∂X++2¯θI\n˙a∂¯θ˙a\nI, (3.34a)\nπ−=i∂X−+2θI\na∂θa\nI, (3.34b)\nπi=i∂Xi+(∂¯θIγiθI)−(¯θIγi∂θI), (3.34c)\nda\nI=pa\nI−i∂X+θa\nI−i∂Xi(¯γiθI)a+∂¯θ˙b\nJ¯θJ\n˙bθa\nI−¯θ˙b\nI∂¯θJ\n˙bθa\nJ+¯θ˙b\nI¯θJ\n˙b∂θa\nJ,(3.34d)\n¯d˙a\nI= ¯p˙a\nI−i∂X−¯θ˙a\nI−i∂Xi(γiθI)˙a+∂θb\nJθJ\nb¯θ˙a\nI−θb\nI∂θJ\nb¯θ˙a\nJ+θb\nIθJ\nb∂¯θ˙a\nJ,(3.34e)\nwhere we have used the notation\nγi=iσi(i= 1,2,3), γ4= 12, (3.35a)\n¯γi=−iσi(i= 1,2,3),¯γ4= 12, (3.35b)\nwhich are 2 ×2 blocks of the gamma matrices defined in Ref. 5).∗)Their standard index\npositions are ( γi)˙a\nband (¯γi)a˙b. Using these 2 ×2 matrices, we also define\n(γij)a\nb≡ −i\n2(¯γiγj−¯γjγi)a\nb, (3.36a)\n(¯γij)˙a\n˙b≡ −i\n2(γi¯γj−γj¯γi)˙a\n˙b, (3.36b)\n(γijk)˙a\nb≡+1\n6(γi¯γjγk−γi¯γkγj+γj¯γkγi−γj¯γiγk+γk¯γiγj−γk¯γjγi)˙a\nb.(3.36c)\n∗)These are denoted by ˜ γiand˜¯γiin Ref. 5).\n15The Lorentz generators, except for Ni−, can be easily obtained as\nNij=−1\n4Xi∂Xj+1\n4Xj∂Xi+i\n4(θIγijpI)+i\n4(¯θI¯γij¯pI)\n−i\n4(λIγijωI)−i\n4(¯λI¯γij¯ωI)+i\n8(SIγijSI), (3.37a)\nN+−=−1\n4X+∂X−+1\n4X−∂X++1\n2θI\napa\nI−1\n2¯θI\n˙a¯p˙a\nI−1\n2λI\naωa\nI+1\n2¯λI\n˙a¯ω˙a\nI+bc,(3.37b)\nNi+=−1\n4Xi∂X++1\n4X+∂Xi+1\n2(¯θIγipI)−1\n2(¯λIγiωI). (3.37c)\nThe remaining generator Ni−is given by\nNi−=−1\n4Xi∂X−+1\n4X−∂Xi+1\n2(θI¯γi¯pI)−1\n2(λI¯γi¯ωI)\n+πibc\nπ++1\n4iπj(SIγijSI)\nπ+−1\n4∂πi\nπ+−√\n2bc(∂¯θIγiSI)\n(π+)3/2\n−√\n2\n3(∂¯θIγiSJ)SJ\naSa\nI\n(π+)3/2+1√\n2∂π+(∂¯θIγiSI)\n(π+)5/2−iπj(∂¯θI¯γij∂¯θI)\n(π+)2\n−(∂¯θIγi∂θI)\nπ+−8√\n2\n3(∂¯θIγiSJ)∂¯θJ\n˙a∂¯θ˙a\nI\n(π+)5/2+√\n2b(¯λIγiSI)\n(π+)1/2. (3.37d)\nThe integrated generators\nMµν=/contintegraldisplaydz\n2πiNµν(z) (3 .38)\nare BRST invariant, satisfying [ ˜Q,Mµν] = 0, and form the Lorentz algebra, except for\n[Mi−,Mj−]\n=/contintegraldisplaydz\n2πi/parenleftBigg\n−1\n2/parenleftbigg\nδikδjl−1\n2ǫijkl/parenrightbiggπk∂πl−πl∂πk\n(π+)2\n+i\n2(SIγijSI)\nπ+/parenleftbigg1\n2πµπµ\nπ++1\n8SJ\na∂Sa\nJ\nπ++1\n4∂2π+\n(π+)2−∂/parenleftbiggbc\nπ+/parenrightbigg/parenrightbigg\n−i\n4(SIγij∂2SI)\n(π+)2−i\n8(SIγij∂SJ)SJ\naSa\nI\n(π+)2\n+√\n2b/parenleftbiggπi(¯λIγjSI)\n(π+)3/2−πj(¯λIγiSI)\n(π+)3/2+πk(¯λIγijkSI)\n(π+)3/2+i(λIγijSI)\n(π+)1/2/parenrightbigg\n−√\n2bc/parenleftbiggπi(∂¯θIγjSI)\n(π+)5/2−πj(∂¯θIγiSI)\n(π+)5/2+πk(∂¯θIγijkSI)\n(π+)5/2+i(∂θIγijSI)\n(π+)3/2/parenrightbigg\n−1√\n2/parenleftBigg\n3∂πi(∂¯θIγjSI)\n(π+)5/2−3∂πj(∂¯θIγiSI)\n(π+)5/2−4∂π+πi(∂¯θγjSI)\n(π+)7/2\n+4∂π+πj(∂¯θγiSI)\n(π+)7/2−∂π+πk(∂¯θIγijkSI)\n(π+)7/2−i∂π+(∂θIγijSI)\n(π+)5/2/parenrightBigg\n16−2bi∂π+(¯λI¯γij∂¯θI)\n(π+)3+b\n3/parenleftBigg\n8(¯λIγiSI)(∂¯θJγjSJ)\n(π+)2−8(¯λIγjSI)(∂¯θJγiSJ)\n(π+)2\n−4(¯λIγiSJ)(∂¯θJγjSI)\n(π+)2+4(¯λIγjSJ)(∂¯θJγiSI)\n(π+)2−12i(∂¯λI¯γij∂¯θI)\n(π+)2+6i∂π+(¯λI¯γij∂¯θI)\n(π+)3\n−4i(¯λI¯γij∂¯θJ)SJ\naSa\nI\n(π+)2/parenrightBigg\n−√\n2\n3/parenleftBigg\nπi(∂¯θIγjSJ)SJ\naSa\nI\n(π+)5/2−πj(∂¯θIγiSJ)SJ\naSa\nI\n(π+)5/2\n+πk(∂¯θIγijkSJ)SJ\naSa\nI\n(π+)5/2+i(∂θIγijSJ)SJ\naSa\nI\n(π+)3/2/parenrightBigg\n−i(∂¯θI¯γij∂¯θI)\n(π+)2/parenleftbigg\nπ−+SJ\nb∂Sb\nJ\nπ++1\n2∂2logπ+\nπ+−4b∂c\nπ+/parenrightbigg\n+2iπiπk(∂¯θI¯γkj∂¯θI)\n(π+)3+2iπkπj(∂¯θI¯γik∂¯θI)\n(π+)3−2πi(∂¯θIγj∂θI)\n(π+)2+2πj(∂¯θIγi∂θI)\n(π+)2\n−2πk(∂¯θIγijk∂θI)\n(π+)2−i(∂θIγij∂θI)\nπ+−(∂¯θIγiSI)∂(∂¯θJγjSJ)\n(π+)3+(∂¯θIγjSI)∂(∂¯θJγiSJ)\n(π+)3\n−(∂¯θIγiSJ)∂(∂¯θJγjSI)\n(π+)3+(∂¯θIγjSJ)∂(∂¯θJγiSI)\n(π+)3\n+2i(∂¯θI¯γij∂2¯θJ)SJ\naSa\nI\n(π+)3−32b\n3i(¯λI¯γij∂¯θJ)∂¯θJ\n˙a∂¯θ˙a\nI\n(π+)3+i\n3(SIγijSI)SJ\naSa\nK∂¯θK\n˙a∂¯θ˙a\nJ\n(π+)3\n−8√\n2\n3/parenleftBigg\n2πi(∂¯θIγjSJ)∂¯θJ\n˙a∂¯θ˙a\nI\n(π+)7/2−2πj(∂¯θIγiSJ)∂¯θJ\n˙a∂¯θ˙a\nI\n(π+)7/2+πk(∂¯θIγijkSJ)∂¯θJ\n˙a∂¯θ˙a\nI\n(π+)7/2/parenrightBigg\n+2√\n2/parenleftBigg\n(∂¯θIγiSJ)(∂¯θIγj∂θJ)\n(π+)5/2−(∂¯θIγjSJ)(∂¯θIγi∂θJ)\n(π+)5/2−2i(∂θIγijSJ)∂¯θJ\n˙a∂¯θ˙a\nI\n(π+)5/2/parenrightBigg\n+i(∂¯θI¯γij∂¯θI)∂¯θJ\n˙a∂2¯θ˙a\nJ\n(π+)4−2i(∂¯θI¯γij∂2¯θJ)∂¯θJ\n˙a∂¯θ˙a\nI\n(π+)4+8\n3i(∂¯θI¯γij∂¯θI)∂¯θJ\n˙a∂¯θ˙a\nKSK\naSa\nJ\n(π+)4/parenrightBigg\n.\n(3.39)\nIn particular, we have\n[Mi−,Mj−]\n=/contintegraldisplaydz\n2πi/parenleftBigg\n−1\n2/parenleftbigg\nδikδjl−1\n2ǫijkl/parenrightbiggπk∂πl−πl∂πk\n(π+)2\n+i\n2(SIγijSI)\nπ+/parenleftbigg1\n2πµπµ\nπ++1\n8SJ\na∂Sa\nJ\nπ++1\n4∂2π+\n(π+)2−∂/parenleftbiggbc\nπ+/parenrightbigg/parenrightbigg\n−i\n4(SIγij∂2SI)\n(π+)2−i\n8(SIγij∂SJ)SJ\naSa\nI\n(π+)2/parenrightBigg\n+O(∂θ)+O(λ).(3.40)\n17On the other hand, we find\n{/contintegraldisplaydz\n2πicT0,/contintegraldisplaydz\n2πi/parenleftbigg\n−i\n2b(SIγijSI)\nπ+/parenrightbigg\n}\n=/contintegraldisplaydz\n2πi/parenleftBigg\ni\n2(SIγijSI)\nπ+/parenleftBigg\n1\n2πµπµ\nπ++1\n2SJ\na∂Sa\nJ\nπ+−1\n4∂2π+\n(π+)2+7\n4(∂π+)2\n(π+)3−∂/parenleftbiggbc\nπ+/parenrightbigg/parenrightBigg\n−3\n4i(SIγij∂2SI)\n(π+)2/parenrightBigg\n, (3.41)\nwhere, as above, T0is the∂θ-independent part of T. Then, repeating the same argument as\nin the previous section, we find that the commutator is not BRST exa ct.\n§4. Conclusions and discussion\nIn this paper, we have shown that the D= 4 and 6 double-spinor (DS) superstrings\ndo not possess the full Lorentz symmetry, as in the light-cone and semi-light-cone gauge\nquantizations of lower-dimensional Green-Schwarz superstrings .\nWe have emphasized that the modification of the energy-momentum tensor is a com-\nmon procedure employed to preserve quantum conformal invarian ce in the semi-light-cone\ngauge quantization, even in the critical case. One can rewrite the lo garithmic term of the\nenergy-momentum tensor (2 .15) or, more generally, (2 .16) in the usual linear-dilaton form\nby bosonization. Owing to the relation\n∂X+(z)X−(w)∼2\nz−w, (4.1)\nwe can identify them as a βγ-system. Therefore, we define\n∂X+(z) =γ(z) =eφ−χ(x), (4.2a)\nX−(z) = 2β(z) = 2∂χe−φ−χ(x), (4.2b)\nwhereγ(z)β(w)∼1\nz−w,φ(z)φ(w)∼ −log(z−w) andχ(z)χ(w)∼+log(z−w). Plugging\nthese into (2 .16), we obtain\nTX+X−(z) =−1\n2(∂φ)2+/parenleftbigg1\n2+ξ/parenrightbigg\n∂2φ+1\n2(∂χ)2+/parenleftbigg1\n2−ξ/parenrightbigg\n∂2χ, (4.3)\nwhereξ=7\n8(D= 4),3\n4(D= 6) and1\n2(D= 10). Therefore, the modification of the\nenergy-momentum tensor can be regarded as a change of the bac kground from flat to linear-\ndilaton, although the dilaton is only linear with respect to the special b osonized coordinates.\nThis way of viewing the modification is consistent with that in recent wo rks on the relation\n18between the lower-dimensional PS and non-critical superstrings.12)It is also interesting that\ntheχfield becomes a normal scalar in the critical ( D= 10) case. However, the meaning of\nthis observation is yet unclear.\nWe showed in Ref. 5) that the physical spectra of the D= 4 andD= 6 DS superstrings\ncoincide with those of the pure-spinor (PS) formalisms in the same nu mbers of dimensions.\nLet us now compare the Lorentz generators given in Refs. 6) and 7 ) and ours obtained in\nthe DS formalism. In four dimensions, the necessary similarity trans formations relating the\nBRST charges of the two D= 4 theories are5)\nX=−1\n4/contintegraldisplaydz\n2πicˇ¯D˙2\n˜λ2, (4.4)\nY=−1\n2/contintegraldisplaydz\n2πiS¯Slogπ+, (4.5)\nZ=/contintegraldisplaydz\n2πi/parenleftBigg\ni√\n2¯d˙1¯S+∂θ2∂¯θ˙2\nπ+/parenrightBigg\n. (4.6)\nThen, the BRST charge ˜Qis transformed to\n(eZeYeX)˜Q(eZeYeX)−1=Q+δb+δ, (4.7)\nQ=/contintegraldisplaydz\n2πiλαdα, (4.8)\nδb=−4/contintegraldisplaydz\n2πiλ2¯λ˙2b, (4.9)\nδ=√\n2i/contintegraldisplaydz\n2πi¯λ˙1S, (4.10)\nwhereδbandδanti-commute with Qand have trivial cohomologies of the BRST quartets\n(b,c;¯λ˙2,¯ω˙2) and (S,¯S;¯λ˙1,¯ω˙1) (where ¯ ω˙αis the field conjugate to ¯λ˙α). One can alternatively\ndecouple λαinstead of ¯λ˙α. Taking the quotients with respect to the Hilbert space of these\nBRST trivial fields leaves precisely the D= 4 PS Hilbert space with the BRST charge Q\nproposed in Refs. 6) and 7).\nTheD= 4 PS superstring has an anomaly-free set of level-1 Lorentz curr ents. If they are\nsimilarity-transformed back to the DS theory by using the above X,YandZ, they do not\ncoincide with the Lorentz generators we considered in the previous section. This is obvious,\nbecause the Lorentz generators in the PS formalism do not act on t he BRST-quartet fields\ndecoupled through the similarity transformations. This can also be v erified by an explicit\ncalculation. Thus, we conclude that, although the generators of t he PS formalism realize a\nrepresentation of the D= 4 Lorentz group on the PS fields, they are not directly related to\nthe symmetries of the DS Lagrangian. A similar statement holds in the D= 6 case.\n19Acknowledgements\nTheworkofHKissupportedinpartbyaGrant-in-AidforScientificRe search(No.19540284)\nand a Grant-in-Aid for the 21st Century COE “Center for Diversity and Universality in\nPhysics”, while the work of SM is supported by a Grant-in-Aid for Scie ntific Research\n(No.16540273) from the Ministry of Education, Culture, Sports, S cience and Technology\n(MEXT) of Japan.\nReferences\n1) N. Berkovits, J. High Energy Phys. 04(2000), 018; hep-th/0001035.\n2) Y. Aisaka and Y. Kazama, J. 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High Energy\nPhys.0703(2007), 091; hep-th/0605118.\n20"    },    {        "title": "2309.15171v2.Qualitative_properties_of_solutions_to_a_nonlinear_transmission_problem_for_an_elastic_Bresse_beam.pdf",        "content": "Qualitative properties of solutions to a nonlinear\ntransmission problem for an elastic Bresse beam\nTamara Fastovska1,2,∗, Dirk Langemann3and Iryna Ryzhkova1\nJanuary 23, 2024\n1Department of Mathematics and Computer Science, V.N. Karazin\nKharkiv National University, Kharkiv, Ukraine\n2Institut f¨ ur Mathematik, Humboldt-Universit¨ at zu Berlin, Berlin, Germany\n3Institut f¨ ur Partielle Differentialgleichungen,Technische Universit¨ at Braun-\nschweig, Braunschweig, Germany\nAbstract\n1\nWe consider a nonlinear transmission problem for a Bresse beam, which\nconsists of two parts, damped and undamped. The mechanical damp-\ning in the damped part is present in the shear angle equation only,\nand the damped part may be of arbitrary positive length. We prove\nwell-posedness of the corresponding PDE system in energy space and\nestablish existence of a regular global attractor under certain conditions\non nonlinearities and coefficients of the damped part only. Moreover, we\nstudy singular limits of the problem when l→0 orl→0 simultaneously\nwith ki→+∞and perform numerical modelling for these processes.\n2 Keywords:\nBresse beam, transmission problem, global attractor, singular limit\n1arXiv:2309.15171v2  [math.AP]  22 Jan 20243 Introduction\nIn this paper we consider a contact problem for the Bresse beam. Originally\nthe mathematical model for homogeneous Bresse beams was derived in [ 4].\nWe use the variant of the model described in [ 20, Ch. 3]. Let the whole beam\noccupies a part of a circle of length Land has the curvature l=R−1. We\nconsider the beam as a one-dimensional object and measure the coordinate\nxalong the beam. Thus, we say that the coordinate xchanges within the\ninterval (0 , L). The parts of the beam occupying the intervals (0 , L0) and\n(L0, L) consist of different materials. The part lying in the interval (0 , L0) is\npartially subjected to a structural damping (see Figure 1). The Bresse system\nFigure 1: Composite Bresse beam.\ndescribes evolution of three quantities: transversal displacement, longitudinal\ndisplacement and shear angle variation. We denote by φ,ψ, and ωthe\ntransversal displacement, the shear angle variation, and the longitudinal\ndisplacement of the left part of the beam lying in (0 , L0). Analogously, we\ndenote by u,v, and wthe transversal displacement, the shear angle variation,\nand the longitudinal displacement of the right part of the beam occupying\nthe interval ( L0, L). We assume the presence of the mechanical dissipation in\nthe equation for the shear angle variation for the left part of the beam. We\nalso assume that both ends of the beam are clamped. Nonlinear oscillations\n2of the composite beam can be described by the following system of equations\nρ1φtt−k1(φx+ψ+lω)x−lσ1(ωx−lφ) +f1(φ, ψ, ω ) =p1(x, t), (1)\nβ1ψtt−λ1ψxx+k1(φx+ψ+lω) +γ(ψt) +h1(φ, ψ, ω ) =r1(x, t), x∈(0, L0), t >0,\n(2)\nρ1ωtt−σ1(ωx−lφ)x+lk1(φx+ψ+lω) +g1(φ, ψ, ω ) =q1(x, t), (3)\nand\nρ2utt−k2(ux+v+lw)x−lσ2(wx−lu) +f2(u, v, w ) =p2(x, t), (4)\nβ2vtt−λ2vxx+k2(ux+v+lw) +h2(u, v, w ) =r2(x, t), x ∈(L0, L), t >0,\n(5)\nρ2wtt−σ2(wx−lu)x+lk2(ux+v+lw) +g2(u, v, w ) =q2(x, t), (6)\nwhere ρj, βj, kj, σj, λjare positive parameters, fj, gj, hj:R3→Rare\nnonlinear feedbacks, pj, qj, rj: (0, L)×R3→Rare known external loads,\nγ:R→Ris a nonlinear damping. The system is subjected to the Dirichlet\nboundary conditions\nφ(0, t) =u(L, t) = 0 , ψ(0, t) =v(L, t) = 0 , ω(0, t) =w(L, t) = 0 ,(7)\nthe transmission conditions\nφ(L0, t) =u(L0, t), ψ(L0, t) =v(L0, t), ω(L0, t) =w(L0, t),(8)\nk1(φx+ψ+lω)(L0, t) =k2(ux+v+lw)(L0, t), (9)\nλ1ψx(L0, t) =λ2vx(L0, t), (10)\nσ1(ωx−lφ)(L0, t) =σ2(wx−lu)(L0, t), (11)\nand supplemented with the initial conditions\nφ(x,0) = φ0(x), ψ(x,0) = ψ0(x), ω(x,0) = ω0(x), (12)\nφt(x,0) = φ1(x), ψ t(x,0) = ψ1(x), ω t(x,0) = ω1(x), (13)\n3u(x,0) = u0(x), v(x,0) = v0(x), w (x,0) = w0(x), (14)\nut(x,0) = u1(x), v t(x,0) = v1(x), w t(x,0) = w1(x). (15)\nOne can observe patterns in the problem which appear to have physical\nmeaning:\nQi(ξ, ζ, η ) =ki(ξx+ζ+lη) are shear forces ,\nNi(ξ, ζ, η ) =σi(ηx−lξ) are axial forces ,\nMi(ξ, ζ, η ) =λiζxare bending moments\nfor damped ( i= 1) and undamped ( i= 2) parts respectively. Later we will\nuse them to rewrite the problem in a compact and physically natural form.\nThe paper is devoted to the well-posedness and long-time behaviour of\nthe system (1)-(15). Our main goal is to establish conditions under which\nthe assumed amount of dissipation is sufficient to guarantee the existence of\na global attractor.\nThe paper is organized as follows. In Section 2 we represent functional\nspaces and pose the problem in an abstract form. In Section 3 we prove\nthat the problem is well-posed and possesses strong solutions provided\nnonlinearities and initial data are smooth enough. Section 4 is devoted\nto the main result on the existence of a compact attractor. The nature\nof dissipation prevents us from proving dissipativity explicitly, thus we\nshow that the corresponding dynamical system is of gradient structure and\nasymptotically smooth. We establish the unique continuation property by\nmeans of the observability inequality obtained in [ 27] to prove the gradient\nproperty. The compensated compactness approach is used to prove the\nasymptotic smoothness. In Section 5 we show that solutions to (1)-(15)tend\nto solutions to a transmission problem for the Timoshenko beam when l→0\nand to solutions to a transmission problem for the Euler-Bernoulli beam\nwhen l→0 and ki→ ∞ as well as perform numerical modelling of these\nsingular limits.\n44 Preliminaries and Abstract formulation\n4.1 Spaces and notations\nLet us denote\nΦ1= (φ, ψ, ω ),Φ2= (u, v, w ),Φ = (Φ1,Φ2).\nThus, Φ is a six-dimensional vector of functions. Analogously,\nFj= (fj, gj, hj) :R3→R3, F = (F1, F2),\nPj= (pj, qj, rj) : [(0 , L)×R+]3→R3, P = (P1, P2),\nRj=diag{ρj, βj, ρj}, R =diag{ρ1, β1, ρ1, ρ2, β2, ρ2},\nΓ(Φ t) = (0 , γ(ψt),0,0,0,0),\nwhere j= 1,2. The static linear part of the equation system can be formally\nrewritten as\nAΦ =\n−∂xQ1(Φ1)−lN1(Φ1)\n−∂xM1(Φ1) +Q1(Φ1)\n−∂xN1(Φ1) +lQ1(Φ1)\n−∂xQ2(Φ2)−lN2(Φ2)\n−∂xM2(Φ2) +Q2(Φ2)\n−∂xN2(Φ2) +lQ2(Φ2)\n. (16)\nThen transmission conditions (8)-(11) can be written as follows\nΦ1(L0, t) = Φ2(L0, t),\nQ1(Φ1(L0, t)) =Q2(Φ2(L0, t)),\nM1(Φ1(L0, t)) =M2(Φ2(L0, t)),\nN1(Φ1(L0, t)) =N2(Φ2(L0, t)).\nThroughout the paper we use the notation ||·||for the L2-norm of a function\nand (·,·) for the L2-inner product. In these notations we skip the domain,\non which functions are defined. We adopt the notation || · || L2(Ω)only when\n5domain is not evident. We also use the same notations || · || and (·,·) for\n[L2(Ω)]3.\nTo write our problem in an abstract form we introduce the following spaces.\nFor the velocities of the displacements we use the space\nHv={Φ = (Φ1,Φ2) : Φ1∈[L2(0, L0)]3,Φ2∈[L2(L0, L)]3}\nwith the norm\n||Φ||2\nHv=||Φ||2\nv=2X\nj=1||p\nRjΦj||2,\nwhich is equivalent to the standard L2-norm.\nFor the beam displacements we use the space\nHd={Φ∈Hv: Φ1∈[H1(0, L0)]3,Φ2∈[H1(L0, L)]3,\nΦ1(0, t) = Φ2(L, t) = 0 ,Φ1(L0, t) = Φ2(L0, t)\t\nwith the norm\n||Φ||2\nHd=||Φ||2\nd=2X\nj=1\u0000\n||Qj(Φj)||2+||Nj(Φj)||2+||Mj(Φj)||2\u0001\n.\nThis norm is equivalent to the standard H1-norm. Moreover, the equivalence\nconstants can be chosen independent of lforlsmall enough (see [ 24], Remark\n2.1). If we set\nΨ(x) =(\nΦ1(x), x ∈(0, L0),\nΦ2(x), x ∈[L0, L)\nwe see that there is isomorphism between Hdand [ H1\n0(0, L)]3.\n4.2 Abstract formulation\nThe operator A:D(A)⊂Hv→Hvis defined by formula (16), where\n6D(A) ={Φ∈Hd: Φ1∈H2(0, L0),Φ2∈H2(L0, L), Q1(Φ1(L0, t)) =Q2(Φ2(L0, t)),\nN1(Φ1(L0, t)) =N2(Φ2(L0, t)), M1(Φ1(L0, t)) =M2(Φ2(L0, t))}\nArguing analogously to Lemmas 1.1-1.3 from [ 23] one can prove the following\nlemma.\nLemma 4.1. The operator Ais positive and self-adjoint. Moreover,\n(A1/2Φ, A1/2B) =1\nk1(Q1(Φ1), Q1(B1)) +1\nσ1(N1(Φ1), N1(B1)) +1\nλ1(M1(Φ1), M1(B1))+\n1\nk2(Q2(Φ2), Q2(B2)) +1\nσ2(N2(Φ2), N2(B2)) +1\nλ2(M2(Φ2), M2(B2))\n(17)\nandD(A1/2) =Hd⊂Hv.\nThus, we can rewrite equations (1)-(6) in the form\nRΦtt+AΦ + Γ(Φ t) +F(Φ) = P(x, t), (18)\nboundary conditions (7) in the form\nΦ1(0, t) = Φ2(L, t) = 0 , (19)\nand transmission conditions (8)-(11) can be written as\nΦ1(L0, t) = Φ2(L0, t), (20)\nQ1(Φ1(L0, t)) =Q2(Φ2(L0, t)), (21)\nM1(Φ1(L0, t)) =M2(Φ2(L0, t)), (22)\nN1(Φ1(L0, t)) =N2(Φ2(L0, t)). (23)\nInitial conditions have the form\nΦ(x,0) = Φ 0(x), Φt(x,0) = Φ 1(x). (24)\nWe use H=Hd×Hvas a phase space.\n75 Well-posedness\nIn this section we study strong, generalized and variational (weak) solutions\nto (18)-(24).\nDefinition 5.1. Φ∈C(0, T;Hd)TC1(0, T;Hv)such that Φ(x,0) = Φ 0(x),\nΦt(x,0) = Φ 1(x)is said to be a strong solution to (18)-(24) if\n•Φ(t)lies in D(A)for almost all t;\n•Φ(t)is an absolutely continuous function with values in HdandΦt∈\nL1(a, b;Hd)for0< a < b < T ;\n•Φt(t)is an absolutely continuous function with values in HvandΦtt∈\nL1(a, b;Hv)for0< a < b < T ;\n•equation (18) is satisfied for almost all t.\nDefinition 5.2. Φ∈C(0, T;Hd)TC1(0, T;Hv)such that Φ(x,0) = Φ 0(x),\nΦt(x,0) = Φ 1(x)is said to be a generalized solution to (18)-(24) if there\nexists a sequence of strong solutions Φ(n)to(18)-(24) with the initial data\n(Φ(n)\n0,Φ(n)\n1)and right hand side P(n)(x, t)such that\nlim\nn→∞max\nt∈[0,T]\u0010\n||Φ(n)(·, t)−Φ(·, t)||d+||Φ(n)\nt(·, t)−Φt(·, t)||v\u0011\n= 0.\nWe also need a definition of a variational solution. We use six-dimensional\nvector-functions B= (B1, B2),Bj= (βj, γj, δj) from the space\nFT={B∈L2(0, T;Hd), Bt∈L2(0, T;Hv), B(T) = 0}\nas test functions.\nDefinition 5.3. Φis said to be a variational (weak) solution to (18)-(24) if\n•Φ∈L∞(0, T;Hd),Φt∈L∞(0, T;Hv);\n•Φsatisfies the following variational equality for all B∈FT\n8−TZ\n0(RΦt, Bt)(t)dt−(RΦ1, B(0)) +ZT\n0(A1/2Φ, A1/2B)(t)dt+\nZT\n0(Γ(Φ t), B)(t)dt+ZT\n0(F(Φ), B)(t)dt−ZT\n0(P, B)(t)dt= 0; (25)\n•Φ(x,0) = Φ 0(x).\nNow we state a well-posedness result for problem (18)-(24).\nTheorem 5.4 (Well-posedness) .Let\nfi, gi, hi:R3→Rare locally Lipschitz i.e.\n|fi(a)−fi(b)| ≤L(K)|a−b|,provided |a|,|b| ≤K; (N1)\nthere exist Fi:R3→Rsuch that (fi, hi, gi) =∇Fi;\nthere exists δ >0such that Fj(a)≥ −δfor all a∈R3; (N2)\nP∈L2(0, T;Hv); (R1)\nand the nonlinear dissipation satisfies\nγ∈C(R)and non-decreasing , γ(0) = 0 . (D1)\nThen for every initial data Φ0∈Hd,Φ1∈Hvand time interval [0, T]there\nexists a unique generalized solution to (18)-(24) with the following properties:\n•every generalized solution is variational;\n•energy inequality\nE(T) +ZT\n0(γ(ψt), ψt)dt≤ E(0) +ZT\n0(P(t),Φt(t))dt (26)\n9holds, where\nE(t) =1\n2h\n||R1/2Φt(t)||2+||A1/2Φ(t)||2i\n+LZ\n0F(Φ(x, t))dx\nand\nF(Φ(x, t)) =(\nF1(φ(x, t), ψ(x, t), ω(x, t)), x ∈(0, L0),\nF2(u(x, t), v(x, t), w(x, t)), x ∈(L0, L).\n•If, additionally, Φ0∈D(A),Φ1∈Hdand\n∂tP(x, t)∈L2(0, T;Hv) (R2)\nthen the generalized solution is also strong and satisfies the energy\nequality.\nProof. The proof essentially uses monotone operators theory. It is rather\nstandard by now (see, e.g., [ 9]), so in some parts we give only references to\ncorresponding arguments. However, we give some details which demonstrate\nthe peculiarity of 1D problems.\nStep 1. Abstract formulation. We need to reformulate problem (18)-(24)as\na first order problem. Let us denote\nU= (Φ,Φt), U 0= (Φ 0,Φ1)∈H=Hd×Hv,\nTU= \nI0\n0R−1! \n0−I\nA0!\nU+ \n0\nΓ(Φ t)!\n.\nConsequently, D(T) =D(A)×Hd⊂H. In what follows we use the notations\nB(U) = \nI0\n0R−1! \n0\nF(Φ)!\n,P(x, t) = \n0\nP(x, t)!\n.\n10Thus, we can rewrite problem (18)-(24) in the form\nUt+TU+B(U) =P, U (0) = U0∈H. (27)\nStep 2. Existence and uniqueness of a local solution. Here we use Theorem 7.2\nfrom [9]. For the reader’s convenience we formulate it below.\nTheorem 5.5 ([9]).Consider the initial value problem\nUt+TU+B(U) =f, U (0) = U0∈H. (28)\nSuppose that T:D(T)⊂H→His a maximal monotone mapping, 0∈ T0\nandB:H→His locally Lipschitz, i.e. there exits L(K)>0such that\n||B(U)−B(V)||H≤L(K)||U−V||H,||U||H,||V||H≤K.\nIfU0∈D(T),f∈W1\n1(0, t;H)for all t >0, then there exists tmax≤ ∞ such\nthat(28) has a unique strong solution Uon(0, tmax).\nIfU0∈D(T),f∈L1(0, t;H)for all t >0, then there exists tmax≤ ∞ such\nthat(28) has a unique generaized solution Uon(0, tmax).\nIn both cases\nlim\nt→tmax||U(t)||H=∞provided tmax<∞.\nFirst, we need to check that Tis a maximal monotone operator. Mono-\ntonicity is a direct consequence of Lemma 4.1 and (D1).\nTo prove Tis maximal as an operator from HtoH, we use Theorem 1.2\nfrom [ 3, Ch. 2]. Thus, we need to prove that Range (I+T) =H. Let\nz= (Φ z,Ψz)∈Hd×Hv. We need to find y= (Φ y,Ψy)∈D(A)×Hd=D(T)\nsuch that\n−Ψy+ Φ y= Φ z,\nAΦy+ Ψ y+ Γ(Ψ y) = Ψ z,\n11or, equivalently, find Ψ y∈Hdsuch that\nM(Ψy) =1\n2AΨy+1\n2AΨy+ Ψ y+ Γ(Ψ y) = Ψ z−AΦz= Θ z\nfor an arbitrary Θ z∈H′\nd=D(A1/2)′. Naturally, due to Lemma 4.1 Ais a\nduality map between HdandH′\nd, thus the operator Mis onto if and only\nif1\n2AΨy+ Ψ y+ Γ(Ψ y) is maximal monotone as an operator from Hdto\nH′\nd. According to Corollary 1.1 from [ 3, Ch. 2], this operator is maximal\nmonotone if1\n2Ais maximal monotone (it follows from Lemma 4.1) and\nI+ Γ(·) is monotone, bounded and hemicontinuous from HdtoH′\nd. The last\nstatement is evident for the identity map, now let’s prove it for Γ.\nMonotonicity is evident. Due to the continuity of the embedding H1(0, L0)⊂\nC(0, L0) in 1D every bounded set XinH1(0, L0) is bounded in C(0, L0) and\nthus due to (D1) Γ(X) is bounded in C(0, L0) and, consequently, in L2(0, L0).\nTo prove hemicontinuity we take an arbitrary Φ = ( φ, ψ, ω, u, v, w )∈Hd, an\narbitrary Θ = ( θ1, θ2, θ3, θ4, θ5, θ6)∈Hdand consider\n(Γ(Ψ y+tΦ),Θ) =ZL0\n0γ(ψy(x) +tψ(x))θ2(x)dx,\nwhere Ψ y= (φy, ψy, ωy, uy, vy, wy). Since ψy+tψ→ψy,ast→0 inH1(0, L0)\nand in C(0, L0), we obtain that γ(ψy(x) +tϕ(x))→γ(ψy(x)) as t→0 for\nevery x∈[0, L0], and has an integrable bound from above due to (D1).\nThis implies γ(ψy(x) +tϕ(x))→γ(ψy(x)) in L1(0, L0) ast→0 . Since\nθ2∈H1(0, L0)⊂L∞(0, L0), then\n(Γ(Ψ y+tΦ),Θ)→(Γ(Ψ y),Θ), t→0.\nHemicontinuity is proved.\nFurther, we need to prove that Bis locally Lipschitz on H, that is, Fis\nlocally Lipschitz from HdtoHv. The embedding H1/2+ε(0, L)⊂C(0, L)\nand (N1) imply\n|Fj(eΦj(x))−Fj(bΦj(x))| ≤C(max(||eΦ||d,||bΦ||d))||eΦj−bΦj||1 (29)\n12for all x∈[0, L0], ifj= 1 and for all x∈[L0, L], ifj= 2. This, in turn,\ngives us the estimate\n||F(eΦ)−F(bΦ)||v≤C(max(||eΦ||d,||bΦ||d))||eΦ−bΦ||d.\nThus, all the assumptions of Theorem 5.5 are satisfied and existence of a\nlocal strong/generalized solution is proved.\nStep 3. Energy inequality and global solutions. It can be verified by direct\ncalculations, that strong solutions satisfy energy equality. Using the same\narguments, as in proof of Proposition 1.3 [11], and (D1) we can pass to the\nlimit and prove (26) for generalized solutions.\nLet us assume that a local generalised solution exists on a maximal interval\n(0, tmax),tmax<∞. Then (26) implies E(tmax)≤ E(0). Since due to (N2)\nc1||U(t)||H≤ E(t)≤c2||U(t)||H,\nwe have ||U(tmax)||H≤C||U0||H. Thus, we arrive to a contradiction which\nimplies tmax=∞.\nStep 4. Generalized solution is variational (weak). We formulate the\nfollowing obvious estimate as a lemma for future use.\nLemma 5.6. Let(N1) holds and eΦ,bΦare two weak solutions to (18)-\n(24) with the initial conditions (eΦ0,eΦ1)and(bΦ0,bΦ1)respectively. Then the\nfollowing estimate is valid for all x∈[0, L], t > 0andϵ∈[0,1/2):\n|Fj(eΦj(x, t))−Fj(bΦj(x, t))| ≤C(max(||(eΦ0,eΦ1)||H,||(bΦ0,bΦ1)||H))||eΦj(·, t)−bΦj(·, t)||1−ϵ, j = 1,2.\nProof. The energy inequality and the embedding H1/2+ε(0, L)⊂C(0, L)\nimply that for every weak solution Φ\nmax\nt∈[0,T],x∈[0,L]|Φ(x, t)| ≤C(||Φ0||d,||Φ1||v).\nThus, using (N1) and (29), we prove the lemma.\nEvidently, (25)is valid for strong solutions. We can find a sequence of\n13strong solutions Φ(n), which converges to a generalized solution Φ strongly\ninC(0, T;Hd), and Φ(n)\ntconverges to Φ tstrongly in C(0, T;Hv). Using\nLemma 5.6, we can easily pass to the limit in nonlinear feedback term in\n(25). Since the test function B∈L∞(0, T;Hd)⊂L∞((0, T)×(0, L)), we\ncan use the same arguments as in the proof of Proposition 1.6 [ 11] to pass\nto the limit in the nonlinear dissipation term. Namely, we can extract from\nΦ(n)\nta subsequence that converges to Φ talmost everywhere and prove that\nit converges to Φ tstrongly in L1((0, T)×(0, L)).\nRemark 1. In space dimension greater then one we do not have the embed-\nding H1(Ω)⊂C(Ω), therefore, we need to assume polynomial growth of the\nderivative of the nonlinearity to obtain estimates similar to Lemma 5.6.\n6 Existence of attractors.\nIn this section we study long-time behaviour of solutions to problem (18)-(24)\nin the framework of dynamical systems theory. From Theorem 5.4 we have\nCorollary 1. Let, additionally to conditions of Theorem 5.4, P(x, t) =P(x).\nThen (18)-(24) generates a dynamical system (H, S t)by the formula\nSt(Φ0,Φ1) = (Φ( t),Φt(t)),\nwhere Φ(t)is the weak solution to (18)-(24) with initial data (Φ0,Φ1).\nTo establish the existence of the attractor for this dynamical system we\nuse Theorem 6.8 below, thus we need to prove the gradientness and the\nasymptotic smoothness as well as the boundedness of the set of stationary\npoints.\n6.1 Gradient structure\nIn this subsection we prove that the dynamical system generated by (18)-(24)\npossesses a specific structure, namely, is gradient under some additional\nconditions on the nonlinearities.\n14Definition 6.1 ([8,10,12]).LetY⊆Xbe a positively invariant set of\n(X, S t).\n•a continuous functional L(y)defined on Yis said to be a Lyapunov\nfunction of the dynamical system (X, S t)on the set Y, if a function\nt7→L(Sty)is non-increasing for any y∈Y.\n•the Lyapunov function L(y)is said to be strict onY, if the equality\nL(Sty) =L(y) for all t >0implies Sty=yfor all t >0;\n•A dynamical system (X, S t)is said to be gradient , if it possesses a\nstrict Lyapunov function on the whole phase space X.\nThe following result holds true.\nTheorem 6.2. Let, additionally to the assumptions of Corollary 1, the\nfollowing conditions hold\nf1=g1= 0, h 1(φ, ψ, ω ) =h1(ψ), (N3)\nf2, g2, h2∈C1(R3), (N4)\nγ(s)s >0for all s̸= 0. (D2)\nThen the dynamical system (H, S t)is gradient.\nProof. We use as a Lyapunov function\nL(Φ(t)) =L(t) =1\n2\u0010\n||R1/2Φt(t)||2+||A1/2Φ(t)||2\u0011\n+LZ\n0F(Φ(x, t))dx+(P,Φ(t)).\n(30)\nEnergy inequality (26) implies that L(t) is non-increasing. The equality\nL(t) =L(0) together with (D2) imply that ψt(t)≡0 on [0 , T]. We need\nto prove that Φ( t)≡const , which is equivalent to Φ( t+h)−Φ(t) = 0 for\nevery h >0. In what follows we use the notation Φ( t+h)−Φ(t) =Φ(t) =\n(φ,ψ,ω,u,v,w)(t) .\nStep 1. Let us prove that Φ1≡0. In this step we use the distribution theory\n(see, e.g., [ 5]) because some functions involved in computations are of too low\n15smoothness. Let us set the test function B= (B1,0) = ( β1, γ1, δ1,0,0,0).\nThen Φ(t) satisfies\n−TZ\n0(R1Φ1\nt, Bt)(t)dt−(R1(Φ1\nt(h)−Φ1\n1), B1(0))+\nTZ\n0\u00141\nk1(Q1(Φ1), Q1(B1))(t)dt+1\nσ1(N1(Φ1), N1(B1))(t)\u0015\n+\nTZ\n0(h1(ψ(t+h))−h1(ψ(t)), γ1(t))dt= 0.\nThe last term equals to zero due to (N3) and ψ(t)≡const .\nSetting in turn B= (0, γ1,0,0,0,0),B= (0,0, δ1,0,0,0),B= (β1,0,0,0,0,0)\nwe obtain\nφx+lω= 0 almost everywhere on (0 , L0)×(0, T),\n(31)\nρ1ωtt−lσ1(ωx−lφ)x= 0 almost everywhere on (0 , L0)×(0, T),\n(32)\nρ1φtt−σ1(ωx−lφ) = 0 in the sense of distributions on (0 , L0)×(0, T).\n(33)\nThese equalities imply\nφttx= 0,ωtt= 0 in the sense of distributions . (34)\nSimilar to regular functions, if partial derivative of a distribution equals to\nzero, then the distribution ”does not depends” on the corresponding variable\n(see [5, Ch. 7], Example 2.) That is,\nωt=c1(x)×1(t) in the sense of distributions\n16However, Theorem 5.4 implies that ωtis a regular distribution, thus, we can\ntreat the equality above as the equality almost everywhere. Furthermore,\nω(x, t) =ω(x,0) +Zt\n0c1(x)dτ=ω(x,0) +tc1(x).\nSince||ω(·, t)|| ≤Cfor all t∈R+,c1(x) must be zero. Thus,\nω(x, t) =c2(x), (35)\nwhich together with (31) implies\nφx=−lc2(x),\nφ(x, t) =φ(0, t)−lxZ\n0c2(y)dy=c3(x),\nφtt= 0.\nThe last equality together with (31),(33)and boundary conditions (19)give\nus that φ,ωare solutions to the following Cauchy problem (with respect to\nx):\nωx=lφ,\nφx=−lω,\nω(0, t) =φ(0, t) = 0 .\nConsequently, ω≡φ≡0.\nStep 2. Let us prove, that u≡v≡w≡0. Due to (N4), we can use the\nTaylor expansion of the difference F2(Φ2(t+h))−F2(Φ2(t)) and thus ( u,v,w)\n17satisfies on (0 , T)×(L0, L)\nρ2utt−k2uxx+gu(∂xΦ2,Φ2) +∇f2(ζ1,h(x, t))·Φ2= 0, (36)\nβ2vtt−λ2vxx+gv(∂xΦ2,Φ2) +∇h2(ζ2,h(x, t))·Φ2= 0, (37)\nρ2wtt−σ2wxx+gw(∂xΦ2,Φ2) +∇g2(ζ3,h(x, t))·Φ2= 0 (38)\nu(L0, t) =v(L0, t) =w(L0, t) = 0 , (39)\nu(L, t) =v(L, t) =w(L, t) = 0 , (40)\nk2(ux+v+lw)(L0, t) = 0 , (41)\nvx(L0, t) = 0 , σ 2(wx−lu)(L0, t) = 0 , (42)\nΦ2(x,0) = Φ2(x, h)−Φ2\n0,Φ2\nt(x,0) = Φ2\nt(x, h)−Φ2\n1, (43)\nwhere gu, gv, gware linear combinations of ux, vx, wx, u, v, w with the constant\ncoefficients, ζj,h(x, t) are 3D vector functions which components lie between\nu(x, t+h) and u(x, t),v(x, t+h) and v(x, t),w(x, t+h) and w(x, t) respectively.\nThus, we have a system of linear equations on ( L0, L) with overdetermined\nboundary conditions. L2-regularity of ux, vx, wxon the boundary for solutions\nto a linear wave equation was established in [ 21], thus, boundary conditions\n(41)-(42) make sense.\nIt is easy to generalize the observability inequality [ 27, Th. 8.1] for the case\nof the system of the wave equations.\nTheorem 6.3 ([27] ).For the solution to problem (36)-(43) the following\nestimate holds:\nZT\n0[|ux|2+|vx|2+|wx|2](L0, t)dt≥C(E(0) + E(T)),\nwhere\nE(t) =1\n2\u0000\n||ut(t)||2+||vt(t)||2+||wt(t)||2+||ux(t)||2+||vx(t)||2+||wx(t)||2\u0001\n.\nTherefore, if conditions (41),(42)hold true, then u=v=w= 0. The\ntheorem is proved.\n186.2 Asymptotic smoothness.\nDefinition 6.4 ([8,10,12]).A dynamical system (X, S t)is said to be\nasymptotically smooth if for any closed bounded set B⊂Xthat is positively\ninvariant ( StB⊆B) one can find a compact set K=K(B)which uniformly\nattracts B, i. e. sup{distX(Sty,K) :y∈B} →0ast→ ∞ .\nIn order to prove the asymptotical smoothness of the system considered\nwe rely on the compactness criterion due to [ 17], which is recalled below in\nan abstract version formulated in [12].\nTheorem 6.5. [12] Let (St, H)be a dynamical system on a complete metric\nspace Hendowed with a metric d. Assume that for any bounded positively\ninvariant set BinHand for any ε >0there exists T=T(ε, B)such that\nd(STy1, STy2)≤ε+ Ψ ε,B,T(y1, y2), yi∈B, (44)\nwhere Ψε,B,T(y1, y2)is a function defined on B×Bsuch that\nlim inf\nm→∞lim inf\nn→∞Ψε,B,T(yn, ym) = 0\nfor every sequence yn∈B. Then (St, H)is an asymptotically smooth\ndynamical system.\nTo formulate the result on the asymptotic smoothness of the system\nconsidered we need the following lemma.\nLemma 6.6. Let assumptions (D1) hold. Let moreover, there exists a\npositive constant Msuch that\nγ(s1)−γ(s2)\ns1−s2≤M, s 1, s2∈R, s1̸=s2. (D3)\nThen for any ε >0there exists Cε>0such that\n\f\f\f\f\f\fL0Z\n0(γ(ξ1)−γ(ξ2))ζdx\f\f\f\f\f\f≤ε∥ζ∥2+CεL0Z\n0(γ(ξ1)−γ(ξ2))(ξ1−ξ2)dx (45)\n19for any ξ1, ξ2, ζ∈L2(0, L0).\nThe proof is similar to that given in [12, Th.5.5]).\nTheorem 6.7. Let assumptions of Theorem 5.4, (D3), and\nm≤γ(s1)−γ(s2)\ns1−s2, s 1, s2∈R, s1̸=s2. (D4)\nwithm > 0hold. Let, moreover,\nk1=σ1 (46)\nρ1\nk1=β1\nλ1. (47)\nThen the dynamical system (H, S t)generated by problem (1)–(11) is asymp-\ntotically smooth.\nProof. In this proof we perform all the calculations for strong solutions and\nthen pass to the limit in the final estimate to justify it for weak solutions.\nLet us consider strong solutions ˆU(t) = ( ˆΦ(t),ˆΦt(t)) and ˜U(t) = ( ˜Φ(t),˜Φt(t))\nto problem (1)–(11)with initial conditions ˆU0= (ˆΦ0,ˆΦ1) and ˜U0= (˜Φ0,˜Φ1)\nlying in a ball, i.e. there exists R >0 such that\n∥˜U0∥H+∥ˆU0∥H≤R. (48)\nDenote U(t) =˜U(t)−ˆU(t) and U0=˜U0−ˆU0. Obviously, U(t) is a weak\n20solution to the problem\nρ1φtt−k1(φx+ψ+lω)x−lσ1(ωx−lφ) +f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω) = 0\n(49)\nβ1ψtt−λ1ψxx+k1(φx+ψ+lω) +γ(˜ψt)−γ(ˆψt) +h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω) = 0\n(50)\nρ1ωtt−σ1(ωx−lφ)x+lk1(φx+ψ+lω) +g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω) = 0\n(51)\nρ2utt−k2(ux+v+lw)x−lσ2(wx−lu) +f2(˜u,˜v,˜w)−f2(ˆu,ˆv,ˆw) = 0\n(52)\nβ2vtt−λ2vxx+k2(ux+v+lw) +h2(˜u,˜v,˜w)−h2(ˆu,ˆv,ˆw) = 0 , (53)\nρ2wtt−σ2(wx−lu)x+lk2(ux+v+lw) +g2(˜u,˜v,˜w)−g2(ˆu,ˆv,ˆw) = 0\n(54)\nwith boundary conditions (7),(8)–(11)and the initial conditions U(0) =\n˜U0−ˆU0. It is easy to see by the energy argument that\nE(U(T)) +TZ\ntL0Z\n0(γ(˜ψs)−γ(ˆψs))ψsdxds =E(U(t)) +TZ\ntH(ˆU(s),˜U(s))ds,\n(55)\nwhere\nH(ˆU(t),˜U(t)) =L0Z\n0(f1( ˆφ,ˆψ,ˆω)−f1( ˜φ,˜ψ,˜ω))φtdx+L0Z\n0(h1( ˆφ,ˆψ,ˆω)−h1( ˜φ,˜ψ,˜ω))ψtdx\n+L0Z\n0(g1( ˆφ,ˆψ,ˆω)−g1( ˜φ,˜ψ,˜ω))ωtdx+LZ\nL0(f2(ˆu,ˆv,ˆw)−f2(˜u,˜v,˜w))utdx\n+LZ\nL0(h2(ˆu,ˆv,ˆw)−h2(˜u,˜v,˜w))vtdx+LZ\nL0(g2(ˆu,ˆv,ˆw)−g2(˜u,˜v,˜w))wtdx,\n(56)\n21and\nE(t) =E1(t) +E2(t), (57)\nhere\nE1(t) =ρ1L0Z\n0ω2\ntdxdt +ρ1L0Z\n0φ2\ntdxdt +β1L0Z\n0ψ2\ntdx+σ1L0Z\n0(ωx−lφ)2dx+\n+k1L0Z\n0(φx+ψ+lω)2dx+λ1L0Z\n0ψ2\nxdx(58)\nand\nE2(t) =ρ2L0Z\n0w2\ntdxdt +ρ2L0Z\n0u2\ntdxdt +β2L0Z\n0v2\ntdx+σ2L0Z\n0(wx−lu)2dx+\n+k2L0Z\n0(ux+v+lw)2dx+λ2L0Z\n0v2\nxdx. (59)\nIntegrating in (55) over the interval (0 , T) we come to\nTE(U(T))+TZ\n0TZ\ntL0Z\n0(γ(˜ψs)−γ(ˆψs))ψsdxdsdt =TZ\n0E(U(t))dt+TZ\n0TZ\ntH(ˆU(s),˜U(s))dsdt.\n(60)\nNow we estimate the first term in the right-hand side of (60). In what follows\nwe present formal estimates which can be performed on strong solutions.\nStep 1. We multiply equation (51)byωandx·ωxand sum up the results.\nAfter integration by parts with respect to twe obtain\n22ρ1TZ\n0L0Z\n0ωtxωtxdxdt +ρ1TZ\n0L0Z\n0ω2\ntdxdt +σ1TZ\n0L0Z\n0(ωx−lφ)xxωxdxdt\n+σ1TZ\n0L0Z\n0(ωx−lφ)xωdxdt −k1lTZ\n0L0Z\n0(φx+ψ+lω)xωxdxdt\n−k1lTZ\n0L0Z\n0(φx+ψ+lω)ωdxdt −TZ\n0L0Z\n0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))(xωx+ω)dxdt\n=ρ1L0Z\n0ωt(x, T)xωx(x, T)dx+ρ1L0Z\n0ωt(x, T)ω(x, T)dx\n−ρ1L0Z\n0ωt(x,0)xωx(x,0)dx−ρ1L0Z\n0ωt(x,0)ω(x,0)dx. (61)\nIntegrating by parts with respect to xwe get\nρ1TZ\n0L0Z\n0ωtxωtxdxdt =−ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n2TZ\n0ω2\nt(L0, t)dt (62)\nand\n23σ1TZ\n0L0Z\n0(ωx−lφ)xxωxdxdt−k1lTZ\n0L0Z\n0(φx+ψ+lω)xωxdxdt\n=σ1TZ\n0L0Z\n0(ωx−lφ)xx(ωx−lφ)dxdt +σ1lTZ\n0L0Z\n0(ωx−lφ)xxφdxdt\n−k1lTZ\n0L0Z\n0(φx+ψ+lω)xωxdxdt =−σ1\n2TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+σ1L0\n2TZ\n0(ωx−lφ)2(L0, t)dt−σ1lTZ\n0L0Z\n0(ωx−lφ)φdxdt\n−2σ1lTZ\n0L0Z\n0(ωx−lφ)x(φx+ψ+lω)dxdt +σ1lTZ\n0L0Z\n0(ωx−lφ)x(ψ+lω)dxdt\n−σ1lL0TZ\n0(ωx−lφ)(L0, t)φ(L0, t)dt−k1l2TZ\n0L0Z\n0(φx+ψ+lω)xφdxdt.\n(63)\nAnalogously,\nσ1TZ\n0L0Z\n0(ωx−lφ)xωdxdt =−σ1TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+σ1TZ\n0(ωx−lφ)(L0, t)ω(L0, t)dt−lσ1TZ\n0L0Z\n0(ωx−lφ)φdxdt. (64)\nIt follows from Lemma 5.6, energy relation (26), and property (N2) that\nTZ\n0L0Z\n0|g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω)|2dxdt≤C(R, T) max\nt∈[0,T]∥Φ(·, t)∥2\nH1−ϵ,0< ϵ < 1/2.\n(65)\n24Therefore, for every ε >0\n\f\f\f\f\f\fTZ\n0L0Z\n0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))(xωx+ω)dxdt\f\f\f\f\f\f≤εTZ\n0∥ωx−lφ∥2dt+C(ε, R, T )lot,\n(66)\nwhere we use the notation\nlot= max\nt∈[0,T](∥φ(·, t)∥2\nH1−ϵ+∥ψ(·, t)∥2\nH1−ϵ+∥ω(·, t)∥2\nH1−ϵ\n+∥u(·, t)∥2\nH1−ϵ+∥v(·, t)∥2\nH1−ϵ+∥w(·, t)∥2\nH1−ϵ),0< ϵ < 1/2.(67)\nSimilar estimates hold for nonlinearities g2,fi,hi,i= 1,2.\nWe note that for any η∈H1(0, L0) (or analogously η∈H1(L0, L))\nη(L0)≤sup\n(0,L0)|η| ≤C∥η∥H1−ϵ,0< ϵ < 1/2. (68)\nSince due to (46)\n2σ1l\f\f\f\f\f\fTZ\n0L0Z\n0(ωx−lφ)x(φx+ψ+lω)dxdt\f\f\f\f\f\f\n≤σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt + 16k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt,\nthe following estimate can be obtained from (61)– (66)\nρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n2TZ\n0ω2\nt(L0, t)dt+13σ1L0\n8TZ\n0(ωx−lφ)2(L0, t)dt\n≤13σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt+17k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt+C(R, T)lot\n+C(E(0) + E(T)),(69)\n25where C >0.\nStep 2. Multiplying equation (51)byωand ( x−L0)·ωxand arguing as\nabove we come to the estimate\nρ1\n2TZ\n0L0Z\n0ω2\ntdxdt+13σ1L0\n8TZ\n0(ωx−lφ)2(0, t)dt≤13σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+ 17k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +C(R, T)lot+C(E(0) + E(T)).(70)\nSumming up estimates (69)and(70)and multiplying the result by1\n2we get\nρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n4TZ\n0ω2\nt(L0, t)dt+3σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt\n+3σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt≤13σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+ 17k1l2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +C(R, T)lot+C(E(0) + E(T)).(71)\nStep 3. Next we multiply equation (49)by−1\nl(ωx−lφ), equation (51)by\n1\nlφx, summing up the results and integrating by parts with respect to twe\narrive at\n26ρ1\nlTZ\n0L0Z\n0φt(ωtx−lφt)dxdt +k1\nlTZ\n0L0Z\n0(φx+ψ+lω)x(ωx−lφ)dxdt\n+σ1TZ\n0L0Z\n0(ωx−lφ)2dxdt−1\nlTZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))(ωx−lφ)dxdt\n+ρ1\nlTZ\n0L0Z\n0ωtφtxdxdt +σ1\nlTZ\n0L0Z\n0(ωx−lφ)xφxdxdt\n−k1TZ\n0L0Z\n0(φx+ψ+lω)φxdxdt−TZ\n0L0Z\n0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))φxdxdt\n=ρ1\nlL0Z\n0φt(x, T)(ωx−lφ)(x, T)dx−ρ1\nlL0Z\n0φt(x,0)(ωx−lφ)(x,0)dx\n+ρ1\nlL0Z\n0ωt(x, T)φx(x, T)dx−ρ1\nlL0Z\n0ωt(x,0)φx(x,0)dx.\n(72)\nIntegrating by parts with respect to xwe obtain\n\f\f\f\f\f\fρ1\nlTZ\n0L0Z\n0φtωtxdxdt +ρ1\nlTZ\n0L0Z\n0ωtφtxdxdt\f\f\f\f\f\f=\f\f\f\f\f\fρ1\nlTZ\n0φt(L0, t)ωt(L0, t)dt\f\f\f\f\f\f\n≤ρ1L0\n8TZ\n0ω2\nt(L0, t)dt+2ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt.\n(73)\nTaking into account (46) we get\n27k1\nlTZ\n0L0Z\n0(φx+ψ+lω)x(ωx−lφ)dxdt +σ1\nlTZ\n0L0Z\n0(ωx−lφ)xφxdxdt\n=k1\nlTZ\n0(φx+ψ+lω)(L0, t)(ωx−lφ)(L0, t)dt−k1\nlTZ\n0(φx+ψ+lω)(0, t)(ωx−lφ)(0, t)dt\n+k1\nlTZ\n0L0Z\n0ψx(ωx−lφ)dxdt+σ1TZ\n0L0Z\n0(ωx−lφ)2dxdt+σ1lTZ\n0L0Z\n0(ωx−lφ)φdxdt.\n(74)\nUsing the estimates\n\f\f\f\f\f\fk1\nlTZ\n0(φx+ψ+lω)(L0, t)(ωx−lφ)(L0, t)dt\f\f\f\f\f\f\n≤4k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt,\n\f\f\f\f\f\fk1\nlTZ\n0L0Z\n0ψx(ωx−lφ)dxdt\f\f\f\f\f\f≤4k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt\nand (72)–(74) we infer\n2815σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt≤ρ1TZ\n0L0Z\n0φ2\ntdxdt+ 2k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+4k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt+4k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+σ1L0\n8TZ\n0(ωx−lφ)2(L0, t)dt\n+4k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt+σ1L0\n8TZ\n0(ωx−lφ)2(0, t)dt\nρ1L0\n8TZ\n0ω2\nt(L0, t)dt+2ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+C(R, T)lot+C(E(0) + E(T)).\n(75)\nAdding (75) to (71) we obtain\nσ1\n4TZ\n0L0Z\n0(ωx−lφ)2dxdt+ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt+ρ1L0\n8TZ\n0ω2\nt(0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt≤ρ1TZ\n0L0Z\n0φ2\ntdxdt+k1(2+17 l2L2\n0)TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+4k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+4k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+4k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +2ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+C(R, T)lot+C(E(0) + E(T)).\n(76)\nStep 4. Now we multiply equation (49)by−16\nl2L2\n0xφxand−16\nl2L2\n0(x−L0)φx\nand sum up the results. After integration by parts with respect to twe get\n2916ρ1\nl2L2\n0TZ\n0L0Z\n0φtxφtxdxdt +16ρ1\nl2L2\n0TZ\n0L0Z\n0φt(x−L0)φtxdxdt\n+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)xxφxdxdt+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)x(x−L0)φxdxdt\n+16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)xφxdxdt +16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(x−L0)φxdxdt\n−16\nl2L2\n0TZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))(2x−L0)φxdxdt\n=16ρ1\nl2L2\n0L0Z\n0φt(x, T)(2x−L0)φx(x, T)dx−16ρ1\nl2L2\n0L0Z\n0φt(x, T)(2x−L0)φx(x, T)dx.\n(77)\nIt is easy to see that\n16ρ1\nl2L2\n0TZ\n0L0Z\n0φtxφtxdxdt +16ρ1\nl2L2\n0TZ\n0L0Z\n0φt(x−L0)φtxdxdt\n=−16ρ1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt+8ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt\n(78)\nand\n3016k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)xxφxdxdt+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)x(x−L0)φxdxdt\n=−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +8k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+8k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)xx(ψ+lω)dxdt\n−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)x(x−L0)(ψ+lω)dxdt\n=−16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +8k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+8k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt−16k1\nl2L0TZ\n0(φx+ψ+lω)(L0, t)(ψ+lω)(L0, t)dt\n+32k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)(ψ+lω)dxdt++16k1\nlL2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)(ωx−lφ)dxdt\n+16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt+16k1\nL2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)φdxdt.\n(79)\nMoreover,\n16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)xφxdxdt +16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(x−L0)φxdxdt\n=16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(2x−L0)(φx+ψ+lω)dxdt\n31−16σ1\nlL2\n0TZ\n0L0Z\n0(ωx−lφ)(2x−L0)(ψ+lω)dxdt. (80)\nCollecting (77)–(80) and using the estimates\n\f\f\f\f\f\f32k1\nlL2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)(ωx−lφ)dxdt\f\f\f\f\f\f\n≤σ1\n8TZ\n0L0Z\n0(ωx−lφ)2dxdt +2046k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nand\n\f\f\f\f\f\f16k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt\f\f\f\f\f\f\n≤k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +64k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nwe come to\n7k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+7k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n(81)\n+8ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt≤16ρ1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt+2150k1\nl2L2\n0TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt+3σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt+C(R, T)lot+C(E(0)+E(T)).\n(82)\nAdding (82) to (76) we arrive at\n32σ1\n16TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+3k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+3k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+6ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt≤ρ1\u0012\n1 +16\nl2L2\n0\u0013TZ\n0L0Z\n0φ2\ntdxdt\n+k1\u0012\n2 + 17 l2L2\n0+2150\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+5k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt +C(R, T)lot+C(E(0) + E(T)).(83)\nStep 5. Next we multiply equation (49)by−\u0010\n1 +18\nl2L2\n0\u0011\nφand integrate by\nparts with respect to t\nρ1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0φ2\ntdxdt +k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)xφdxdt\n+lσ1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(ωx−lφ)φdxdt−\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))φdxdt =\nρ1\u0012\n1 +18\nl2L2\n0\u0013L0Z\n0(φt(x, T)φ(x, T)−φt(x,0)φ(x,0))dx. (84)\nSince\n33k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)xφdxdt =−k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0(φx+ψ+lω)(L0, t)φ(L0, t)dt\n+k1\u0012\n1 +18\nl2L2\n0\u0013TZ\n0(φx+ψ+lω)(ψ+lω)dxdt (85)\nwe obtain the estimate\nρ1\u0012\n1 +17\nl2L2\n0\u0013TZ\n0L0Z\n0φ2\ntdxdt≤k1\u0012\n2 +18\nl2L2\n0\u0013TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+σ1\n32TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+C(R, T)lot+C(E(0) + E(T)).(86)\nSumming up (83) and (86) we get\n34σ1\n32TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n2TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+2k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+2k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+6ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt\n≤k1\u0012\n4 + 17 l2L2\n0+2200\nl2L2\n0\u0013TZ\n0(φx+ψ+lω)2dxdt\n+6k1\nl2TZ\n0L0Z\n0ψ2\nxdxdt+C(R, T)lot+C(E(0)+E(T)).\n(87)\nStep 6. Next we multiply equation (50)byC1(φx+ψ+lω) and equation (49)\nbyC1β1\nρ1ψx, where C1= 2(6 + 17 l2L2\n0+2200\nl2L2\n0). Then we sum up the results\nand integrate by parts with respect to t. Taking into account (46),(47)we\ncome to\n35−β1C1TZ\n0L0Z\n0φtψtxdxdt−λ1C1TZ\n0L0Z\n0(φx+ψ+lω)xψxdxdt\n−lC1λ1TZ\n0L0Z\n0(ωx−lφ)ψxdxdt+C1β1\nρ1TZ\n0L0Z\n0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))ψxdxdt\n−β1C1TZ\n0L0Z\n0ψt(φxt+ψt+lωt)dxdt−λ1C1TZ\n0L0Z\n0ψxx(φx+ψ+lω)dxdt\n+k1C1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +C1TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))(φx+ψ+lω)dxdt\n+C1TZ\n0L0Z\n0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))(φx+ψ+lω)dxdt =β1C1L0Z\n0φt(x,0)ψx(x,0)dx\n−β1C1L0Z\n0φt(x, T)ψx(x, T)dx+β1C1L0Z\n0ψt(x,0)(φx+ψ+lω)(x,0)dx\n−β1C1L0Z\n0ψt(x, T)(φx+ψ+lω)(x, T)dx. (88)\nIntegrating by parts with respect to xwe get\n\f\f\f\f\f\fβ1C1TZ\n0L0Z\n0φtψtxdxdt +β1C1TZ\n0L0Z\n0ψt(φxt+lωt)dxdt\f\f\f\f\f\f\n≤\f\f\f\f\f\fβ1C1TZ\n0φt(L0, t)ψt(L0, t)dt+β1C1lTZ\n0L0Z\n0ψtωtdxdt\f\f\f\f\f\f≤ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt\n+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt+ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +β2\n1C2\n1l2\nρ1TZ\n0L0Z\n0ψ2\ntdxdt (89)\nand\n36\f\f\f\f\f\fλ1C1TZ\n0L0Z\n0(φx+ψ+lω)xψxdxdt +λ1C1TZ\n0L0Z\n0ψxx(φx+ψ+lω)dxdt\f\f\f\f\f\f\n=\f\f\f\f\f\fλ1C1TZ\n0(φx+ψ+lω)(L0, t)ψx(L0, t)dt−λ1C1TZ\n0(φx+ψ+lω)(0, t)ψx(0, t)dt\f\f\f\f\f\f\n≤k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt.(90)\nMoreover,\n\f\f\f\f\f\flC1λ1TZ\n0L0Z\n0(ωx−lφ)ψxdxdt\f\f\f\f\f\f≤σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt+16l2C2\n1λ2\n1\nσ1TZ\n0L0Z\n0ψ2\nxdxdt.\n(91)\nIt follows from Lemma (6.6) with ε=k1C1\n4\n\f\f\f\f\f\fC1TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))(φx+ψ+lω)dxdt\f\f\f\f\f\f\n≤k1C1\n4TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt (92)\nConsequently, collecting (88)–(92) we obtain\n37C1k1\n2TZ\n0L0Z\n0(φx+ψ+lω)2dxdt≤σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt\n+20l2C2\n1λ2\n1\nσ1TZ\n0L0Z\n0ψ2\nxdxdt +C1\u0012\nβ1+β2\n1l2\nρ1\u0013TZ\n0L0Z\n0ψ2\ntdxdt+\nk1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt\n+ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt+ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(93)\nCombining (93) with (87) we get\n38σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+5ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt + 2k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\n≤\u00126k1\nl2+20l2C2\n1λ2\n1\nσ1\u0013TZ\n0L0Z\n0ψ2\nxdxdt +C1\u0012\nβ1+β2\n1l2\nρ1\u0013TZ\n0L0Z\n0ψ2\ntdxdt\n+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt+β2\n1C2\n1l2L0\n4TZ\n0ψ2\nt(L0, t)dt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(94)\nStep 7. Our next step is to multiply equation (50)by−C2xψx−C2(x−L0)ψx,\nwhere C2=l2λ1C2\n1\nk1. After integration by parts with respect to twe obtain\nβ1C2TZ\n0L0Z\n0ψtxψxtdxdt+β1C2TZ\n0L0Z\n0ψt(x−L0)ψxtdxdt\n39+λ1C2TZ\n0L0Z\n0ψxxxψxdxdt +λ1C2TZ\n0L0Z\n0ψxx(x−L0)ψxdxdt\n−k1C2TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt−C2TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))(2x−L0)ψxdxdt\n+TZ\n0L0Z\n0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))(2x−L0)ψxdxdt\n=β1C2L0Z\n0ψt(x, T)(2x−L0)ψx(x, T)dx−β1C2L0Z\n0ψt(x,0)(2x−L0)ψx(x,0)dx.\n(95)\nAfter integration by parts with respect to xwe get\nβ1C2TZ\n0L0Z\n0ψtxψxtdxdt +β1C2TZ\n0L0Z\n0ψt(x−L0)ψxtdxdt\n=−β1C2TZ\n0L0Z\n0ψ2\ntdxdt +β1C2L0\n2TZ\n0ψ2\nt(L0, t)dt(96)\nand\nλ1C2TZ\n0L0Z\n0ψxxxψxdxdt +λ1C2TZ\n0L0Z\n0ψxx(x−L0)ψxdxdt\n=λ1C2L0\n2TZ\n0ψ2\nx(L0, t)dt+λ1C2L0\n2TZ\n0ψ2\nx(0, t)dt−λ1C2TZ\n0L0Z\n0ψ2\nxdxdt. (97)\nFurthermore,\n40\f\f\f\f\f\fk1C2TZ\n0L0Z\n0(φx+ψ+lω)(2x−L0)ψxdxdt\f\f\f\f\f\f\n≤k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt +k1C2\n2L2\n0\n4TZ\n0L0Z\n0ψ2\nxdxdt. (98)\nBy Lemma (6.6) with ε=k1C22L2\n0\n4we have\n\f\f\f\f\f\fC2TZ\n0L0Z\n0ψt(2x−L0)ψxdxdt\f\f\f\f\f\f≤k1C2\n2L2\n0\n4TZ\n0L0Z\n0ψ2\nxdxdt+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt.\n(99)\nAs a result of (95)– (99) we obtain the estimate\nβ1C2L0\n2TZ\n0ψ2\nt(L0, t)dt+λ1C2L0\n2TZ\n0ψ2\nx(L0, t)dt+λ1C2L0\n2TZ\n0ψ2\nx(0, t)dt\n≤k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt+\u0000\nk1C2\n2L2\n0+λ1C2\u0001TZ\n0L0Z\n0ψ2\nxdxdt+β1C2TZ\n0L0Z\n0ψ2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(100)\nSumming up (94) and (100) and using (47) we infer\n41σ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n+5ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt +k1TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nl2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt\n+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt≤\u00126k1\nl2+20l2C2\n1λ2\n1\nσ1+λ1C2+k1C2\n2L2\n0\u0013TZ\n0L0Z\n0ψ2\nxdxdt\n+\u0012\n(C1+C2)β1+C1β2\n1l2\nρ1\u0013TZ\n0L0Z\n0ψ2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(101)\nStep 8. Now we multiply equation (50)byC3ψ, where C3=2\nλ1\u0010\n6k1\nl2+20l2C2\n1λ2\n1\nσ1+λ1C2+k1C2\n2L2\n0\u0011\nand integrate by parts with respect to t\n42−C3β1TZ\n0L0Z\n0ψ2\ntdxdt−λ1C3TZ\n0L0Z\n0ψxxψdxdt +k1C3TZ\n0L0Z\n0(φx+ψ+lω)ψdxdt\n+C3TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψdxdt +C3TZ\n0L0Z\n0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))ψdxdt\n(102)\n=C3β1L0Z\n0ψt(x,0)ψ(x,0)dx−C3β1L0Z\n0ψt(x, T)ψ(x, T)dx\n(103)\nAfter integration by parts we infer the estimate\nλ1C3TZ\n0L0Z\n0ψ2\nxdxdt≤k1\n2TZ\n0(φx+ψ+lω)2dxdt+C3β1TZ\n0L0Z\n0ψ2\ntdxdt+l2L0λ2\n1C2\n1\n8k1TZ\n0ψ2\nx(L0, t)dt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(104)\nCombining (104) with (101) we obtain\nσ1\n64TZ\n0L0Z\n0(ωx−lφ)2dxdt +ρ1\n4TZ\n0L0Z\n0ω2\ntdxdt +ρ1L0\n8TZ\n0ω2\nt(L0, t)dt\n+σ1L0\n16TZ\n0(ωx−lφ)2(L0, t)dt+σ1L0\n16TZ\n0(ωx−lφ)2(0, t)dt\n+k1\nl2L0TZ\n0(φx+ψ+lω)2(L0, t)dt+k1\nl2L0TZ\n0(φx+ψ+lω)2(0, t)dt\n43+5ρ1\nl2L0TZ\n0φ2\nt(L0, t)dt+1\nl2L2\n0TZ\n0L0Z\n0φ2\ntdxdt +k1\n2TZ\n0L0Z\n0(φx+ψ+lω)2dxdt\nl2L0λ2\n1C2\n1\n8k1TZ\n0ψ2\nx(L0, t)dt+l2L0λ2\n1C2\n1\n4k1TZ\n0ψ2\nx(0, t)dt+β2\n1C2\n1l2L0\n4ρ1TZ\n0ψ2\nt(L0, t)dt\n+\u00126k1\nl2+20l2C2\n1λ2\n1\nσ1+λ1C2+k1C2\n2L2\n0\u0013TZ\n0L0Z\n0ψ2\nxdxdt\n≤\u0012\n(C1+C2)β1+C1β2\n1l2\nρ1+C3β1\u0013TZ\n0L0Z\n0ψ2\ntdxdt\n+CTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(105)\nStep 9. Consequently, it follows from (105) and assumption (D4) for any\nl >0 where exist constants Mi,i={1,3}(depending on l) such that\nTZ\n0E1(t)dt+TZ\n0B1(t)dt≤M1TZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt\n+M2(R, T)lot+M3(E(T) +E(0)),(106)\nwhere\nB1(t) =TZ\n0(ωx−lφ)2(L0, t)dt+TZ\n0(φx+ψ+lω)2(L0, t)dt+TZ\n0ψ2\nx(L0, t)dt\n+TZ\n0ω2\nt(L0, t)dt+TZ\n0ψ2\nt(L0, t)dt+TZ\n0φ2\nt(L0, t)dt.(107)\nStep 10. Finally, we multiply equation (52)by (x−L)ux, equation (53)by\n(x−L)vx, and (54)by (x−L)wx. Summing up the results and integrating\n44by parts with respect to twe arrive at\n−ρ2TZ\n0LZ\nL0ut(x−L)utxdxdt−k2TZ\n0LZ\nL0(ux+v+lw)x(x−L)uxdxdt\n−lσ2TZ\n0LZ\nL0(wx−lu)(x−L)uxdxdt+TZ\n0LZ\nL0(f2(˜u,˜v,˜w)−f2(ˆu,ˆv,ˆw))(x−L)uxdxdt\n−β2TZ\n0LZ\nL0vt(x−L)vxtdxdt−λ2TZ\n0LZ\nL0vxx(x−L)vxdxdt\n+k2TZ\n0LZ\nL0(ux+v+lw)(x−L)vxdxdt+TZ\n0LZ\nL0(h2(˜u,˜v,˜w)−h2(ˆu,ˆv,ˆw))(x−L)vxdxdt\n−ρ2TZ\n0LZ\nL0wt(x−L)wxtdxdt−σ2TZ\n0LZ\nL0(wx−lu)x(x−L)wxdxdt\n+lk2TZ\n0LZ\nL0(ux+v+lw)(x−L)wxdxdt+TZ\n0LZ\nL0(g2(˜u,˜v,˜w)−g2(ˆu,ˆv,ˆw))(x−L)wxdxdt =\n−ρ2LZ\nL0(x−L)((utux)(x, T)−(utux)(x,0))dx−β2LZ\nL0(x−L)((vtvx)(x, T)−(vtvx)(x,0))dx\n−ρ2LZ\nL0(x−L)((wtwx)(x, T)−(wtwx)(x,0))dx. (108)\nAfter integration by parts with respect to xwe infer\n45−ρ2TZ\n0LZ\nL0ut(x−L)utxdx−β2TZ\n0LZ\nL0vt(x−L)vxtdxdt−ρ2TZ\n0LZ\nL0wt(x−L)wxtdxdt\n=ρ2\n2TZ\n0LZ\nL0u2\ntdx+β2\n2TZ\n0LZ\nL0v2\ntdxdt +ρ2\n2TZ\n0LZ\nL0w2\ntdxdt\n−ρ2(L−L0)\n2TZ\n0u2\nt(L0)dt−β2(L−L0)\n2TZ\n0v2\nt(L0)dt−ρ2(L−L0)\n2TZ\n0w2\nt(L0)dt\n(109)\nand\n−k2TZ\n0LZ\nL0(ux+v+lw)x(x−L)uxdxdt−lσ2TZ\n0LZ\nL0(wx−lu)(x−L)uxdxdt\n−λ2TZ\n0LZ\nL0vxx(x−L)vxdxdt +k2TZ\n0LZ\nL0(ux+v+lw)(x−L)vxdxdt\n46−σ2TZ\n0LZ\nL0(wx−lu)x(x−L)wxdxdt+lk2TZ\n0LZ\nL0(ux+v+lw)(x−L)wxdxdt =\n−k2TZ\n0LZ\nL0(ux+v+lw)x(x−L)(ux+v+lw)dxdt\n−σ2TZ\n0LZ\nL0(wx−lu)x(x−L)(wx−lu)dxdt−λ2TZ\n0LZ\nL0vxx(x−L)vxdxdt\n−lσ2(L−L0)TZ\n0(wx−lu)(L0)u(L0)dt+k2(L−L0)TZ\n0(ux+v+lw)(L0)v(L0)dt\n+lk2(L−L0)TZ\n0(ux+v+lw)(L0)w(L0)dt=\n−k2(L−L0)\n2TZ\n0(ux+v+lw)2(L0)dt+k2\n2TZ\n0LZ\nL0(ux+v+lw)2dxdt\n+σ2\n2TZ\n0LZ\nL0(wx−lu)2dxdt−σ2(L−L0)\n2TZ\n0(wx−lu)2(L0)dt\n+λ2\n2TZ\n0LZ\nL0v2\nxdxdt−λ2(L−L0)\n2TZ\n0v2\nx(L0)dt−lσ2(L−L0)TZ\n0(wx−lu)(L0)u(L0)dt\n+k2(L−L0)TZ\n0(ux+v+lw)(L0)v(L0)dt\n+lk2(L−L0)TZ\n0(ux+v+lw)(L0)w(L0)dt.(110)\nConsequently, it follows from (108) –(110) that for any l >0 where exist\n47constants M4, M5, M6>0 such that\nTZ\n0E2(t)dt≤M4TZ\n0B2(t)dt+M5(R, T)lot+M6(E(T) +E(0)),(111)\nwhere\nB2(t) =TZ\n0(wx−lu)2(L0, t)dt+TZ\n0(ux+v+lw)2(L0, t)dt+TZ\n0v2\nx(L0, t)dt\n+TZ\n0w2\nt(L0, t)dt+TZ\n0v2\nt(L0, t)dt+TZ\n0u2\nt(L0, t)dt.(112)\nThen, due to transmission conditions (8)–(11) there exist δ, M 7, M8>0\n(depending on l), such that\nTZ\n0E(t)dt≤δTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt+M7(R, T)lot+M8(E(T)+E(0)).\n(113)\nIt follows from (55) that there exists C >0 such that\nTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt≤C\nE(0) +TZ\n0|H(ˆU(t),˜U(t))|dt\n.(114)\nBy Lemma 5.6 we have that for any ε >0 there exists C(ε, R)>0 such that\nTZ\n0|H(ˆU(t),˜U(t))|dt≤εTZ\n0L0Z\n0E(t)dxdt +C(ε, R, T )lot. (115)\nCombining (115) with (114) we arrive at\nTZ\n0L0Z\n0(γ(˜ψt)−γ(ˆψt))ψtdxdt≤CE(0) + C(R, T)lot. (116)\n48Substituting (116) into (113) we obtain\nTZ\n0E(t)dt≤C(R, T)lot+C(E(T) +E(0)) (117)\nfor some C, C(R, T)>0.\nOur remaining task is to estimate the last term in (60).\n\f\f\f\f\f\fTZ\n0TZ\ntH(ˆU(s),˜U(s))dsdt\f\f\f\f\f\f≤TZ\n0E(t)dt+T3C(R)lot. (118)\nThen, it follows from (60) and (118) that\nTE(T)≤CTZ\n0E(t)dt+C(T, R)lot. (119)\nThen the combination of (119) with (117) leads to\nTE(T)≤C(R, T)lot+C(E(T) +E(0)). (120)\nChoosing Tlarge enough one can obtain estimate (44)which together with\nTheorem 6.5 immediately leads to the asymptotic smoothness of the system.\n6.3 Existence of attractors.\nThe following statement collects criteria on existence and properties of\nattractors to gradient systems.\nTheorem 6.8 ([10,12]).Assume that (H, S t)is a gradient asymptotically\nsmooth dynamical system. Assume its Lyapunov function L(y)is bounded\nfrom above on any bounded subset of Hand the set WR={y:L(y)≤R}\nis bounded for every R. If the set Nof stationary points of (H, S t)is\nbounded, then (St, H)possesses a compact global attractor. Moreover, the\n49global attractor consists of full trajectories γ={U(t) :t∈R}such that\nlim\nt→−∞distH(U(t),N) = 0 and lim\nt→+∞distH(U(t),N) = 0 (121)\nand\nlim\nt→+∞distH(Stx,N) = 0 for any x∈H; (122)\nthat is, any trajectory stabilizes to the set Nof stationary points.\nNow we state the result on the existence of an attractor.\nTheorem 6.9. Let assumptions of Theorems 6.2, 6.7, hold true, moreover,\nlim inf\n|s|→∞h1(s)\ns>0, (N5)\n∇F2(u, v, w )(u, v, w )−a1F2(u, v, w )≥ −a2, a i≥0.\nThen, the dynamical system (H, S t)generated by (1)-(11)possesses a compact\nglobal attractor Apossessing properties (121) ,(122) .\nProof. In view of Theorems 6.2, 6.7, 6.8 our remaining task is to show the\nboundedness of the set of stationary points and the set WR={Z:L(Z)≤R},\nwhere Lis given by (30).\nThe second statement follows immediately from the structure of function\nLand property (N5).\nThe first statement can be easily shown by energy-like estimates for\nstationary solutions taking into account (N5).\n7 Singular Limits on finite time intervals\n7.1 Singular limit l→0\nLet the nonlinearities fj, hj, gjare such that\nf1(φ, ψ, ω ) =f1(φ, ψ), h 1(φ, ψ, ω ) =h1(φ, ψ), g1(φ, ψ, ω ) =g1(ω),\nf2(u, v, w ) =f2(u, v), h 2(u, v, w ) =h2(u, v), g 2(u, v, w ) =g2(w).(N6)\n50If we formally set l= 0 in (18)-(24), we obtain the contact problem for a\nstraight Timoshenko beam\nρ1φtt−k1(φx+ψ)x+f1(φ, ψ) =p1(x, t), (x, t)∈(0, L0)×(0, T),\n(123)\nβ1ψtt−λ1ψxx+k1(φx+ψ) +γ(ψt) +h1(φ, ψ) =r1(x, t), (x, t)∈(0, L0)×(0, T),\n(124)\nρ2utt−k2(ux+v)x+f2(u, v) =p2(x, t), (x, t)∈(L0, L)×(0, T),\n(125)\nβ2vtt−λ2vxx+k2(ux+v) +h2(u, v) =r2(x, t), (x, t)∈(L0, L)×(0, T),\n(126)\nφ(0, t) =ψ(0, t) = 0 , u(L, t) =v(L, t) = 0 , (127)\nφ(L0, t) =u(L0, t), ψ(L0, t) =v(L0, t), (128)\nk1(φx+ψ)(L0, t) =k2(ux+v)(L0, t), λ1ψx(L0, t) =λ2vx(L0, t), (129)\nand an independent contact problem for wave equations\nρ1ωtt−σ1ωxx+g1(ω) =q1(x, t), (x, t)∈(0, L0)×(0, T),\n(130)\nρ2wtt−σ2wxx+g2(w) =q2(x, t), (x, t)∈(L0, L)×(0, T),\n(131)\nσ1ωx(L0, t) =σ2wx(L0, t), ω(L0, t) =w(L0, t), (132)\nw(L, t) = 0 , ω(0, t) = 0 . (133)\nThe following theorem gives an answer, how close are solutions to (18)-(24)\nto the solution of decoupled system (123)-(133) when l→0.\nTheorem 7.1. Assume that the conditions of Theorem 5.4, (D3) and(N6)\nhold. Let Φ(l)be the solution to (18)-(24) with the fixed land the initial data\nΦ(x,0) = ( φ0, ψ0, ω0, u0, v0, w0)(x),Φt(x,0) = ( φ1, ψ1, ω1, u1, v1, w1)(x).\n51Then for every T >0\nΦ(l)∗⇀(φ, ψ, ω, u, v, w ) inL∞(0, T;Hd)asl→0,\nΦ(l)\nt∗⇀(φt, ψt, ωt, ut, vt, wt) inL∞(0, T;Hv)asl→0,\nwhere (φ, ψ, u, v )is the solution to (123) -(129) with the initial conditions\n(φ, ψ, u, v )(x,0) = ( φ0, ψ0, u0, v0)(x),(φt, ψt, ut, vt)(x,0) = ( φ1, ψ1, u1, v1)(x),\nand(ω, w)is the solution to (130) -(133) with the initial conditions\n(ω, w)(x,0) = ( ω0, w0)(x),(ωt, wt)(x,0) = ( ω1, w1)(x).\nThe proof is similar to that of Theorem 3.1 [ 24] for the homogeneous\nBresse beam with obvious changes, except for the limit transition in the\nnonlinear dissipation term. For the future use we formulate it as a lemma.\nLemma 7.2. Let(D3) holds. Then\nZT\n0ZL0\n0γ(ψ(l)(x, t))γ1(x, t)dxdt→ZT\n0ZL0\n0γ(ψ(x, t))γ1(x, t)dxdt asl→0\nfor every γ1∈L2(0, T;H1(0, L0)).\nProof. Since (D1) and (D3) hold |γ(s)| ≤Ms, therefore\n||γ(ψ(l))||L∞(0,T;L2(0,L0))≤C(||ψ(l)||L∞(0,T;L2(0,L0))).\nThus, due to Lemmas 4.1, 5.6 the sequence\nRΦ(l)\ntt=AΦ(l)+ Γ(Φ(l)\nt) +F(Φ(l)) +P\nis bounded in L∞(0, T;H−1(0, L)) and we can extract from Φ(l)\ntta subse-\nquence, that converges ∗-weakly in L∞(0, T;H−1(0, L)). Thus,\nΦ(l)\nt→Φtstrongly in L2(0, T;H−ε(0, L)), ε > 0.\n52Consequently,\n\f\f\f\fZT\n0ZL0\n0(γ(ψ(l)(x, t))−γ(ψ(x, t)))γ1(x, t)dxdt\f\f\f\f≤\nC(L)ZT\n0ZL0\n0|ψ(l)(x, t)−ψ(x, t)||γ1(x, t)|dxdt→0.\nWe perform numerical modelling for the original problem with l=\n1,1/3,1/10,1/30,1/100,1/300,1/1000 and the limiting problem ( l= 0) with\nthe following values of constants ρ1=ρ2= 1, β1=β2= 2, σ1= 4,\nσ2= 2,λ1= 8,λ2= 4,L= 10, L0= 4 and the right-hand sides\np1(x) = sin x, r 1(x) =x, q 1(x) = sin x, (134)\np2(x) = cos x, r 2(x) =x+ 1, q 2(x) = cos x. (135)\nIn this subsection we consider the nonlinearities with the potentials\nF1(φ, ψ, ω ) =|φ+ψ|4− |φ+ψ|2+|φψ|2+|ω|3,\nF2(u, v, w ) =|u+v|4− |u+v|2+|uv|2+|w|3.\nConsequently, the nonlinearities have the form\nf1(φ, ψ, ω ) = 4( φ+ψ)3−2(φ+ψ) + 2φψ2, f 2(u, v, w ) = 4( u+v)3−2(u+v) + 2uv2,\nh1(φ, ψ, ω ) = 4( φ+ψ)3−2(φ+ψ) + 2φ2ψ, h 2(u, v, w ) = 4( u+v)3−2(u+v) + 2u2v,\ng1(φ, ψ, ω ) = 3|ω|ω, g 2(u, v, w ) = 3|w|w.\nFor modelling we choose the following (globally Lipschitz) dissipation\nγ(s) =\n\n1\n100s3,|s| ≤10,\n10s, |s|>10\n53and the following initial data:\nφ(x,0) =−3\n16x2+3\n4x, u (x,0) = 0 ,\nψ(x,0) =−1\n12x2+7\n12x, v (x,0) =−1\n6x+5\n3,\nω(x,0) =1\n16x2−1\n4x, w (x,0) =−1\n12x2+7\n6x−10\n3,\nφt(x,0) =x\n4, u t(x,0) =−1\n6(x−10),\nψt(x,0) =x\n4, v t(x,0) =−1\n6(x−10),\nωt(x,0) =x\n4, w t(x,0) =−1\n6(x−10).\nFigures 2-7 show the behavior of solutions when l→0 for the chosen\ncross-sections of the beam.\n0 1 2 3 4 5 6 7 8-2-1.5-1-0.500.51\nFigure 2: Transversal displacement of the beam, cross-section x= 2.\n7.2 Singular limit ki→ ∞, l→0\nThe singular limit for the straight Timoshenko beam ( l= 0) as ki→+∞is\nthe Euler-Bernoulli beam equation [ 19, Ch. 4]. We have a similar result for\nthe Bresse composite beam when ki→ ∞ , l→0.\nTheorem 7.3. Let the assumptions of Theorem 5.4, (N6) and(D3) hold.\n540 1 2 3 4 5 6 7 8-2-1.5-1-0.500.511.5Figure 3: Transversal displacement of the beam, cross-section x= 6.\n0 1 2 3 4 5 6 7 8-2-1.5-1-0.500.51\nFigure 4: Shear angle variation of the beam, cross-section x= 2.\nMoreover,\n(φ0, u0)∈\b\nφ0∈H2(0, L0), u0∈H2(L0, L), φ0(0) = u0(L) = 0 ,\n∂xϕ0(0) = ∂xu0(L) = 0 , ∂xφ0(L0, t) =∂xu0(L0, t)};(I1)\nψ0=−∂xφ0, v0=−∂xu0; (I2)\n(φ1, u1)∈ {φ1∈H1(0, L0), u1∈H1(L0, L), φ1(0) = u1(L) = 0 , φ1(L0, t) =u1(L0, t)};\n(I3)\nω0=w0= 0; (I4)\nh1, h2∈C1(R2); (N6)\nr1∈L∞(0, T;H1(0, L0)), r2∈L∞(0, T;H1(L0, L)),\nr1(L0, t) =r2(L0, t)for almost all t >0.(R3)\n550 1 2 3 4 5 6 7 8-2-1.5-1-0.500.511.5Figure 5: Shear angle variation of the beam, cross-section x= 6.\n0 1 2 3 4 5 6 7 800.20.40.60.81\nFigure 6: Longitudinal displacement of the beam, cross-section x= 2.\nLetk(n)\nj→ ∞ ,l(n)→0asn→ ∞ , and Φ(n)be weak solutions to (18)-(24)\nwith fixed k(n)\nj, l(n)and the same initial data\nΦ(x,0) = ( φ0, ψ0, ω0, u0, v0, w0)(x),Φt(x,0) = ( φ1, ψ1, ω1, u1, v1, w1).\nThen for every T >0\nΦ(n)∗⇀(φ, ψ, ω, u, v, w ) inL∞(0, T;Hd)asn→ ∞ ,\nΦ(n)\nt∗⇀(φt, ψt, ωt, ut, vt, wt) inL∞(0, T;Hv)asn→ ∞ ,\nwhere\n560 1 2 3 4 5 6 7 800.511.522.53Figure 7: Longitudinal displacement of the beam, cross-section x= 6.\n•(φ, u)is a weak solution to\nρ1φtt−β1φttxx+λ1φxxxx−γ′(−φtx)φtxx+∂xh1(φ,−φx) +f1(φ,−φx) =\np1(x, t) +∂xr1(x, t),(x, t)∈(0, L0)×(0, T),\n(136)\nρ2utt−β2uttxx+λ2uxxxx+∂xh2(u,−ux) +f2(u,−ux) =\np2(x, t) +∂xr2(x, t),(x, t)∈(L0, L)×(0, T),\n(137)\nφ(0, t) =φx(0, t) = 0 , u(L, t) =ux(L, t) = 0 , (138)\nφ(L0, t) =u(L0, t), φx(L0, t) =ux(L0, t), λ1φxx(L0, t) =λ2uxx(L0, t),\n(139)\nλ1φxxx(L0, t)−β1φttx(L0, t) +h1(φ(L0, t),−φx(L0, t)) +γ(−φtx(L0, t)) =\nλ2uxxx(L0, t)−β2uttx(L0, t) +h2(u(L0, t),−ux(L0, t)),\n(140)\nwith the initial conditions\n(φ, u)(x,0) = ( φ0, u0)(x),(φt, ut)(x,0) = ( φ1, u1)(x).\n•ψ=−φx, v=−ux;\n57•(ω, w)is the solution to\nρ1ωtt−σ1ωxx+g1(ω) =q1(x, t),(x, t)∈(0, L0)×(0, T),(141)\nρ2wtt−σ2wxx+g2(w) =q2(x, t),(x, t)∈(L0, L)×(0, T),(142)\nω(0, t) = 0 , w(L, t) = 0 , (143)\nσ1ωx(L0, t) =σ2wx(L0, t), ω(L0, t) =w(L0, t) (144)\nwith the initial conditions\n(ω, w)(x,0) = (0 ,0),(ωt, wt)(x,0) = ( ω1, w1)(x).\nProof. The proof uses the idea from [ 19, Ch. 4.3] and differs from it mainly\nin transmission conditions. We skip the details of the proof, which coincide\nwith [19].\nEnergy inequality (26) implies\n∂t(φ(n), ψ(n), ω(n), u(n), v(n), w(n)) bounded in L∞(0, T;Hv),(145)\nψ(n)bounded in L∞(0, T;H1(0, L0)),(146)\nv(n)bounded in L∞(0, T;H1(L0, L)) (147)\nω(n)\nx−l(n)φ(n)bounded in L∞(0, T;L2(0, L0)),(148)\nw(n)\nx−l(n)u(n)bounded in L∞(0, T;L2(L0, L)),(149)\nk(n)\n1(φ(n)\nx+ψ(n)+l(n)ω(n)) bounded in L∞(0, T;L2(0, L0)),(150)\nk(n)\n2(u(n)\nx+v(n)+l(n)w(n)) bounded in L∞(0, T;L2(L0, L)),(151)\nThus, we can extract subsequences which converge in corresponding spaces\nweak-∗. Similarly to [19] we have\nφ(n)\nx+ψ(n)+l(n)ω(n)∗⇀0 in L∞(0, T;L2(0, L0)),\ntherefore\nφx=−ψ.\n58Analogously,\nux=−v.\n(146)-(151) imply\nω(n)∗⇀ ω inL∞(0, T;H1(0, L0)), w(n)∗⇀ w inL∞(0, T;H1(L0, L)),\n(152)\nφ(n)∗⇀ φ inL∞(0, T;H1(0, L0)), u(n)∗⇀ u inL∞(0, T;H1(L0, L)).\n(153)\nThus, the Aubin’s lemma gives that\nΦ(n)→Φ strongly in C(0, T; [H1−ε(0, L0)]3×[H1−ε(L0, L)]3) (154)\nfor every ε >0 and then\n∂xφ0+ψ0+l(n)ω0→0 strongly in H−ε(0, L0),\nThis implies that\n∂xφ0=−ψ0, ω 0= 0.\nAnalogously,\n∂xu0=−v0, w 0= 0.\nLet us choose a test function of the form B= (β1,−β1\nx,0, β2,−β2\nx,0)∈FT\nsuch that β1\nx(L0, t) =β2\nx(L0, t) for almost all t. Due to (152) -(154) and\nLemma 7.2 we can pass to the limit in variational equality (25)asn→ ∞ .\nThe same way as in [ 19, Ch. 4.3] we obtain that the limiting functions φ, u\nare of higher regularity and satisfy the following variational equality\n59ZT\n0ZL0\n0\u0000\nρ1φtβ1\nt−β1φtxβ1\ntx\u0001\ndxdt +ZT\n0ZL\nL0\u0000\nρ2utβ2\nt−β1utxβ2\ntx\u0001\ndxdt−\nZL0\n0\u0000\nρ1(φtβ1\nt)(x,0)−β1(φtxβ1\ntx)(x,0)\u0001\ndx+ZL\nL0\u0000\nρ2(utβ2\nt)(x,0)−β1(utxβ2\ntx)(x,0)\u0001\ndx+\nZT\n0ZL0\n0λ1φxxβ1\nxxdxdt+ZT\n0ZL\nL0λ2uxxβ2\nxxdxdt−ZT\n0ZL0\n0γ′(−φxt)φtxxβ1dxdt+\nZT\n0ZL0\n0\u0000\nf1(φ,−φx)β1−h1(φ,−φx)β1\nx\u0001\ndxdt+ZT\n0ZL\nL0\u0000\nf2(u,−ux)β2−h2(u,−ux)β2\nx\u0001\ndxdt =\nZT\n0ZL0\n0\u0000\np1β1−r1β1\nx\u0001\ndxdt +ZT\n0ZL\nL0\u0000\np2β2−r2β2\nx\u0001\ndxdt. (155)\nProvided φ, uare smooth enough, we can integrate (155) by parts with\nrespect to x, tand obtain\nZT\n0ZL0\n0(ρ1−β1∂xx)φttβ1dxdt +ZT\n0ZL\nL0(ρ2−β2∂xx)uttβ2dxdt+\nZT\n0[β1φttx(t, L0)−β2uttx(t, L0)]β1(t, L0)dt+\nZT\n0ZL0\n0λ1φxxxxβ1dxdt +ZT\n0ZL\nL0λ2uxxxxβ2dxdt+\nZT\n0[λ1φxx−λ2uxx] (t, L0)β1\nx(t, L0)dt−ZT\n0[λ1φxxx−λ2uxxx] (t, L0)β1(t, L0)dt−\nZT\n0ZL0\n0γ′(−φxt)φxxtβ1dxdt−ZT\n0γ(−φxt(L0, t))β1(L0, t)+\nZT\n0ZL0\n0(f1(φ,−φx) +∂xh1(φ,−φx))β1dxdt+ZT\n0ZL\nL0(f2(u,−ux) +∂xh2(u,−ux))β2dxdt+\nZT\n0(h2(u(L0, t),−ux(L0, T))−h1(φ(L0, t),−φx(L0, T)))β1(L0, t)dt=\nZT\n0ZL0\n0(p1+∂xr1)β1dxdt+ZT\n0ZL\nL0(p2+∂xr2)β2dxdt+ZT\n0[r2(t, L0)−r1(t, L0)]β1(t, L0)dt.\n(156)\nRequiring all the terms containing β1(L0, t),β1\nx(L0, t) to be zero, we get\ntransmission conditions (139) -(137) . Equations (136) -(137) are recovered\n60from the variational equality (156).\nProblem (141)-(144) can be obtained in the same way.\nWe perform numerical modelling for the original problem with the initial\nparameters\nl(1)= 1, k(1)\n1= 4, k(1)\n2= 1;\nWe model the simultaneous convergence l→0 and k1, k2→ ∞ in the\nfollowing way: we divide lby the factor χand multiply k1, k2by the factor\nχ. Calculations performed for the original problem with\nχ= 1, χ= 3, χ= 10, χ= 30, χ= 100 , χ= 300\nand the limiting problem (136) -(140) . Other constants in the original problem\nare the same as in the previous subsection and we choose the functions in\nthe right-hand side of (134)-(135) as follows:\nr1(x) =x+ 4, r2(x) = 2 x.\nThe nonlinear feedbacks are\nf1(φ, ψ, ω ) = 4 φ3−2φ, f 2(u, v, w ) = 4 u3−8u,\nh1(φ, ψ, ω ) = 0 , h 2(u, v, w ) = 0 ,\ng1(φ, ψ, ω ) = 3|ω|ω, g 2(u, v, w ) = 6|w|w.\nWe use linear dissipation γ(s) =sand choose the following initial displace-\nment and shear angle variation\nφ0(x) =−13\n640x4+6\n40x2−23\n40x2,\nu0(x) =41\n2160x4−68\n135x3+823\n180x2−439\n27x+520\n27.\nψ0(x) =−\u0012\n−13\n160x3+27\n40x2−23\n20x\u0013\n,\n61v0(x) =−\u001241\n540x3−68\n45x2+823\n90x−439\n27\u0013\n.\nand set\nω0(x) =w0(x) = 0 .\nWe choose the following initial velocities\nφ1(x) =−1\n32x3+3\n16x2, u 1(x) =1\n108x3−7\n36x2+10\n9x−25\n27,\nω1(x) =ψ1(x) =3\n5x,\nw1(x) =v1(x) =−2\n5x+ 4.\nThe double limit case appeared to be more challenging from the point\nof view of numerics, then the case l→0. The numerical simulations of\nthe coupled system in equations (1)-(7)including the interface conditions\nin(8)-(11)were done by a semidiscretization of the functions ϕ, ψ, ω, u, v, w\nwith respect to the position xand by using an explicit scheme for the time\nintegration. That allows to choose the discretized values at grid points near\nthe interface in a separate step so that they obey the transmission conditions.\nIt was necessary to solve a nonlinear system of equations for the six functions\nat three grid points (at the interface, and left and right of the interface) in\neach time step. Any attempt to use a full implicit numerical scheme led to\nextremely time-expensive computations due to the large nonlinear system\nover all discretized values which was to solve in each time step. On the\nother hand, increasing k1, k2increase the stiffness of the system of ordinary\ndifferential equations which results from the semidiscretization, and the CFL-\nconditions requires small time steps — otherwise numerical oscillations occur.\nFigures 8-13 present smoothed numerical solutions, particularly necessary\nfor large factors χ, e. g. χ= 300. When the parameters k1, k2are large,\nthe material of the beam gets stiff, and so does the discretized system of\ndifferential equations. Nevertheless the oscillations are still noticeable in the\ngraph. The observation that the factor χcannot be arbitrarily enlarged,\nunderlines the importance of having the limit problem for χ→ ∞ in(1)-(15).\n620 1 2 3 4 5 6 7 8-1-0.500.511.52Figure 8: Transversal displacement of the beam, cross-section x= 2.\n0 1 2 3 4 5 6 7 8-4-2024\nFigure 9: Transversal displacement of the beam, cross-section x= 6.\n8 Discussion\nThere is a number of papers devoted to long-time behaviour of linear ho-\nmogeneous Bresse beams (with various boundary conditions and dissipation\nnature). If damping is present in all three equations, it appears to be sufficient\nfor the exponential stability of the system without additional assumptions\non the parameters of the problems (see, e.g., [2]).\nThe situation is different if we have a dissipation of any kind in one or\ntwo equations only. First of all, it matters in which equations the dissipation\nis present. There are results on the Timoshenko beams [ 25] and the Bresse\n630 1 2 3 4 5 6 7 8-20246Figure 10: Shear angle variation of the beam, cross-section x= 2.\n0 1 2 3 4 5 6 7 80510\nFigure 11: Shear angle variation of the beam, cross-section x= 6.\nbeams [ 13] that damping in only one of the equations does not guarantee the\nexponential stability of the whole system. It seems that for the Bresse system\nthe presence of the dissipation in the shear angle equation is necessary for the\nstability of any kind. To get the exponential stability, one needs additional\nassumptions on the coefficients of the problem, usually, the equality of the\npropagation speeds:\nk1=σ1,ρ1\nk1=β1\nλ1.\nOtherwise, only polynomial (non-uniform) stability holds (see, e.g., [ 1] for\nmechanical dissipation and [ 13] for thermal dissipation). In [ 6] analogous\n640 1 2 3 4 5 6 7 8-3-2-10123Figure 12: Longitudinal displacement of the beam, cross-section x= 2.\n0 1 2 3 4 5 6 7 8-3-2-1012\nFigure 13: Longitudinal displacement of the beam, cross-section x= 6.\nresults are established in case of nonlinear damping.\nIf dissipation is present in all three equations of the Bresse system,\ncorresponding problems with nonlinear source forces of local nature possesses\nglobal attractors under the standard assumptions for nonlinear terms (see,\ne.g., [ 24]). Otherwise, nonlinear source forces create technical difficulties and\nmay cause instability of the system. To the best of our knowledge, there is\nno literature on such cases.\nIn the present paper we study a transmission problem for the Bresse\nsystem.\nTransmission problems for various equation types have already had\n65some history of investigations. One can find a number of papers concern-\ning their well-posedness, long-time behaviour and other aspects (see, e.g.,\n[26] for a nonlinear thermoelastic/isothermal plate, or [ 14] for the Euler-\nBernoulli/Timoshenko beam and [ 15] for the full von Karman beam). Prob-\nlems with localized damping are close to transmission problems. In the recent\nyears a number of such problems for the Bresse beams were studied in, e.g.,\n[24,6]. To prove the existence of attractors in this case a unique continuation\nproperty is an important tool, as well as the frequency method.\nThe only paper we know on a transmission problem for the Bresse system\nis [28]. The beam in this work consists of a thermoelastic (damped) and elastic\n(undamped) parts, both purely linear. Despite the presence of dissipation\nin all three equations for the damped part, the corresponding semigroup is\nnot exponentially stable for any set of parameters, but only polynomially\n(non-uniformly) stable. In contrast to [ 28], we consider mechanical damping\nonly in the equation for the shear angle for the damped part. However, we\ncan establish exponential stability for the linear problem and existence of an\nattractor for the nonlinear one under restrictions on the coefficients in the\ndamped part only. The assumption on the nonlinearities can be simplified in\n1D case (cf. e.g. [16]).\nConflict of Interest Statement\nThe research was conducted in the absence of any commercial or financial\nrelationships that could be construed as a potential conflict of interest.\nAcknowledgements\nThe research is supported by the Volkswagen Foundation project ”From\nModeling and Analysis to Approximation”. The first and the third authors\nwere also successively supported by the Volkswagen Foundation project\n”Dynamic Phenomena in Elasticity Problems” at Humboldt-Universit¨ at zu\nBerlin, Funding for Refugee Scholars and Scientists from Ukraine.\n66References\n[1]Fatiha Alabau-Boussouira and Jaime E. Munoz-Rivera and Dilberto\nda S. Almeida Junior, Stability to weak dissipative Bresse system,\nJournal of Mathematical Analysis and Applications, 374 (2011), 481-\n498, https://doi.org/10.1016/j.jmaa.2010.07.046.\n[2]Dilberto da S. Almeida Junior and M L Santos, Numerical exponential\ndecay to dissipative Bresse system, Journal of Applied Mathematics,\n2010, Art. ID 848620, 17 pp.\n[3]Viorel Barbu, Nonlinear Semigroups and Differential Equations in\nBanach Spaces, Nordhof, 1976.\n[4]J. A. C. Bresse, Cours de Mechanique Appliquee, Mallet Bachelier,\nParis, 1859.\n[5] R.P. Kanwal, Generalized functions. Theory and applications,\nBirkh¨ auser, 2004.\n[6]Wenden Charles and J.A. Soriano and Fl´ avio A. Falc˜ ao Nascimento\nand J.H. Rodrigues, Decay rates for Bresse system with arbitrary\nnonlinear localized damping, Journal of Differential Equations, 255\n(2013), 2267-2290.\n[7]Igor Chueshov, Strong solutions and attractor of the von Karman\nequations (in Russian), Mathematicheskii Sbornik, 181 (1990) , 25–36.\n[8]Igor Chueshov, Introduction to the Theory of Infinite-Dimensional\nDissipative Systems, Acta, Kharkiv, 2002.\n[9]Igor Chueshov and Matthias Eller and Irena Lasiecka, On the attractor\nfor a semilinear wave equation with critical exponent and nonlinear\nboundary dissipation, Communications in Partial Differential Equations,\n27 (2002), 1901–1951.\n67[10] Igor Chueshov and Tamara Fastovska and Iryna Ryzhkova, Quasistabil-\nity method in study of asymptotical behaviour of dynamical systems, J.\nMath. Phys. Anal. Geom., 15 (2019), 448-501.10.15407/mag15.04.448.\n[11] Igor Chueshov and Irena Lasiecka, Long-time dynamics of von Kar-\nman semi-flows with non-linear boundary/interior damping, Journal of\ndifferential equation, 233 (2007), 42–86.\n[12] Igor Chueshov and Irena Lasiecka, Long-time behavior of second order\nevolution equations with nonlinear damping, Memoirs of AMS ,912,\nAMS, Providence, RI, 2008.\n[13] F. Dell’Oro, Asymptotic stability of thermoelastic systems of Bresse\ntype, J. Differential Equations, 258 (2015), 3902-3927.\n[14] Tamara Fastovska, Decay rates for Kirchhoff-Timoshenko transmission\nproblems, Communications on Pure and Applied Analysis, 12 (2013),\n2645-2667, 10.3934/cpaa.2013.12.2645.\n[15] Tamara Fastovska, Global attractors for a full von Karman beam\ntransmission problem” Communications on Pure and Applied Analysis,\n22 (2023), 1120-1158.\n[16] T. Fastovska, Attractor for a composite system of nonlinear wave and\nthermoelastic plate equations, Visnyk of Kharkiv National University\n70 (2014), 4-35.\n[17] AKh. Khanmamedov, Global attractors for von Karman equations\nwith nonlinear dissipation, Journal of Mathematical Analysis and\nApplications 318 (2016), 92-101, 10.1016/j.jmaa.2005.05.031.\n[18] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in\nnonlinear dynamic elasticity-full von Karman systems, Prog. Nonlinear\nDiffer. Equ. Appl. vol.50, Birkh¨ auser, Basel, 2002, 197-216.\n[19] J.E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadel-\nphia, PA, 1989.\n68[20]J. E. Lagnese and G. Leugering and E. J. P. G. Schmidt, Modeling, Anal-\nysis and Control of Dynamic Elastic Multi-Link Structures, Birkh auser,\nBoston, 1994.\n[21] Irena Lasiecka and Roberto Triggiani, Regularity of Hyperbolic Equa-\ntions Under L2(0, T;L2(Γ))-Dirichlet Boundary Terms, Appl. Math.\nOptim., 10 (1983), 275-286.\n[22] J.-L. Lions and E. Magenes, Probl´ emes aux limites non homog´ enes et\napplications, Vol 1, Dunod, Paris, 1968.\n[23] Weijiu Liu and Graham H. Williams, Exact Controllability for Prob-\nlems of Transmission of the Plate Equation with Lower-order Terms,\nQuarterly of Applied Math., 58 (2000), 37-68.\n[24] To Fu Ma and Rodrigo Nunes Monteiro, Singular Limit and Long-Time\nDynamics of Bresse Systems, SIAM Journal on Mathematical Analysis,\n49 (2017), 2468-2495, 10.1137/15M1039894.\n[25] Jaime E. Mu˜ noz Rivera, Reinhard Racke, Mildly dissipative nonlinear\nTimoshenko systems—global existence and exponential stability, Journal\nof Mathematical Analysis and Applications, 276 (2002), 10.1016/S0022-\n247X(02)00436-5.\n[26] M. Potomkin, A nonlinear transmission problem for a compound plate\nwith thermoelastic part, Mathematical Methods in the Applied Sciences,\n35 (2012), 530-546.\n[27] Roberto Triggiani and P. F. Yao, Carleman Estimates with No Lower-\nOrder Terms for General Riemann Wave Equations. Global Uniqueness\nand Observability in One Shot, Appl. Math. Optim., 46 (2002), 331-375.\n[28]W. Youssef, Asymptotic behavior of the transmission problem of the\nBresse beam in thermoelasticity, Z. Angew. Math. Phys., 73 (2022),\n10.1007/s00033-022-01797-7.\n69"    },    {        "title": "2009.01107v1.Developments_in_Lorentz_and_CPT_Violation.pdf",        "content": "arXiv:2009.01107v1  [hep-ph]  2 Sep 2020Proceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n1\nDevelopments in Lorentz and CPT Violation\nV. Alan Kosteleck´ y\nPhysics Department, Indiana University\nBloomington, IN 47405, USA\nThis talk at the CPT’19 meeting outlines a few recent develop ments in Lorentz\nand CPT violation, with particular attention to results obt ained by researchers\nat the Indiana University Center for Spacetime Symmetries.\n1. Introduction\nMotivated by the prospect of minuscule observable effects arising f rom\nPlanck-scale physics, searches for Lorentz and CPT violation have made\nimpressive advances in recent years.1The scope of ongoing efforts pre-\nsented at the CPT’19meeting indicates that this rapid pace ofdevelo pment\nwill continue unabated, with experiments achieving sensitivities to Lo rentz\nviolation that are orders of magnitude beyond present capabilities a nd pro-\nviding unprecedented probes of the CPT theorem. Substantial th eoretical\nadvances in the subject are also being made, and the prospects ar e excel-\nlent for completing a comprehensive description of possible effects o n all\nforces and particles and for achieving a broad understanding of th e under-\nlying mathematical structure in the near future. In this talk, I sum marize\nsome basics of the subject and outline a few recent results obtaine d at the\nIndiana University Center for Spacetime Symmetries (IUCSS).\n2. Basics\nNocompellingexperimentalevidenceforLorentzorCPTviolationhas been\nreported to date, so any effects are expected to involve only tiny d evia-\ntions from the physics of General Relativity (GR) and the Standard Model\n(SM). In studying the subject, it is therefore desirable to work wit hin a\ntheoretical description of Lorentz and CPT violation that is both mo del in-\ndependent and includes all possibilities consistent with the structur e of GR\nand the SM. The natural context for a description of this type is eff ective\nfield theory.2Therealisticandcoordinate-independenteffectivefield theoryProceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n2\nfor Lorentz and CPT violation is known as the Standard-Model Exte nsion\n(SME).3,4It can be obtained by incorporating all coordinate-independent\nand Lorentz-violating terms in the action for GR coupled to the SM. T hese\nterms alsodescribe generalrealistic CPTviolation,3,5and they arecompat-\nible with either spontaneous or explicit Lorentz violation in an underlyin g\nunified theory such as strings.6\nThe SME action incorporates Lorentz-violating operators of any m ass\ndimension d, with the minimal SME defined to include the subset of op-\nerators of renormalizable dimension d≤4. A given SME term is formed\nas the observer-scalar contraction of a Lorentz-violating opera tor with a\ncoefficient for Lorentz violation that acts as a background coupling to con-\ntrol observableeffects. The propagationand interactionsofeac h species are\nmodified and can vary with momentum, spin, and flavor. All minimal-SME\nterms3,4and many nonminimal terms7–9have been explicitly constructed.\nThe resulting experimental signals are expected to be suppressed either di-\nrectly or through a mechanism such as countershading via naturally small\ncouplings.10Impressive constraints on SME coefficients from many experi-\nments have been obtained.1The generality of the SME framework insures\nthese constraints apply to any specific Lorentz-violating model th at is con-\nsistent with realistic effective field theory.\nThe geometryofLorentzviolationis an interestingissue for explora tion.\nIf the Lorentz violation is spontaneous, then the geometry can re main Rie-\nmann or Riemann-Cartan4and the phenomenology incorporates Nambu-\nGoldstone modes.11However, if the Lorentz violation is explicit, then the\ngeometry cannot typically be Riemann and is conjectured to be Finsle r in-\nstead.4Support for this idea has grown in recent years,12,13but a complete\ndemonstration is lacking at present.\n3. Developments from the IUCSS\nIn the past three years, developments from the IUCSS have prima rily in-\nvolved the quark, gauge, and gravity sectors. In the quark sect or, direct\nconstraints on minimal-SME coefficients can be extracted using neut ral-\nmeson oscillations,14and numerous experiments on K,D,Bd, andBs\nmixing have achieved high sensitivities to CPT-odd effects on the u,d,s,c,\nandbquarks.15Recent work reveals that nonminimal quark coefficients at\nd= 5 provide numerous independent measures of CPT violation,16many\nof which are experimentally unconstrained to date. The tquark decays\ntoo rapidly to hadronize, but t-tpair production and single- tproductionProceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n3\nare sensitive to t-sector coefficients and are the subject of ongoing exper-\nimental analyses.17High-energy studies of deep inelastic scattering and\nDrell-Yan processes also offer access to quark-sector coefficient s,18and cor-\nresponding data analyses are being pursued. Another active line of reason-\ning adapts chiral perturbation theory to relate quark coefficients to hadron\ncoefficients,19yielding further tests of Lorentz and CPT symmetry.\nIn the gauge sector, the long-standing challenge of constructing all non-\nabelianLorentz-violatingoperatorsatarbitrary dhasrecentlybeensolved.8\nThe methodology yieldsall matter-gaugecouplings, so the full Lore ntz-and\nCPT-violating actions for quantum electrodynamics, quantum chro mody-\nnamics, and related theories are now available for exploration. Cons traints\non photon-sector coefficients continue to improve.20Signals of Lorentz vio-\nlation arising in clock-comparison experiments at arbitrary dhave recently\nbeen studied,21revealing complementary sensitivities from fountain clocks,\ncomagnetometers, ion traps, lattice clocks, entangled states, a nd antimat-\nter. These various advances suggest excellent prospects for fu ture searches\nfor Lorentz and CPT violation in the gauge and matter sectors.\nIn the gravity sector, all operators modifying the propagation of the\nmetric perturbation hµν, including ones preservingorviolating Lorentz and\ngauge invariance, have been classified and constructed.9Many of the corre-\nsponding coefficients are unexploredbut could be measured via grav itation-\nwave and astrophysical observations. A general methodology ex ists for an-\nalyzing Lorentz-violation searches in experiments on short-range gravity,9\nand constraints on certain coefficients with dup to eight have now been\nobtained.22Workin progressfurther extends gravity-sectortests to matt er-\ngravitycouplingsatarbitrary d.23ResultsfromtheSMEcanalsobeapplied\nto constrain hypothesized Lorentz-invariant effects whenever t hese lead to\nnonzero background values for vector or tensor objects. This id ea recently\nyielded the first experimental constraints on all components of no nmetric-\nity.24At the foundational level, further confirmation of the correspon dence\nbetween the SME and Finsler geometry has been established via the c on-\nstruction of all Finsler geometries for spin-independent Lorentz- violating\neffects.13The scope and breadth of all these results augurs well for future\nadvances in the gravity sector on both theoretical and experimen tal fronts.\nAcknowledgments\nThis work was supported in part by U.S. D.o.E. grant DE-SC0010120 a nd\nby the Indiana University Center for Spacetime Symmetries.Proceedings of the Eighth Meeting on CPT and Lorentz Symmetr y (CPT’19), Indiana University, Bloomington, May 12–16, 2019\n4\nReferences\n1. V.A. Kosteleck´ y and N. Russell, arXiv:0801.0287v13 (20 20).\n2. See, e.g., S. Weinberg, Proc. Sci. CD 09, 001 (2009).\n3. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev.\nD58, 116002 (1998).\n4. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n5. O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002).\n6. V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 39, 683 (1989); V.A. Kost-\neleck´ y and R. Potting, Nucl. Phys. B 359, 545 (1991); Phys. Rev. D 51, 3923\n(1995); V.A. Kosteleck´ y and R. Lehnert, Phys. Rev. D 63, 065008 (2001).\n7. V.A. Kosteleck´ y and M. Mewes, Phys. Rev. D 80, 015020 (2009); Phys. Rev.\nD85, 096005 (2012); Phys. Rev. D 88, 096006 (2013); Y. Ding and V.A.\nKosteleck´ y, Phys. Rev. D 94, 056008 (2016).\n8. V.A. Kosteleck´ y and Z. Li, Phys. Rev. D 99, 056016 (2019).\n9. Q.G. Bailey et al., Phys. Rev. D 91, 022006 (2015); V.A. Kosteleck´ y and M.\nMewes, Phys. Lett. B 757, 510 (2016); Phys. Lett. B 766, 137 (2017); Phys.\nLett. B779, 136 (2018).\n10. V.A.Kosteleck´ yand J.D. Tasson, Phys.Rev.Lett. 102, 010402 (2009); Phys.\nRev. D83, 016013 (2011).\n11. R. Bluhm and V.A. Kosteleck´ y, Phys. Rev. D 71, 065008 (2005); V.A. Kost-\neleck´ y and R. Potting, Gen. Rel. Grav. 37, 1675 (2005); Phys. Rev. D 79,\n065018 (2009); B. Altschul et al., Phys. Rev. D 81, 065028 (2010).\n12. M. Schreck, Phys. Lett. B 793, 70 (2019); D. Colladay and P. McDonald,\nPhys. Rev. D 85, 044042 (2012); V.A. Kosteleck´ y, Phys. Lett. B 701, 137\n(2011); V.A. Kosteleck´ y and N. Russell, Phys. Lett. B 693, 2010 (2010).\n13. B.R. Edwards and V.A. Kosteleck´ y, Phys. Lett. B 786, 319 (2018).\n14. V.A. Kosteleck´ y, Phys. Rev. Lett. 80, 1818 (1998).\n15. K.R.Schubert, arXiv:1607.05882; R.Aaij et al., Phys.Rev.Lett. 116, 241601\n(2016); V.M. Abazov et al., Phys. Rev. Lett. 115161601 (2015); D. Babusci\net al., Phys. Lett. B 730, 89 (2014).\n16. B.R. Edwards and V.A. Kosteleck´ y, Phys. Lett. B 795, 620 (2019).\n17. V.M. Abazov et al., Phys. Rev. Lett. 108, 261603 (2012); M.S. Berger et al.,\nPhys. Rev. D 93, 036005 (2016); A. Carle et al., arXiv:1908.11256.\n18. V.A. Kosteleck´ y et al., Phys. Lett. B 769, 272 (2017); JHEP 04, 143 (2020).\n19. R.Kamand et al., Phys. Rev. D 95, 0556005 (2017); Phys. Rev.D 97, 095027\n(2018); B. Altschul and M.R. Schindler, Phys. Rev. D 100, 075031 (2019);\nJ.P. Noordmans et al., Phys. Rev. C 94, 025502 (2016).\n20. L. Pogosian et al., Phys. Rev. D 100, 023407 (2019); F. Kislat, Symmetry\n10, 596 (2018); F. Kislat and H. Krawczynski, Phys. Rev. D 95, 083013\n(2017); J.J. Wei et al., Ap. J.842, 115 (2017).\n21. V.A. Kosteleck´ y and A.J. Vargas, Phys. Rev. D 98, 036003 (2018).\n22. C.G. Shao et al., Phys. Rev. Lett. 122, 011102 (2019); Phys. Rev. Lett. 117,\n071102 (2016).\n23. V.A. Kosteleck´ y and Z. Li, arXiv:2008.12206.\n24. J. Foster et al., Phys. Rev. D 95, 084033 (2017)."    },    {        "title": "1812.07994v1.Gravitational_quasinormal_modes_of_black_holes_in_Einstein_aether_theory.pdf",        "content": "arXiv:1812.07994v1  [gr-qc]  18 Dec 2018Gravitational quasinormal modes of black holes in Einstein -aether theory\nChikun Ding∗\nDepartment of Physics, Hunan University of Humanities,\nScience and Technology, Loudi, Hunan 417000, P. R. China\nKey Laboratory of Low Dimensional Quantum Structures and Qu antum Control of Ministry of Education,\nand Synergetic Innovation Center for Quantum Effects and Appl ications,\nHunan Normal University, Changsha, Hunan 410081, P. R. Chin a\nAbstract\nThe local Lorentz violation (LV) in gravity sector should sh ow itself in derivation of the char-\nacteristic quasinormal modes (QNMs) of black hole mergers f rom their general relativity case. In\nthis paper, I study QNMs of the gravitational field perturbat ions to Einstein-aether black holes\nand, at first compare them to those in Schwarzschild black hol e, and then some other known LV\ngravity theories. By comparing to Schwarzschild black hole , the first kind aether black holes have\nlarger damping rate and the second ones have lower damping ra te. And they all have smaller real\noscillation frequency of QNMs. By comparing to some other LV theories, the QNMs of the first\nkind aether black hole are similar to that of the QED-extensi on limit of standard model extension,\nnon-minimal coupling to Einstein’s tensor and massive grav ity theories. While as to the second kind\naether black hole, they are similar to those of the noncommut ative gravity theories and Einstein-\nBorn-Infeld theories. These similarities may imply that LV in gravity sector and LV in matter sector\nhave some intrinsic connections.\nPACS numbers: 04.50.Kd, 04.70.Dy, 04.30.-w\nI. INTRODUCTION\nLIGO (Laser Interferometer Gravitational wave Observatory) has detected gravitational wave(GW) from a\nbinary black hole coalescence for the fourth time, on August 14, 20 17 (GW170814) [1]. It provides a direct\nconfirmationfortheexistenceofablackholeand, confirmsthatbla ckholemergersarecommonintheuniverse.\nThe black-hole binary systems are intrinsically strongly gravitating o bjects that curve spacetime dramatically,\nand the detections of GW from them give us opportunity and an ideal tool to stress test general relativity\n(GR) [2]. Some of the detections are used to test alternative theor ies of gravity where Lorentz invariance (LI)\nis broken which affects the dispersion relation for GW [3]. For the first time, they used GW170104 to put\nupper limits on the magnitude of Lorentz violation (LV) tolerated by t heir data and found that the bounds are\nimportant. Sotiriou [4] argued that when higher order corrections to dispersion relation of GW are present,\nthere will be a scalar excitation which travels at a speed different fro m that of the standard GW polarizations\nor light. Hence, a smoking-gun observation of LV would be the direct detection of this scalar wave.\nLorentz invariance(LI) is one of the fundamental principles of GR a nd the standard model(SM) of particles\nand fields. Why consider LV? Because LI may not be an exact symmet ry at all energies [5], particularly when\none considering quantum gravityeffect, it should not be applicable. T hough both GR and SM based on LI and\nthe background of spacetime, they handle their entities in profoun dly different manners. GR is a classical field\ntheory in curved spacetime that neglects all quantum properties o f particles; SM is a quantum field theory in\n∗Email: dingchikun@163.com; Chikun Ding@huhst.edu.cn2\nflat spacetime that neglects all gravitational effects of particles. For collisions of particles of 1030eV energy\n(energy higher than Planck scale), the gravitational interactions predicted by GR are very strong and gravity\nshould not be negligible[6]. So in this very high energy scale, one have to consider merging SM with GR in a\nsingle unified theory, known as ”quantum gravity”, which remains a c hallenging task. Lorentz symmetry is a\ncontinuous spacetime symmetry and cannot exist in a discrete spac etime. Therefore quantization of spacetime\nat energies beyond the Planck energy, Lorentz symmetry is invalid a nd one should reconsider giving up LI.\nThere are some phenomena of LV. On the SM side, there is an a prioriunknown physics at high-energy scales\nthat could lead to a spontaneous breaking of LI by giving an expecta tion value to certain non-SM fields that\ncarry Lorentz indices[7]. LI also leads to divergences in quantum field theory which can be cured with a short\ndistance of cutoff that breaks it [8]. On the GR side, astrophysical observations suggest that the high-energy\ncosmic rays above the Greisen-Zatsepin-Kuzmin cutoff are a result of LV[9].\nThus, the study of LV is a valuable tool to probe the foundations of modern physics. These studies include\nLV in the neutrino sector [10], the standard-model extension [11], L V in the non-gravity sector [12], and LV\neffect on the formation of atmospheric showers [13]. Einstein-aeth er theory can be considered as an effective\ndescription of Lorentz symmetry breaking in the gravity sector an d has been extensively used in order to\nobtain quantitative constraints on Lorentz-violating gravity[14].\nIn Einstein-aether theory, the background tensor fields uabreak the Lorentz symmetry only down to a\nrotation subgroup by the existence of a preferred time direction a t every point of spacetime. The introduction\nof the aether vector allows for some novel effects, e.g., matter fie lds can travel faster than the speed of\nlight [15], dubbed superluminal particle. Due the existence of the sup erluminal gravitational modes, so the\ncorresponding light-cones can be completely flat, and the causality is more like that of Newtonian theory[16].\nIt is the universal horizons that can trap excitations traveling at a rbitrarily high velocities. Recently, two\nexact charged black hole solutions and their Smarr formula on univer sal horizons in 4- and 3-dimensional\nspacetime were found by Ding et al[17, 18]. Constraints on Einstein-aether theory were studied by Oo stet al\n[19] and, gravitational wave studied by Gong et al[20] after GW170817. Other studies on universal horizons\ncan be found in [21].\nIn Ref. [22], Ding et alstudied Hawking radiation from the charged Einstein aether black ho le and found\nthat i)the universalhorizonseemstobe noroleontheprocessofr adiatingluminal orsubluminalparticlesand,\nits temperature is dependant on z,TUH= (z−1)κuh/zπ, wherezcharacterizes the species of the particles;\nwhile ii) the Killing horizon seems to be no role on superluminal particle rad iation. Since up to date, the\nparticles with speed higher than vacuum light speed aren’t yet found , we here consider only subluminal or\nluminal particles perturbation to these LV black holes. In 2007, Kon oplyaet al[23] studied the gravitational\nperturbations of the non-reduced Einstein aether black holes and found that both the real part and the\nabsolute imaginary part of QNMs increase with the aether coefficient c1.\nRecently, Ding studied the scalar and electromagnetic perturbatio ns of the first and second Einstein-\naether black holes, and found [24] that their characteristics are s imilar to that of another LV model—the\nQED(quantum electrodynamics)-extension limit of SME (SM extensio n) [43]. What is about their gravita-\ntional perturbations? There are two types of perturbations of a black hole: adding fields to the black hole\nspacetime or perturbing the black hole metric itself. Then the scalar and electromagnetic perturbations are\nof the first type. After the coalescence of a binary black holes or t he formation of a black hole by collapse,\nthe black hole is in a perturbed state,\ngµν=g0\nµν+δgµν, (1.1)\nwhere the metric g0\nµνis of the nonperturbed black hole when all perturbations have been damped. This is the\nsecond type — gravitational perturbation which is important for em itting gravitational waves. In the linear\n[43] For SME, see Appendix in the reference [24]3\napproximation, the perturbations δgµνare supposed to be much less than the background δgµν≪g0\nµν. The\nbackground g0\nµνcan be Schwarzschild or Einstein-aether black hole solutions.\nIn this paper, I study the gravitational QNMs for two kinds of Einst ein-aether black holes and compare\nthem to Schwarzschild black hole to find some derivations. And I also c ompare them to black holes in some\nLV theories to find some connections between these theories. The plan of rest of our paper is organized as\nfollows. In Sec. II, I review brieflythe Einstein aether blackholes an d the sixth orderWKB(Wentzel-Kramers-\nBrillouin)method. InSec. III,IadopttothesixthorderWKBmetho dandobtaintheperturbationfrequencies\nof the first kind Einstein aether black holes. In Sec. IV, I discuss th e QNMs for the second kind Einstein\naether black hole. In Sec. V, I present a summary. In Appendix, I b riefly introduce some Lorentz violating\ntheories, i.e., nonminimal coupling, massive gravity, noncommutative and Einstein-Born-Infeld theories.\nII. EINSTEIN AETHER BLACK HOLES AND WKB METHOD\nThe general action for the Einstein-aether theory can be constr ucted by assuming that: (1) it is general\ncovariant; and(2)it isafunction ofonlythespacetimemetric gabandaunit timelikevector ua, andinvolvesno\nmore than two derivatives of them, so that the resulting field equat ions are second-order differential equations\nofgabandua. Then, the Einstein aether theory to be studied in this paper is desc ribed by the action [25],\nS=/integraldisplay\nd4x√−g/bracketleftBig1\n16πGæ(R+Læ)/bracketrightBig\n, (2.1)\nwhereGæis the aether gravitational constant, Læis the aether Lagrangian density\n−Læ=Zab\ncd(∇auc)(∇bud)−λ(u2+1) (2.2)\nwith\nZab\ncd=c1gabgcd+c2δa\ncδb\nd+c3δa\ndδb\nc−c4uaubgcd, (2.3)\nwhereci(i= 1,2,3,4) are coupling constants of the theory. The aether Lagrangian d ensity is therefore the\nsum of all possible terms for the aether field uaup to mass dimension two, and the constraint term λ(u2+1)\nwith the Lagrange multiplier λimplementing the normalization condition u2=−1. The equations of motion,\nobtained by varying the action (2.1) with respect to gab,ua,λ, can be found in Ref. [17, 25].\nTherearea numberoftheoreticaland observationalboundson t he coupling constants ci[14, 19, 26, 27], e.g.,\nrequiring stability and absence of gravitational Cherenkov radiatio n for theoretical constraint, Solar System\nand cosmological observations, etc. The strong field such as binar y pulsar gives more stringent constraints\non the couplings, c13/lessorsimilar0.03, wherec13≡c1+c3, and so on. But this analysis of constraints is only valid in\nthe small coupling region ci≪1. Under the condition that sufficiently large sensitivity and large cou plings,\nthe Einstein-aether terms can in principle dominate over the GR ones [26]. Recently, Oost et alfind that the\nGW170817 and GRB170817 events provides much more severe cons traint that |c13|<10−15[19]. However, I\nhere don’t concern these severe constraints and would draw our a ttention on the LV gravity theory itself for\ntheoretical interest. Therefore from this point of view, I impose t he following theoretical constraints [28],\n0≤c14<2,2+c13+3c2>0,0≤c13<1. (2.4)\nThe static, spherically symmetric metric for Einstein-aether black h ole spacetime can be written in the form\nds2=−f(r)dt2+dr2\nf(r)+r2(dθ2+sin2θdφ2). (2.5)\nThere are two kinds of exact solutions [17, 28]. In the first case c14= 0, c123/negationslash= 0 (termed the first kind aether\nblack hole), the metric function is\nf(r) = 1−2M\nr−I/parenleftBig2M\nr/parenrightBig4\n, I=27c13\n256(1−c13). (2.6)4\nIf the coefficient c13= 0, then it reduces to Schwarzschild black hole. The quantity Mis the mass of the\nblack hole spacetime. In this case, the aether field has no contribut ion to black hole mass. In the second case\nc14/negationslash= 0, c123= 0 (termed the second kind aether black hole), the metric function is\nf(r) = 1−2M\nr−J/parenleftBigM\nr/parenrightBig2\n, J=c13−c14/2\n1−c13. (2.7)\nHerec13≥c14/2 (cf. 4.26 in [17]). In this case, the aether field contributes spacet ime mass as Mæ=\n−c14MADM/2[17], where MADMisKomarmass. Ifthecoefficient c13=c14/2,italsoreducestoSchwarzschild\nblack hole.\nTo gravitational field perturbations, we shall neglect small pertur bations of aether field, keeping only linear\nperturbations of Ricci tensor for simplicity. According to Chandra sekhar designations [29], the general form\nof the perturbed metric is\nds2=e2νdt2−e2ψ(dφ−σdt−qrdr−qθdθ)2−e−2µ2dr2−e−2µ3dθ2, (2.8)\nwheree2ν=e2µ2=f(r),e2µ3=r2,e2ψ=r2sin2θandσ=qr=qθ= 0 for non-perturbed case. The pertur-\nbations will lead to non-vanishing values of σ,qr,qθand increments in ν,µ2,µ3,ψ, which are corresponding\nto axial and polar perturbations, respectively. Here we shall cons ider the axial type ones. The perturbation\nequation reads\nr4∂\n∂r/parenleftBigf(r)\nr2∂Q\n∂r/parenrightBig\n+sin3θ∂\n∂θ/parenleftBig1\nsin3θ∂Q\n∂θ/parenrightBig\n−r2\nf(r)∂2Q\n∂t2= 0, (2.9)\nwhere\nQ(t,r,θ) =eiωtQ(r,θ), Q(r,θ) =r2f(r)sin3θQrθ, Qrθ=qr,θ−qθ,r. (2.10)\nFurther with Q(r,θ) =rΨ(r)C−2/3\nl+2, it can be reduced to Schrodinger wave-like equations:\nd2Ψ\ndr2∗+[ω2−V(r)]Ψ = 0, dr∗=f(r)dr, (2.11)\nfor gravitational field Ψ. The effective potentials take the form as:\nV=f(r)/bracketleftbigg(l+2)(l−1)+2f(r)\nr2−1\nrdf(r)\ndr/bracketrightbigg\n. (2.12)\nThe effective potentials Vdepend on the value r, angular quantum number (multipole momentum) land the\naether coefficient c13.\nFrom the potential formula (2.12) and the metric function (2.6), th e effective potential for the first kind\naether black hole is\nV=/parenleftbig\n1−2M\nr/parenrightbig/bracketleftbiggl(l+1)\nr2−6M\nr3/bracketrightbigg\n+16M4I\nr6/bracketleftbigg18M\nr+96IM4\nr4−l(l+1)−6/bracketrightbigg\n, (2.13)\nwhere the first two terms are Schwarzschild potential, the rests a re the aether modified terms, shown in Fig.\n1.\nIn Fig. 1, it is the effective potential of gravitationalfield perturba tions near the first kind aether black hole.\nObviously, if c13= 0, it can be reduced to those of the Schwarzschild black hole. The p eak value gets lower\nand the turning point shifts to right with c13increasing. This potential behavior is similar to that of some\nother Lorentz violating theories. In the theory of coupling to Einst ein’s tensor of Reissner-Norstr¨ om black\nhole (see Appendix A), the coupling parameter ηdecreases the peak value of scalar field potential for all l>0\n[30]. In the massive gravity theory (see Appendix B), the scalar cha rgeˆSdecreases the peak value of scalar\nfield potential and shifts its turning point to right [31]. In the Einstein -Born-Infeld theory (see Appendix D),5\n0 2 4 6 8 10r/Slash1M0.050.100.150.20Vg/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0.0\n/GothicL/Equal2\n0 2 4 6 8 10r/Slash1M0.10.20.30.4Vg/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0\n/GothicL/Equal3\n0 2 4 6 8 10r/Slash1M0.10.20.30.40.50.60.7Vg/LParen1r/RParen1\nc13/Equal0.8c13/Equal0.6c13/Equal0.0\n/GothicL/Equal4\nFIG. 1: The figures are the effective potentials of gravitatio nal field perturbation Vgnear the first kind aether black\nhole (M= 1) with different angular quantum number land coefficients c13.\nthe Born-Infeld scale parameter bdecreases the peak value of scalar and gravitational field potentia l for alll\n[32]. In the QED-extension limit of SME theory (see Appendix in Ref. [2 4]), Lorentz violation vector bµalso\ndecreases the peak value of Dirac field potential and shifts its turn ing point to right [33]. These properties of\nthe potential will imply that the quasinormal modes possess some diff erent behavior from those of GR case\nand, some similarities between these Lorentz violating theories.\nFrom the potential formula (2.12) and the metric function (2.7), th e effective potential for the second kind\naether black hole is\nV=/parenleftbig\n1−2M\nr/parenrightbig/bracketleftbiggl(l+1)\nr2−6M\nr3/bracketrightbigg\n+M2J\nr4/bracketleftbigg14M\nr+4JM2\nr2−l(l+1)−4/bracketrightbigg\n, (2.14)\nwhere the the first two terms are Schwarzschild potential, the res ts are the aether modified terms, shown in\nFig. 2.\n0 2 4 6 8 10r/Slash1M0.050.100.150.20Vg/LParen1r/RParen1\nc13/Equal0.78c13/Equal0.60c13/Equal0.01\n/GothicL/Equal2\n0 2 4 6 8 10r/Slash1M0.10.20.30.4Vg/LParen1r/RParen1\nc13/Equal0.78c13/Equal0.60c13/Equal0.01\n/GothicL/Equal3\n0 2 4 6 8 10r/Slash1M0.10.20.30.40.50.60.7Vg/LParen1r/RParen1\nc13/Equal0.78c13/Equal0.60c13/Equal0.01\n/GothicL/Equal4\nFIG. 2: The figures are the effective potentials of gravitatio nal field perturbation Vgnear the second kind aether black\nhole (M= 1) with different coefficients c13and fixed coefficient c14= 0.02.\nIn Fig. 2, it is the effective potential of the gravitational field pertu rbation near the second kind aether\nblack hole. It is easy to see that for all l, the peak value of the potential barrier gets lower with c13increasing\njust like the first kind aether black hole and decreasing more quickly. Although both kinds of black holes have\nsimilar potential with coupling constant c13, their QNM behaviors with c13will be different with each other\npartially.\nThe Schr¨ odinger-like wave equation (2.11) with the effective poten tial (2.12) containing the lapse function\nf(r) related to the Einstein-aether black holes is not solvable analytically . Since then I now use the sixth-\norder WKB approximation method to evaluate the quasinormal mode s of gravitational field perturbation to\nthe first and second kind aether black holes. The third-order WKB s emianalytic method has been proved to\nbe accurate up to around one percent for the real and the imagina ry parts of the quasinormal frequencies for\nlow-lying modes with n < l[34, 35]. The sixth-order WKB method shows more accurate than t hird-order\none [36]. Due to its considerable accuracy for lower lying modes, this m ethod has been used extensively in\nevaluating quasinormal frequencies of various black holes. In the s ixth approximation, the formula for the\ncomplex quasinormal frequencies is given by\nω2=V0+Λ2−i/radicalbig\n−2V′′\n0(α+Λ3+Λ4+Λ5+Λ6), (2.15)6\nTABLE I: The lowest overtone ( n= 0) quasinormal frequencies of the gravitational field in th e first kind aether black\nhole spacetime.\nc13Mω(l= 2) Mω(l= 3) Mω(l= 4) Mω(l= 5) Mω(l= 6)\n0.00 0.373619-0.088891 i0.599443-0.092703 i0.809178-0.094164 i1.012300-0.094871 i1.212010-0.095266 i\n0.15 0.371123-0.089949 i0.595985-0.093676 i0.804653-0.095142 i1.006720-0.095852 i1.205390-0.096250 i\n0.30 0.367765-0.091235 i0.591354-0.094848 i0.798599-0.096326 i0.999260-0.097042 i1.196530-0.097444 i\n0.45 0.363006-0.092838 i0.584808-0.096284 i0.790050-0.097785 i0.988731-0.098514 i1.184040-0.098924 i\n0.60 0.355705-0.094918 i0.574769-0.098074 i0.776938-0.099617 i0.972582-0.100371 i1.164880-0.100795 i\n0.75 0.342863-0.097769 i0.557015-0.100290 i0.753718-0.101918 i0.943965-0.102720 i1.130900-0.103173 i\n0.90 0.311907-0.101800 i0.513073-0.102246 i0.696009-0.104045 i0.872663-0.104961 i1.046130-0.105484 i\nwhere\nΛ2=1\n8/parenleftBigV(4)\n0\nV′′\n0/parenrightBig/parenleftBig1\n4+α2/parenrightBig\n−1\n288/parenleftBigV(3)\n0\nV′′\n0/parenrightBig2\n(7+60α2),\nΛ3=α\n−2V′′\n0/bracketleftBig5\n6912/parenleftBigV(3)\n0\nV′′\n0/parenrightBig4/parenleftBig\n77+188α2/parenrightBig\n−1\n384/parenleftBigV′′′2\n0V(4)\n0\nV′′3\n0/parenrightBig\n(51+100α2)\n+1\n2304/parenleftBigV(4)\n0\nV′′\n0/parenrightBig2\n(67+68α2)1\n288/parenleftBigV′′′\n0V(5)\n0\nV′′2\n0/parenrightBig\n(19+28α2)\n−1\n288/parenleftBigV(6)\n0\nV′′\n0/parenrightBig\n(5+4α2)/bracketrightBig\n, (2.16)\nand\nα=n+1\n2, V(m)\n0=dmV\ndrm∗/vextendsingle/vextendsingle/vextendsingle\nr∗(rp), (2.17)\nthe constants Λ 4,Λ5,Λ6are from Ref. [36][44], nis overtone number and rpis the turning point value of\npolar coordinate rat which the effective potential (2.12) reaches its maximum. Substit uting the effective\npotentialV(2.12) into the formula above, we can obtain the quasinormal frequ encies for the gravitational\nfield perturbations to Einstein-aether black holes. In the next sec tions, we obtain the quasinormal modes for\nboth kinds of Einstein-aether black holes and analyze their propert ies.\nIII. QUASINORMAL MODES FOR THE FIRST KIND AETHER BLACK HOLE\nIn this section, I study the gravitational field perturbations to th e first kind Einstein aether black hole. The\nperturbation frequencies are shown in Tab. I and Figs. from 3 to 5.\nBehaviors with fixed l,nandc13. Tab. I shows the real and the absolute imaginary parts of freque n-\ncies are both lower than the electromagnetical and scalar fields per turbations obtained in Ref. [24], i.e.,\ngravitational <electromagnetical <scalar. It shows that gravitational perturbations have lowest da mping rates\nand smallest oscillation frequencies.\nBehaviors with different l. Tab. I shows that, for fixed c13, both the real part and the absolute imaginary\npart of frequencies increase with the angular quantum number l. For large l, the imaginary parts approach a\nfixed value. These properties are similar to the usual black holes and , also shown in Fig. 4. These increasing\nbehaviors are similar to the electromagnetical field perturbations, but different from scalar field ones whose\nabsolute imaginary part decrease with l. Tab I also shows the derivations from Schwarzschild black hole. For\n[44] The relation between Qithere and here VisQi=−V(i)\n0.7\nn/Equal0,/GothicL/Equal2,c13is from0.96 to 0c13/Equal0.96 0.93\n0.90\n0.87\n0.28 0.30 0.32 0.34 0.360.0900.0920.0940.0960.0980.1000.102\nReΩ/MinusImΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4\n/GothicL/Equal5,c13is from0.96 to 0.00.96\n0.96\n0.96\n0.96\n0.96\n0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.0\nReΩ/MinusImΩ\nFIG. 3: The relationship between the real and imaginary part s of quasinormal frequencies of the gravitational field in\nthe background of the first kind aether black hole with the dec reasing of c13.\nn/Equal0\n/GothicL/Equal2/GothicL/Equal3/GothicL/Equal4/GothicL/Equal5/GothicL/Equal6\n0.0 0.2 0.4 0.6 0.80.40.60.81.01.2\nc13ReΩn/Equal0\n/GothicL/Equal2/GothicL/Equal3/GothicL/Equal4/GothicL/Equal5/GothicL/Equal6\n0.0 0.2 0.4 0.6 0.80.0900.0950.1000.105\nc13/MinusImΩ\nFIG.4: Thereal(left)andimaginary (right)partsofquasin ormalfrequenciesofthegravitational fieldinthebackgrou nd\nof the first kind aether black hole with different c13.\nl= 2, the decrease in Re ωis about from 0 .7 percent to 17 percents, while the increase in −Imωis about from\n1 percent to 15 percents, and may be detected by new generation of gravitational antennas. It will help us to\nseek LV information in nature in low energy scale.\nBehaviors with different overtone numbers n. Fig. 3 and Fig. 5 show that the real parts decrease and\nthe absolute imaginary ones increase with n, which are the same as those of the scalar and electromagnetic\nperturbations [24].\nBehaviors with different aether constants c13. For the fixed angular number l, or overtone number nwith\ndifferentc13(smallc13), Tab. I, Fig. 3 and Fig. 5 show that the real parts of frequencies decrease, and the\nabsolute imaginary ones increase with the small c13firstly toc13= 0.9 and then decrease on the contrary,\nwhich are similar to the scalar and electromagnetic perturbations [2 4]. It is different from that of the non-\nreduced aether black hole [23], where both all increase with c1. By comparing to Reissner-Norstr¨ om black\nhole, Fig. 3 and 4 show us a similar behavior that the absolute imaginary part of frequencies increases for\nsmall parameter c13orQ, and then decrease in the region of large parameter [37]. The only diff erence is that\nthe real part decreases here for all c13and increases there for all Q.\nBy comparing to some other LV models — nonminimal coupling theory to Eistein’s tensor, massive gravity\ntheory and the QED-extension limit of SME, the above gravitational field QNMs properties with c13are\nsimilar to those of them: the scalar field QNMs with coupling constant η[30], the scalar field QNMs with the\nscalar charge ˆS[31] and Dirac field QNMs with LV coefficient bµ[33], respectively. These similarities are that\nthe real part decreases while the absolute imaginary part increase s with the given LV coefficient η,ˆSandb.\nIn the theory of QED-extension limit of SME, local LV coefficient bµis introduced in a matter sector,\nwhile in Einstein-aether theory, the nonminimal coupling theory and m assive gravity, local LV is in a gravity8\nn/Equal0\nn/Equal1\nn/Equal2\nn/Equal3\nn/Equal4\n/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.60.70.80.91.0\nc13ReΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4\n/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0\nc13/MinusImΩ\nFIG.5: Thereal(left)andimaginary(right)partsofquasin ormal frequenciesofthegravitational fieldinthebackgrou nd\nof the first kind aether black hole with different c13.\nTABLE II: The lowest overtone ( n= 0) quasinormal frequencies of the gravitational field in th e second kind aether\nblack hole spacetime with fixed c14= 0.02.\nc13Mω(l= 2) Mω(l= 3) Mω(l= 4) Mω(l= 5) Mω(l= 6)\n0.01 0.373619-0.088891 i0.599443-0.092703 i0.809178-0.094164 i1.012300-0.094871 i1.212010-0.095266 i\n0.15 0.363508-0.087996 i0.583585-0.091775 i0.787938-0.093240 i0.985828-0.093950 i1.180390-0.094348 i\n0.30 0.350114-0.086611 i0.562563-0.090330 i0.759762-0.091795 i0.950709-0.092508 i1.138430-0.092909 i\n0.45 0.332795-0.084518 i0.535354-0.088129 i0.723264-0.089587 i0.905196-0.090301 i1.084040-0.090703 i\n0.60 0.309073-0.081156 i0.498024-0.084570 i0.673136-0.086006 i0.842654-0.086713 i1.009280-0.087113 i\n0.75 0.273208-0.075123 i0.441421-0.078143 i0.597022-0.079520 i0.747622-0.080205 i0.895630-0.080593 i\n0.90 0.205326-0.061015 i0.333580-0.063088 i0.451685-0.064270 i0.565957-0.064866 i0.678232-0.065208 i\nsector. For the former, LV matter perturbs to LI black hole — Sch warzschild black hole and produces QNMs.\nFor the latter, LI matter perturbs to LV black hole — Einstein-aeth er black hole, the modified Reissner-\nNorstr¨ om black hole and massive gravity black hole, and then produ ces QNMs. These similarities between\ndifferent backgrounds may imply some common property of LV coeffic ient on QNMs, i.e., in presence of LV,\nthe perturbation field oscillation damps more rapidly, and its period be comes longer. They also motivate us\nto the further theoretical study on the possible intrinsic connect ions between them.\nIV. QUASINORMAL MODES FOR THE SECOND KIND AETHER BLACK HOLE\nIn this section, I study the gravitational field perturbations to th e second kind of Einstein-aether black hole\nwith fixedc14= 0.02. Their behaviors are shown in Tab. II and Figs. from 6 to 8.\nThel-dependant behaviors. Tab. II shows that, for fixed c13, both the real and the absolute imaginary\nparts of frequencies increase with the angular quantum number l. For large l, the imaginary parts approach\na fixed value which is similar to the first kind aether black hole and, also s hown in Fig. 7. Tab II also shows\nthe derivations from Schwarzschild black hole. For l= 2, the decrease in Re ωis about from 3 percents to 45\npercents, while the decrease in −Imωis about from 1 percent to 31 percents, both are bigger than the fi rst\nkind aether black hole, and may be detected by new generation of gr avitational antennas.\nThen-dependant behaviors. For different overtone numbers n, Fig. 6 and Fig. 8 show that the real parts\ndecrease and the absolute imaginary ones increase with n, which is the same as that of the first kind aether\nblack hole.\nThec13-dependant behaviors. For the fixed angular number l, or overtone number n, Tab. II, Fig. 6 and\nFig. 7 show that both the real and the absolute imaginary parts of f requencies all decrease with c13increasing,\nwhich are completely different from that of the non-reduced aethe r black hole [23], or partially different from9\nn/Equal0,/GothicL/Equal2,c13is from0.96 to 0\nc13/Equal0.960.930.900.87\n0.15 0.20 0.25 0.30 0.350.050.060.070.080.09\nReΩ/MinusImΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4/GothicL/Equal5,c13is from0.96 to 0.03\n0.960.960.960.960.96\n0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.8\nReΩ/MinusImΩ\nFIG. 6: The relationship between the real and imaginary part s of quasinormal frequencies of the gravitational field in\nthe background of the second kind aether black hole with the d ecreasing of c13.\nn/Equal0/GothicL/Equal2/GothicL/Equal3/GothicL/Equal4/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0\nc13ReΩn/Equal0/GothicL/Equal2/GothicL/Equal3/GothicL/Equal4/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.050.060.070.080.09\nc13/MinusImΩ\nFIG.7: Thereal(left)andimaginary (right)partsofquasin ormalfrequenciesofthegravitational fieldinthebackgrou nd\nof the second kind aether black hole with different c13.\nthose of the first kind aether black hole. And they are the same as t hose of the scalar and electromagnetic\nperturbations obtained in Ref. [24]. This property of both decreas es is similar to that of the noncommutative\nSchwarzschild black hole [38] and Einstein-Born-Infeld black hole [32 ].\nIt is interesting that the three kinds of black holes’ masses/charg e are all modified. For the second kind\naetherblackhole, theaetherfield contributesthe spacetimemass asMæ=−c14MADM/2[17], where MADMis\nits Komar mass. For the noncommutative Schwarzschild black hole, t he pointlike mass is replaced by smeared\nmass distribution so that there is no singularity at the origin r= 0. For the Einstein-Born-Infeld black hole,\nthe pointlike charge is replaced by non-linear distribution also leads to no singularity the electromagnetic\nfield at the origin. Is Einstein-Born-Infeld theory also Lorentz bre aking? It is an open issue that needs to be\nstudied in the future.\nNoncommutative theory has a strong quantum gravitymotivation. The similarities of QNMs between it and\nthe second kind aether black hole show us that Lorentz symmetry s hould be given up in a potential quantum\ngravity.\nV. SUMMARY\nThe local Lorentz violation in the gravity sector should show itself in r adiative processes around black holes.\nAnd the similar behaviors of QNMs with the corresponding parameter s in some LV theories can lead us to\nfurther understand them.\nIn this paper, I study on QNMs of the gravitational field perturbat ions to Einstein-aether black holes.10\nn/Equal0\nn/Equal1n/Equal2\nn/Equal3\nn/Equal4\n/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.40.50.60.70.80.91.0\nc13ReΩ\nn/Equal0n/Equal1n/Equal2n/Equal3n/Equal4/GothicL/Equal5\n0.0 0.2 0.4 0.6 0.80.00.20.40.60.8\nc13/MinusImΩ\nFIG. 8: The relationship between the real and imaginary part s of quasinormal frequencies of the gravitational field in\nthe background of the second kind aether black hole with the d ecreasing of c13\nIn Einstein-aether theory, Lorentz symmetry is broken by the ex istence of the aether field ua. This aether\nfield doesn’t affect the spacetime mass of the first kind aether black hole, but contributes mass as Mæ=\n−c14MADM/2 in the second kind aether black hole spacetime.\nTo the effective potential, when c13increases, the turning points for both kinds of blackholes alwaysbe come\nlarger, and the value of their peak becomes lower. However, their Q NMs are different from each other that\nshow their complexity. For the first kind aether black hole, the real part of gravitational QNMs becomes\nsmaller with all c13increase. The absolute value of imaginary part of QNMs becomes bigg er with small c13\nincrease, and then decreases with big c13. For the second kind aether black hole, both decrease with c13. For\nthe non-reduced aether hole [23], both real part and the absolute value of imaginary part of QNMs increase\nwithc1.\nFirstly by comparing to Schwarzschild black hole, the first kind aethe r black holes have larger damping rate\nand the second ones have lower damping rate. They all have smaller r eal oscillation frequency of QNMs. If\nthe breaking of Lorentz symmetry is not very small, the derivation o f QNMs from Schwarzschild values might\nbe observed in the near future gravitational wave events and, de tected by new generation of gravitational\nantennas.\nSecondly, by comparing to some other LV gravity theories, the pro perties of QNM behaviors for the first\nkind Einstein-aether black hole with c13are similar to the behaviors of the scalar field QNMs with coupling\nconstantη[30], the scalar field QNMs with the scalar charge ˆS[31] and Dirac field QNMs with LV coefficient\nb[33]. These similarities between different backgrounds may imply some c onnections between Einstein-aether\ntheory, the non-minimal coupling theory, massive gravity theory a nd the QED-extension limit of SME, i.e.,\nLV in gravity sector and LV in matter sector will make quasinormal rin ging of black holes damping more\nrapidly and its period becoming longer.\nThe properties of QNM behaviors for the second Einstein-aether b lack hole with c13are similar to the scalar\nand gravitational field QNMs of Einstein-Born-Infeld black holes with the Born-Infeld parameter b[32] and,\nthe scalar, gravitational, electromagnetic and Dirac fields QNMs of n oncommutative Schwarzschild black hole\n[38] with the spacetime noncommutative parameter ϑ. In this case, LV in gravity sector affects spacetime\nmass and will make quasinormal ringing of black holes damping more slow ly and its period becoming longer.\nNoncommutative theory has a strong quantum gravity motivation. The similarities of QNMs between it and\nthe second kind aether black hole show us that Lorentz symmetry s hould be given up in a model merging SM\nand GR.\nThirdly by comparing to Ref. [24], the real and the absolute imaginar y parts of QNMs are lower than the\nelectromagnetical and scalar fields perturbations, i.e., gravitation al<electromagnetical <scalar. For fixed c13,\nlike the electromagnetical field, both the real part and the absolut e imaginary part of frequencies increase\nwithl.11\nAcknowledgments\nThe author would like to thank Emanuele Berti, Jiliang Jing, Songbai Ch en, R. A. Konoplya and Kai Lin\nfor enthusiastic helps. This work was supported by the National Na tural Science Foundation of China (grant\nNo. 11247013), Hunan Provincial Natural Science Foundation of C hina (grant No. 2015JJ2085),and the fund\nunder grant No. QSQC1708.\nAppendix A: Non-minimal coupling theory\nScalar fields are relatively simple, which allows us to probe the detailed f eatures of the more complicated\nphysical system. The non-minimal coupling between scalar field and h igher order terms in the curvature\nnaturally rise to improve the early inflationary models and could contr ibute to solve the dark matter problem.\nDinget alextended this coupling theory to dynamical gravity [39]. Chen et alfound that the coupling theory\nbetween the kinetic term of scalar field ψand the Einstein’s tensor Gµν\nS=/integraldisplay\nd4x√−g/bracketleftbigR\n16πG+1\n2gµν∂µψ∂νψ+η\n2Gµν∂µψ∂νψ/bracketrightbig\n(A1)\nis Lorentz violation [40], where ηis a coupling constant. This coupling modifies Reissner-Nordstr¨ oms pacetime\nas\nds2=/parenleftbigg\n1−ηQ2\nr4/parenrightbigg/bracketleftbigg\n−f(r)dt2+1\nf(r)dr2/bracketrightbigg\n+/parenleftbigg\n1+ηQ2\nr4/parenrightbigg\nr2(dθ2+sin2θdφ2), (A2)\nwheref(r) = 1−2M/r+Q2/r2. In 2010, Chen et alstudied the dynamical evolution of a scalar field coupling\nto Einstein’s tensor in Reissner-Nordstr¨ om spacetime [30].\nAppendix B: Massive gravity\nTo explain the accelerated expansion of the Universe without dark e nergy and dark matter components, one\ncan employ spontaneous breaking Lorentz symmetry by condensa tes of four scalar fields φ0andφicoupled to\ngravity. Then the gravitons acquire a mass, which is similar to the Higg s mechanism [31]. The action is\nS=/integraldisplay\nd4x√−g/bracketleftbigR\n16π+Λ4F(X,Wij)/bracketrightbig\n, (B1)\nwhereRis spacetime curvature scalar, XandWijare defined by the four scalar fields φ0andφias\nX=∂µφ0∂µφ0\nΛ4, Wij=∂µφi∂µφj\nΛ4−∂µφi∂µφ0∂νφj∂νφ0\nΛ8X. (B2)\nHere the constant Λ has the dimension of mass; φ0= Λ2[t+h(r)] andφi= Λ2xiwhich are called Goldstone\nfields. There is a static spherically symmetric solution in which the metr ic function is\nf(r) = 1−2M\nr−ˆS\nrλ, (B3)\nwhereˆSis a scalar charge, λis a positive constant. When the scalar charge ˆS= 0, Schwarzschild solution\nis recovered. And in the limit λ→ ∞, it approaches to Schwarzschild behavior in the large rregion. With\nthis metric function, Fernando et alstudied the QNMs of spherically symmetric black hole [31] in the massive\ngravity theory.12\nAppendix C: Noncommutative gravity theory\nIn quantization of spacetime to construct quantum gravity theor y, a common conjecture is that the algebra\nof spacetime coordinates is actually noncommutative. The most fam iliar form of Noncommutative gravity\ntheory is that the spacetime coordinates acquire the commutative relation\n[xµ, xν] =iϑµν, (C1)\nwhereϑµνis a real, antisymmetric, and constant tensor which determines the fundamental cell discretization\nof spacetime much in the same way as Planck constant /planckover2pi1discretizes the phase space [ xi,pj] =i/planckover2pi1δij. Lorentz\nsymmetry is intrinsically broken by virtue of nonzero ϑµν[41]. By using ϑµν=ϑdiag(ǫ1,...,ǫD/2), Nicolini et\nalderived a noncommutative Schwarzschild black hole solution by a metr ic function\nf(r) = 1−4M\nr√πγ(3/2,r2/4ϑ), (C2)\nwhereγ(r,ϑ) is the lower incomplete Gamma function and ϑis a real constant denotes spacetime noncom-\nmutativity. It leads to the mass distribution m(r) = 2Mγ(3/2,r2/4ϑ)/√π, whereMis the total mass of the\nsource. In another word, it is that the pointlike mass Mis replaced by smeared mass mwhich leads to no\nsingularity at origin r= 0, i.e., its spacetime curvature scalar\nR|r=0=4M√πϑ3/2. (C3)\nWhenϑ→0, a Schwarzschild black hole is recovered. With this metric function, Jun Liang studied QNMs of\na noncommutative geometry inspired Schwarzschild black hole [38].\nAppendix D: Einstein-Born-Infeld gravity theory\nBorn-Infeld theory plays an important role in string theory: it arise s naturally in open superstrings and in\nD-branes. Einstein-Born-Infeld black hole solutions may also play a r ole in understanding the black hole in\nthe deformed Ho˘ rava-Lifshitz gravity[42]. Is Einstein-Born-Inf eld theory also Lorentz breaking? It is an open\nissue that needs to be studied in the future. In Born-Infeld nonline ar electrodynamics theory, the action is\nS=/integraldisplay\nd4x√−g/bracketleftbiggR\n16πG+1\nb/parenleftBig\n1−/radicalbig\n1+2bFµνFµν/parenrightBig/bracketrightbigg\n, (D1)\nwhereFµνis the electromagnetic field strength. For the spherically symmetric case, the electric field is\nE(r) =Q//radicalbigg\nr4+b2Q2\n4, (D2)\nwherebis a Born-Infeld parameter of dimensions length square. Then it can avoid the singularity at the\nposition of a pointlike charge. The metric function of static charged black hole solution in Einstein-Born-\nInfeld theory is\nf(r) = 1−2M\nr+r2\n6b−1\nr/integraldisplay/radicalbigg\nQ2\nb+r4\n4b2dr, (D3)\nwhich is regular at the origin. In the limit b→0[45], the Reissner-Nordstr¨ om solution is recovered,\nf(r) = 1−2M\nr+Q2\nr2. (D4)\n[45] Here brelates to βin ref. [32] by b= 1/4β2.13\nWith this metric function, Fernando and Chen et alstudied the QNMs of spherically symmetric Einstein-\nBorn-Infeld black hole[32].\n[1] B. P. 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B 685318 (2010)."    },    {        "title": "1610.00654v1.Self_dual_solitons_in_a__CPT__odd_and_Lorentz_violating_gauged__O_3___sigma_model.pdf",        "content": "arXiv:1610.00654v1  [hep-th]  3 Oct 2016Self-dual solitons in a CPT-odd and Lorentz-violating gauged O(3)sigma model\nR. Casana,∗C. F. Farias,†M. M. Ferreira, Jr.,‡and G. Lazar§\n1Departamento de F´ ısica, Universidade Federal do Maranh˜ a o, 65080-805, S˜ ao Lu´ ıs, Maranh˜ ao, Brazil.\nWe have performed a complete study of self-dual configuratio ns in a CPT-odd and Lorentz-\nviolating gauged O(3) nonlinear sigma model. We have consistently implemente d the Bogomol’nyi-\nPrasad-Sommerfield (BPS) formalism and obtained the corres pondent differential first-order equa-\ntions describing electrically charged self-dual configura tions. The total energy and magnetic flux\nof the vortices, besides being proportional to the winding n umber, also depend explicitly on the\nLorentz-violating coefficients belonging to the sigma secto r. The total electrical charge is propor-\ntional to the magnetic flux such as it occurs in Chern-Simons m odels. The Lorentz violation in the\nsigma sector allows one to interpolate between Lorentz-vio lating versions of some sigma models: the\ngaugedO(3) sigma model and the Maxwell-Chern-Simons O(3) sigma model. The Lorentz violation\nenhances the amplitude of the magnetic field and BPS energy de nsity near the origin, augmenting\nthe deviation in relation to the solutions deprived of Loren tz violation.\nPACS numbers: 11.10.Lm,11.27.+d,12.60.-i, 74.25.Ha\nI. INTRODUCTION\nIn the early 1960s Gell-Mann and Levy [ 1], based on\nthe works of Schwinger [ 2] and Polkinghorne [ 3], con-\nstructed a renormalizable field theory for the new scalar\nmesonsσwith null isotopic spin, being this theory called\nthe sigma model. These authors considered the possi-\nbility of modifying the sigma model by supposing it as\na composite of the pion field rather than an elementary\none describing a new particle. This was the arising of\nthe nonlinear sigma model (NL σM), which featured an\nO(4) symmetry initially. Later, the model was endowed\nwith anO(3) symmetry, the O(3) NLσM [4], started to\ngain special attention and was widely applied to study\ndifferent aspects of field theory and condensed matter\nphysics[5]. TheO(3) NLσM is also interesting because\nit provides topological solitons whose equations are ex-\nactly integrable in the Bogomol’nyi-Prasad-Sommerfield\n(BPS) limit [ 6]. The solutions of these BPS equations\ndescribe a map from a spherical surface that represents\nthe two-dimensional physical space to a spherical surface\nin the internal field space, being classified according to\nthe second homotopy group Π 2(S2) =Z. This model,\nhowever, has a serious inconvenience because the solu-\ntions are scale invariant preventing one from describing\nlocalizedparticles[ 7]. Awayforbreakingthescaleinvari-\nance is gauging the U(1) subgroup and providing a spon-\ntaneous symmetry breaking potential. Such mechanism\nwas suggested in Ref. [ 8], where a Maxwell term con-\ntrolling the gauge field dynamics was introduced, so that\ntopologicalsolitonswith arbitrarymagnetic flux were en-\ngendered. A similar mechanism was implemented in Ref.\n[9] with the Abelian Chern-Simons in the gauge sector,\n∗Electronic address: rodolfo.casana@gmail.com\n†Electronic address: cffarias@gmail.com\n‡Electronic address: manojr.ufma@gmail.com\n§Electronic address: gzsabito@gmail.comimplying topological and nontopological solitons. These\ntwo approaches produce topological solitons which are\ninfinitely degenerate in a given topological sector. Such\ndegenerescence was circumvented in Refs. [ 10,11] by\nmeans of the introduction of a self-interacting potential\npossessing a symmetry breaking minima. This poten-\ntial induces a new topology in which the infinite circle\nof physical space is mapped on the equatorial circle of\nthe internal space so that the solitons are now classified\nby the first homotopy group Π 1(S1) =Z. TheO(3)\nsigma model has also been analyzed with the gauge field\ndynamics ruled by both the Maxwell and Chern-Simons\nterms [12,13]. In the same context, but with a nonmini-\nmally coupled gauge field, charged BPS soliton solutions\nwere found [ 14], revealing a magnetic flux quantized only\nfor topological solitons.\nInvestigation of topological defects in different theo-\nretical contexts has been a sensitive matter in the latest\nyears. Among the new frameworks are the CPT- and\nLorentz-violating (LV) field theories. The violation of\ntheCPTand Lorentz symmetries is a theoretical possi-\nbility that has been extensively investigated since 1996,\nmainly in the framework of the standard model exten-\nsion (SME) [ 15,16], with several repercussions [ 17–19].\nIn LV theories, the formation of defects was considered\nin many situations, for example, in solitons generated\nby scalar fields [ 20], general defects modified by ten-\nsor fields [ 21], Abelian monopoles [ 22] and oscillons [ 23].\nBPS vortex solutions in Abelian Higgs models including\nLorentz-violating terms have been extensively examined\nin Refs. [ 24–29]. The effects of CPT-even LV terms on\nthe topological BPS configurations of the gauged O(3)\nsigma model were recently studied in Ref. [ 30], with\nthe achievement of a generalization of the results of the\ngaugedO(3) sigma models studied in Refs. [ 10,11].\nIn this manuscript, we analyze the self-dual structure\nof the gauged O(3) sigma model provided with CPT-odd\nandCPT-even Lorentz-violating terms. More specifi-\ncally, we have modified the dynamics of the Maxwell sec-\ntor with the CPT-odd Carroll-Field-Jackiw term [ 31],2\nwhereas a CPT-even LV term was included in the O(3)\nsigma sector. The suitable projection of this Lagrangian\nin (1+2)-dimensions yields a model similar to Maxwell-\nChern-Simons gauged O(3) sigma model (MCS σM) in-\ncluding LV terms in the scalar sigma sector. Next, we\nhave implemented the BPS formalism and written the\nself-dual equationsdescribingthe self-dualconfigurations\nof thisCPT-odd and LV model. To assess the repercus-\nsion of the LV terms on the self-dual configurations of\nthe gauged O(3) sigma model, we have found the rota-\ntionallysymmetric vortexsolutions. Thesesolutionspos-\nsess finite energy proportional to the topological charge.\nThe LV model provides some interesting limits connect-\ning different versionsof the gauge O(3)sigma model. The\nprofiles are obtained numerically and compared with the\nones in the total absence of Lorentz-violation in order\nto highlight the LV effects. Finally, we present our final\nremarks and conclusions.\nII. ACPT-ODD AND LORENTZ-VIOLATING\nGAUGED O(3)SIGMA MODEL\nThe (1+2)-dimensional gauged O(3) sigma model\n(MσM) introduced in Refs. [ 8,10,11] is defined by the\nfollowing Lagrangian density\nLMσM=−1\n4FµνFµν+1\n2Dµ/vectorφ·Dµ/vectorφ−U(/vectorφ),(1)\nwhereAµisthe gaugefield and Fµν=∂µAν−∂νAµisthe\nAbelian strength tensor field. The field /vectorφ= (φ1,φ2,φ3)\nis a triplet of real scalar fields constituting a vector in\nthe internal space, with fixed norm /vectorφ·/vectorφ= 1, describ-\ning theO(3) NLσM. The coupling between the Abelian\ngauge and sigma field is ruled by the minimal covariant\nderivative\nDµ/vectorφ=∂µ/vectorφ−Aµˆn3×/vectorφ, (2)\nwith ˆn3being a unitary vector along the 3-direction\nin the internal scalar field space, while U(/vectorφ)is the self-\ninteracting potential.\nIn order to include Lorentz violation, we consider the\n(1+3)-dimensional version of model ( 1) supplementing\nit with the CPT-odd Carroll-Field-Jackiw (CFJ) term\nin the gauge sector and a CPT-even term in the sigma\nfield sector. This way, the CPT-odd Lagrangian density\ndescribing the proposed Lorentz-violatingsigma model is\nL=−1\n4FµνFµν−1\n4ǫµνρσ(kAF)µAνFρσ (3)\n+1\n2Dµ/vectorφ·Dµ/vectorφ+1\n2(kφφ)µνDµ/vectorφ·Dν/vectorφ−U.\nThefour-vector( kAF)αistheCFJbackgroundwithmass\ndimension +1. The dimensionless tensor ( kφφ)µνis real\nand symmetric, containing the LV and CPT-even pa-\nrameters in the sigma sector. The potential Udescribessome convenient interaction producing self-dual configu-\nrations, still to be determined.\nFrom the Lagrangian density ( 3), the equation of mo-\ntion for the gauge field reads\n∂νFνµ+1\n2ǫµαρσ(kAF)αFρσ=jµ, (4)\nwhere the conserved current density in this LV frame-\nwork,\njµ= [gµν+(kφφ)µν]ˆn3·/parenleftBig\n/vectorφ×Dν/vectorφ/parenrightBig\n,(5)\nis the counterpart of the one in absence of Lorentz vio-\nlation,jµ= ˆn3·/parenleftBig\n/vectorφ×Dµ/vectorφ/parenrightBig\n, displayed in Ref. [ 11]. The\nequation of motion for the sigma field is\n[gµν+(kφφ)µν]DµDν/vectorφ=/parenleftbigg\n/vectorφ·∂U\n∂/vectorφ/parenrightbigg\n/vectorφ−∂U\n∂/vectorφ(6)\n+[gµν+(kφφ)µν]/parenleftBig\n/vectorφ·DµDν/vectorφ/parenrightBig\n/vectorφ.\nFrom now on we are interested in the topological self-\ndual configurations arising from the (1+2)-dimensional\nversion of the model ( 3). For such a purpose to be ful-\nfilled, we implement a projection procedure doing ∂3/vectorφ=\n0,A3= 0,∂3Aµ= 0 (µ= 0,1,2). Besides, it is necessary\nto impose ( kφφ)0i= 0. In this way, from ( 4), the planar\nstationary Gauss law is\n∂j∂jA0−(kAF)3B= [1+(kφφ)00][(φ1)2+(φ2)2]A0,(7)\nand the planar stationary Ampere law reads\nǫij∂jB−(kAF)3ǫij∂jA0=−[δij−(kφφ)ij]ˆn3·(/vectorφ×Dj/vectorφ),\n(8)\nwhereB≡B3=F12defines the magnitude of the mag-\nnetic field along the z-axis and Latin indexes run over\ni,j= 1,2. Here, we point out that the magnetic field\nB=F12is effectively a planar or ”scalar” version of\nthe 3D-magnetic field Bk=ǫkmnFmn/2. Both equations\nclearly show that the LV coefficient ( kAF)3is responsible\nforthecouplingbetweenelectricandmagneticsectors,al-\nlowing,in principle, theoccurrenceofelectricallycharged\nconfigurations. The ( kAF)3coefficient in both the Gauss\nandAmperelawsplaysasimilarroleoftheChern-Simons\nmass in the (1 + 2)-dimensional Maxwell-Chern-Simons\ngaugedO(3) sigma model (MCS σM) [12,13], defined by\nLMCSσM =−1\n4FαβFαβ−1\n4κǫβρσAβFρσ\n+1\n2Dα/vectorφ·Dα/vectorφ+1\n2∂µΨ∂µΨ (9)\n−1\n2[(φ1)2+(φ2)2]Ψ2−U(|φ|,Ψ),\nwhereκis the Chern-Simons mass, U(|φ|,Ψ) is the self-\ndual potential and Ψ is a neutral field. The introduction3\nof this neutral scalar field is a well established proce-\ndure for a consistent description of self-dual configura-\ntions, and it was first reported in the context of Maxwell-\nChern-Simons-Higgs models [ 32] based in supersymmet-\nric arguments. It was also successfully implemented in\nother subsequent extensions [ 33], including to describe\ncharged topological configurations in Lorentz-violating\nmodels [25,29,30]. This Lagrangiandensity wasinitially\nproposed (but not solved) in Ref. [ 12], once its focus was\nin the Chern-Simons O(3) gauge sigma model. It was\nalso addressed in Ref. [ 13], where the self-duality of the\nMCSσM was considered, but without developing the nu-\nmerical solutions for the self-dual configurations. Here,\nthe numerical solutions for MCS σM will be achieved,\nsolved and used as a suitable background for physical\ncomparisons.\nA consistent description of the self-dual configura-\ntions carrying electric field in (1+3) dimensions also re-\nquires the introduction of the neutral scalar field appear-\ning Lagrangian ( 9), being given by the following (1+3)-\ndimensional model:\nL=−1\n4FαβFαβ−1\n4ǫαβρσ(kAF)αAβFρσ\n+1\n2Dα/vectorφ·Dα/vectorφ+1\n2(kφφ)αβDα/vectorφ·Dβ/vectorφ\n+1\n2∂µΨ∂µΨ−1\n2[1+(kφφ)00][(φ1)2+(φ2)2]Ψ2\n−U(|φ|,Ψ), (10)\nwhereU(|φ|,Ψ) is a convenient potential providing\nchargedBPSconfigurations. The(1+2)-dimensionalpro-\njection of the Lagrangian density ( 10) engenders a modi-\nfied version of the MCS σM model ( 9) with the CFJ term\nbecoming ( kAF)3ǫβρσAβFρσ, and the coefficient ( kAF)3\nplaying the role of the Chern-Simons mass ( κ). The dif-\nference rests in the presence of LV terms in the sigma\nsector.\nIII. THE BPS FORMALISM\nWe are interested in finding first-order self-dual dif-\nferential (BPS) equations whose solutions are minimum\nenergy configurations that also solve the second-order\nEuler-Lagrange equations. To carry out the BPS pro-\ncedure, we first write the stationary energy of the (1+2)-\ndimensional version of the model ( 10),\nE=/integraldisplay\nd2x/braceleftbigg1\n2/bracketleftBig\nδij−(kφφ)ij/bracketrightBig\nDi/vectorφ·Dj/vectorφ\n+1\n2B2+U+1\n2(∂jA0)2+1\n2(∂jΨ)2\n+1\n2/bracketleftbig\n1+(kφφ)00/bracketrightbig/bracketleftBig\n(φ1)2+(φ2)2/bracketrightBig\n(A0)2\n+1\n2/bracketleftbig\n1+(kφφ)00/bracketrightbig/bracketleftBig\n(φ1)2+(φ2)2/bracketrightBig\nΨ2/bracerightbigg\n,(11)which is positive definite as long as\nδjk−(kφφ)jk>0,(kφφ)00>−1. (12)\nIn order to implement the BPS procedure, we define\n˜Dk/vectorφ=MkjDj/vectorφ,which allows one to write\n/bracketleftBig\nδij−(kφφ)ij/bracketrightBig\nDi/vectorφ·Dj/vectorφ=˜Dk/vectorφ·˜Dk/vectorφ (13)\n=MkiMkjDi/vectorφ·Dj/vectorφ,\nwhere the Mijare the elements of the matrix Menglob-\ning the spatial LV parameters of the sigma sector, being\ndefined as\nMkiMkj=δij−(kφφ)ij. (14)\nBy introducing the identity,\n1\n2˜Dk/vectorφ·˜Dk/vectorφ=1\n4/parenleftBig\n˜Dj/vectorφ±ǫjm/vectorφטDm/vectorφ/parenrightBig2\n(15)\n∓(detM)φ3B±(detM)ǫik∂i(Akφ3)\n±(detM)/vectorφ·/parenleftBig\n∂1/vectorφ×∂2/vectorφ/parenrightBig\n,\nthe energy ( 11) is expressed as\nE=/integraldisplay\nd2x/braceleftbigg1\n4/parenleftBig\n˜Dj/vectorφ±ǫjm/vectorφטDm/vectorφ/parenrightBig2\n+1\n2/parenleftBig\nB∓√\n2U/parenrightBig2\n+1\n2(∂jA0±∂jΨ)2\n1\n2/bracketleftbig\n1+(kφφ)00/bracketrightbig/bracketleftBig\n(φ1)2+(φ2)2/bracketrightBig\n[A0±Ψ]2\n±(detM)/bracketleftBig\n/vectorφ·/parenleftBig\n∂1/vectorφ×∂2/vectorφ/parenrightBig\n+ǫik∂i(Akφ3)/bracketrightBig\n±B√\n2U∓(detM)φ3B∓(∂jΨ)(∂jA0)\n∓/bracketleftbig\n1+(kφφ)00/bracketrightbig/bracketleftBig\n(φ1)2+(φ2)2/bracketrightBig\nA0Ψ/bracerightBig\n.(16)\nUsing the Gauss law ( 7), the last row reads as\n∓Ψ∂j∂jA0±(kAF)3BΨ, (17)\nso that the energy reads\nE=/integraldisplay\nd2x/braceleftbigg1\n4/parenleftBig\n˜Dj/vectorφ±ǫjm/vectorφטDm/vectorφ/parenrightBig2\n+1\n2/parenleftBig\nB∓√\n2U/parenrightBig2\n+1\n2(∂jA0±∂jΨ)2\n+1\n2/bracketleftbig\n1+(kφφ)00/bracketrightbig/bracketleftBig\n(φ1)2+(φ2)2/bracketrightBig\n[A0±Ψ]2(18)\n±(detM)/bracketleftBig\n/vectorφ·/parenleftBig\n∂1/vectorφ×∂2/vectorφ/parenrightBig\n+ǫik∂i(Akφ3)/bracketrightBig\n±B/bracketleftBig√\n2U−(detM)φ3+(kAF)3Ψ/bracketrightBig\n∓∂j(Ψ∂jA0)/bracerightBig\n.\nIn the fifth row, one requires the factor multiplying the\nmagnetic field to be null, which leads to the BPS poten-\ntial\nU=1\n2[(detM)φ3−(kAF)3Ψ]2.(19)4\nThe integration of the expression in the fourth row of\nEq. (18),\nT0=(detM)\n4π/integraldisplay\nd2x/bracketleftBig\n/vectorφ·/parenleftBig\n∂1/vectorφ×∂2/vectorφ/parenrightBig\n+ǫik∂i(Akφ3)/bracketrightBig\n,\n(20)\nprovidesthetopologicalchargeofthemodel, whichshows\ndependence on the Lorentz-violating coefficients belong-\ning to the sigma sector [ 30]. As it was also reported in\nRef. [30], the topological conserved current is\nKµ=(detM)\n8πǫµαβ/bracketleftBig\n/vectorφ·/parenleftBig\nDα/vectorφ×Dβ/vectorφ/parenrightBig\n+Fαβφ3/bracketrightBig\n,(21)\nwhose component K0,whenever integrated over the\nspace, yields the conserved topological charge ( 20). By\nconsidering the fields Ψ and A0going to zero at infinity,\nin the fifth row of Eq. ( 18) the integration of the term\n∂j(Ψ∂jA0) gives null contribution to the energy. Thus,\nthe energy of the solutions becomes\nE= 4πT0 (22)\n+/integraldisplay\nd2x/braceleftbigg1\n2(B∓[(detM)φ3−(kAF)3Ψ])2\n+1\n4/parenleftBig\n˜Dj/vectorφ±ǫjm/vectorφטDm/vectorφ/parenrightBig2\n+1\n2(∂jA0±∂jΨ)2\n+1\n2/bracketleftbig\n1+(kφφ)00/bracketrightbig/bracketleftBig\n(φ1)2+(φ2)2/bracketrightBig\n[A0±Ψ]2/bracerightbigg\n.\nThis equation allows us to establish that the energy has\na lower bound given by\nE≥ ±4πT0, (23)\nattainedwheneverthefieldssatisfythefollowingself-dual\nor BPS equations,\n˜Dj/vectorφ±ǫjm/vectorφטDm/vectorφ= 0, (24)\nB=±[(detM)φ3−(kAF)3Ψ], (25)\n∂iA0±∂iΨ = 0, (26)\nA0±Ψ = 0. (27)\nThe condition Ψ = ∓A0saturates the two last equa-\ntions and the self-dual charged configurations are de-\nscribed by\n˜Dj/vectorφ±ǫjm/vectorφטDm/vectorφ= 0, (28)\nB=±(detM)φ3+(kAF)3A0, (29)\nwith the modified Gauss law\n∂j∂jA0−(kAF)3B= [1+(kφφ)00][(φ1)2+(φ2)2]A0.(30)It is clear that, for null Lorentz-violating parameters,\nwe recover the BPS equations of the gauged O(3) sigma\nmodel (1). On the other hand, if we set to be null only\nthe Lorentz-violating parameters of the sigma sector we\nrecuperate the self-dual equations of the Maxwell-Chern-\nSimonsO(3) sigma model ( 9) with (kAF)3=κ. More-\nover, the Gauss law ( 30) implies the proportionality re-\nlation,\nQ=(kAF)3\n1+(kφφ)00Φ, (31)\nbetween the total charge ( Q) of the self-dual configura-\ntions and the total magnetic flux (Φ),\nQ=−/integraldisplay\nd2x[(φ1)2+(φ2)2]A0,(32)\nΦ =/integraldisplay\nd2x B. (33)\nOur analysis will also compare the Lorentz-violating\nsolutions with the profiles of the corresponding models\nwithout Lorentz violation at all, provided by the La-\ngrangian densities ( 1) and (9). The self-dual equation\nfor the gauge O(3) sigma model ( 1) are given by\nDj/vectorφ±ǫjm/vectorφ×Dm/vectorφ= 0, (34)\nB=±φ3, (35)\nwhereas for the MCS σM model ( 9), the BPS equations\nand Gauss law describing self-dual configurations read as\nDj/vectorφ±ǫjm/vectorφ×Dm/vectorφ= 0, (36)\nB=±φ3+κA0, (37)\n∂j∂jA0−κB= [(φ1)2+(φ2)2]A0,(38)\nrespectively. Forthislattercase,thechargeandmagnetic\nflux fulfill, Q=κΦ.\nIn the sequel we study the axially symmetrical self-\ndual solutions describing electrically charged vortices in\nCPT-odd Lorentz-violating framework, comparing them\nwith solutions of the usual models preserving Lorentz\ninvariance.\nIV. AXIALLY SYMMETRICAL SELF-DUAL\nCHARGED VORTICES\nFortheenergytobefinite, the field /vectorφshouldgoasymp-\ntotically to one of the minimum configurations of the po-\ntential, stated in Eq. ( 19). This is reached following5\ntheAnsatzintroduced in Ref. [ 30] for axially symmetric\nvortices in the presence of Lorentz violation,\nφ1= sing(r)cos/parenleftBign\nΛθ/parenrightBig\n, φ2= sing(r)sin/parenleftBign\nΛθ/parenrightBig\n,\n(39)\nφ3= cosg(r), Aθ=−1\nr/bracketleftBig\na(r)−n\nΛ/bracketrightBig\n, A0=A0(r),\nwith the radial functions, g(r),a(r) andA0(r) being well\nbehavedandsatisfyingthefollowingboundaryconditions\n(see Sec. IVA):\ng(0) = 0 , a(0) =n\nΛ, A′\n0(0) = 0,\n(40)\ng(∞) =π\n2, a(∞) = 0, A0(∞) = 0,\nwhich are compatible with the vacuum configurations of\nthe potential for r→ ∞, while providing consistent so-\nlutions at r= 0. The non-null integer nis the wind-\ning number of the self-dual vortices. The constant Λ is\ndefined in terms of the Lorentz-violating parameters be-\nlonging to the sigma sector,\nΛ =/radicalBigg\n1−(kφφ)θθ\n1−(kφφ)rr. (41)\nIn theAnsatz(39), the magnetic field Breads\nB(r) =−a′\nr, (42)\nwhere (′) stands for the radial derivative. The BPS equa-\ntions(28)and(29), projectedonthe Ansatz(39), become\ng′=±Λa\nrsing, (43)\n−a′\nr=±ηcosg+(kAF)3A0, (44)\nwhereas the Gauss law ( 30) reads\nA′′\n0+A′\n0\nr−(kAF)3B=ηΛ∆A0sin2g,(45)\nwith the parameters ∆ and ηgiven by\n∆ =1+(kφφ)00\nηΛ, (46)\nη= detM=/radicalBig/bracketleftbig\n1−(kφφ)θθ/bracketrightbig/bracketleftbig\n1−(kφφ)rr/bracketrightbig\n.(47)\nWe use the BPS equations and the Gauss law to ex-\npress the BPS energy density as\nEBPS=B2+ηΛ/parenleftBiga\nrsing/parenrightBig2\n+ηΛ∆(A0sing)2+(A′\n0)2,\n(48)\nwhich is positive definite because η,Λ,∆>0.\nReplacing the Ansatz(39) and the boundary condi-\ntions (40) in Eq. ( 20), the resulting topological charge\nis\nT0=n\n2η\nΛ. (49)Moreover, under boundary conditions ( 40), the mag-\nnetic flux ( 33) and the electric charge ( 32) become\nΦ = 2πn\nΛ, (50)\nQ= 2π(kAF)3\nηΛ∆n\nΛ, (51)\nbeing both proportional to the winding number, n.\nThe topological charge ( 49) and the magnetic flux ( 50)\ndiffer from the Lorentz symmetric ones, T0=n/2 and\nΦ = 2πn, by the LV factors η,Λ. The self-dual vortices\nof the Lorentz-symmetric MCS σM model are described\nby\ng′=±a\nrsing, (52)\n−a′\nr=±cosg+κA0,(53)\nA′′\n0+A′\n0\nr−κB=A0sin2g, (54)\nwhile the correspondent BPS energy density is\nEBPS=B2+/parenleftBiga\nrsing/parenrightBig2\n+(A0sing)2+(A′\n0)2.(55)\nWe also write self-dual the equations for the neutral\nvortices of the Lorentz-symmetric gauged O(3) sigma\nmodel,\ng′=±a\nrsing, (56)\n−a′\nr=±cosg, (57)\nwhose BPS energy density is\nEBPS=B2+/parenleftBiga\nrsing/parenrightBig2\n. (58)\nA. Behavior of the profiles at boundaries\nWe study the behavior of the solutions at boundaries\nby solving the BPS equations and the Gauss law ( 45) at\nthe limits r→0 andr→ ∞. Close to the origin, we\nobtain the following expansions\ng(r)≈Gnrn+···, (59)\na(r)≈n\nΛ−[ev2+(kAF)3A0(0)]\n2er2+···,(60)\nA0(r)≈A0(0)+[ev2+(kAF)3A0(0)]\n4(kAF)3r2+···\n(61)\nWe observe that Eq. ( 60) justifies the use of the mod-\nifiedAnsatz(40) and the boundary condition for a(0).\nThe constant A0(0) in (61) is determined numerically for6\neveryn. Moreover, from ( 61) it is clear that the electric\nfield must be null at origin, i.e., A′(0) = 0, as stated in\nEq. (40).\nAtr→ ∞, the profiles present the following asymp-\ntotic behavior:\ng(r)≈π\n2−C∞e−mr\n√r+···, (62)\na(r)≈mC∞\nΛ√re−mr+···, (63)\nA0(r)≈C∞/parenleftbig\nm2−ηΛ/parenrightbig\nΛ(kAF)3e−mr\n√r+···,(64)\nwhereC∞is a positive constant determined numerically.\nThus, the profiles behave in a similar way to the vortices\nof Abrikosov-Nielsen-Olesen [ 34]. The parameter mis\nreal and positive,\nm=1\n2/radicalbigg\n(kAF)2\n3+ηΛ/parenleftBig\n1+√\n∆/parenrightBig2\n(65)\n−1\n2/radicalbigg\n(kAF)2\n3+ηΛ/parenleftBig\n1−√\n∆/parenrightBig2\n,\nwhich is associated to the mass of the self-dual bosons\nand to the extension of the defect.\nNow we analyze the behavior of the profiles by study-\ning some values of the Lorentz-violating parameters in-\nvolved in the bosonic mass. For fixed ( kAF)3, the behav-\niorofthevortexprofilesisgovernedbytheLVcoefficients\nbelonging only to the sigma sector. In this scenario we\ncananalyzetwosituationsofinterest: the firstoneoccurs\nwhen ∆ = 1, providing\nm=1\n2/radicalBig\n(kAF)2\n3+4ηΛ−1\n2|(kAF)3|.(66)\nThis is the mass the MCS σM [12],[13] would have in the\npresence of Lorentz violation only in the sigma sector.\nIndeed, for η= Λ = 1 (absence of LV terms in the sigma\nsector), it becomes\nm=1\n2/radicalBig\n(kAF)2\n3+4−1\n2|(kAF)3|,(67)\nthe same mass of the Maxwell-Chern-Simons O(3) sigma\nmodel [13].\nThe second regime to be highlighted happens when ∆\ntakes sufficiently large values [∆ ≫(kAF)3]:\nm→/radicalbig\nηΛ, (68)\ncorresponding to the bosonic mass of the gauged O(3)\nsigma model of Ref. [ 30] with Lorentz violation in the\nsigma sector (only).\nAnother interesting limit occurs by fixing the Lorentz-\nviolating parameters of the sigma sector and considering\nsufficiently large values of ( kAF)3, that is\nm→ηΛ√\n∆\n(kAF)3, (69)FIG. 1: Sigma field g(r) profiles.\nwhich corresponds to the mass the self-dual vortices of\nthe Chern-Simons O(3) sigma model of Ref. [ 9],[12]\nwould possess considering Lorentz violation only in the\nsigma sector. In this limit, taking ( kφφ)µν= 0 or\nη= Λ = ∆ = 1 leads to the Lorentz symmetric mass\n(1/(kAF)3).\nB. Numerical analysis\nBelow, we depict the profiles obtained from numeri-\ncal solutions of Eqs. ( 43)-(45), under boundary condi-\ntions (40), for winding number n= 1. We have fixed\nthe Lorentz violating parameters Λ = 1 .25,η= 2 and\n(kAF)3= 1.5, allowing the parameter ∆ to be free. Be-\ncause the BPS energy density ( 48) is positive definite for\n∆>0, we consider two regions: 0 <∆<1 (blue lines)\nand ∆>1 (orange lines), in which the behavior of the\nsolutions are different.\nThere are two interesting values or limits of ∆ that\nallow one to recover the behavior of two known mod-\nels in the presence of LV coefficients. The first one is\n∆ = 1 (solid magenta line), whose profiles correspond to\nMaxwell-Chern-Simons O(3) sigma model with Lorentz\nviolation only in the sigma sector [see comment after Eq.\n(66)]. The second one is the limit ∆ → ∞(solid black\nline), yieldingthe gauged O(3) sigmamodel with Lorentz\nviolation only in the sigma sector [ 30] [see comment af-\nter Eq. ( 68)]. We also have depicted the symmetric\nprofiles corresponding to: (i) the Maxwell-Chern-Simons\ngaugedO(3)sigma model ( 9) (green solid lines) with κ=\n(kAF)3= 1.5and(kφφ)µν= 0,(ii)thegauged O(3)sigma\nmodel (1) (red solid lines) which corresponds to the total\nabsence of Lorentz-violation [( kAF)µ= 0,(kφφ)µν= 0].\nThese two cases will serve as comparation basis to the\nLorentz-violating profiles.\nFigure1depicts the profiles of the sigma field. For7\nFIG. 2: Vector potential a(r) profiles.\n0<∆<1, the profiles are more spread and saturate\nthe asymptotic value π/2 more slowly when ∆ →0 (blue\nlines). On the other hand, for ∆ >1, the profiles are\nprogressively narrower for growing ∆ (orange lines), the\nmaximum tightness being achieved in the limit ∆ → ∞\n(solid black line). Thus, the profiles for ∆ >1 are\nconfined between the models described by ∆ = 1 and\n∆→ ∞.\nAsimilardescriptioncanbe givenfortheprofilesofthe\nvector field a(r), presented in Fig. 2. It is worthwhile to\nnote that the gauge field value at the origin changes from\na(0) =n(in the absence of LV) to a(0) =n/Λ (in the\npresence of LV in the sigma sector), which is evident in\nthis graphic.\nFigure3depicts the profiles for the scalar potential.\nFor0<∆<1(bluelines), theprofilesaremoreextended\nand with greater intensity at the origin. The influence\nof the CFJ parameter, ( kAF)3, is more pronounced when\n∆→0, while for ∆ >1 (orange lines) its effect becomes\nnegligible for increasing values of ∆. So, for large values\nof∆ the profilesbecomesmallerand smaller, overlapping\nthe horizontal axis in the limit ∆ → ∞(solid black line).\nIt means that the vortices become electrically neutral\nsuch as the ones of the usual Lorentz-invariant gauged\nO(3) sigma model, in this limit.\nFigure4describes the behavior for the electric field.\nForn= 1, the maximum electric field amplitude is\nreached for some value ∆∗such that 0 .5<∆∗<1. For\n0<∆<1, the profiles become radially more spread out\nfor decreasing ∆ values, i.e., ∆ →0. On the other hand,\nfor ∆>1,the profilesarelocated closerto the origin, be-\ning narrower, with their amplitude decaying rapidly for\nincreasing values of ∆. In the limit ∆ → ∞, the electric\nfield disappears, which agreeswith electrically uncharged\nvortices.\nFigure5showstheprofilesforthemagneticfield,which\narelumpscenteredattheoriginfor n= 1. For0 <∆<1\n(blue lines), the profiles are more spread out and theirFIG. 3: Scalar potential A0(r) profiles.\nFIG. 4: Electric field El(r) =−A′\n0(r) profiles.\namplitude at the origin decreases continuously when ∆\ngoes to zero. For ∆ >1 (orange lines), the profiles be-\ncome narrower and attain higher amplitudes for progres-\nsively increasing ∆. Nevertheless, the maximum narrow-\nness and amplitude are reached in the limit ∆ → ∞\n(solid black line). Similarly, as it occurs with the sigma\nand vector fields, the magnetic field profiles are located\nbetween the models defined by ∆ = 1 and ∆ → ∞. The\nmagnetic field at the origin, B(0)can be increased or re-\nduced in relation to the Lorentz symmetric case.\nForn= 1, the profiles of the BPS energy density (see\nFig.6) are lumps centered at the origin, such as the ones\nof the magnetic field. For increasing values of ∆, the am-8\nFIG. 5: Magnetic field B(r) profiles.\nFIG. 6: Energy density εBPS(r) profiles.\nplitude growsat the origin, the vortexcorebecomesmore\nlocalized than the one of the Lorentz symmetric counter-\npart (solid green or red lines). Similarly to the magnetic\nfield case, the maximum amplitude and narrownessoccur\nin the limit ∆ → ∞.\nByanalyzingtheprofilesofthe magneticfieldandBPS\nenergydensityfor n >1and∆finite, whentherotational\nsymmetryisconsidered,aringlikeprofileissetout,whose\nvalues at origin and maximum amplitude increase with\n∆. The ringlike structure of the magnetic field mimics\nthe behavior of the ones in models endowed with the\nChern-Simons term in the gauge sector, modifying the\nvalue and behavior at and near the origin, however.V. REMARKS AND CONCLUSIONS\nWe have developed a comprehensive study about the\nelectricallychargedself-dual configurationsofthe gauged\nO(3) nonlinear sigma model supplemented with a CPT-\nodd LV term in the gauge sector and a CPT-even LV\nterm in the sigma sector. We have verified that, for\nsupporting charged BPS solutions, the original model\nmust be modified by introducing a neutral scalar field\nwith appropriate dynamics, in the same manner as it\nhappens with the Maxwell-Chern-Simons-Higgs model\nor the Maxwell-Chern-Simons O(3) sigma model. We\nhave managed to implement the Bogomol’nyi-Prasad-\nSommerfield formalism, finding the first order differen-\ntial equations describing self-dual charged configurations\nwhose total energy is proportional to the topological\ncharge of the model, which gains LV contributions be-\nlonging to the sigma sector. We have also observed that\nthe total electric charge and the total magnetic flux are\nrelated to each other such as it is shown in Eq. ( 31).\nThese charged BPS configurations can be considered as\nclassicalsolutionsrelatedtoanextended supersymmetric\ntheory [35] in a Lorentz-violating framework.\nIn particular, we have made an analysis of the axi-\nallysymmetricvortexsolutionsoftheself-dualequations,\ndemonstrating that the total BPS energy, the magnetic\nflux and the electric charge are quantized (proportional\nto the winding number) and also proportional to the LV\ncoefficients introduced in the sigma sector. A remarkable\nfeature is that, choosing some limits for the LV parame-\nters, it is possible to reproduce other gauged sigma mod-\nels in the presence of Lorentz violation, like the Maxwell-\nChern-Simons O(3) sigmamodel (∆ = 1)and the gauged\nO(3)sigmamodel(∆ → ∞)orChern-Simons O(3)sigma\nmodel [for very large values of ( kAF)3], all them modified\nby Lorentz violation only in the sigma sector. In general,\nLorentz violation engenders altered solutions in relation\nto the MCS σM or MσM profiles, as explicitly depicted\nin Figs. 1–6. More specifically, LV affects the behavior\nof the magnetic field and BPS energy density at the ori-\ngin and near the origin : the amplitude augments with\n∆, reaching its maximum deviation in the limit ∆ → ∞\n, while the width decreases, yielding more compact and\nlocalized vortex profiles (for large values of ∆). Thus,\nthe LV defects have amplitude more pronounced near\nthe origin and are much more localized than the Lorentz\ninvariant solutions. 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Xu\nState Key Laboratory of Nuclear Physics, Peking University , Beijing, China\n(Dated: July 6, 2021)\nMagneto-solid-mechanical model of two-component, core-c rust, paramagnetic neutron star re-\nsponding to quake-induced perturbation by differentially r otational, torsional, oscillations of crustal\nelectron-nuclear solid-state plasma about axis of magneti c field frozen in the immobile paramag-\nnetic core is developed. Particular attention is given to th e node-free torsional crust-against-core\nvibrations under combined action of Lorentz magnetic and Ho oke’s elastic forces; the damping is\nattributedtoNewtonian force of shear viscose stresses in c rustal solid-state plasma. Thespectral for-\nmulae for the frequency and lifetime of this toroidal mode ar e derived in analytic form and discussed\nin the context of quasi-periodic oscillations of the X-ray o utburst flux from quaking magnetars. The\napplication of obtained theoretical spectra to modal analy sis of available data on frequencies of os-\ncillating outburst emission suggests that detected variab ility is the manifestation of crustal Alfv´ en’s\nseismic vibrations restored by Lorentz force of magnetic fie ld stresses.\nPACS numbers: 94.30.cq, 97.60.Jd\nI. INTRODUCTION\nThe investigations of neutron star seismic vibrations\noffer unique opportunity of studying their internal struc-\nture, solid-mechanical and electrodynamical properties\nof superdense degenerate matter. The most conspicu-\nous feature of these non-convective solid stars is the ca-\npability of accommodating magnetic fields of extremely\nhigh intensity[1] that serve as a chief promoter of their\nobservable electromagnetic activity. The absence of nu-\nclear energy sources in these final stage (FS) stars sug-\ngests that their magnetic fields are definitely not gener-\nated by persistent current-carrying flows in self-exciting\ndynamo processes, as is the case of liquid main-sequence\n(MS) stars. It seems quite likely, therefore, that stability\nto spontaneous decay[2] of fossil magnetic fields of iso-\nlated neutron stars[3] is maintained by permanent mag-\nnetization of neutron-dominated (poorly conducting) de-\ngenerate Fermi-matter. Such an understanding has been\nlaid at the base of paramagnetic neutron star model[4–\n7]. In this model the degenerate Fermi-matter of non-\nrelativistic neutrons (whose degeneracy pressure with-\nstands the pressureofself-gravity)is regardedas being in\n∗Also at Joint Institute for Nuclear Research, Dubna, Russiathepermanentlymagnetizedstateoffield-inducedPauli’s\nparamagnetic saturation which is characterized by align-\nment of spin magnetic moments of neutrons along the\naxis of frozen-in magnetic field. The most striking dy-\nnamical manifestation of spin paramagnetic polarization\nof non-conducing neutron matter is that such a mat-\nter can transmit perturbations by transverse magneto-\nmechanical waves; in such a wave the vector-fields of\nmagnetization and material displacements undergo cou-\npled differentially rotational vibrations traveling along\nthe axis of magnetic field. In a spherical mass of param-\nagnetic neutron star this unique feature of field-induced\nspin magnetic polarization of neutron matter is mani-\nfested in that such a star can undergo solely torsional\nvibrations about axis of its dipole magnetic moment.\nBased on this finding, it was argued in above works\nthat the model of paramagnetic neutron star execut-\ning torsional axisymmetric vibrations, weakly damped\nby nuclear matter viscosity, is able to explain long pe-\nriodic ([5 < P <12] s – non-typical to young neutron\nstars) pulsed character of magnetar radiation (both, Soft\nGammaRepeatersandAnomalousX-rayPulsars)inseis-\nmically quiescent regime of their emission, as being pro-\nduced by torsional vibrations, rather than rotation as is\nthe case of radio pulsars.\nRecent years have seen a resurgence of interest2\nin torsional vibrations of magnetars, prompted by\nobservations[8–10] of quasi-periodic oscillations (QPOs)\nduring the outburst flare from SGR 0525-66, SGR 1806-\n20, and SGR 1900+14. The statistics of X-ray burst of\nSGRs exhibits typical for earthquakes features[11]. It is\nbelieved, therefore, that detected QPOs are of seismic\norigin. Particular attention in this development of mag-\nnetar asteroseismologyhas been paid to the following set\nof data on QPO frequencies[12]\nSGR1806 −20 : 18,26,29,92,150,625,1840; (1)\nSGR1900+14 : 28 ,54,84,155[Hz]. (2)\nThe correspondingperiods are substantially shorter than\nthe above mentioned periods of seismically quiescent\npulsed emission. In works[13–17] motivated by this dis-\ncovery, several models of post-quake vibrational relax-\nation of above magnetars have been investigated. Partic-\nular attention has been given to the regime of node-free\nor nodeless shear axisymmetric vibrations. This regime\nis interesting in its own right because such vibrations\nhave been and still are poorly investigated in theoretical\nasteroseismologyof both solid FS stars and such solid ce-\nlestial objects as Earth-likeplanets[18, 19]. In particular,\nin works[13–15], a case of the elastic-force-driven node-\nless shear oscillations, both torsional– 0tℓand spheroidal\n–0sℓ, entrapped in the crust of finite depth ∆ Rhas been\nstudied in some details with remarkable inference that\ndipole overtones of spheroidal and torsion vibrations of\ncrust against immobile core exhibit features generic to\nGoldstone soft modes. On the other hand, one can prob-\nablycastdoubtonargumentsofthemodelpresumingthe\ndominant role of solid-mechanical Hooke’s force of elastic\nstresses because such interpretation rests on poorly jus-\ntifiable assumption about dynamically passive role of an\nultra strong magnetic field, that is, that the field frozen\nin the star remains unaltered in the process of vibrations.\nBearing this in mind and assuming that the presence\nof charged particles in neutron-dominated stellar mat-\nter imparts to it the properties of electric conductor, in\nworks[16, 17], the post-quake relaxation of above magne-\ntarshasbeenstudiedinthemodelofperfectlyconducting\nsolid star executing global torsional vibrations restored\nby joint action of Lorentz force of magnetic field stresses\nand Hooke’s force of solid-mechanical elastic stresses.\nIt was found that such a model provides fairly reason-\nable account of general trends in QPO frequencies for\nall data from SGR 1900+14 and for SGR 1900+14 from\nthe range 30 ≤ν≤200 Hz, but faces serious difficul-\nties in interpreting low-frequency vibrations with ν= 18\nandν= 26 Hz in data from SGR 1806-20. Also, the\nmodel of global torsional vibrations leaves some uncer-\ntainties regarding the nature of vibrations with ν= 625\nandν= 1840 Hz. This last issue has been scrutinized\nin recent work[17] from the standpoint of a solid star\nmodel with non-homogeneous poloidal magnetic field of\nwell-known Ferraro’s form. And it was found that these\nhigh-frequency QPOs can be properly explained as being\nproduced by very high overtones of node-free torsionalAlfv´ en oscillations.\nP ARAMAGNETIC□NEUTRON□STAR\nFIG. 1. (Color online) The internal constitution of two-\ncomponent, core-crust, model of paramagnetic neutron star .\nThe massive core is considered as a poorly conducting per-\nmanent magnet composed of degenerate Fermi-gas of non-\nrelativistic neutrons in the state of Pauli’s paramagnetic satu-\nration caused byfield-inducedalignment ofspin magnetic mo -\nment of neutrons along the axis of uniform internal and dipo-\nlar external magnetic field frozen in the star on the stage of\ngravitational collapse of its MS progenitor. A highly condu ct-\ning metal-like material of the neutron star crust, composed\nof nuclei embedded in the super dense degenerate Fermi-gas\nof relativistic electrons, is regarded as electron-nuclea r solid-\nstate magneto-active plasma capable of sustaining Alfv´ en os-\ncillations.\nAs a logical extension of above line of investigation, in\nthis paper we consider in some details a case of node-\nfree torsional vibrations locked in the crust with focus\non toroidal Alfv´ en mode. In so doing we work from the\ntwo-component model of paramagnetic neutron star, pic-\ntured in Fig.1, whose crust and core materials are re-\ngarded as endowed with substantially different electro-\ndynamic properties. The immobile massive core, pri-\nmarily consisting of degenerate neutron matter in the\nabove described permanently magnetized state of Pauli’s\nparamagnetic saturation, is regarded as a main source\nof magnetic field of the star crust. This implies that\nthe core material is just incapable of sustaining Alfv´ en\nvibrations which owe their existence to extremely large\n(effectively infinite) electrical conductivity of matter[20–\n22]. The micro-composition of crust, which is dominated3\nby nuclei embedded in degenerate Fermi- gas of rela-\ntivistic electrons, suggests that its metal-like material\npossesses properties of perfectly conducting solid-state\nplasma. Such a view suggests that seismic stability of\nthe star to quake-induced tectonic displacements of crust\nagainst core is primarily determined by well-known effect\nof magnetic (magnet-metal) cohesion mediated by mag-\nnetic field lines which operate as a super-hard piles en-\ndowing the core-crust construction of neutron star with\nsupplementary (to gravity forces) stiffness of magnetic\nnature. The intermediate layer between core and crust\n(the inner crust whose density several times less than the\ncoredensity) ismost likelycomposedofquasi-bosonmat-\nter of paired neutrons. But it is highly unlikely that such\nquasi- boson matter is capable of undergoing phase tran-\nsition to the Bose-Einstein Condensation (BEC) which\nis characterized by vanishingly small pressure. This sug-\ngests that BEC state of paired neutrons, if exist, can\nonly insufficiently contribute to the total mass budget of\nneutron star – a compact object in which pressureof self-\ngravity is brought to equilibrium by degeneracy Fermi-\npressure of non-relativistic neutrons in the core and rela-\ntivistic electrons in the crust. In seismo-dynamics of the\nparamagnetic neutron star under consideration the inner\ncrust is thought of as operating like a lubricant facilitat-\ning differentially rotational shear displacements of crust\nrelative to much denser matter of massive core. From\nthe view point of this core-crust model, the star-quake is\nthough ofasimpulsive releaseofenergyofmagnetic core-\ncrust cohesion (by means of disruption of magnetic field\nlines on the core-crust interface) resulting in the crust\nfracturing by revealed magnetic stresses. In this paper\nwe focus, however, not on dynamics of quake, but on\nthe post-quakevibrational relaxation of the star, namely,\non node-free torsional oscillations of crustal solid-state\nplasma about axis of magnetic field frozen in an immo-\nbile paramagnetic core. In section 2, a brief outline is\ngiven of theory of solid-magnetics appropriate for the\nperfectlyconductingviscoelasticcontinuousmediumper-\nvaded by a magnetic field. In section 3, the spectral\nformulas for the frequency and lifetime of differentially\nrotational, torsional, nodeless vibrations of the crust re-\nstored by combined action of magnetic Lorentz and elas-\ntic Hooke’s forces are obtained. In section 4, the com-\nputed frequency spectra are used for the forward astero-\nseismic analysis of the fast oscillations of X-ray outburst\nfrom above mentioned magnetars. The obtained results\nare briefly summarized in section 5.\nII. GOVERNING EQUATIONS OF\nSOLID-MAGNETICS\nIt is generally realized today that seismic vibrations\nof superdense matter of non-convective FS-stars (white\ndwarfs, pulsars and quark stars) can be properly de-\nscribed by equations of solid-mechanical theory of vis-\ncoelastic continuous media[23–28]. In what follows wedeal with the shear differentially rotational fluctuations\nof viscoelastic crustal matter of density ρwhich are de-\nscribed by quake-induced material displacements ui(ba-\nsic variable of solid-mechanics). The non-compressional\ncharacter of vibrations under consideration implies that\nδρ=−ρ∇kuk= 0. With this in mind, the govern-\ning equations of solid-magnetics (solid-mechanical coun-\nterpart of equations of magneto-fluid-mechanics) can be\nwritten in the form\nρ¨ui=∇kτik+∇kσik+∇kπik,∇kuk= 0 (3)\npresuming that Hooke’s elastic stresses σikand Newton’s\nviscous stresses πikare described by linear constitutive\nequations\nσik= 2µuik, uik=1\n2[∇iuk+∇kui],(4)\nπik= 2η˙uik,˙uik=1\n2[∇i˙uk+∇k˙ui] (5)\nwhereµstands for the shear modulus, ηfor shear viscos-\nity anduikis the tensor of shear strains or deformations.\nThe central to our further discussion is the tensor of fluc-\ntuating magnetic field stresses\nτik=1\n4π[BiδBk+BkδBi−BjδBjδik],(6)\nδBi=∇k[uiBk−ukBi] (7)\nAs in our previous works[16, 29], we consider model with\nhomogeneous internal magnetic field whose components\nin spherical polar coordinates read\nBr=Bcosθ, B θ=−Bsinθ, B φ= 0 (8)\nand external dipolar magnetic field is described by B=\n∇ ×A, whereA= [0,0,Aφ= ms/r2] is the vector po-\ntential with the standard parametrization of the dipole\nmagnetic moment m s= (1/2)BR3of star of radius R\nand byBis understood the magnetic field intensity at\nits magnetic poles, B=Bp.\nA. The energy method\nThis method of computing frequency of shear vibra-\ntions rests on the equation of energy balance\n∂\n∂t/integraldisplayρ˙u2\n2dV=−/integraldisplay\n[τik+σik+πik] ˙uikdV(9)\nwhich is obtained by scalar multiplication of equation\nof magneto-solid-mechanics (3) with ˙ uiand integration\nover the volume of seismogenic layer. From the technical\npoint of view, the shear character of material distortions\nbrought about by forces under consideration owes its ori-\ngin to the symmetric form of stress-tensors in terms of\nwhich these forces are expressed. It is this last feature\nof solid-mechanical elastic stresses and magnetic field\nstresses that endows the solid-state plasma pervaded by4\nhomogeneous magnetic field with the capability of re-\nsponding to non-compression perturbation by reversal\nshear vibrations (which are not accompanied by fluctua-\ntions in density). The physical significance and practical\nusefulness of the energy method under consideration is\nthat it can be efficiently utilized not only in the study of\nnon-radial seismic vibrations of neutron stars, but also\ncan be applied to the study of more wide class of solid\ndegenerate stars like white dwarfs stars[30, 31] and ul-\ntra dense quark-matter stars[32] whose material is most\nlikely in the solid aggregate state[33, 34]. At this point it\nseems appropriate to mention here theoretical investiga-\ntions of vibration properties of atomic nuclei (thought of\nasultrafinepiecesofcontinuousnuclearmatter)inwhich\nit has been found that nuclear giant-resonant excitations\n(fundamental vibration modes generic to all nuclei of pe-\nriodicchart)areproperlydescribedintermsofspheroidal\nand torsionalelastic vibrationsofa solid sphere[35]. This\nsuggests that degenerate nucleon Fermi-matter, regarded\nas continuous medium, can be thought of as a strained\nFermi-solid, rather than flowing Fermi-liquid.\nThe key idea of this method consists in using of the\nfollowing separable form of material displacements\nui(r,t) =ai(r)α(t) (10)\nwhereai(r) is the time-independent solenoidal field and\namplitude α(t) carries information about temporal evo-\nlution of fluctuations. Thanks to this form of ui, all the\nabovetensorsof fluctuating stressesand strainstake sim-\nilar separable form\nτik(r,t) = [˜τik(r)−1\n2˜τjj(r)δik]α(t), (11)\n˜τik(r) =1\n4π[Bi(r)bk(r)+Bk(r)bi(r)], (12)\nbi(r) =∇k[ai(r)Bk(r)−ak(r)Bi(r)], (13)\nσik(r) = 2µaik(r)α(t), πik(r) = 2ηaik(r)˙α(t),(14)\naik(r) =1\n2[∇iak(r)+∇kai(r)]. (15)\nOn inserting (10)-(15) in the integral equation of energy\nbalance(9) wearriveatequationfor α(t) havingthe well-\nfamiliar form\ndE\ndt=−2F,E=M˙α2\n2+Kα2\n2,F=D˙α2\n2(16)\nM¨α+D˙α+Kα= 0, (17)\nα(t) =α0exp(−t/τ)cos(Ωt), (18)\nΩ2=ω2/bracketleftbig\n1−(ωτ)−2/bracketrightbig\n, ω2=K\nM, τ=2M\nD.(19)\nwhere the inertia M, viscous friction Dand stiffness Kof damped oscillator are given by\nM=/integraldisplay\nρ(r)ai(r)ai(r)dV,K=Ke+Km,(20)\nKe= 2/integraldisplay\nµ(r)aik(r)aik(r)dV, (21)\nKm=/integraldisplay\n˜τik(r)aik(r)dV, (22)\nD= 2/integraldisplay\nη(r)aik(r)aik(r)dV. (23)\nAll the above equations valid for arbitrary volume occu-\npied by magneto-active solid-state plasma whose density,\nshearmodulus and shearviscosityarearbitraryfunctions\nof position. As in our previous works, here we confine\nour computations to the case of uniform profile of these\nlaterparameters. Andbecausethepurposeofourpresent\nstudy is the frequency spectrum of node-free torsion vi-\nbrations about magnetic axis of the star, several com-\nments should be made regarding the axisymmetric field\nof material displacements awhich is taken in one and\nthe same shape in computing parameter of inertia M\nand spring constants of both solid-mechanical Keand\nmagneto-mechanical Kmstiffness.\nFrom the physical point of view, the main argument\njustifying the use of one and the same field of mate-\nrial displacements in computing frequency spectra of tor-\nsional vibrations driven by forces of elastic and magnetic\nfield stresses rests on general statement of continuum-\nmechanical theories of magneto-active perfectly con-\nducting continuous media – magnetic field pervading\n(both liquid-state and solid-state) plasmas imparts to\nsuch a medium a supplementary portion of elasticity\nwhich is manifested in its capability of transmitting non-\ncompressional mechanical perturbations by transverse\nAlfv´ en waves. Such a view is substantiated by the com-\nmonly known fact that transverse wavein incompressible\ncontinuous medium is the feature of material oscillatory\nbehavior which is generic to elastic solid, not an incom-\npressible flowing liquid. The transverse hydromagnetic\nwave propagating, along the lines of constant magnetic\nfieldBfrozen-in the perfectly conducting medium, with\nAlfv´ en speed vA= [2PB/ρ]1/2(wherePB=B2/8πis the\nmagnetic field pressure), is characterized by dispersion\nequation ω=vAkwhich is similar to that for transverse\nwave of shear mechanical displacements, ω=ctk, travel-\ning in an elastic solid with the speed ct= [µ/ρ]1/2where\nµis the shear modulus (which has physical dimension of\npressure). The profound discussion of analogy between\noscillatory behavior of incompressible perfectly conduct-\ning plasmas and elastic solid (similarity of transverse hy-\ndromagnetic wavein incompressible perfectly conducting\nmedia and transverse wave of shear mechanical displace-\nments in an elastic solid) can be found in monographs of\nChandrasekhar[20] (section Alfv´ en waves), and more ex-\ntensively this issue is discussed in monograph of Alfv´ en\nand F¨ altammar[21]. Regarding the difference between\nnode-free oscillatory behavior of solid sphere and spher-\nical mass of an incompressible liquid it is appropriate5\nto note that the liquid sphere is able to sustain solely\nspheroidal node-free vibrations of fluid velocity. The\ncanonical example is the Kelvin fundamental mode of\noscillating fluid velocity in a heavy spherical mass of in-\ncompressible homogeneous liquid restored by forces rep-\nresented as gradient of pressure and gradient of potential\nof self-gravity[36]. In the meantime, the node-free vibra-\ntions of solid sphere restored by elastic force (represented\nas divergence of shear mechanical stresses) are charac-\nterized by two eigenmodes. Namely, the even-parity\nspheroidal mode of non-rotational vibrations of material\ndisplacements and the odd-parity torsional mode of dif-\nferentiallyrotationalvibrations,theprobleminwhichthe\nvery notion of the torsion vibration mode has come into\nexistence[13]. Inourstudies focusislaidonpoorlyinves-\ntigated regime of node-free vibrationsin which solenoidal\nfield of material displacements, ∇·a= 0, obeys the vec-\ntor Laplace equation, ∇2a= 0. In this regime the in-\nstantaneous material displacements are described by the\ntoroidal field of the form a=Aℓ∇×[rrℓPℓ(cosθ)]. Sub-\nstituting this field in the above given integrals for inertia\nMand solid-mechanicalstiffness Keand integratingover\nthe entire volume of oscillating star we have found in our\npreviousstudies that elastic-force-drivennode-free global\ntorsionalvibrations, M¨α+Keα= 0, are characterizedby\nthe frequency spectrum ωe(ℓ) = [Ke/M]1/2of the form\nω2\ne(ℓ) =ω2\ne[(2ℓ+3)(ℓ−1)], (24)\nωe=ct\nR, ct=/radicalbigg¯µ\n¯ρ. (25)\nwhereωeis the natural unit of frequency of shear elastic\nvibrations and ctis the speed of transverse wave in the\nelastic solid characterized by average shear modulus ¯ µ\nand average density ¯ ρ.\nThe above line of argument about similarity between\noscillatory behavior of elastic solid and magneto-active\nplasma suggests that perfectly conducting matter of neu-\ntronstarpervadedbyhomogeneousmagneticfield should\nbe able to sustain the Lorentz-force-driven differentially\nrotational seismic vibrations (triggered by quake) about\nmagnetic axis in which oscillating field of material dis-\nplacements has one and the same form as in the above\noutlined Hooke’s-force-driven torsional vibrations. Ad-\nhering to this assumption and making use of the above\nnode-free toroidal field, a, as a trial function for comput-\ningMandKmwe found in[16, 17] that torsional vibra-\ntionsrestoredbymagneticLorentzforce, M¨α+Kmα= 0,\nare characterized by the frequency spectrum\nω2\nm(ℓ) =ω2\nA/bracketleftbigg\n(ℓ2−1)2ℓ+3\n2ℓ−1/bracketrightbigg\n, (26)\nωA=vA\nR, vA=B√4π¯ρ(27)\nwhereωAis the natural unit of frequency of Alfv´ en vi-\nbrations and vAis the speed of Alfv´ en wave. The prac-\ntical usefulness of outlined computations with one and\nthe same trial toroidal field of displacements is that al-\nlowsusto assessthe relativeroleofrestoringHooke’sandLorentz forces in torsional seismic vibrations of neutrons\nstars which are responsible, as is believed, for the fast os-\ncillations of X-ray flares from quaking magnetars. With\nall above in mind, one of the main purposes of present\npaper is to make such an assessment in a mathematically\nconsistent fashion for the torsional node-free vibrations\nentrapped in the crust, the problem which is considered,\nto the best of our knowledge, for the first time.\nB. Material displacements in torsional mode of\ncrust-against-core nodeless vibrations\nThe above equations of the energy method show that\nthe main trial function of the frequency spectrum com-\nputation is the toroidal field of instantaneous displace-\nments. Because one of the main our purposes here is to\nassess the relative role of elastic and magnetic forces in\nquake-induced torsion nodeless vibrations of magnetars,\nwe again adopt all the above arguments regarding the\nchoice of this field in the form of the general solution to\nvector Laplace equation.\nIt is convenient to start with the rate of material dis-\nplacements which is described by general formula of ro-\ntational motions\nδv(r,t) =˙u(r,t) = [Ω(r,t)×r], (28)\nΩ(r,t) = [∇×δv(r,t)] = [∇×˙u(r,t)] =˙Φ(r,t)(29)\nHowever, unlike a case of rigid-body rotation, in which\nthe angular velocity is a constant vector, in a solid mass\nundergoing axisymmetric differentially rotational vibra-\ntions the angular velocity Ω(r,t) is the vector-function\nof position which can be represented as\nΩ(r,t) =˙Φ(r,t) =φ(r) ˙α(t),φ(r) = [∇×a(r)](30)\nIn the regime of node-free vibrations in question, a(r)\nis described by the divergence-free odd-parity, axial,\ntoroidal field which is one of two harmonic solenoidal\nfieldsoffundamentalbasis[20]obeyingthevectorLaplace\nequation ∇2a= 0. Thisfield canbe expressedin termsof\ngeneral solution of the scalar Laplace equation as follows\na(r) =at(r) =∇×[rχ(r)] = [∇χ(r)×r] (31)\n∇2χ(r) = 0, (32)\nχ(r) = [Aℓrℓ+Bℓr−ℓ−1]Pℓ(ζ), ζ= cosθ(33)\nwherePℓ(ζ) is the Legendre polynomial of multipole de-\ngreeℓ. It follows that the angular field φ(r) is the\npoloidal vector field\nφ(r) = [∇×at(r)] =∇×∇× [rχ(r)] (34)\n=∇[Aℓrℓ+Bℓr−ℓ−1]Pℓ(ζ), (35)\nAℓ=Aℓ(ℓ+1),Bℓ=Bℓℓ. (36)\nAnd this field is irrotational, ∇×φ(r) = 0.\nAs was stated, we study a model of differentially rota-\ntional vibrations of peripheral finite-depth crust against\nimmobile core. In this case, the arbitrary constants Aℓ6\nFIG. 2. (Color online) Material displacements of crustal ma t-\nter about the dipole magnetic moment axis of paramagnetic\nneutronstar undergoingnodeless differentially rotationa l, tor-\nsional, vibrations in quadrupole and octupole overtones.\nandBℓcan be uniquely eliminated from two boundary\nconditions: (i) on the core-crust interface r=Rc\nuφ|r=Rc= 0 (37)\nand (ii) on the star surface r=R\nuφ|r=R= [Φ×R]φ|r=R, (38)\nΦ=α(t)∇ˆnPℓ(ζ),R=erR (39)\nwhere\n∇ˆn=1\nR∇Ω,∇Ω=/bracketleftbigg\neθ∂\n∂θ+eφ1\nsinθ∂\n∂φ/bracketrightbigg\n.(40)\nThe no-slip condition on the core-crust interface, r=Rc,\nreflects the fact that the amplitude of differentially rota-\ntional oscillations gradually decreases down to the star\ncenter and turns into zero on the core. The boundary\ncondition on the star surface, r=R, is dictated by\nsymmetry of the general toroidal solution of the vector\nLaplace equation which then is tested to reproduce the\nmoment of inertia of a rigidly rotating solid star[15, 27].\nThe above boundary conditions lead to the coupled alge-\nbraic equations\nAℓRℓ−1\nc+BℓR−ℓ−2\nc= 0,AℓRℓ+BℓR−ℓ−1=R(41)\nwhose solutions are\nAℓ=Nℓ,Bℓ=−NℓR2ℓ+1\nc,Nℓ=Rℓ+2\nR2ℓ+1−R2ℓ+1c.(42)\nIn spherical polar coordinates, the nodeless toroidal field\nhas only one non-zero azimuthal component\nar= 0, aθ= 0, (43)\naφ=/bracketleftbigg\nAℓrℓ+Bℓ\nrℓ+1/bracketrightbigg\n(1−ζ2)1/2dPℓ(ζ)\ndζ.\nThe snapshot of material node-free displacements in the\ncrust undergoing torsional oscillations against immobile\ncore of paramagnetic neutron star under consideration\nis pictured in Fig.2 for quadrupole, ℓ= 2, and octupole\nℓ= 3, overtones of this axial mode. The adopted firstboundary condition (37) implying that all stresses (elas-\ntic, magnetic and viscous) vanish on the core-crust inter-\nface suggests that quake-induce perturbation sets in the\nnode-free torsional motions only a finite-depth crustal\nregion, whereas central undisturbed region of the star\nremains at rest. In the next section the result of an-\nalytic computations are presented in the form showing\nthat spectral formulas for toroidal modes entrapped in\nthe crust are reduced to the above presented ones, equa-\ntions (24) and (26), for the global oscillations (in the en-\ntire volume of the star) when core radius tends to zero;\nthis fact is regarded as a test justifying mathematical\ncorrectness of presented computations.\nIII. SPECTRAL FORMULAE FOR THE\nFREQUENCY AND LIFETIME\nThe computation of integrals defining mass parame-\nterM, parameter of vibrational rigidity Kand viscous\nfrictionD[which has been presented in some details\nelsewhere[13, 27] are quite lengthy but straightforward\nand, therefore, are not presented here. The mass param-\neter can be conveniently represented in the form\nM= 4πρR5ℓ(ℓ+1)\n(2ℓ+1)(2ℓ+3)m(ℓ), (44)\nm(ℓ) = (1−λ2ℓ+1)−2(45)\n×/bracketleftbig\n1−(2ℓ+3)λ2ℓ+1+\n+(2ℓ+1)2\n2ℓ−1λ2ℓ+3−2ℓ+3\n2ℓ−1λ2(2ℓ+1)/bracketrightbigg\n,\nλ=Rc\nR= 1−h, h=∆R\nR, (46)\n∆R=R−Rc,0≤λ <1. (47)\nTheλ-terms in the aboveand foregoingequationsemerge\nas result of integration along the radial coordinate from\nradius of the core-crust interface r=Rcto the star ra-\ndius,r=R. The integral coefficient of viscous friction is\ngiven by\nD= 4πηR3ℓ(ℓ2−1)\n2ℓ+1d(ℓ), (48)\nd(ℓ) = (1−λ2ℓ+1)−1/bracketleftbigg\n1−(ℓ+2)\n(ℓ−1)λ2ℓ+1/bracketrightbigg\n.(49)\nFor the lifetime we obtain\nτ(0tℓ) =2τν\n(2ℓ+3)(ℓ−1)m(ℓ)\nd(ℓ), τν=R2\nν, ν=η\nρ.(50)\nIn Fig.3, the fractional lifetime is plotted as a function\nof multipole degree ℓwith indicated values of fractional\ndepthsh= ∆R/R. It shows that the higher ℓthe shorter\nlifetime. It is easy to see that in the limit, λ= (Rc/R)→\n0, we regain the spectral formula for lifetime of global\ntorsional nodeless vibrations of solid star[7, 13]\nτ(0tℓ) =2¯τν\n(2ℓ+3)(ℓ−1),¯τν=R2\n¯ν,¯ν=¯η\n¯ρ.(51)7\n1 5 10 15 20\nl0.000.010.101.00τ(0tl)/τνh = 0.2\nh = 0.4\nh = 0.6Torsion vibration mode\nFIG. 3. (Color online) The fractional lifetime of torsion no de-\nless oscillations of the neutron star crust damped by force o f\nviscous shear stresses as a function of multipole degree ℓcom-\nputed at indicated values of the fractional depth hof periph-\neral seismogenic layer.\nin which by ¯ τνis understood, in this latter case, the av-\nerage kinematic viscosity of the star matter as a whole;\nthe extensive discussion of this transport coefficient can\nbe found in[37]. Forthe node-freetorsionaloscillationsof\nsolidstar,thelastequationhasoneandthesamephysical\nsignificance as the well-known Lamb formula does for the\ntime of viscous damping of spheroidal node-free vibra-\ntions which in the context of neutron star pulsations has\nbeen extensivelydiscussed in[38]. Regardingthe problem\nunder consideration we cannot see, however, how the ob-\ntained formulae can be applied to observational data on\nQPOs in SGRs. Nonetheless, their practical usefulness is\nthat theycanbe utilized in the study ofamorewideclass\nof solid celestial objects such as Earth-likeplanets[18, 19]\nand white dwarf stars[39].\nFrom above it is clear that the integral coefficient of\nelastic rigidity Keof torsional vibrations has analytic\nform similar to that for coefficient of viscous friction D,\nnamely\nKe= 4πµR3ℓ(ℓ2−1)\n2ℓ+1ke(ℓ), (52)\nke(ℓ) = (1−λ2ℓ+1)−1/bracketleftbigg\n1−(ℓ+2)\n(ℓ−1)λ2ℓ+1/bracketrightbigg\n.(53)\nThefrequencyasafunctionofmultipoledegree ℓofnode-\nfree elastic vibrations in question νe(ℓ) (measured in Hz\nand related to angular frequency as ωe(ℓ) = 2πνe(ℓ) =\nKe/M) is given by\nν2\ne(ℓ) =ν2\ne[(2ℓ+3)(ℓ−1)]ke(ℓ)\nm(ℓ), (54)\nωe= 2πνe=ct\nR, ct=/radicalbiggµ\nρ, λ= 1−h, h=∆R\nR.(55)\nItiseasytoseethatinthelimit λ→0, weregainspectral\nformula for the frequency of global torsional oscillations,\nhaving the form of equation (54) with ke(ℓ) =m(ℓ) =\n1. Understandably that in this latter case all material\ncharacteristics belong to the star as a whole.0204060νm(l)/νA\nh = 0.2\nh = 0.4\nh = 0.6Alfven toroidal mode\n0 10 20 30 40\nl0.00.20.40.60.8Pm(l)/PA\nh = 0.2\nh = 0.4\nh = 0.6\nFIG. 4. (Color online) Fractional frequency and period\nof nodeless torsional magneto-solid-mechanical oscillat ions,\ntoroidal Alfv´ en mode – 0at\nℓ, entrapped in the neutron star\ncrust as functions of multipole degree ℓcomputed at indi-\ncated values of the fractional depth hof peripheral seismo-\ngenic layer. The value h= 1 corresponds to global torsional\noscillations excited in the entire volume of the star. Here\nνA=ωA/2π, where ωA=vA/RwithvA=B/√4πρbeing\nthe velocity of Alfv´ en wave in crustal matter of density ρand\nPA= 2π/ωA.\nThe magneto-mechanical stiffness of Alfv´ en vibrations\nKmcan conveniently be written as\nKm=B2R3ℓ(ℓ2−1)(ℓ+1)\n(2ℓ+1)(2ℓ−1)km(ℓ) (56)\nkm(ℓ) = (1−λ2ℓ+1)−2/braceleftbigg\n1+3λ2ℓ+1\n(ℓ2−1)(2ℓ+3)×(57)\n×/bracketleftbigg\n1−1\n3ℓ(ℓ+2)(2ℓ−1)λ2ℓ+1/bracketrightbigg/bracerightbigg\n.\nThis leads to the following two-parametric spectral for-\nmula\nν2\nm(ℓ) =ν2\nA/bracketleftbigg\n(ℓ2−1)2ℓ+3\n2ℓ−1/bracketrightbiggkm(ℓ)\nm(ℓ),(58)\nωA= 2πνA=vA\nR, vA=B√4πρ. (59)\nIn the forward asteroseismic analysis of QPO data rely-\ning on this latter spectral formula, the Alfv´ en frequency,\nνAand the fractional depth of seismogenic zone, h, are\nregarded as free parameters which are adjusted so as\nto reproduce general trends in the observed QPOs fre-\nquencies. In Fig.4, the fractional frequencies and periods\nof this toroidal Alfv´ en mode as functions of multipole\ndegreeℓare plotted with indicated values of fractional\ndepth of the seismogenic layer h. Remarkably, the lowest8\novertone of global oscillations is of quadrupole degree,\nℓ= 2, whereas for vibrations locked in the crust, the\nlowest overtone is of dipole degree, ℓ= 1, as is clearly\nseen in Fig.5. This suggests that dipole vibration can be\nthought of as Goldstone’s soft mode whose most conspic-\nuouspropertyis that the mode disappears(the frequency\ntends to zero) when key parameter regulating the depth\nof seismogenic zone λ→0.\n0 0.2 0.4 0.6 0.8 1\nh0.011100ω/ωA\nl=1\nl=2\nl=3\nl=10\nl=50\nFIG. 5. (Color online) Fractional frequency of nodeless tor -\nsional Alfv´ enoscillations ofindicated overtones ℓas afunction\nof the fractional depth hof peripheral seismogenic layer. The\nvanishing of dipole overtone in the limit of h→1, the case\nwhen entire mass of neutron star sets in torsional oscillati ons,\nsuggests that dipole vibration possesses property typical to\nGoldstone’s soft modes .\nIV. APPLICATION TO QPOS IN THE\nOUTBURST X-RAY FLUX FROM SGR 1806-20\nAND SGR 1900+14\nThe basic physics underlying current understanding of\ninterconnection between quasi-periodic oscillations of de-\ntectedelectromagneticfluxandvibrationsofneutronstar\nhas been recognized long ago[23, 25]. Owing to the ef-\nfect of strong flow-field coupling, which is central to the\npropagation of Alfv´ en waves, the quake induced pertur-\nbation excites coupled vibrations of perfectly conduct-\ning solid-state plasma of the crust (as well as gaseous\nplasma of magnetar corona expelled from the surface by\noutburst) and frozen-in lines of magnetic field. Outside\nthe star the vibrations of magnetic field lines are cou-\npled with oscillations of gas-dust plasma expelled from\nthe star surface by quake. And it is these fluctuations of\nouter lines of magnetic field, operating like transmitters\nof beams of charged particles producing coherent (curva-\nture and/or synchrotron) high-energy radiation, are de-\ntected as QPOs of light curves of the SGRs giant flares.\nIn applying the obtained spectral formulae to the fre-\nquencies of detected QPOs we examine two scenarios,\nnamely, when quake-induced torsional vibrations are re-\nstored by joint action of Lorentz magnetic and Hooke’s\nelastic forces and when oscillations are of pure Alfv´ en’snature, that is, produced by torsional seismic vibra-\ntions of crust against core under the action of solely one\nLorentz force of magnetic field stresses.\nA. Crust vibrations driven by combined action of\nLorentz magnetic and Hooke’s elastic forces\nIn this case, the asteroseismic analysis of detected\nQPOs rests on the three-parametric spectral formula\nν2(ℓ)[νA,νe,h] =ν2\nm(ℓ)[νA,h]+ν2\ne(ℓ)[νe,h].(60)\nThe suggested theoretical ℓ-pole specification of the de-\ntectedfrequenciesispresentedinFig. 6forSGR1900+14\nand in Fig. 7 and Fig. 8, exhibiting remarkable correla-\ntion between depth of seismogenic zone and fundamen-\ntal frequencies of magnetic and elastic oscillations - the\nlarger ∆ R, the higher basic frequencies of Alfv´ enic νA\nand elastic νevibrations. It is seen from computations\nfor SGR 1806-20, that reasonable fit of data can be at-\ntained with h= 0.2 (for the star model with radius 20\nkm, ∆R= 2 km) and with h= 0.4 (∆R= 5 km).\n0 5 10 15 20050100150200ν, Hzνdata\nh = 0.4\n0 5 10 15 20 25 30 35\nl050100150200ν, Hzνdata\nh = 0.2νe = 5.1 Hz, νA = 6.1 Hz\nνe = 2.4 Hz, νA = 3.5 HzSGR 1900+14\nSGR 1900+14\nFIG. 6. (Color online) Theoretical fit of the QPOs frequency\nin the X-ray flux from SGR 1900+14 on the basis of three-\nparametric theoretical spectrum of frequency of torsional seis-\nmic vibrations in the crustal region of indicated fractiona l\ndepth.\nIt is worth emphasizing that at above values of h,\nthe obtained here tree-parametric spectral formula much\nbetter match the data as compared to that for global,\nin the entire volume, vibrations studied in[16]. On this\nground we conclude, if the detected QPOs are produced\nby seismic vibrations of peripheral region of the star un-\nder coherent action of Lorentz and Hooke’s forces, then\nthe depth of seismogenic layer ∆ Rshould be quite large,\nsomewhere in the range 0 .2R <∆R <0.4R.9\n0 5 10 15 20 25 30050100150200ν, Hzνdata          SGR 1806-20\nh = 0.2\n100 150 200 250 300 350 400\nl500100015002000ν, Hzνdata          SGR 1806-20\nh = 0.2νe = 3.4 Hz, νA = 5.0 Hz\nνe = 3.4 Hz, νA = 5.0 Hz\nFIG. 7. (Color online) The same as Fig.6, but for SGRs 1806-\n20 with h=0.2.\n0 5 10 15 20050100150200ν, Hzνdata          SGR 1806-20\nh = 0.4\n50 100 150 200 250\nl5001000150020002500ν, Hzνdata          SGR 1806-20\nh = 0.4νe = 7.2 Hz, νA = 8.6 Hz\nνe = 7.2 Hz, νA = 8.6 Hz\nFIG. 8. (Color online) The same as Fig.6, but for SGRs 1806-\n20 with h=0.4.\nB. Lorentz-force-driven vibrations of crustal\nsolid-state plasma\nIt seems appropriate to note, that pure Alfv´ en oscilla-\ntions ofcrustalelectron-nuclearsolid-stateplasmaabout\naxis of magnetic field frozen in the immobile core have\nbeen studied some time ago[29] in the context of search-\ning for fingerprints of post-glitch vibrational behavior of\nradio pulsars. In the problem under consideration, one\ncan use one and the same spectral formula for the ℓ-pole\nspecification of detected QPOs which in above notationsis written as\nν2(ℓ) =ν2\nm(ℓ)[vA,h]. (61)\n0 5 10 15\nl0100200300ν, Hzνdata\nAlfven toroidal mode0100200240ν, Hzνdata\nAlfven toroidal mode\nνA = 9.4 Hz, h = 0.6νA = 9.8 Hz, h = 0.6SGR 1900+14\nSGR 1806-20\nFIG. 9. (Color online) Theoretical description (lines) of d e-\ntected QPO frequencies (symbols) in the X-ray flux during\nthe flare of SGRs 1806-20 and SGR 1900+14 as overtones of\npure Alfv´ en torsional nodeless oscillations of crustal ma gneto-\nactive plasma under the action of solely Lorentz restoring\nforce.\nThe results presented in Fig.9 and Fig.10 show that at\nindicated input parameters, i.e., the Alfv´ en frequency νA\nand the fractional depth of seismogenic layer h= ∆R/R,\nthe model too adequately reproduces general trends in\nthe data with fairly reasonable ℓ-pole specification of\novertones pointed out by integer numbers along x-axis.\nIt is seen that the low-frequency QPOs in data for SGR\n1806-20, are interpreted as dipole and quadrupole over-\ntones:ν(0at\n1) = 18 and ℓ(0at\n2) = 26 Hz. And the high-\nfrequency kilohertz vibrations with 627 Hz and 1870 Hz\nare unambiguously specified as high-multipole overtones:\nν(0at\nℓ=42) = 627 Hz and ν(0at\nℓ=122) = 1870 Hz. However,\nin this latter scenario of Lorentz-force-dominated vibra-\ntionsthe best fit ofdataisattained atfairlylargevalueof\nfractional depth, h= 0.6, which is much larger than the\nexpected depth of the crust. In our opinion, this result\nmaybe regardedasindication to that the detected QPOs\nare formed by coherent vibrations of crustal solid-state\nplasma and plasma of magnetar corona.\nV. CONCLUDING REMARKS\nEver since identification of pulsars with rapidly rotat-\ning neutron stars it has been argued[40, 41] that two\nkey properties of these compact objects – (i) the degen-\neracy of neutron (non-conducting) Fermi-matter whose\npressure opposes the pressure of self-gravity, and (ii) a\nhighly stable to decay super-strong magnetic fields – can\nbe reconciled, if poorly conducting neutron-dominated\nstellar matter, constituting the neutron star cores, has10\n0 50 100 150\nl1010010005000ν, Hzνdata\nAlfven toroidal mode\nνA = 9.4 Hz, h = 0.6SGR 1806-20\nFIG. 10. (Color online) Same as Fig.9 but for SGRs 1806-20.\nbeen brought to gravitational equilibrium in the per-\nmanently magnetized state. The most plausible is the\nstate of Pauli’s paramagnetic saturation with spin mag-\nnetic momentsofneutronspolarizedalongthe axisoffos-\nsil field inherited from massive progenitor and amplified\nin magnetic-flux-conserving core- collapse supernova[6].\nThis idea is central to the considered two-component,\ncore-crust, model of paramagnetic neutron star whose\nless dense and highly conducting, metal-like, material\nof the crust is considered as a solid-state, electron-\nnuclear, plasma pervaded by frozen in the core magnetic\nfield. This difference between electrodynamic properties\nof core and crust matter (permanently magnetized non-\nconducting core and perfectly conducting non- magnetic\ncrust) suggests that magnetic cohesion between massive\ncore (permanent magnet) and crust (metal-like material)\nshould plays central role in seismic activity of the star.\nWorking from such an understanding, we have computed\nfrequency spectra of node-free torsional oscillations of\ncrust against immobile core under the action Lorentz\nand Hooke restoring forces and damped by Newtonian\nviscous force. As a trial function of oscillating mate-rial displacements we have used the node-free toroidal\nfield computed from vector Laplace equation. The ob-\ntained spectral formulas are of some interest in their own\nright because they can be applied to more wide class of\ncelestial objects. In this work we applied the obtained\nanalytic frequency spectra to magnetars, highly magne-\ntized quaking neutron starswhose bursting seismic activ-\nity is commonly associated with release of magnetic field\nstresses. Focus was laid on forward asteroseismic anal-\nysis of fast X-ray flux oscillations during the giant flare\nof SGR 1900+14 and SGR 1806-20 and, thus, assuming\nthat these oscillations are produced by torsion vibrations\nof crustal solid-state plasma about axis of dipole mag-\nnetic field frozen in the immobile permanently magne-\ntized core. In so doing we have investigated two cases\nof post-quake vibrational relaxation of the star, depend-\ning on restoring forces. In first case, the analysis of data\nhas been based on assumption that detected QPOs owe\ntheir existence to node-free torsional vibrations of crust\nagainst core restored by joint action of Lorentz magnetic\nand Hooke’s elastic forces. 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Trimble, Stellar Interio rs\n(Springer, Berlin, 2004)\n[40] J. D. Anand, P. Bhattacharjee, S. N. Biswas, M. Hasan,\nPhys. Rev. D 23, 316 (1981)\n[41] G. A. Shulman, Sov. Astron., 35, 50 (1991)"    },    {        "title": "1811.00216v1.Hereditary_effects_of_exponentially_damped_oscillators_with_past_histories.pdf",        "content": "arXiv:1811.00216v1  [math.DS]  1 Nov 2018Hereditary effects of exponentially damped oscillators\nwith past histories\nJian Yuan✩1,∗\nCollege of Mathematic and Information Science, Shandong In stitute of Business and\nTechnology, Yantai 264005, P.R.China\nGuozhong Xiu✩1, Bao Shi, Liying Wang\nInstitute of System Science and Mathematics, Naval Aeronau tical University, Yantai\n264001, P.R.China\nAbstract\nHereditary effects of exponentially damped oscillators with past hist ories are\nconsidered in this paper. Nonviscously damped oscillators involve her editary\ndamping forces which depend on time-histories of vibrating motions v ia con-\nvolution integrals over exponentially decaying functions. As a result , this\nkind of oscillators are said to have memory. In this work, initialization f or\nnonviscously damped oscillators is firstly proposed. Unlike the classic al vis-\ncously damped ones, information of the past history of response v elocity is\nnecessary to fully determine the dynamic behaviors of nonviscously damped\noscillators. Then, initialization response of exponentially damped osc illators\nis obtained to characterize the hereditary effects on the dynamic r esponse.\nAt last, stability of initialization response is proved and the hereditar y effects\nare shown to gradually recede with increasing of time.\nKeywords: Nonviscously damping model, exponentially damped\noscillators, initialization problems, hereditary effects\n✩These authors contributed equally to this work.\n∗Corresponding author. Email: yuanjianscar@gmail.com\nPreprint submitted to Elsevier November 2, 20181. Introduction\nViscoelastic materials have seen broad applications in vibration contr ol\nengineering due to their high damping capacity [1]. The modeling of con-\nstitutive relations of viscoelastic materials is a fundamental task fo r analysis\nand design of viscoelastically damped structures. However, it is also a chal-\nlenging work because the mechanical behavior of viscoelastic mater ials is\nhighly dependent on various factors such as time, temperature, t he vibrating\nfrequency and so on [2].\nThe integral constitutive models are derived based on the materials prop-\nerties of stress relaxation and creep. The stress relaxation func tions and\ncreep functions are memory and hereditary kernels in the integral constitu-\ntive equations. They can be expressed by a series of exponential f unctions\n[3, 4], power-law functions [5], Mittage-Leffler functions [5], or other types\nof functions. The integral constitutive models are superior to the differential\nones in many aspects: (i) the fading memory property can be chara cterized\nand time-history of loading acting on materials can be recorded; (ii) t he\nstress relaxation functions or creep functions are easily and direc tly obtained\nvia experiment data fitting; (iii) other factors such as temperatur e and the\nageing affects can be conveniently considered and included in the mod el.\nThe integral constitutive relations can be readily used as damping mo dels\nin viscoelastically damped structures. Equations of motion of such s ystems\nare a set of coupled second-order integro-differential equations . The presence\nof the ”integral” term makes the vibration analysis and control des ign more\ncomplicated than the classical ones. The integral type damping mod els may\nbealsocalled nonviscously damping models andthecorresponding osc illators\nare called nonviscously damped oscillators.\nResearches on nonviscously damped oscillators are mainly concentr ated\non two types: one is the exponentially damped oscillators where the d amp-\ning force are expressed by exponentially fading memory kernel; the other is\nthe fractional-order oscillators where the viscoelastic relaxation f unctions are\ncharacterized by power-law functions or Mittage-Leffler function s. Adhikari\nand his colleagues have systematically investigated the structural dynam-\nics with exponentially damped models, including dynamics of exponentia lly\ndamped single-degree-of-freedom and multi-degree-of-freedo m systems, iden-\ntification and quantification of damping in [6-8]. They have discovered that,\nunlike the classical viscously damped oscillators, an exponentially dam ped\nsingle-degree-of-freedom oscillator has three eigenvalues. The c omplex con-\n2jugate pair of roots corresponds to the vibration motion and the t hird one\ncorresponds to a purely dissipative motion which is always non-oscillat ory\nin nature. The dynamics of exponentially damped oscillators is govern ed\nby both of the viscous damping factor and the nonviscous damping f actor.\nFor the dynamic analysis of multi-degree-of-freedom systems, th ey have de-\nveloped a state-space approach using additional dissipation coord inates and\nthe configuration space method. It was shown that the characte ristic equa-\ntion for an Ndegree-of-freedom system is more than Nand the modes are\ndivided into elastic modes and nonviscous modes. In [9], an analytical s o-\nlution using modal superposition and two numerical solutions on the b asis\nof finite element formulations are developed for the analysis of an ex ponen-\ntially damped solid rod. In [10], a method has been proposed to calculat e\neigensolution derivatives for the nonviscously damped systems. In [11], a\nstate-space method has been proposed to identify modal and phy sical pa-\nrameters for the nonviscously damped systems using recorded tim e-histories\nof the dynamic response. A closed-form approximation expression of the\neigenvalues for non-viscous, non-proportional vibrating system has been de-\nrived in [12]. The forced vibration response to an arbitrary forced e xcitation\nhas been obtained in [4].\nFractional-order oscillators are another type of nonviscously dam ped os-\ncillators. They are under extensive investigations for the last thre e decades.\nTheconstitutivemodelsinvolvingfractional-orderderivativeshave beenviewed\nmore accurate and concise than other ones, because the relaxat ion functions\nof viscoelastic materials can be perfectly fitted from experimental data by\nusing power-law functions or Mittage-Leffler functions [13, 14]. As a result,\nresearches onfractionaloscillatorshavebeenexpected tobeap romising work\nfor structural dynamics analysis and control design. However, t he fractional\ndifferential equations of motions are difficult to deal with due to the p resence\nof the weakly singular kernels. Studies on dynamic responses of fra ctional-\norder oscillators have been reviewed in [15]. Asymptotically steady st ate\nbehavior of fractional oscillators has been presented in [16, 17]. Th e crite-\nria for the existence and the behavior of solutions on the basis of fu nctional\nanalytic approach have been obtained in [18-20], and the impulsive res ponse\nfunction for the linear single-degree-of-freedom fractional osc illator has been\nderived. The asymptotically steady state response of fractional oscillators\nwith more than one fractional derivatives have been analyzed in [21]. Based\non the energy storage and dissipation properties of the Caputo fr actional-\norder derivatives, the expression of mechanical energy in single-d egree-of-\n3freedom fractional oscillators has been determined and energy re generation\nand dissipation during the vibratory motion have been obtained in [22]. Vi-\nbration controls have been designed using sliding mode control tech nique\nand adaptive control technique for single-degree-of-freedom f ractional oscil-\nlators, multi-degree-of-freedom fractional oscillators, and fra ctional Duffing\noscillators in [23, 24].\nThe damping forces in nonviscously damped oscillators depend on the\npast history of motion via convolution integrals over relaxation func tions. As\na result, the dynamics of the nonviscously damped oscillators is said t o have\nmemory. However, the memory and hereditary properties of frac tional-order\nsystems have been neglected andambiguous for a long time [25]. Cons idering\nthe memory effect and prehistory of fractional oscillators, the his tory effects\nand initialization problems for fractionally damped vibration equations have\nbeen proposed in [26-30]. The stability of initialization response has be en\nproved based on the unit impulse response function and the Lyapun ov sta-\nbility theorem in [31].\nThis paper investigates the hereditary effects on the dynamics of t he\nexponentially damped oscillators with past histories. We first declare that\nknowledge of the equations of motion, along with the initial displaceme nt\nand velocity is insufficient to determine the dynamics behaviors. The in itial\nconditions for the systems should also contain the past history of r esponse\nvelocity. Then we obtain the initialization response of exponentially da mped\noscillators, which characterizes the hereditary effects of the hist ory of vibra-\ntion on the dynamic response. At last, we prove that hereditary eff ects on\nthe initialization response recede to zero with increasing of time.\n2. Initialization for nonvicously damped oscillators\nTheintegralconstitutiverelationsofviscoelasticmaterialsarerep resented\nby the following integro-differential equation of Volterra type:\nσ(t) =/integraldisplayt\n−∞G(t−τ) ˙ε(τ)dτ (1)\nwhereσ(t) is the stress, ε(t) is the stain, G(t) is the stress relaxation func-\ntion. Thelower terminal intheintegral is −∞because thestress ofviscoelas-\ntic materials is dependent on all the time histories of the strain [2]. Whe n\nthe integral constitutive equation (1) is applied to model the dynam ics of\n4structures incorporated with viscoelastic dampers, the equation of motion is\nm¨x(t)+c/integraldisplayt\n−∞G(t−τ) ˙x(τ)dτ+kx(t) =f(t) (2)\nwheremis the mass, kis the stiffness, cis the damping coefficient, f(t) is\nthe external force acting on the system.\nThe integral term in Eq. (2) makes the dynamic models different from\nthe classical ones. It contains not only the information of vibrating displace-\nmentx(t) andvelocity ˙ x(t), but also thetime-histories ofvelocity ˙ x(t). This\nimplies that, unlike the viscously damped systems, the equation of mo tion,\nthe instantaneous displacement and velocity are insufficient to pred ict the\ndynamic behaviors. Time-histories of motion should be added to initial con-\nditions to fully determine the dynamics of nonvicously damped oscillato rs.\nAs a result, the dynamic equation with past history is described as\n\n\nm¨x(t)+c/integraltextt\n−∞G(t−τ) ˙x(τ)dτ+kx(t) =f(t), t>0\nx(0) =x0,˙x(0) =v0\n˙x(t) =v(t),−∞<t<0(3)\nwheret= 0 is the initial time and the lower terminal in the integral t=−∞\nis the starting time of vibration. In reality it is more reasonable to set the\nstarting time as t=−a, which means that the system is at quiescent before\nt=−aand begins vibrating at t=−a.\nTo be more familiar with the dynamic systems with memory, the integra l\nterm in Eq.(3) can be separated into two part:\n/integraldisplayt\n−aG(t−τ) ˙x(τ)dτ=/integraldisplay0\n−aG(t−τ) ˙x(τ)dτ+/integraldisplayt\n0G(t−τ) ˙x(τ)dτ\nThe first part characterizes the hereditary effects of the histor ies of motion\non the system dynamics, which is denoted as ψ(t):\nψ(t) =/integraldisplay0\n−aG(t−τ) ˙x(τ)dτ=/integraldisplay0\n−aG(t−τ)v(τ)dτ (4)\nIt contains the time histories of motion, acts as an internal force a nd\ninfluence the behavior of dynamic systems after the initial time t= 0.Eq.(3)\ncan be rewritten as/braceleftBigg\nm¨x(t)+c/integraltextt\n0G(t−τ) ˙x(τ)dτ+kx(t) =f(t)−cψ(t), t>0\nx(0) =x0,˙x(0) =v0(5)\n5Remark 1. In studies of Adhikari and his colleagues [6], vibrating motions\nfrom past histories have not been taken into account. In this case , the equa-\ntion of motion is\nm¨x(t)+c/integraldisplayt\n0G(t−τ) ˙x(τ)dτ+kx(t) =f(t)\nThe initial values contain only the initial displacement x(0) and the initial\nvelocity ˙x(0).\nRemark 2. In the integral constitutive relations of viscoelastic materials\n(1) and the corresponding dynamic equation of viscoelastically/non viscously\ndamped oscillators (2), G(t) can be fitted by many types of decaying func-\ntions,suchastheexponentialfunctions, thepower-lawfunction s, theMittage-\nLeffler functions, and so on. In the following sections, we concentr ate on the\ncase of exponentially decaying functions G(t) =µe−µt,µ >0. The corre-\nsponding oscillators are called exponentially damped oscillators.\n3. Hereditary effects on the dynamic response\nNow we are ready to study the initialization response of exponentially\ndamped oscillators with memories, which characterizes the heredita ry effects\nof the vibrating motion from the past history. For this purpose, we will not\ntake into account the external acting force f(t) and set it to be zero. In this\ncase, the equation of motion with initial condition is\n/braceleftBigg\nm¨x(t)+c/integraltextt\n0G(t−τ) ˙x(τ)dτ+kx(t) =−cψ(t), t>0\nx(0) =x0,˙x(0) =v0(6)\nwhereG(t) =µe−µt,µ>0. Taking Laplace transform of Eq.(6), one derives\nm/parenleftbig\ns2¯x(s)−sx0−v0/parenrightbig\n+c/parenleftbiggµs\nµ+s¯x(s)−µ\nµ+sx0/parenrightbigg\n+k¯x(s) =−c¯ψ(s) (7)\nwhere ¯x(s) is the Laplace transform of x(t),¯ψ(s) is the Laplace transform\nofψ(t). After rearranging Eq.(7), one has\n/parenleftbigg\nms2+cµs\nµ+s+k/parenrightbigg\n¯x(s) =−c¯ψ(s)+msx0+cµ\nµ+sx0+mv0(8)\n6We denote that ¯d(s) =ms2+cµs\nµ+s+kand¯h(s) =1\n¯d(s).\nFrom Eq.(8), the solution of ¯ x(s) can be derived as\n¯x(s) =−c¯h(s)¯ψ(s)+mx0s¯h(s)+cµx0¯h(s)\ns+µ+mv0¯h(s) (9)\nTaking the inverse Laplace transform of Eq.(9), one derives\nx(t) =−c/integraldisplayt\n0h(t−τ)ψ(τ)dτ+mx0˙h(t)\n+cµx0/integraldisplayt\n0h(t−τ)e−µτdτ+mv0h(t) (10)\nwhereh(t) is inverse Laplace transform of ¯h(s).\nNext we determine the expression of h(t). It has been shown in [6] that ¯d(s)\nhas zeros at s=sj,j= 1,2,3:\ns1=−α+βi,s2=−α−βi,s1=−γ,\nwhereα,β,γ> 0.\nFurthermore, ¯h(s)can be expressed in the pole-residue form as\n¯h(s)=3/summationdisplay\nj=1Rj\ns−sj\nwhereRjare the residues and calculated as\nRi= Res\ns=sj¯h(s) = lim\ns→sj(s−sj)¯h(s) =1\nlim\ns→sjms2+cµs\nµ+s+k\ns−sj=1\n∂¯d(s)\n∂s|s=sj\nAs a result, h(t) can be obtained by taking Laplace transform of ¯h(s):\nh(t) =L−1/braceleftbig¯h(s)/bracerightbig\n=3/summationdisplay\nj=1Rje−sjt(11)\nSubstituting Eq.(11) into Eq.(10), one derives the initialization respo nse of\nexponentially damped oscillators. It represents the hereditary eff ects of past\nhistories of motions from the starting time at t=−a.\n74. Stability of initialization response\nIt has been shown in Section 2 and Section 3 the effects of past histo ries\nof motions on the setting of initial conditions and dynamic response. In this\nsection, we proceed to show the hereditary effects onstability of in itialization\nresponse. We prove that this influence will gradually recede with incr easing\nof time. From Eq.(10), it is clear that the initialization response involve s\nfour parts. The last three parts decreased as time increases. No w we prove\nin mathematics that the same holds for the first part.\n−c/integraldisplayt\n0h(t−τ)ψ(τ)dτ\n=−c/integraldisplayt\n0R1es1(t−τ)ψ(τ)dτ−c/integraldisplayt\n0R1es2(t−τ)ψ(τ)dτ\n−c/integraldisplayt\n0R3es3(t−τ)ψ(τ)dτ\n=−c/integraldisplayt\n0R1e−(α−βi)(t−τ)ψ(τ)dτ−c/integraldisplayt\n0R1e−(α+βi)(t−τ)ψ(τ)dτ\n−c/integraldisplayt\n0R3e−γ(t−τ)ψ(τ)dτ\n=−2cR1/integraldisplayt\n0e−α(t−τ)cosβ(t−τ)ψ(τ)dτ−cR3/integraldisplayt\n0e−γ(t−τ)ψ(τ)dτ\n=I1+I2 (12)\nwhere\nI1=−2cR1/integraldisplayt\n0e−α(t−τ)cosβ(t−τ)ψ(τ)dτ (13)\nI2=−cR3/integraldisplayt\n0e−γ(t−τ)ψ(τ)dτ (14)\nSubstituting Eq.(4) into Eq.(13),one has\n|I1|= 2cR1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−α(t−τ)cosβ(t−τ)ψ(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= 2cR1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−α(t−τ)cosβ(t−τ)dτ/integraldisplay0\n−aG(τ−τ1)v(τ1)dτ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle(15)\n8It is reasonable to suppose that the response velocity before initia l time\nis bounded, i.e., |v(t)| ≤M,t∈[−a,0].\nNoting that |cos(βt)| ≤1, Eq.(15) yields\n|I1| ≤2cR1M/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−α(t−τ)dτ/integraldisplay0\n−aG(τ−τ1)dτ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle(16)\nSubstituting G(t) =µe−µtinto Eq.(16) yields\n|I1| ≤2cR1M/parenleftbig\n1−e−µa/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−α(t−τ)−µτdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n= 2cR1M/parenleftbig\n1−e−µa/parenrightbig\ne−αt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e(α−µ)τdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=2cR1M(1−e−µa)\n|α−µ|/vextendsingle/vextendsinglee−µt−e−αt/vextendsingle/vextendsingle\n(17)\nDue to the fact that µ,α>0, it is clear to see that\nlim\nt→∞|I1|= 0. (18)\nSubstituting Eq.(4) into Eq.(14), one has\n|I2|=cR3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−γ(t−τ)ψ(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=cR3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−γ(t−τ)dτ/integraldisplay0\n−aG(τ−τ1)v(τ1)dτ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤McR3/parenleftbig\n1−e−µa/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−γ(t−τ)−µτdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=McR3/parenleftbig\n1−e−µa/parenrightbig\ne−γt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e(γ−µ)τdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=McR3(1−e−µa)\n|γ−µ|/vextendsingle/vextendsinglee−µt−e−γt/vextendsingle/vextendsingle\n(19)\nBecauseµ,γ >0, it is also clear to see that\nlim\nt→∞|I2|= 0 (20)\n9From (12) (18) and (20), we derive that the first part of the initializ ation\nresponse (10) decreases to zero. It is very easy to verify that t he remaining\nthree parts of Eq.(10) also decreases to zero. So, we can conclud e that\nthe initialization response will gradually recede with increasing of time. In\nother words, memories from past histories of motion have no influen ce on the\nstability of exponentially damped oscillators.\n5. Conclusions\nHereditary effects of exponentially damped oscillators with past hist ories\nhave been investigated in this paper. Initial conditions have been pr oposed\nfor nonvicously damped oscillators with past histories. It has been s hown\nthat initial conditions should contain not only the vibration displaceme nt\nand velocity at initial time, but also the time-history of response velo city\nfrom the starting time of vibration. Then, initialization response of e xpo-\nnentially damped oscillators has been obtained to characterize the h ereditary\neffects of past histories on the dynamic response. At last, initializat ion re-\nsponse has been proved to gradually recede with increasing of time, which\nimplies that memories from past histories have no influence on the sta bility\nof exponentially damped oscillators.\nAcknowledgements All the authors acknowledge the valuable suggestions\nfrom the peer reviewers. This work was supported by the National Natural\nScience Foundation of China (Grant No. 11802338).\nReferences\n[1] Ibrahim, R.A., Recent advances in nonlinear passive vibration isolat ors,\nJournal of Sound Vibration 314 (2008) 371-452.\n[2] Shaw, M.T. and W.J. 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Stability of initialization response of fraction al\noscillators, Journal of Vibroengineering, 2016.\n13"    },    {        "title": "2001.03950v1.The_sharp_higher_order_Lorentz__Poincaré_and_Lorentz__Sobolev_inequalities_in_the_hyperbolic_spaces.pdf",        "content": "arXiv:2001.03950v1  [math.FA]  12 Jan 2020The sharp higher order Lorentz–Poincar´ e and\nLorentz–Sobolev inequalities in the hyperbolic spaces\nVan Hoang Nguyen\nJanuary 14, 2020\nAbstract\nIn this paper, we study the sharp Poincar´ e inequality and th e Sobolev inequalities\nin the higher order Lorentz–Sobolev spaces in the hyperboli c spaces. These results\ngeneralize the ones obtained in [ 17] to the higher order derivatives and seem to be\nnew in the context of the Lorentz–Sobolev spaces defined in th e hyperbolic spaces.\n1 Introduction\nForn≥2, let us denote by Hnthe hyperbolic space of dimension n, i.e., a complete, simply\nconnected, n−dimensional Riemmanian manifold having constant sectional curvatu re−1.\nThe aim in this paper is to generalize the main results obtained by the au thor in [17] to\nthe higher order Lorentz–Sobolev spaces in Hn. Before stating our results, let us fix some\nnotation. Let Vg,∇gand ∆ gdenote the volume element, the hyperbolic gradient and the\nLaplace–Beltrami operator in Hnwith respect to the metric grespectively. For higher\norder derivatives, we shall adopt the following convention\n∇m\ng·=/braceleftBigg\n∆m\n2g ifmis even,\n∇g(∆m−1\n2g·) ifmis odd.\nFurthermore, for simplicity, we write |∇m\ng· |instead of |∇m\ng· |gwhenmis odd if no\nconfusion occurs. For 1 ≤p,q <∞, we denote by Lp,q(Hn) the Lorentz space in Hnand\nby/ba∇dbl·/ba∇dblp,qthe Lorentz quasi-norm in Lp,q(Hn). When p=q,/ba∇dbl·/ba∇dblp,pis replaced by /ba∇dbl·/ba∇dblpthe\nEmail:vanhoang0610@yahoo.com .\n2010Mathematics Subject Classification : 26D10, 46E35, 46E30,\nKey words and phrases : Poincar´ e inequality, Poincar´ e–Sobolev inequality, Lorentz–Sob olev space,\nhyperbolic spaces.\n1Lebesgue Lp−norm inHn, i.e.,/ba∇dblf/ba∇dblp= (/integraltext\nHn|f|pdVg)1\npfor a measurable function fonHn.\nThe Lorentz–Sobolev space WmLp,q(Hn) is defined as the completion of C∞\n0(Hn) under\nthe Lorentz quasi-norm /ba∇dbl∇m\ngu/ba∇dblp,q:=/ba∇dbl|∇m\ngu|/ba∇dblp,q. In [17], the author proved the following\nPoincar´ e inequality in W1Lp,q(Hn)\n/ba∇dbl∇gu/ba∇dblq\np,q≥/parenleftbiggn−1\np/parenrightbiggq\n/ba∇dblu/ba∇dblq\np,q,∀u∈W1Lp,q(Hn). (1.1)\nprovided 1 < q≤p. Furthermore, the constant (n−1\np)qin (1.1) is the best possible and is\nnever attained. The inequality ( 1.1) generalizes the result in [ 13] to the setting of Lorentz–\nSobolev space. The first main result in this paper extends the inequa lity (1.1) to the higher\norder Sobolev space WmLp,q(Hn).\nTheorem 1.1. Givenn≥2,m≥1and1< p <∞, let us denote the following constant\nC(n,m,p) =/braceleftBigg\n((n−1)2\npp′)m\n2 ifmis even,\nn−1\np((n−1)2\npp′)m−1\n2ifmis odd,\nwherep′=p\np−1. Then the following Poincar´ e inequality holds in WmLp,q(Hn)\n/ba∇dbl∇m\ngu/ba∇dblq\np,q≥C(n,m,p)q/ba∇dblu/ba∇dblq\np,q, u∈WmLp,q(Hn) (1.2)\nfor any1< p,q < ∞ifmis even, or for any 1< q≤p <∞ifmis odd. Moreover, the\nconstant C(n,m,p)in(1.2)is sharp and is never attained.\nLet us give some comments on Theorem 1.1. The Poincar´ e inequality in the hyperbolic\nspace was proved by Tataru [ 20]\n/integraldisplay\nHn|∇m\ngu|pdVg≥C/integraldisplay\nHn|u|pdVg, u∈C∞\n0(Hn) (1.3)\nfor some constant C >0. The sharp value of constant Cin (1.3) is computed by Mancini\nand Sandeep [ 12] whenp= 2 and by Ngo and the author [ 13] for arbitrary p(see [5] for\nanother proof when m= 1). Theorem 1.1gives an extension of the Poincar´ e inequality\n(1.3) with the sharp constant to the higher order Sobolev spaces WmLp,q(Hn). Similar\nto the case m= 1 established in [ 17], we need an extra condition q≤pwhenmis odd\nto apply the symmetrization argument. The proof of Theorem 1.1follows the idea in the\nproof of Theorem 1 .1 in [13] by using the iterate argument. The main step in the proof is\nto establish the inequality when m= 2. The case m= 1 was already done in [ 17].\nThere have been many improvements of ( 1.3) with the sharp constant in literature. For\nexamples, theinteresting readersmayconsult thepapers[ 3,5,6,12,15]fortheimprovements\nof (1.3) form= 1 by adding the remainder terms concerning to Hardy weights or to the\nLq−norms with p < q≤np\nn−p. For the higher order Sobolev spaces, we refer the readers\n2to the papers of Lu and Yang [ 8,10,11,16]. Especially, in [ 16, Theorem 1 .1] the author\nestablished the following improvement of ( 1.3) forp= 2\n/integraldisplay\nHn|∆gu|2dVg−(n−1)4\n16/integraldisplay\nHn|u|2dVg≥Sn,2/parenleftbigg/integraldisplay\nHn|u|2n\nn−4dVg/parenrightbiggn−4\nn\n, u∈C∞\n0(Hn) (1.4)\nprovided n≥5whereSn,kdenotesthesharpconstantintheSobolevinequalityinEuclidean\nspaceRn\n/integraldisplay\nRn|∇ku|2dx≥Sn,k/parenleftbigg/integraldisplay\nRn|u|2n\nn−2kdx/parenrightbiggn−2k\nn\n, u∈C∞\n0(Rn) (1.5)\nwhenn >2k. Theconstant Sn,1wasfoundoutindependently byTalenti [ 18]andAubin[ 2].\nThe sharp constant Sn,k,k≥2 was computed explicitly by Lieb [ 9] by proving the sharp\nHardy–Littlewood–Sobolev inequality which is the dual version of ( 1.5). In [17] the author\nprovedthefollowinginequality: given n≥4and2n\nn−1≤q≤p < n, thenforany q≤l≤nq\nn−p\nwe have\n/ba∇dbl∇gu/ba∇dblq\np,q−/parenleftbiggn−1\np/parenrightbiggq\n/ba∇dblu/ba∇dblq\np,q≥Sq\nn,p,q,l/ba∇dblu/ba∇dblq\np∗,l,∀u∈W1Lp,q(Hn),(1.6)\nwherep∗=np\nn−p, and\nSn,p,q,l=\n\n/bracketleftbigg\nn1−q\nlσq\nnn/parenleftBig\n(n−p)(l−q)\nqp/parenrightBigq+q\nl−1\nS/parenleftBig\nlq\nl−p,q/parenrightBig/bracketrightbigg1\nq\nifq < l≤nq\nn−p,\nn−p\npσ1\nnn ifl=q,\nwhereσndenotes the volume of unit ball in RnandS/parenleftbiglq\nl−p,q/parenrightBig\nis the sharp constant in the\nSobolev inequality with fractional dimension (see [ 14]). It is interesting that the constant\nSn,p,q,lin (1.6) is sharp and coincides with the sharp constant in the Lorentz–Sob olev type\ninequality in Euclidean space Rn,\n/ba∇dbl∇u/ba∇dblq\np,q≥Sq\nn,p,q,l/ba∇dblu/ba∇dblq\np∗,l.\nThe previous inequality was proved by Alvino [ 1] forl=q≤p < nand by Cassani, Ruf\nand Tarsi [ 7] forl=q≥p. Our next aim is to improve the inequality ( 1.2) in spirit of\n(1.4) and (1.6).\nTheorem 1.2. Letn > m≥1be integers, 1< p <n\nmand2n\nn−1≤q <∞. Suppose, in\naddition, that q≤pifmis odd. Then there holds\n/ba∇dbl∇m\ngu/ba∇dblq\np,q−C(n,m,p)q/ba∇dblu/ba∇dblq\np,q≥S(n,m,p)q/ba∇dblu/ba∇dblq\np∗m,q, u∈WmLp,q(Hn),(1.7)\nwherep∗\ni=np\nn−ip,i= 0,1,2,...,and\nS(n,m,p) =\n\nσm\nnn/producttextk−1\ni=0n(n−2p∗\n2i)\np∗\n2i(p∗\n2i)′ ifm= 2k,k≥1\nσm\nnnn−p\np/producttextk\ni=1n(n−2p∗\n2i−1)\np∗\n2i−1(p∗\n2i−1)′ifm= 2k+1,k≥1.\n3The rest of this paper is organized as follows. In Section §2, we recall some facts on\nthe hyperbolic spaces and the non-increasing spherically symmetric rearrangement in the\nhyperbolic spaces. We also prepare some auxiliary results which are im portant in the proof\nof our main results. Theorem 1.1is proved in Section §3 while the Section §4 is devoted\nto prove Theorem 1.2.\n2 Preliminaries\nWe start this section by briefly recalling some basis facts on the hype rbolic spaces and the\nLorentz–Sobolev space defined in the hyperbolic spaces. Let n≥2, a hyperbolic space\nof dimension n(denoted by Hn) is a complete , simply connected Riemannian manifold\nhaving constant sectional curvature −1. There are several models for the hyperbolic space\nHnsuch as the half-space model, the hyperboloid (or Lorentz) model and the Poincar´ e\nball model. Notice that all these models are Riemannian isometry. In t his paper, we are\ninterested in the Poincar´ e ball model of the hyperbolic space since this model is very useful\nfor questions involving rotational symmetry. In the Poincar´ e ball model, the hyperbolic\nspaceHnis the open unit ball Bn⊂Rnequipped with the Riemannian metric\ng(x) =/parenleftBig2\n1−|x|2/parenrightBig2\ndx⊗dx.\nThe volume element of Hnwith respect to the metric gis given by\ndVg(x) =/parenleftBig2\n1−|x|2/parenrightBign\ndx,\nwheredxis the usual Lebesgue measure in Rn. Forx∈Bn, letd(0,x) denote the geodesic\ndistance between xand the origin, then we have d(0,x) = ln(1+ |x|)/(1−|x|). Forρ >0,\nB(0,ρ) denote the geodesic ball with center at origin and radius ρ. If we denote by ∇\nand ∆ the Euclidean gradient and Euclidean Laplacian, respectively as well as/an}b∇acketle{t·,·/an}b∇acket∇i}htthe\nstandard scalar product in Rn, then the hyperbolic gradient ∇gand the Laplace–Beltrami\noperator ∆ ginHnwith respect to metric gare given by\n∇g=/parenleftBig1−|x|2\n2/parenrightBig2\n∇,∆g=/parenleftBig1−|x|2\n2/parenrightBig2\n∆+(n−2)/parenleftBig1−|x|2\n2/parenrightBig\n/an}b∇acketle{tx,∇/an}b∇acket∇i}ht,\nrespectively. For a function u, we shall denote/radicalbig\ng(∇gu,∇gu) by|∇gu|gfor simplifying\nthe notation. Finally, for a radial function u(i.e., the function depends only on d(0,x)) we\nhave the following polar coordinate formula\n/integraldisplay\nHnu(x)dx=nσn/integraldisplay∞\n0u(ρ)sinhn−1(ρ)dρ.\nIt is now known that the symmetrization argument works well in the s etting of the\nhyperbolic. It is the key tool in the proof of several important ineq ualities such as the\n4Poincar´ e inequality, the Sobolev inequality, the Moser–Trudinger in equality in Hn. We\nshall see that this argument is also the key tool to establish the main results in the present\npaper. Let us recall some facts about the rearrangement argum ent in the hyperbolic space\nHn. A measurable function u:Hn→Ris called vanishing at the infinity if for any t >0\nthe set{|u|> t}has finite Vg−measure, i.e.,\nVg({|u|> t}) =/integraldisplay\n{|u|>t}dVg<∞.\nFor such a function u, its distribution function is defined by\nµu(t) =Vg({|u|> t}).\nNotice that t→µu(t) is non-increasing and right-continuous. The non-increasing rear -\nrangement function u∗ofuis defined by\nu∗(t) = sup{s >0 :µu(s)> t}.\nThe non-increasing, spherical symmetry, rearrangement funct ionu♯ofuis defined by\nu♯(x) =u∗(Vg(B(0,d(0,x)))), x∈Hn.\nIt is well-known that uandu♯have the same non-increasing rearrangement function (which\nisu∗). Finally, the maximal function u∗∗ofu∗is defined by\nu∗∗(t) =1\nt/integraldisplayt\n0u∗(s)ds.\nEvidently, u∗(t)≤u∗∗(t).\nFor 1≤p,q <∞, the Lorentz space Lp,q(Hn) is defined as the set of all measurable\nfunction u:Hn→Rsatisfying\n/ba∇dblu/ba∇dblLp,q(Hn):=/parenleftbigg/integraldisplay∞\n0/parenleftBig\nt1\npu∗(t)/parenrightBigqdt\nt/parenrightbigg1\nq\n<∞.\nIt is clear that Lp,p(Hn) =Lp(Hn). Moreover, the Lorentz spaces are monotone with\nrespect to second exponent, namely\nLp,q1(Hn)/subsetnoteqlLp,q2(Hn),1≤q1< q2<∞.\nThe functional u→ /ba∇dblu/ba∇dblLp,q(Hn)is not a norm in Lp,q(Hn) except the case q≤p(see [4,\nChapter 4, Theorem 4 .3]). Ingeneral, it is aquasi-norm which turns out tobe equivalent to\nthe norm obtained replacing u∗by its maximal function u∗∗in the definition of /ba∇dbl·/ba∇dblLp,q(Hn).\nMoreover, as a consequence of Hardy inequality, we have\n5Proposition 2.1. Givenp∈(1,∞)andq∈[1,∞). Then for any function u∈Lp,q(Hn)\nit holds\n/parenleftbigg/integraldisplay∞\n0/parenleftBig\nt1\npu∗∗(t)/parenrightBigqdt\nt/parenrightbigg1\nq\n≤p\np−1/parenleftbigg/integraldisplay∞\n0/parenleftBig\nt1\npu∗(t)/parenrightBigqdt\nt/parenrightbigg1\nq\n=p\np−1/ba∇dblu/ba∇dblLp,q(Hn).(2.1)\nFor 1≤p,q <∞and an integer m≥1, we define the m−th order Lorentz–Sobolev\nspaceWmLp,q(Hn) by taking the completion of C∞\n0(Hn) under the quasi-norm\n/ba∇dbl∇m\ngu/ba∇dblp,q:=/ba∇dbl|∇m\ngu|/ba∇dblp,q.\nIt is obvious that WmLp,p(Hn) =Wm,p(Hn) them−th order Sobolev space in Hn. In [17],\ntheauthorestablishedthefollowingP´ olya–Szeg¨ oprincipleinthefir storderLorenz–Sobolev\nspacesW1Lp,q(Hn) which generalizes the classical P´ olya–Szeg¨ o principle in the hyper bolic\nspace.\nTheorem 2.2. Letn≥2,1≤q≤p <∞andu∈W1Lp,q(Hn). Thenu♯∈W1Lp,q(Hn)\nand\n/ba∇dbl∇gu♯/ba∇dblp,q≤ /ba∇dbl∇gu/ba∇dblp,q.\nForr≥0, define\nΦ(r) =n/integraldisplayr\n0sinhn−1(s)ds, r≥0,\nand letFbe the function such that\nr=nσn/integraldisplayF(r)\n0sinhn−1(s)ds, r≥0,\ni.e.,F(r) = Φ−1(r/σn). It was proved in [ 17, Lemma 2 .1] that\nsinhq(n−1)(F(t))≥/parenleftbiggt\nσn/parenrightbiggqn−1\nn\n+/parenleftbiggn−1\nn/parenrightbiggq/parenleftbiggt\nσn/parenrightbiggq\n, t≥0, (2.2)\nprovided q≥2n\nn−1. Moreover, we have the following result.\nProposition 2.3. Letn≥2. Then it holds\nsinhn(F(t))>t\nσn, t >0. (2.3)\nProof.Indeed, for ρ >0,we have\nn/integraldisplayρ\n0sinhn−1(s)ds < n/integraldisplayρ\n0sinhn−1(s)cosh(s)ds= sinhn(ρ).\nTakingρ=F(t),t >0 we obtain ( 2.3).\n6Proposition 2.4. Letn≥2, then the function\nϕ(t) =t\nsinhn−1(F(t))\nis strictly increasing on (0,∞), and\nlim\nt→∞ϕ(t) =nσn\nn−1.\nProof.Sincet/mapsto→F(t) is strictly increasing function, then it is enough to prove that the\nfunction\nη(ρ) =/integraltextρ\n0sinhn−1(s)ds\nsinhn−1(ρ)\nis strictly increasing on (0 ,∞). Indeed, we have\nη′(ρ) = 1−(n−1)cosh(ρ)/integraltextρ\n0sinhn−1(s)ds\nsinhn(ρ)\n=1\nsinhn(ρ)/parenleftbigg\nsinhn(ρ)−(n−1)cosh(ρ)/integraldisplayρ\n0sinhn−1(s)ds/parenrightbigg\n=:ξ(ρ)\nsinhn(ρ),\nand\nξ′(ρ) = cosh( ρ)sinhn−1(ρ)−(n−1)sinh(ρ)/integraldisplayρ\n0sinhn−1(s)ds.\nForρ >0, it holds\n(n−1)/integraldisplayρ\n0sinhn−1(s)ds <(n−1)/integraldisplayρ\n0sinhn−2(s)cosh(s)ds= sinhn−1(ρ),\nhere we use cosh( s)>sinh(s) fors >0. Therefore, we get\nξ′(ρ)>sinhn−1(ρ)(cosh(ρ)−sinh(ρ))>0,\nforρ >0. Consequently, we have ξ(ρ)> ξ(0) = 0 for ρ >0. Hence, η′(ρ)>0 forρ >0\nwhich implies that ηis strictly increasing function on (0 ,∞). By L’Hospital rule, we have\nlim\nρ→∞η(ρ) = lim\nρ→∞sinhn−1(ρ)\n(n−1)sinhn−2(ρ)cosh(ρ)=1\nn−1\nwhich yields the desired limit in this proposition.\nIntherest ofthissection, we shall frequently using thefollowing on e-dimensional Hardy\ninequality\n7Lemma 2.5. Let1< q < p. Then for any absolutely continuous function uin(0,∞)such\nthatlimt→∞|u(t)|tp−q\nq= 0, it holds\n/integraldisplay∞\n0|u′(t)|qtp−1dt≥/parenleftbiggp−q\nq/parenrightbiggq/integraldisplay∞\n0|u(t)|qtp−q−1dt. (2.4)\nLetu∈C∞\n0(Hn) andf=−∆gu. It was proved by Ngo and the author (see [ 13,\nProposition 2 .2]) that\nu∗(t)≤v(t) :=/integraldisplay∞\ntsf∗∗(s)\n(nσnsinhn−1(F(s)))2ds, t > 0. (2.5)\nThe following results are important in the proof of Theorem 1.1and Theorem 1.2.\nProposition 2.6. Letn≥2,p∈(1,n)andq∈(1,∞). Then it holds\n/ba∇dbl∆gu/ba∇dblq\np,q≥/parenleftbiggp−1\npnσ1\nnn/parenrightbiggq/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq(1\np−1\nn)−1dt.(2.6)\nFurthermore, if q≥2n\nn−1then we have\n/ba∇dbl∆gu/ba∇dblq\np,q−/parenleftbigg(n−1)(p−1)\np/parenrightbiggq/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt\n≥/parenleftbiggp−1\npnσ1\nnn/parenrightbiggq/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq(1\np−1\nn)−1dt. (2.7)\nProof.We have\nv′(t) =−tf∗∗(t)\n(nσnsinhn−1(F(t)))2,\nand hence\n/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq(1\np−1\nn)−1dt=/integraldisplay∞\n0(f∗∗(t))qtqn−1\nn\n(nσnsinhn−1(F(t)))qtq\np−1dt.(2.8)\nUsing (2.3) and (2.1) we obtain\n/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq(1\np−1\nn)−1dt≤1\nnqσq\nnn/integraldisplay∞\n0(f∗∗(t))qtq\np−1dt\n≤1\nnqσq\nnn/parenleftbiggp\np−1/parenrightbiggq/integraldisplay∞\n0(f∗(t))qtq\np−1dt\n=/parenleftBigg\n1\nnσ1\nnnp\np−1/parenrightBiggq\n/ba∇dbl∆gu/ba∇dblq\np,q,\nas wanted ( 2.6).\n8We next prove ( 2.7). We notice that\n/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt=/integraldisplay∞\n0(f∗∗(t))qtq\n(nσnsinhn−1(F(t)))qtq\np−1dt\nThis equality together with ( 2.8), the fact q≥2n\nn−1and the inequality ( 2.1) implies\n(n−1)q/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt\n+nqσq\nnn/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq(1\np−1\nn)−1dt\n≤/integraldisplay∞\n0(f∗∗(t))q/parenleftBigg/parenleftbiggt\nσnsinhn(F(t))/parenrightbiggqn−1\nn\n+/parenleftbiggn−1\nn/parenrightbiggq/parenleftbiggt\nσnsinhn−1((F(t))/parenrightbiggq/parenrightBigg\ntq\np−1dt\n≤/integraldisplay∞\n0(f∗∗(t))qtq\np−1dt\n≤/parenleftbiggp\np−1/parenrightbiggq/integraldisplay∞\n0(f∗(t))qtq\np−1dt\n=/parenleftbiggp\np−1/parenrightbiggq\n/ba∇dbl∆gu/ba∇dblq\np,q\nas wanted ( 2.7).\nProposition 2.7. Letn≥2,p∈(1,n)andq≥2n\nn−1. Then we have\n/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt≥/parenleftbiggn−1\np/parenrightbiggq/integraldisplay∞\n0|v(t)|qtq\np−1dt\n+nqσq\nnn/integraldisplay∞\n0|v′(t)|qtq(1\np−1\nn)+q−1dt,(2.9)\nand\n/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq(1\np−1\nn)−1dt≥/parenleftbigg(n−1)(n−p)\nnp/parenrightbiggq/integraldisplay∞\n0|v(t)|qtq(1\np−1\nn)−1dt\n+nqσq\nnn/integraldisplay∞\n0|v′(t)|qtq(1\np−2\nn)+q−1dt,(2.10)\nProof.Ifq≥2n\nn−1then by using ( 2.2) we have\n/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt≥(n−1)q/integraldisplay∞\n0|v′(t)|qtq\np+q−1dt\n+nqσq\nnn/integraldisplay∞\n0|v′(t)|qtq(1\np−1\nn)+q−1dt,\n9Using the one dimensional Hardy inequality ( 2.4), we have\n/integraldisplay∞\n0|v′(t)|qtq\np+q−1dt≥/parenleftbigg1\np/parenrightbiggq/integraldisplay∞\n0|v(t)|qtq\np−1dt.\nCombining these two inequalities proves the inequality ( 2.9).\nSinceq≥2n\nn−1then by using again ( 2.2), we get\n/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq(1\np−1\nn)−1dt≥(n−1)q/integraldisplay∞\n0|v′(t)|qtq(1\np−1\nn)+q−1dt\n+nqσq\nnn/integraldisplay∞\n0|v′(t)|qtq(1\np−2\nn)+q−1dt.\nUsing the one dimensional Hardy inequality ( 2.4), we have\n/integraldisplay∞\n0|v′(t)|qtq(1\np−1\nn)+q−1dt≥/parenleftbiggn−p\nnp/parenrightbiggq/integraldisplay∞\n0|v(t)|qtq(1\np−1\nn)−1dt.\nCombining these two inequalities proves the inequality ( 2.10).\nCombining Propositions 2.6and2.7, we obtain\nTheorem 2.8. Letn≥2. Ifp∈(1,n)andq∈(1,∞). For any u∈C∞\n0(Hn)we define v\nby(2.5). Then we have\n/ba∇dbl∆gu/ba∇dblq\np,q≥/parenleftBigg\nn2σ2\nnn\np′/parenrightBiggq/integraldisplay∞\n0|v′(t)|qtq(1\np−2\nn)+q−1dt, (2.11)\nwherep′=p/(p−1). In particular, if p∈(1,n\n2)then it holds\n/ba∇dbl∆gu/ba∇dblq\np,q≥/parenleftbiggn(n−2p)\npp′σ2\nnn/parenrightbiggq\n/ba∇dblu/ba∇dblq\np∗\n2,q. (2.12)\nFurthermore, if p∈(1,n)andq≥2n\nn−1then we have\n/ba∇dbl∆gu/ba∇dblq\np,q−C(n,2,p)q/ba∇dblu/ba∇dblq\np,q≥/parenleftBigg\nn2σ2\nnn\np′/parenrightBiggq/integraldisplay∞\n0|v′(t)|qtq(1\np−2\nn)+q−1dt.(2.13)\nIn particular, if p∈(1,n\n2)andq≥2n\nn−1then we have\n/ba∇dbl∆gu/ba∇dblq\np,q−C(n,2,p)q/ba∇dblu/ba∇dblq\np,q≥/parenleftbiggn(n−2p)\npp′σ2\nnn/parenrightbiggq\n/ba∇dblu/ba∇dblq\np∗\n2,q, u∈C∞\n0(Hn).(2.14)\nIt is worthing to mention here that in the Euclidean space Rn, an analogue of the\ninequality ( 2.12) was proved by Tarsi (see [ 19, Theorem 2]).\n10Proof.Letu∈C∞\n0(Hn) andvbe defined by ( 2.5). We know that u∗≤v, then\n/ba∇dblu/ba∇dblq\np,q≤/integraldisplay∞\n0v(t)qtq\np−1dt,and/ba∇dblu/ba∇dblq\np∗\n2,q≤/integraldisplay∞\n0v(t)qtq\np∗\n2−1dt. (2.15)\nThe inequality ( 2.11) is a consequence of ( 2.6) and (2.2). The inequality ( 2.12) is conse-\nquence of ( 2.11), the one dimensional Hardy inequality ( 2.4)\n/integraldisplay∞\n0|v′(t)|qtq(1\np−2\nn)+q−1dt≥/parenleftbiggn−2p\nnp/parenrightbiggq/integraldisplay∞\n0|v(t)|qtq(1\np−2\nn)−1dt.\nand the second inequality in ( 2.15).\nTo prove ( 2.13), we first notice by the first inequality in ( 2.15) that\n/ba∇dbl∆gu/ba∇dblq\np,q−C(n,2,p)q/ba∇dblu/ba∇dblq\np,q≥ /ba∇dbl∆gu/ba∇dblq\np,q−C(n,2,p)q/integraldisplay∞\n0v(t)qtq\np−1dt.\nHence, it holds\n/ba∇dbl∆gu/ba∇dblq\np,q−C(n,2,p)q/ba∇dblu/ba∇dblq\np,q\n≥/ba∇dbl∆gu/ba∇dblq\np,q−/parenleftbiggn−1\np′/parenrightbiggq/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt\n+/parenleftbiggn−1\np′/parenrightbiggq/parenleftbigg/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt−/parenleftbiggn−1\np/parenrightbiggq/integraldisplay∞\n0v(t)qtq\np−1dt/parenrightbigg\n≥/ba∇dbl∆gu/ba∇dblq\np,q−/parenleftbiggn−1\np′/parenrightbiggq/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt, (2.16)\nhere we use q≥2n\nn−1and the inequality ( 2.9). Using again the assumption q≥2n\nn−1and\nthe inequalities ( 2.7) and (2.10), we obtain\n/ba∇dbl∆gu/ba∇dblq\np,q−/parenleftbiggn−1\np′/parenrightbiggq/integraldisplay∞\n0|v′(t)|q(nσnsinhn−1(F(t)))qtq\np−1dt\n≥/parenleftBigg\nn2σ2\nnn\np′/parenrightBiggq/integraldisplay∞\n0|v′(t)|qtq(1\np−2\nn)+q−1dt. (2.17)\nCombining the estimates ( 2.16) and (2.17) proves ( 2.13). The inequality ( 2.14) follows\nfrom (2.13) and the one dimensional Hardy inequality ( 2.4)\n/integraldisplay∞\n0|v′(t)|qtq(1\np−2\nn)+q−1dt≥/parenleftbiggn−2p\nnp/parenrightbiggq/integraldisplay∞\n0|v(t)|qtq(1\np−2\nn)−1dt.\n113 Proof of Theorem 1.1\nIn this section, we prove Theorems 1.1.\nProof of Theorem 1.1.In the case m= 1, Theorem 1.1was already proved in [ 17]. So, we\nwill only consider the case m≥2. We divide the proof into three cases as follows.\nCase 1: m= 2.Ifp∈(1,n) andq≥2n\nn−1then (1.2) follows from ( 2.13). In the\nfollowing, we will give a proof of ( 1.2) for any p,q∈(0,∞). By the density, it is enough\nto prove ( 1.2) for function u∈C∞\n0(Hn),u/ne}ationslash≡0. Letf=−∆guandvbe defined by ( 2.5).\nWe first notice that\n/integraldisplay∞\n0|v′(t)|qtq\np+q−1dt=/integraldisplay∞\n0/parenleftbiggt\nnσnsinhn−1(F(t))/parenrightbigg2q\nf∗∗(t)qtq\np−1dt.\nBy Lemma 2.4, we have\nt\nnσnsinhn−1(F(t))<1\nn−1, t >0.\nThis inequality together with the inequality ( 2.1) implies\n/integraldisplay∞\n0/parenleftbiggt\nnσnsinhn−1(F(t))/parenrightbigg2q\nf∗∗(t)qtq\np−1dt≤/parenleftbiggp′\n(n−1)2/parenrightbiggq/integraldisplay∞\n0f∗(t)qtq\np−1dt\n=/parenleftbiggp′\n(n−1)2/parenrightbiggq\n/ba∇dbl∆gu/ba∇dblq\np,q.\nBy the one dimensional Hardy inequality and the first inequality in ( 2.15), we have\n/integraldisplay∞\n0|v′(t)|qtq\np+q−1dt≥1\npq/integraldisplay∞\n0v(t)qtq\np−1dt≥1\npq/ba∇dblu/ba∇dblq\np,q.\nCombining two previous inequalities, we obtain ( 1.2).\nCase 2:m= 2k,k≥1.This case follows from the Case 1and the iteration argument.\nCase 3:m= 2k+1,k≥1.Sinceq≤p, then it was proved in [ 17, Theorem 1 .1] that\n/ba∇dbl∇g∆k\ngu/ba∇dblq\np,q≥/parenleftbiggn−1\np/parenrightbiggq\n/ba∇dbl∆k\ngu/ba∇dblq\np,q.\nWe now apply the Case 2to obtain the desired result.\nWe next check the sharpness of the constant C(n,m,p) in (1.2). From Proposition 2.4,\nwe see that for any ǫ >0 there exists a >0 such that\n(n−1)s < nσ nsinhn−1(F(s))≤(1+ǫ)(n−1)s,\nfor anys≥a. ForR > a, let us define the function\nfR(s) =\n\na−1\np ifs∈(0,a),\ns−1\np ifs∈[a,R),\nR−1\npmax{2−s/R,0}ifs≥R.\n12Noticethat fRisanonnegative, continuous, non-increasing function. Following[ 13,Section\n2.2], we define two sequences of functions {vR,i}i≥0and{gR,i}i≥1as follows:\n(i) First, we set vR,0=fR,\n(ii) then in terms of vR,i, we define gR,i+1as the maximal function of vR,i, i.e.\ngR,i+1(t) =1\nt/integraldisplayt\n0vR,i(s)ds,\n(iii) and finally in terms of gR,i+1, we define vR,i+1as follows\nvR,i+1(t) =/integraldisplay∞\ntsgR,i+1(s)\n(nσnsinhn−1(F(s)))2ds,\nfori= 0,1,2,...\nNote that vR,iandgR,iare positive, non-increasing functions. Following the proof of [ 13,\nProposition 2 .1], we can prove the following result.\nProposition 3.1. For anyi≥1, there exist function hR,iandwR,isuch that\nvR,i=hR,i+wR,i,/integraldisplay∞\n0|wR,i|qtq\np−1dt≤C\nand\n1\n(1+ǫ)2i/parenleftbiggpp′\n(n−1)2/parenrightbiggi\nfR≤hR,i≤/parenleftbiggpp′\n(n−1)2/parenrightbiggi\n,\nwhere we use Cto denote various constants which are independent of R.\nProof.Let us define the operator Tacting on functions von [0,∞) by\nT(v)(t) =/integraldisplay∞\nts\n(nσn(sinh(F(s)))2/parenleftbigg1\ns/integraldisplays\n0v(r)dr/parenrightbigg\nds.\nWe shall prove that\n/integraldisplay∞\n0|T(v)(t)|qtq\np−1dt≤/parenleftbiggpp′\n(n−1)2/parenrightbiggq/integraldisplay∞\n0|v(t)|qtq\np−1dt. (3.1)\nIndeed, it is enough to prove ( 3.1) for nonnegative function vsuch that\n/integraldisplay∞\n0|v(t)|qtq\np−1dt <∞.\nWe claim that\nlim\nt→0T(v)(t)t1\np= 0 = lim\nt→∞T(v)(t)t1\np. (3.2)\n13For anyǫ >0 there exists t0>0 such that/integraltextt0\n0|v(t)|qtq\np−1dt≤ǫq. Fors≤t0, by using\nH¨ older inequality, we have\n1\ns/integraldisplays\n0v(r)dr≤1\ns/parenleftbigg/integraldisplays\n0v(r)qrq\np−1dr/parenrightbigg1\nq/parenleftbigg/integraldisplays\n0r1\nq−1−q\np(q−1)dr/parenrightbiggq−1\nq\n≤Cǫs−1\np.\nThis together with the inequality nσnsinhn−1(F(s))>(n−1)simplies for t≤t0that\nT(v)(t) =/parenleftBig/integraldisplayt0\nt+/integraldisplay∞\nt0/parenrightBigs\n(nσn(sinh(F(s)))2/parenleftbigg1\ns/integraldisplays\n0v(r)dr/parenrightbigg\nds\n≤Cǫ/integraldisplayt0\nts−1−1\npds+C/parenleftbigg/integraldisplay∞\n0v(r)qrq\np−1dr/parenrightbigg1\nq/integraldisplay∞\nt0s−1−1\npds\n≤Cǫ(t−1\np−t−1\np\n0)+Ct−1\np\n0/parenleftbigg/integraldisplay∞\n0v(r)qrq\np−1dr/parenrightbigg1\nq\n.\nThis estimate yields\nlimsup\nt→0T(v)(t)t1\np≤Cǫ.\nSinceǫ >0 is arbitrary, then the first limit in ( 3.2) is proved.\nSimilarly, for any ǫ >0 there exists t1>0 such that/integraltext∞\nt1|v(t)|qtq\np−1dt < ǫq. Hence, for\ns≥t1, by using H¨ older inequality we get\n/integraldisplays\n0v(r)dr=/integraldisplayt1\n0v(r)dr+/integraldisplays\nt1v(r)dr≤C/parenleftbigg/integraldisplay∞\n0|v(t)|qtq\np−1dt/parenrightbigg1\nq\nt1−1\np\n1+Cǫs1−1\np.\nConsequently, for any t≥t1we get\nT(v)(t)≤C/integraldisplay∞\nt/parenleftBigg/parenleftbigg/integraldisplay∞\n0|v(t)|qtq\np−1dt/parenrightbigg1\nq\nt1−1\np\n1s−2+ǫs−1−1\np/parenrightBigg\nds\n≤C/parenleftbigg/integraldisplay∞\n0|v(t)|qtq\np−1dt/parenrightbigg1\nq\nt1−1\np\n1t−1+Cǫt−1\np.\nThis estimate implies\nlimsup\nt→∞T(v)(t)t1\np≤Cǫ.\nSinceǫ >0 is arbitrary, then the second limit in ( 3.2) is proved.\nUsing the integration by parts, the claim ( 3.2) and the inequality nσnsinhn−1(F(t))>\n(n−1)t, we have\n/integraldisplay∞\n0T(v)(t)qtq\np−1dt=p\nq/integraldisplay∞\n0T(v)(t)q(tq\np)′dt\n=p/integraldisplay∞\n0T(v)(t)q−1/parenleftbiggt\nnσnsinhn−1(F(t))/parenrightbigg2/parenleftbigg1\nt/integraldisplayt\n0v(s)ds/parenrightbigg\ntq\np−1dt\n≤p\n(n−1)2/integraldisplay∞\n0T(v)(t)q−1/parenleftbigg1\nt/integraldisplayt\n0v(s)ds/parenrightbigg\ntq\np−1dt.\n14An easy application of H¨ older inequality implies\n/integraldisplay∞\n0T(v)(t)qtq\np−1dt≤/parenleftbiggp\n(n−1)2/parenrightbiggq/integraldisplay∞\n0/parenleftbigg1\nt/integraldisplayt\n0v(s)ds/parenrightbiggq\ntq\np−1dt.\nThe inequality ( 3.1) follows from the previous inequality and the Hardy inequality ( 2.1).\nThus, with the help of ( 3.1), we can using the induction argument to prove this propo-\nsition by establishing the result for vR,1. In fact, the decomposition for vR,1is already\nproved in the proof of Proposition 2 .1 in [13]. The estimate\n/integraldisplay∞\n0|wR,1|qtq\np−1dt≤C,\nis proved by the same way of the estimate/integraltext∞\n0|wR,1|pdt≤C.\nWe are now ready to check the sharpness of C(n,m,p). The case m= 1 was done\nin [17]. Hence, we only consider the case m≥2. We first consider the case m= 2k,k≥1.\nDefine\nuR(x) =vR,k(Vg(B(0,d(0,x)))).\nIt is clear that ( −∆g)kuR(x) =fR(Vg(B(0,d(0,x)))). Hence, there hold\n/ba∇dbl∆k\nguR/ba∇dblq\np,q=/integraldisplay∞\n0fR(t)qtq\np−1dt=p\nq+lnR\na+/integraldisplay2\n1(2−s)qsq\np−1ds,\nand\n/ba∇dbluR/ba∇dblp,q=/parenleftbigg/integraldisplay∞\n0vq\nR,ktq\np−1dt/parenrightbigg1\nq\n≥/parenleftbigg/integraldisplay∞\n0hq\nR,ktq\np−1dt/parenrightbigg1\nq\n−/parenleftbigg/integraldisplay∞\n0|wR,k|qtq\np−1dt/parenrightbigg1\nq\n≥1\n(1+ǫ)2k)/parenleftbiggpp′\n(n−1)2/parenrightbiggk/parenleftbigg/integraldisplay∞\n0fq\nRtq\np−1dt/parenrightbigg1\nq\n−C\n≥1\n(1+ǫ)2k)/parenleftbiggpp′\n(n−1)2/parenrightbiggk/parenleftbiggp\nq+lnR\na+/integraldisplay2\n1(2−s)qsq\np−1ds/parenrightbigg1\nq\n−C.\nThese estimates imply\nlimsup\nR→∞/ba∇dbl∆k\nguR/ba∇dblq\np,q\n/ba∇dbluR/ba∇dblq\np,q≤(1+ǫ)2kqC(n,2k,p)k,\nfor anyǫ >0. This proves the sharpness of C(n,2k,p). We next consider the case\nm= 2k+1,k≥1. Define\nuR(x) =vR,k(Vg(B(0,d(0,x)))).\n15It is clear that ( −∆g)kuR(x) =fR(Vg(B(0,d(0,x)))). It was shown in the proof of Theorem\n1.1 in [17] (the sharpness of C(n,1,p)) that\nlimsup\nR→∞/ba∇dbl∇g∆k\nguR/ba∇dblq\np,q\n/ba∇dbl∆kguR/ba∇dblq\np,q≤(1+ǫ)q(n−1)q\npq.\nThis together with the estimate in the case m= 2kimplies\nlimsup\nR→∞/ba∇dbl∇g∆k\nguR/ba∇dblq\np,q\n/ba∇dbluR/ba∇dblq\np,q≤limsup\nR→∞/ba∇dbl∇g∆k\nguR/ba∇dblq\np,q\n/ba∇dbl∆k\nguR/ba∇dblq\np,qlimsup\nR→∞/ba∇dbl∆k\nguR/ba∇dblq\np,q\n/ba∇dbluR/ba∇dblq\np,q\n≤(1+ǫ)(2k+)qC(n,2k+1,p)q,\nfor anyǫ >0. This proves the sharpness of C(n,2k+1,p).\nThe proof of Theorem 1.1is then completely finished.\n4 Proof of Theorem 1.2\nThis section is addressed to prove Theorem 1.2. The proof uses the results from Theorem\n2.8and [17, Theorem 1 .2].\nProof of Theorem 1.2.The case m= 1 was already proved in [ 17]. So, we only consider\nthe case m≥2. We divide the proof into two cases as follows.\nCase 1: m= 2k,k≥1.The case k= 1 follows from ( 2.14). Fork≥2, by using\nTheorem 1.1, we have\n/ba∇dbl∆k\ngu/ba∇dblq\np,q−C(n,2k,p)q/ba∇dblu/ba∇dblq\np,q≥ /ba∇dbl∆k\ngu/ba∇dblq\np,q−C(n,2,p)q/ba∇dbl∆k−1\ngu/ba∇dblq\np,q.\nApplying the inequality ( 2.14) to the right hand side of the previous inequality, we obtain\n/ba∇dbl∆k\ngu/ba∇dblq\np,q−C(n,2k,p)q/ba∇dblu/ba∇dblq\np,q≥/parenleftbiggn(n−2p)\npp′σ2\nnn/parenrightbiggq\n/ba∇dbl∆k−1\ngu/ba∇dblq\np∗\n2,q.\nBy iterating the inequality ( 2.12), we then have\n/ba∇dbl∆k\ngu/ba∇dblq\np,q−C(n,2k,p)q/ba∇dblu/ba∇dblq\np,q≥/parenleftBigg\nσ2k\nnnk−1/productdisplay\ni=0n(n−2p∗\n2i)\np∗\n2i(p∗\n2i)′/parenrightBiggq\n/ba∇dblu/ba∇dblq\np∗\n2k,q,\nas wanted ( 1.7).\nCase 1:m= 2k+1,k≥1.In this case, we have\n/ba∇dbl∇g∆k\ngu/ba∇dblq\np,q−C(n,2k+1,p)q/ba∇dblu/ba∇dblq\np,q≥ /ba∇dbl∇g∆k\ngu/ba∇dblq\np,q−/parenleftbiggn−1\np/parenrightbiggq\n/ba∇dbl∆k\ngu/ba∇dblq\np,q.\nSinceq≤pwe then have from Theorem 1 .2 in [17] that\n/ba∇dbl∇g∆k\ngu/ba∇dblq\np,q−/parenleftbiggn−1\np/parenrightbiggq\n/ba∇dbl∆k\ngu/ba∇dblq\np,q≥/parenleftbiggn−p\npσ1\nnn/parenrightbiggq\n/ba∇dbl∆k\ngu/ba∇dblq\np∗,q.\n16Hence, it holds\n/ba∇dbl∇g∆k\ngu/ba∇dblq\np,q−C(n,2k+1,p)q/ba∇dblu/ba∇dblq\np,q≥/parenleftbiggn−p\npσ1\nnn/parenrightbiggq\n/ba∇dbl∆k\ngu/ba∇dblq\np∗,q.\nBy iterating the inequality ( 2.12), we then have\n/ba∇dbl∇g∆k\ngu/ba∇dblq\np,q−C(n,2k+1,p)q/ba∇dblu/ba∇dblq\np,q≥/parenleftBigg\nσ2k+1\nnnn−p\npk/productdisplay\ni=1n(n−2p∗\n2i−1)\np∗\n2i−1(p∗\n2i−1)′/parenrightBiggq\n/ba∇dblu/ba∇dblq\np∗\n2k+1,q,\nas desired ( 1.7).\nReferences\n[1] A. Alvino. Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. A\n(5), 14(1):148–156, 1977.\n[2] T.Aubin. Probl` emes isop´ erim´ etriques etespaces deSobolev .J. Differential Geometry ,\n11(4):573–598, 1976.\n[3] R. D. Benguria, R. L. Frank, and M. Loss. The sharp constant in the Hardy-\nSobolev-Maz’ya inequality in the three dimensional upper half-space .Math. Res.\nLett., 15(4):613–622, 2008.\n[4] C. Bennett and R. Sharpley. Interpolation of operators , volume 129 of Pure and\nApplied Mathematics . Academic Press, Inc., Boston, MA, 1988.\n[5] E. Berchio, L. D’Ambrosio, D. Ganguly, and G. Grillo. Improved Lp-Poincar´ e inequal-\nities on the hyperbolic space. Nonlinear Anal. , 157:146–166, 2017.\n[6] E. Berchio, D. Ganguly, and G. Grillo. Sharp Poincar´ e-Hardy and Poincar´ e-Rellich\ninequalities on the hyperbolic space. J. Funct. Anal. , 272(4):1661–1703, 2017.\n[7] D.Cassani, B.Ruf, andC.Tarsi. Equivalentandattainedversion ofHardy’sinequality\ninRn.J. Funct. Anal. , 275(12):3303–3324, 2018.\n[8] Q. Hong. Sharp constant in third-order hardysobolevmazya ine quality in the half\nspace of dimension seven. Int. Math. Res. Not. IMRN, in press , 2019.\n[9] E. H. Lieb. Sharp constants in the Hardy-Littlewood-Sobolev an d related inequalities.\nAnn. of Math. (2) , 118(2):349–374, 1983.\n[10] G. Lu and Q. Yang. Green’s functions of paneitz and gjms opera tors on hyper-\nbolic spaces and sharp hardy-sobolev-maz’ya inequalities on half spa ces.preprint,\narXiv:1903.10365 , 2019.\n17[11] G. Lu and Q. Yang. Paneitz operators on hyperbolic spaces and high order hardy-\nsobolev-maz’ya inequalities on half spaces. Amer. J. Math. , 141(6):1777–1816, 2019.\n[12] G. Mancini and K. Sandeep. On a semilinear elliptic equation in Hn.Ann. Sc. Norm.\nSuper. Pisa Cl. Sci. (5) , 7(4):635–671, 2008.\n[13] Q. A. Ngˆ o and V. H. Nguyen. Sharp constant for Poincar´ e-t ype inequalities in the\nhyperbolic space. Acta Math. Vietnam. , 44(3):781–795, 2019.\n[14] V. H. Nguyen. Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half-\nspaces via mass transport and consequences. Proc. Lond. Math. Soc. (3) , 111(1):127–\n148, 2015.\n[15] V. H. Nguyen. The sharp Poincar´ e-Sobolev type inequalities in t he hyperbolic spaces\nHn.J. Math. Anal. Appl. , 462(2):1570–1584, 2018.\n[16] V. H. Nguyen. Second order Sobolev type inequalities in the hype rbolic spaces. J.\nMath. Anal. Appl. , 477(2):1157–1181, 2019.\n[17] V. H. Nguyen. The sharp Sobolev type inequalities in the Lorentz –Sobolev spaces in\nthe hyperbolic spaces. preprint, 2019.\n[18] G. Talenti. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) , 110:353–\n372, 1976.\n[19] C. Tarsi. Adams’ inequality and limiting Sobolev embeddings into Zyg mund spaces.\nPotential Anal. , 37(4):353–385, 2012.\n[20] D. Tataru. Strichartz estimates in the hyperbolic space and glo bal existence for the\nsemilinear wave equation. Trans. Amer. Math. Soc. , 353(2):795–807, 2001.\n18"    },    {        "title": "0901.3717v1.Rheological_Interpretation_of_Rayleigh_Damping.pdf",        "content": "RHEOLOGICAL INTERPRETATION OF RAYLEIGH DAMPING \n \nJ.F.Semblat \n \nLaboratoire Central des Ponts et  Chaussées, Eng. Modelling Div., \n58, bd Lefèbvre, 75732 Paris Cedex 15, France \nemail : semblat@lcpc.fr  \n \n \n1. INTRODUCTION \nDamping is defined through various terms [2] such as energy loss per cycle (for cyclic tests), \nlogarithmic decrement (for vibration tests), comp lex modulus, rise-time or spectrum ratio (for \nwave propagation analysis)... For numerical mode ling purposes, another type of damping is \nfrequently used : it is called Rayleigh dampi ng. It is a very convenient way of accounting for \ndamping in numerical models, although the physical  or rheological meaning of this approach \nis not clear. After the definition of Rayleigh damping, we propose a rheo logical inte rpretation \nof Rayleigh damping. \n \n2. RAYLEIGH DAMPING \nRayleigh damping is a classical method to build easily the damping matrix C of a numerical \nmodel [5,7] under the following form : \n \n C =a0 M+a1 K  (1) \n- \nwhere M and K are the mass and stiffness matrices respectively. It is then called Rayleigh \ndamping matrix . C is the sum of two terms : one is pr oportional to mass matrix, the other to \nstiffness matrix. \n \nA more general form was proposed by Caughey [3].  The original form (equation (1)) is very \nconvenient as it can be easily computed. Furthermore, for modal approaches, the Rayleigh (or \nCaughey) damping matrix  is diagonal in the real modes base [4,8]. Damping is therefore \ncalled proportional or classi cal. In case of non proportiona l damping, the complex modes \nhave to be computed (in order to  uncouple the moda l equations).  \n \nConsidering Rayleigh damping [4], the loss factor   can be written as follows : \n  = 2 =  a 0\n+ a 1 (2) \n \nwhere   is the circular frequency and  is the damping ratio. \nOur purpose is to find out a rheological model having th e same attenuation-frequency \ndependence as in the case of Rayleigh damping.  \n3. RHEOLOGICAL INTERPRETATION OF RAYLEIGH DAMPING \nConsidering the relationship between internal friction and frequency for Rayleigh damping, it \nis possible to build a rheological mode l involving the same attenuation-frequency \ndependence. For a linear viscoelastic rheological model of  complex modulus E\n*=ER+i.E I  [2], \nexpression of the quality factor Q is given in the fields of  geophysics and acoustics as \nfollows : \n Q  =  E\nER\nI (3) \n1 For weak to moderate Rayleigh damping, there is  a simple relation between the inverse of the \nquality factor Q-1 and the damping ratio  : \n Q-1  2 (4) \n \nFor Rayleigh damping, the loss factor is infinite  for zero and infinite frequencies. It clearly \ngives the behaviour of the model through instantaneous and long term responses. The rheological model perfectly meeting these requ irements (attenuation-frequency dependence, \ninstantaneous and long term effects) is a part icular type of genera lized Maxwell model. \nFigure (1) gives a schematic of the proposed m odel : it connects, in parallel, a classical \nMaxwell cell to a single dashpot. The generali zed Maxwell model given in figure (1) can be \ndefined through its complex modulus from which we easily derive the i nverse of the quality \nfactor \nQ-1 which takes the same form than the loss factor of Rayleigh damping \n(expression (2)) : it is the sum of two terms, one proportional to frequency and one inversely \nproportional to frequency \n \n \nFig. 1. Proposed generalized Maxwell model and corresponding attenuation curve \n \n4. COMPARISON BETWEEN NUMERICAL AND ANALYTICAL RESULTS \n4.1 Two different approaches \nThe coincidence between Rayleigh damping and the generalized Maxwell model is perfect \nconsidering internal friction (see equation (1) an d figure (1) ). As equation (1) is only valid \nfor moderate values of damping ratio , there is a complete equivalence between both \napproaches since material velocity disp ersion is moderate for such values of . Rayleigh \ndamping and the generalized Maxwell mo del are equivalent for wave propagation \npurposes for small to moderate values of damping ratio . A one-dimensional propagation \ntest is then proposed to demonstrate the coin cidence for moderate values of damping ratio  \nand quantify the discrepancy for higher  values. Rayleigh damping is investigated through a \nfinite element modeling of the problem, wher eas results for generalized Maxwell model are \ndrawn from an analytical description (compl ex wavenumber derive d from complex modulus \n[1,9,10,11]). \n \nThe numerical modeling is performed with CESAR-LCPC : the finite element program \ndeveloped at LCPC and dedicated to civi l engineering problems [6]. We use a one-\ndimensional mesh with linear quadrilateral elem ents and the finite element program performs \n2 a direct time integration. Rayleigh dampin g is involved consider ing expression (1). \nAnalytical approach is based on the one-dimensional wave equation in which the material has viscoelastic properties corresponding to the generalized Maxwell model of figure (1). \nHarmonic solutions of the wave propagation pr oblem in the frequency domain are found first \nand synthetized afterwards into  the time domain [1,9,10,11]. \n \n4.2\n The problem and its parameters \nA one-dimensional wave propagation problem is c hosen to compare numerical and analytical \nresults. The point is to link numerical parameters of Rayleigh damping, that is a0 and a1 (see \neq.(1)), to mechanical parameters of  generalized Maxwell model, that is E, 1 and 2 (see \nfigure (1)). \n \nConsidering eq.(2) and figure (1), these parame ters can be easily related under the following \nform : \n aE\naE012\n12\n12\n\n\n\n.( )\n\n (5) \n \nExpression (5) relates the Rayleigh coefficients ( a0 and a1) to the behaviour parameters ( E, 1 \nand 2) making experimental determination of Rayl eigh coefficients much easier. For this \nnumerical test, the applied loading is a sine-shaped single pulse ( =10000 rad.s-1). Young \nmodulus is E=300 MPa. For finite element model, the time step is t=10-5 s and the elements \ndimension is chosen to minimi ze numerical dispersion effects. \n \n \nFig. 2. Comparison between numerical results (Rayleigh damping) \nand analytical results (generalized Maxwell model) \n \n \n3 The results presented in fig.(2) correspond to six different distances from the source of \nexcitation : 0, 0.1, 0.2, 0.3, 0.4 and 0.5m. Values of Rayleigh coefficients range from a0=40 \nand a1=10-5 (first diagram) to a0=400 and a1=10-4 (last diagram). For such values of Rayleigh \ncoefficients, the corresponding mechanical para meters of the generali zed Maxwell model are \nestimated using eq.(5) : 1=7.5.106 Pa.s, 2=3000 Pa.s (for first diagram). \n \nThe main conclusions draught from  these curves ar e the following : \n for moderate values of damping coefficient (fig.(2)), numerical and analytical results \nperfectly coincide in terms of amplitude reduction and phase delays. It gives a good \nillustration of the theoretical link between Rayleigh damping and the generalized Maxwell model proposed in figure (1) \n for higher values of damping ratio ( >25%), attenuation is str onger for Rayleigh damping \napproach and dispersive phenomen a are different in both cases \n \n5. CONCLUSION \nA rheological model is proposed to be related to classical Rayleigh damping : it is a \ngeneralized Maxwell model with three parame ters (fig.(1)). For moderate damping ( <25%), \nthis model perfectly coincide with Rayleigh da mping approach since internal friction has the \nsame expression in both cases and dispersive phe nomena are negligible. This is illustrated by \nfinite element (Rayleigh damping) and analytic al (generalized Maxwell model) results in a \nsimple one-dimensional case.  \nACKNOWLEDGEMENTS \nThe author is indebted to Pr O. Coussy for his pertinent comments on the preliminary version \nof the manuscript.  \nREFERENCES \n 1. K. Aki and P.G. Richards 1980 Quantitative seismology . San Francisco: Freeman ed.. \n 2. T. Bourbié, O. Coussy and B. Zinszner 1987 Acoustics of porous media . Paris, France: \nTechnip ed.. \n 3. T. Caughey 1960 Journal of Applied Mechanics \n27, 269-271. Classical normal modes in \ndamped linear systems. \n 4. R.W. Clough and J. Penzien J 1993 Dynamics of structures , Mc Graw-Hill. \n 5. T.J.R. Hughes 1987 The finite element method , Prentice-Hall. \n 6. P. Humbert 1989 Bulletin des Laboratoire s des Ponts & Chaussées  (160), 112-115. \nCESAR-LCPC : a general finite  element code (in french). \n 7. M. Liu and D.G. Gorman 1995 Computers & Structures  57(2), 277-285. Formulation of \nRayleigh damping and its extension. \n 8. L. Meirovitch 1986 Elements of vibration analysis , Mc Graw-Hill. \n 9. J.F. Semblat 1995a Soils under dynamic and transient loadings (in french). Paris, France: \nLaboratoire Central des Ponts & Chaussées ed. \n10. J.F. Semblat, M.P. Luong and J.J. Thomas 1995b Proceedings of the  5th SECED Conf. on \nEuropean Seismic Design Practice,  567-575, Chester, UK (Balkema ed.). Drop-ball \narrangement for centrifuge experiments. \n11. J.F. Semblat and M.P. Luong 1998 Journal of Earthquake Engineering  2(1), 147-171, \nWave propagation through soils  in centrifuge testing. \n \n4 "    },    {        "title": "1610.09430v1.Lorentz_Violating_QCD_Corrections_to_Deep_Inelastic_Scattering.pdf",        "content": "arXiv:1610.09430v1  [hep-ph]  28 Oct 2016Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n1\nLorentz-Violating QCD Corrections to Deep Inelastic Scatt ering\nA.R. Vieira\nIndiana University Center for Spacetime Symmetries,\nBloomington, IN 47405, USA\nDepartamento de F´ ısica-ICEx, Universidade Federal de Min as Gerais,\nBelo Horizonte, MG 31.270-901, Brasil\nIn this work, we present CPT- and Lorentz-violating correct ions to observable\nquantities in electron-proton scattering. We also show how the theoretical\nprediction can be used together with data to establish bound s on a coefficient\nfor CPT and Lorentz violation in the QCD sector.\nUnlike the QEDsectorof the SME, the quark and gluonsectorsof th e QCD\nextension are not stringently constrained.1Most of the coefficients of the\nQCD sector are effective and obtained from composite objects. On e reason\nis that the QCD Hilbert space contains baryons and mesons rather t han\nquarks at low energies. Thus, as a first step, we can consider CPT a nd\nLorentz violation (LV) in a process where we can access the quark s truc-\nture of those composite objects, Deep Inelastic Scattering (DIS ). Electron-\nproton (e−P) scattering, for instance, gives us information about QCD and\nthe quark structure of the proton. It is also a high energy proces s so that\nwe can treat the QCD coupling gsperturbatively. The zeroth order, g0\ns, is\nthe so-called partonmodel. Considering the QCD extension,2a Lorentz-\nvolating version of the parton model and its radiative corrections c an be\nobtained from the lagrangian\nLquark=1\n2i(gµν+cµν\nQ)(ψγµ← →Dνψ+2iQfψγµAνψ), (1)\nwhereDµ=∂µ+1\n2igsAµ\niλiis the covariant derivative and cµν\nQis the CPT-\nand Lorentz-violating quark coefficient. In the high-energy limit, th e pho-\nton energy Q2=−q2→∞and we can neglect gs, considering that quarks\nonly interact with the photon by means of their charge Qf.\nThe unpolarized differential cross section of e−Pscattering is\nd2σ\ndxdy=α2y\n(Q2)2LµνImWµν, (2)Proceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n2\nwhereLµν= 2(kµk′ν+kνk′µ−k·k′gµν) is the lepton tensor, y=P·q\nP·k,x=\n−q2\n2P·qis the Bjorken scale and Wµν=i/summationtext\nspins/integraltext\nd4xeiq·x/angbracketleftP|Jµ(x)Jν(0)|P/angbracketright\nis the proton tensor. All the LV information is in the parton-photon cou-\npling,Jµ(x) =Qf¯ψ(x)Γµψ(x). The momenta Pandk(k′) are the proton\nand electron initial (final) energies.\nIn Eq. (2), we divided by the flux factor F= 2s. Some care is re-\nquired in defining F, which is modified by LV.3However, in the present\nsituation, the SME is being considered in the CPT-even quark sector . The\nDIS process assumes that a short-wavelength photon only sees t he quark\nstructure. We do not have to consider LV in Fsince it is defined according\nto the proton initial state and the whole proton would be perceived o nly\nby a long-wavelength photon. Moreover, the proton coefficient cµν\nPis well\nconstrained1and can be neglected compared to the quark one.\nCalculating the explicit form of Wµνis challenging. It represents our\nignorance in the photon-proton interaction. As we stated before , we make\nuse of perturbation theory, where the parton model is the zerot h order\ncontribution to the process. It allows us to rewrite Wµνas\nWµν≈i/integraldisplay\nd4x eiq·x/integraldisplay1\n0dξ/summationdisplay\nfff(ξ)\nξ/angbracketleftqf(ξP)|Jµ(x)Jν(0)|qf(ξP)/angbracketright,(3)\nwhereff(ξ) is the parton distribution function (PDF), the probability of\nfinding a parton fcarrying a momentum ξP.\nIn Eq. (3), there is a sum over flavors. The quark sector of the SM E\nallows a different coefficient for each flavor.2In this case, it is impractical\nto extract the coefficient from the sum over flavors. If we want to consider\none coefficient for each quark, up and down, we find that the up cha rge\nand the two up quarks in the proton make cµν\nUone order of magnitude\nbigger than cµν\nDand so taking only one coefficient is essentially assuming\nthatcµν\nQ≈cµν\nU.\nWhen we take the imaginary part of Wµν, we find that the full fermion\npropagator gives us a delta function corrected by LV, δ(−Q2+ 2ξP·q+\n2cqq+ 2ξ(cqP+cPq) + 2ξ2cPP). Consequently, the Bjorken scale is also\ncorrected by a factor xc=2\nys(xcPq+xcqP+cqq), wheres= 2k·Pand\ncµα\nQpα≡cµp. Therefore, the LV correction can be seen as a tree-level\nviolation of Bjorken scaling. We can also confirm this after computing theProceedings of the Seventh Meeting on CPT and Lorentz Symmet ry (CPT’16), Indiana University, Bloomington, June 20-24, 20 16\n3\ndifferential cross section in Eq. (2)\nd2σ\ndxdy=α2y\nQ4ImW2/bracketleftBig1\n2/parenleftbigg\n1−2\nys(cPq+cqP+2xcPP)/parenrightbigg\ns2(1+(1−y)2)\n−2xyscPP−2M2(ckk′+ck′k)+2s\nx(1−y)ckk+2s(ck′P+cPk′)\n−2s\nxck′k′+2s(1−y)(ckP+cPk)/bracketrightBig\n−α2ys2\n2Q4xcdImW2\ndx(1+(1−y)2),(4)\nwhereW2is one of the proton structure functions. Its derivative comes\nfrom the expansion at first order in cof the whole expression, Im Wc\n2=\n4π\nys/summationtext\nfQ2\nf(x−xc)ff(x−xc).\nWe see that Eq. (4) is symmetric on the cindices as it should be, since\nits antisymmetric part in the lagrangian (1) can be removed by a field\nredefinition. TocompareEq.(4)and W2withdatacollectedataccelerators,\nwe first choose a frame. For instance, this can be the proton rest frame\nfor measurements with the single-arm experiment at SLAC. Howeve r, for\ncurrent data on the DIS cross section measured at HERA, the pro ton is\nnot at rest and has opposite momentum to the initial electron momen tum.\nWe also must consider the sidereal time variation of cµν\nQ, which oscillates as\nthe Earth rotates. Therefore, making a transformation betwee n the Earth\nframe and the canonical Sun-centered frame, we can determine h ow the\nlaboratory components cµν\nQchange with sidereal time.\nWe can then use the data collected on the e−Pcross section to establish\nbounds oncµν\nQ. As presented above, the LV corrections to this cross section\nmanifest themselves as a violation of Bjorken scaling. At tree level, w e can\nverify that the usual SM results for the reduced cross section an dνW2are\nindependent of Q2. The LV correction to these two quantities introduces\na nontrivial dependence on Q2. If we fit the data on W2as a straight line,\ni.e.,νW2(x,Q2) =a+bQ2, the slope bis very small and can be used to\nconstraincµν\nQ. The actual analysis4considers a nontrivial and unknown\ndependence on Q2and it is used to constrain the components of cµν\nQwith\nprecision of 10−5to 10−7.\nReferences\n1.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2016 edition, arXiv:0801.0287v9.\n2. D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D 58, 116002 (1998).\n3. D. Colladay and V.A. Kosteleck´ y, Phys. Lett. B 511, 209 (2001).\n4. E. Lunghi, these proceedings; V.A. Kosteleck´ y, E. Lungh i, and A.R. Vieira,\narXiv:1610.08755."    },    {        "title": "1305.4961v1.Characterization_and_Synthesis_of_Rayleigh_Damped_Elastodynamic_Networks.pdf",        "content": "Manuscript submitted to Website: http://AIMsciences.org\nAIMS' Journals\nVolume X, Number 0X, XX200X pp.X{XX\nAlessandro Gondolo and Fernando Guevara Vasquez\nMathematics Department, University of Utah\n155 S 1400 E RM 233\nSalt Lake City, UT 84112-0090, USA\nCHARACTERIZATION AND SYNTHESIS OF\nRAYLEIGH DAMPED ELASTODYNAMIC NETWORKS\nAbstract. We consider damped elastodynamic networks where the damping\nmatrix is assumed to be a non-negative linear combination of the sti\u000bness\nand mass matrices (also known as Rayleigh or proportional damping). We\ngive here a characterization of the frequency response of such networks. We\nalso answer the synthesis question for such networks, i.e., how to construct\na Rayleigh damped elastodynamic network with a given frequency response.\nOur analysis shows that not all damped elastodynamic networks can be realized\nwhen the proportionality constants between the damping matrix and the mass\nand sti\u000bness matrices are \fxed.\n1.Introduction. The second Newton's law applied to a network of springs, masses,\nand dampers gives\nKu+C_u+Mu=f; (1)\nwhere uis the vector of displacements of the network nodes and fis the vector\nof external forces. The sti\u000bness, damping, and mass matrices are K,C, and M,\nrespectively. These matrices are de\fned in section 2. We are mainly concerned\nwith networks where the damping is proportional or of Rayleigh type , which means\nthat there are some known proportionality constants \u000b;\f\u00150 linking the damping\nmatrix to the sti\u000bness and mass matrices:\nC=\u000bK+\fM: (2)\nThe Rayleigh damping assumption means that:\n(a) The springs have a dashpot in parallel with damping constant proportional to\nthe spring constant. One way of achieving this is to construct springs from\nKelvin-Voigt solids (see e.g., [5, x7.3]), where the damping constant is propor-\ntional to the sti\u000bness constant.\n(b) Each mass lies in a \fxed cavity \flled with a viscous liquid, where the viscosity\nconstant is proportional to the mass, or the cavity size is adjusted to obtain the\nsame e\u000bect (i.e., inversely proportional to the mass).\nThe proportionality constants \u000b(linking the damping to the sti\u000bness) and \f(linking\nthe damping to the mass) are assumed to be the same for all the springs and nodes\nin the network. An example of such a network is given in \fgure 1.\n2000 Mathematics Subject Classi\fcation. Primary: 74B05, 35R02.\nKey words and phrases. Elastodynamic networks, Response function, Damping, Network syn-\nthesis, Proportional damping.\n1arXiv:1305.4961v1  [math-ph]  21 May 20132 A. GONDOLO AND F. GUEVARA VASQUEZ\n1\nFigure 1. An example of a Rayleigh damped network. For the\nterminal nodes the masses are colored in white and for the in-\nternal nodes in black. Each mass is surrounded by a cavity con-\ntaining a viscous liquid. The damping coe\u000ecient of each of the\nlinear dampers is \u000btimes the sti\u000bness constant of the correspond-\ning spring, and the viscous damping coe\u000ecient of each mass is \f\ntimes the corresponding mass.\nWe answer two questions about the (frequency) response of networks with this\nclass of damping. The response is the frequency-dependent linear relationship be-\ntween the displacements and forces at a few terminal, or accessible, nodes. The\n\frst question we answer is the characterization question, i.e., we give the form of\nall possible responses for this particular class of networks. The second question is\nthe synthesis: can we build a network from this class that mimics any admissible\nresponse?\nFor static networks (zero frequency), the characterization and synthesis questions\nwere established by Camar-Eddine and Seppecher [3], as part of a characterization\nof the possible macroscopic behaviors of a static elastic material under a single\ndisplacement \feld. Then, Milton and Seppecher [11] characterized the response of\ndamped elastic networks at a single, non-resonant frequency, and found it is possible\nto build a damped elastic network that mimics a prescribed response at one single\nfrequency . The complete characterization and synthesis for the undamped case is\ndone in [10]: it is shown that the response is a matrix with rational function (of\nfrequency) entries, and that given the response for an undamped elastic network,\nit is possible to build a network that mimics this response for all non-resonant\nfrequencies . The new characterization and synthesis results that we present here\nare a generalization of the results in [10] to damping of Rayleigh type. In particular,\nour results show that the Rayleigh damping model is incomplete, in the sense that\nit cannot describe, by itself, the responses of all possible elastodynamic networks\nwith damping.\nThe characterization and synthesis for elastic networks have been used by Camar-\nEddine and Seppecher [3] to show how to design a (linear) elastic material that\nhas a prescribed response under a single displacement \feld. A similar technique\nexploiting the characterization of networks of resistors was used by Camar-Eddine\nand Seppecher [2] to show how to design a conductor that has a prescribed response\nunder a single voltage \feld. Synthesis results are also known for electrical networks\nwith resistors (Kirchho\u000b's Y\u0000\u0001 theorem, Curtis, Ingerman, and Morrow [6] for\nplanar resistor networks); networks of resistors, capacitors and inductors (FosterRAYLEIGH DAMPED NETWORKS 3\n[7, 8], Bott and Du\u000en [1], Milton and Seppecher [11]); acoustic networks (Milton\nand Seppecher [11]); and an electromagnetic version of elastodynamic networks\n(Milton and Seppecher [12, 13]).\nWe start in section 2 by de\fning the sti\u000bness, mass, and damping matrices for\na damped elastodynamic network and, given only access to a few terminal nodes,\nits frequency response. Then in section 3, we show how the characterization and\nsynthesis results for static elastic networks in [3, 10] can be generalized to massless\nelastic networks with Rayleigh damping. For general Rayleigh damped networks,\nthe characterization result appears in section 4 and the synthesis result in section 5.\nWe \fnish by giving in section 6 the loci of the resonances of Rayleigh damped\nnetwork.\n2.The frequency response of an elastodynamic network with damping.\nThe linearized Hooke's Law for a single spring located between nodes x1andx2\nrelates the forces ai(supported at node xi) to the displacements ui(assuming these\nare small) through the relation\na2=\u0000k1;2(x2\u0000x1)(x2\u0000x1)T\nkx2\u0000x1k2(u2\u0000u1) =\u0000a1:\nLetKbe the sti\u000bness matrix, Cbe the damping matrix, and Mbe a diagonal\nmatrix with the masses of the nodes in the diagonal. In the case of a single spring,\nthe sti\u000bness and mass matrices are\nK=k1;2\u0014n1;2nT\n1;2\u0000n1;2nT\n1;2\n\u0000n1;2nT\n1;2n1;2nT\n1;2\u0015\n;M= diag (m1e;m2e);\nn1;2=x2\u0000x1\nkx2\u0000x1k;\nwhere e= (1;:::; 1)T2Rd,d= 2;3. For a network with Nnodes, the mass\nmatrix is an Nd\u0002Nddiagonal matrix that is de\fned similarly with the nodal\nmassesm1;:::;mN. The sti\u000bness and damping matrices of a network are Nd\u0002Nd\nmatrices obtained by adding the contributions from individual springs or dashpots:\nK=X\nspringijkijNijandC=X\nspringijcijNij (3)\nwherekij(resp.cij) is the spring (resp. damping) constant for the spring (resp.\ndamper) between nodes iandjand\nNij=\u0012\u0002eiej\u0003\u0014\n1\u00001\n\u00001 1\u0015\u0014eT\ni\neT\nj\u0015\u0013\n\n(nijnT\nij): (4)\nHere we used the Kronecker product \nand thei\u0000th canonical basis vector ei2RN.\nAs in the case of a single spring, the vector nij2Rdis a unit length vector with\ndirection xi\u0000xj.\nFor a network of springs, masses, and dashpots, the balance of forces (Newton's\nsecond law) gives a system of ODEs, which is given in (1). Now, recall the Laplace\ntransform:\neu(\u0015) =L[u(t)](\u0015) =Z1\n0e\u0000\u0015tu(t)dt:\nApplying the Laplace transform to (1), the ODE becomes\n(K+\u0015C+\u00152M)eu=ef: (5)4 A. GONDOLO AND F. GUEVARA VASQUEZ\nIn the following, we only work in the Laplace domain; the tilde notation is dropped\nfor simplicity. Our results can also be formulated in the frequency domain by using\nthe Fourier transform instead of the Laplace transform on (1), or simply by setting\n\u0015={!.\nFor the purpose of introducing the problem, we \frst consider the case where all\nnodes have a non-zero mass. Let us partition the nodes of the network into terminal\nnodes (B) and interior nodes ( I). At the interior nodes, the external forces are zero;\nthe displacement of the interior nodes is determined by the displacement of the\nboundary nodes, except at the few frequencies corresponding to the resonances of\nthe system. If subscripts BandIdenote the quantities referring to their respective\nnodes, then the displacements at the boundary nodes are related to the forces at\nthe boundary nodes by\nfB=W(\u0015)uB;\nwhere the displacement to forces map is\nW(\u0015) =KBB+\u0015CBB+\u00152MBB\n\u0000(KBI+\u0015CBI)(KII+\u0015CII+\u00152MII)\u00001(KIB+\u0015CIB);(6)\nwhich is the Schur complement of the IIblock in the matrix K+\u0015C+\u00152M(see\nAppendix A for the de\fnition of the Schur complement and properties).\nRemark 1. We assume throughout this paper that each spring-damper pair oc-\ncupies an arbitrarily small volume surrounding the segment connecting the corre-\nsponding nodes. We also assume that the cavities with viscous \ruid (and hence the\nmasses) can be made arbitrarily small. We need these assumptions to constrain the\ninternal nodes in our synthesis result to be within an \u000f\u0000neighborhood of the convex\nhull of the terminal nodes.\n3.Massless Rayleigh damped networks. First, we characterize the networks\nwith Rayleigh damping and no mass. Later, in section 4, we use such networks to\nsimplify the work for general networks. The theorems in this section are a natural\nextension of the characterization and synthesis in [3, 10] for static networks, i.e.,\nnetworks with springs only. In the static case the response matrix is given by (6)\nwith\u0015= 0. The forces f= (fT\n1;:::;fT\nn)Tat the nodes x1;:::;xnform a balanced\nsystem of forces when\nnX\ni=1fi=0(equilibrium of forces) andnX\ni=1xi\u0002fi=0(equilibrium of torques) ;(7)\nwhere u\u0002v= (u2v3\u0000u3v2;u3v1\u0000u1v3;u1v2\u0000u2v1) ifd= 3 and u\u0002v=u1v2\u0000u2v1\nifd= 2, is the usual cross product. The characterization and synthesis theorems in\n[3, 11, 10] are summarized in the next theorem.\nTheorem 3.1. The response matrix Wof any network of springs is symmetric\npositive semide\fnite where each column is a balanced system of forces at the terminal\nnodes. (Characterization)\nConversely, any matrix Wthat is symmetric positive semide\fnite where each\ncolumn is a balanced system of forces at the terminals can be realized by a network\nof only springs. Moreover the internal nodes of such network can be chosen so as\nto avoid a \fnite number of points and to be within an \u000f\u0000neighborhood of the convex\nhull of the terminal nodes. (Synthesis)RAYLEIGH DAMPED NETWORKS 5\nProof. See e.g. [10, Lemma 2] for the characterization and [10, Theorem 1] for the\nsynthesis.\nTheorem 3.1 can be readily extended to massless Rayleigh damped networks.\nTheorem 3.2. The response of a massless network with Rayleigh damping is of\nthe form\n(1 +\u000b\u0015)W; (8)\nwhere\u000b\u00150,\u0015is the Laplace parameter, and Wis the response of a static network,\ni.e.,Wis symmetric positive semide\fnite with each column being a balanced system\nof forces at the terminals x1;:::;xn(as in theorem 3.1).\nConversely, given any matrix-valued function of \u0015of the form (8)where Wis\nthe response of a static network at the nodes x1;:::;xn, there is a massless Rayleigh\ndamped network with terminals at the nodes x1;:::;xnrealizing it. Moreover the\ninternal nodes of such network can be chosen so as to avoid a \fnite number of points\nand to be within an \u000f\u0000neighborhood of the convex hull of the terminal nodes.\nProof. The forward direction is due to the homogeneity of degree 1 of the Schur\ncomplement (see Appendix A). Let Kbe the sti\u000bness matrix for a static network\nandWthe response matrix of the network at some terminal nodes B. Then the\nfrequency response of the massless network with Rayleigh damping when all the\nnodes are terminals is (1 + \u000b\u0015)K. Then by taking the Schur complement, the\nresponse at the nodes Bis (1 +\u000b\u0015)W.\nTo prove the converse, assume we are given a frequency response of the form\n(1 +\u000b\u0015)W, where Wis the response of a static network. By the second part of\ntheorem 3.1, there is a network of springs with sti\u000bness matrix Kwith response\nWat the nodes B. The internal nodes of this network can be chosen so as to\navoid a \fnite number of points and to be in an arbitrarily small neighborhood\nof the convex hull of the terminals. Then a network that has (1 + \u000b\u0015)Was its\nresponse is the network with response at all the nodes (1 + \u000b\u0015)K, i.e., the network\nobtained from the second part of theorem 3.1 with dashpots in parallel with each\nspring, each dashpot having a damping constant being \u000btimes the sti\u000bness of the\nassociated spring. Here we have used the assumption that each spring and damper\npair occupies an arbitrarily thin segment between the corresponding nodes, so the\nnetwork transformations that do not change the response in [10, x2.3] are still\nvalid.\nRemark 2. Theorem 3.2 is valid for planar networks, i.e., networks for which all\nthe springs lie in a plane. This is because theorem 3.1 is valid for planar networks:\none can always realize the response of a planar network with a planar network.\n4.Characterization of general Rayleigh damped networks. In this section,\nwe give conditions that the response of Rayleigh damped networks needs to satisfy.\nIn general, we may have massless nodes in the network; these are dealt with in\nlemma 4.1, where we use the characterization for the massless case from section 3\nto eliminate any massless interior nodes. Theorem 4.2 is the characterization result\nfor general Rayleigh networks. Then Proposition 1 shows that the conditions of\ntheorem 4.2 are consistent with the characterization at a single frequency found\nby Milton and Seppecher [11] (a condition which means, in physical terms, that\ndamping can only consume energy).6 A. GONDOLO AND F. GUEVARA VASQUEZ\nLet us partition the interior nodes Iinto the set of interior nodes Jwith positive\nmass and the set of massless nodes L. Clearly, MJJis positive de\fnite while\nMLL=0. The following lemma is similar to [10, Lemma 3] and reduces the\ncharacterization problem for networks with massless nodes to networks where each\nnode has mass.\nLemma 4.1. LetKbe aNd\u0002Nd sti\u000bness matrix and let Mbe aNd\u0002Nd\n(diagonal) mass matrix, where N=jB[Ij. The response at the terminals is\nW(\u0015) =eKBB+\u0015eCBB+\u00152MBB\n\u0000(eKBJ+\u0015eCBJ)(eKJJ+\u0015eCJJ+\u00152MJJ)\u00001(eKJB+\u0015eCJB);\nfor all frequencies \u0015for which det(eKJJ+\u0015eCJJ+\u00152MJJ)6= 0. Here the tilde\nmatrices are submatrices of the matrices\neK=\"\neKBBeKBJ\neKJBeKJJ#\n=\u0014KBBKBJ\nKJBKJJ\u0015\n\u0000\u0014KBL\nKJL\u0015\nKy\nLL\u0002KLBKLJ\u0003\nand\neC=\"\neCBBeCBJ\neCJBeCJJ#\n=\u0014CBBCBJ\nCJBCJJ\u0015\n\u0000\u0014CBL\nCJL\u0015\nCy\nLL\u0002CLBCLJ\u0003\n;(9)\nwhere the symbol ydenotes the Moore-Penrose pseudoinverse. Also, eKJJandeCJJ\nare positive semide\fnite.\nProof. The response of the network with terminals B[Jand internal nodes Lis\neK+\u0015eC+\u00152diag ( MBB;MJJ). Equilibrating forces at the nodes J(i.e., taking\nthe Schur complement of the JJblock) gives the desired result. Note that eKJJ\nandeCJJare positive semide\fnite as they are the Schur complement of positive\nsemide\fnite matrices (see e.g., (14)).\nRemark 3. For Rayleigh networks, the elimination of massless nodes made in\nlemma 4.1 preserves the Rayleigh damping structure. If we have Rayleigh damping\nthenC=\u000bK+\fM(at all nodes) and\neC=\u000beK+\fdiag ( MBB;MJJ);\nwhere the tilde matrices are de\fned as in lemma 4.1. Indeed a simple calculation\ngives\neC=\u000b\u0014KBBKBJ\nKJBKJJ\u0015\n\u0000\u000b2\u0014KBL\nKJL\u0015\n(\u000bKLL)y\u0002KLBKLJ\u0003\n+\f\u0014MBB\nMJJ\u0015\n:\nWe can now formulate a characterization of the response at the terminal nodes\nof a Rayleigh damped network.\nTheorem 4.2. Consider a damped mass-spring network with terminals x1;:::;xn\nwith Rayleigh damping, i.e., the damping matrix is of the form C=\u000bK+\fM,\nwhere\u000b;\f\u00150are given. The displacement-to-forces map of any such network is\nof the form:\nW(\u0015) = (1 +\u000b\u0015)A+ (\f\u0015+\u00152)M\u0000pX\nj=1(1 +\u000b\u0015)2R(j)\n\u001bj+\u0015(\u000b\u001bj+\f) +\u00152; (10)\nwhereRAYLEIGH DAMPED NETWORKS 7\ni.R(j)is real symmetric positive semide\fnite and \u001bj>0.\nii.Mis real diagonal positive semide\fnite.\niii.Ais real symmetric positive semide\fnite.\niv. There are at most 2pdistinct poles: namely the roots of the polynomials\nqj(\u0015) =\u001bj+\u0015(\u000b\u001bj+\f) +\u00152;forj= 1;:::;p:\nMoreover, for all roots \u0015\u0003ofqj(\u0015), we have Re(\u0015\u0003)\u00140. This is the second law\nof thermodynamics: damping consumes energy.\nv. The response for \u0015= 0, i.e.,\nW(0) = A\u0000pX\nj=1R(j)\n\u001bj;\nis the response of a static network. The characterization for static elastic\nnetworks [3] states that W(0)must be real symmetric positive semide\fnite\nwith every column being a balanced system of forces (see (7)) at the terminals\nx1;\u0001\u0001\u0001;xn.\nProof. We begin by using lemma 4.1 to remove the massless interior nodes. Let\nI=J[LwhereJare the positive mass nodes and Lare the massless nodes. Let\neKandeCbe as in (9). Then lemma 4.1 gives\nW(\u0015) =eKBB+\u0015eCBB+\u00152MBB\n\u0000(eKBJ+\u0015eCBJ)(eKJJ+\u0015eCJJ+\u00152MJJ)\u00001(eKJB+\u0015eCJB):(11)\nSince MJJis real diagonal positive de\fnite, it has a square root M1=2\nJJ. Now\neKJJis real symmetric positive semide\fnite and so is M\u00001=2\nJJeKJJM\u00001=2\nJJ. Then by\nthe spectral theorem, there exists Uunitary and \u0006real diagonal with nonnegative\nentries so that\nM\u00001=2\nJJeKJJM\u00001=2\nJJ =U\u0006UT:\nConsider the matrix X=M\u00001=2\nJJU, which clearly is invertible and is so that\nXTeKJJX=\u0006andXTMJJX=I;\nwhere Iis the identity matrix (in this case jJj\u0002jJj). The resolvent part of (11)\nbecomes\nQ(\u0015) =eKJJ+\u0015eCJJ+\u00152MJJ\n=X\u0000T(XTeKJJX+\u0015XTeCJJX+\u00152XTMJJX)X\u00001\n=X\u0000T(\u0006+\u0015(\u000b\u0006+\fI) +\u00152I)X\u00001\n=X\u0000TD(\u0015)X\u00001:\nHere we used the result from remark 3, i.e., that eCJJ=\u000beKJJ+\fMJJ. The matrix\nD(\u0015) is diagonal and therefore easily inverted. Then the resolvent [ Q(\u0015)]\u00001can be\nwritten as\n[Q(\u0015)]\u00001=XD(\u0015)\u00001XT\n=Xdiag\u00121\n\u001bi+\u0015(\u000b\u001bi+\f) +\u00152\u0013\nXT;8 A. GONDOLO AND F. GUEVARA VASQUEZ\nwhere\u001biare the diagonal elements of \u0006. Let KBJX= [v1\u0001\u0001\u0001vjJj]. By remark 3\nwe haveeCBB=\u000beKBB+\fMBBandeCBJ=\u000beKBJ. Therefore the response (11)\ncan be written in the form\nW(\u0015) = (1 +\u000b\u0015)eKBB+ (\f\u0015+\u00152)MBB\u0000jJjX\ni=1vivT\ni(1 +\u000b\u0015)2\n\u001bi+\u0015(\u000b\u001bi+\f) +\u00152:\nNotice that the \u001bimay be zero. This is the case when the network has a so\ncalled \\\roppy mode\" (non-zero displacements with zero force required). The same\nargument in [10, Lemma 12] can be used to deal with \roppy modes. Indeed Lemma\n1 in [10] can be used to show that \u001bi= 0 implies vi= 0. By adding up the non-\nzero residues vivT\nithat share the same denominator, we get the symmetric positive\nsemide\fnite residues R(j)of the statement of the theorem. The result follows by\nnoticing that A=eKBBis symmetric positive semide\fnite, MBBis diagonal with\nnonnegative entries and at \u0015= 0, the response\nW(0) = A\u0000pX\nj=1R(j)\n\u001bj;\nis the response of a static elastic network.\nFinally for the roots \u0015(j)\n\u0006of\u001bj+ (\u000b\u001bj+\f)\u0015+\u00152we have\n2 Re\u0015(j)\n+=\u0000(\u000b\u001bj+\f)\u00140:\nHence the resonances lie in the left half plane.\nRemark 4. Setting\u0015= 0 in Theorem 4.2 gives the static or zero frequency result\nin [3]. Setting \u000b=\f= 0 (i.e., no damping) gives the result in [10] for elastodynamic\nnetworks with no damping.\nFor the Laplace parameter \u0015in the imaginary axis (i.e., real frequencies), the\neigenvalues of the imaginary part of the response should have a sign consistent\nwith the energy losses due to damping (again a manifestation of the second law of\nthermodynamics). These inequalities are essential to the single-frequency charac-\nterization and synthesis of Milton and Seppecher [11], and hence these inequalities\nshould also hold in our case. The next proposition shows that these inequalities hold\nautomatically for matrix functions of \u0015satisfying the hypothesis of theorem 4.2.\nProposition 1. Any matrix function of \u0015of the form (10) and with the properties\nof theorem 4.2 is such that\n!ImW({!)\u00150for any!2R;\nwhere the inequality A\u00150for a symmetric matrix A, means Ais positive semi-\nde\fnite.\nProof. Let us rewrite the matrix-valued function W(\u0015) in (10) as\nW(\u0015) =W1(\u0015) +W2(\u0015) +W3(\u0015);RAYLEIGH DAMPED NETWORKS 9\nwhere\nW1(\u0015) = (1 +\u000b\u0015)0\n@A\u0000pX\nj=1R(j)\n\u001bj1\nA= (1 +\u000b\u0015)W(0);\nW2(\u0015) = (\f\u0015+\u00152)M;and\nW3(\u0015) =pX\nj=1\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001bj\n\u001bj+\u0015(\u000b\u001bj+\f) +\u00152\u0015R(j)\n\u001bj:\nTo prove the \fnal result, it is enough to show that !ImWk({!)\u00150,k= 1;2;3. The\n\frst two cases are clear since: !ImW1({!) =\u000b!2W(0)\u00150 because W(0)\u00150;\nand!ImW2({!) =\f!2M\u00150 because Mis diagonal with nonnegative entries.\nSince R(j)=\u001bj\u00150, we have proved the inequality for the third case if for all\n\u001b>0 the function\nf(\u0015) = (1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001b\n\u001b+\u0015(\u000b\u001b+\f) +\u00152\nis such that\n!Imf({!)\u00150;for!2R.\nTo show this inequality, consider the 2 \u00022 complex symmetric matrix\nB(\u0015) =\"\n1 +\u000b\u0015 1 +\u000b\u0015\n1 +\u000b\u0015\u001b+\u0015(\u000b\u001b+\f)+\u00152\n\u001b#\n:\nWe start with !\u00150. Clearly Im B({!)\u00150 since det(Im B({!)) =!\u000b\f=\u001b\u00150\nand all the entries of Im B({!) are non-negative. Then noticing that f(\u0015) is the\nSchur complement of the 2 ;2 entry in B(\u0015) and using the property of the Schur\ncomplement (14), we have that Im B({!)\u00150)Imf({!)\u00150. So we get the result\n!f({!)\u00150 when!\u00150.\nFor!\u00140, the same reasoning holds. We only need to check that Im B({!)\u00140.\nThis is true because all the entries of Im B({!) are non-positive and det(Im B({!)) =\n!\u000b\f=\u001b\u00140. Therefore !f({!)\u00150 when!\u00140, which completes the proof.\n5.Synthesis of general Rayleigh damped networks. The basic building block\nfor the synthesis is a Rayleigh damped network with: a rank one response; a complex\nconjugate pair or a real pair of prescribed resonances; and zero static response. The\nexistence of such a network is stated in lemma 5.1 and proved at the end of this\nsection (it is a similar construction to that in [10, Lemma 12]). The synthesis of a\ngeneral Rayleigh damped network is done in theorem 5.2.\nLemma 5.1. Letx1;:::;xnbe some terminal nodes and let f1;:::;fnbe an ar-\nbitrary system of forces at the corresponding terminals (the forces do not need to\nbe balanced). Then for any \u000b;\f\u00150and\u001b > 0, it is possible to build a Rayleigh\ndamped network with zero static response and with\nW(\u0015) =\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001b\n\u001b+ (\u000b\u001b+\f)\u0015+\u00152\u0015\n\u000bT; (12)\nwhere f= [fT\n1;:::;fT\nn]T. The internal nodes of such a network can be chosen to\navoid a \fnite number of points and to be within an \u000f\u0000neighborhood of the convex\nhull of the terminal nodes.10 A. GONDOLO AND F. GUEVARA VASQUEZ\nTheorem 5.2. Given any matrix-valued function W(\u0015)of the form (10) and sat-\nisfying the conditions of theorem 4.2 with \u000band\f\fxed, there exists a Rayleigh\ndamped network with proportionality constants \u000band\fand response W(\u0015). The\ninternal nodes of such a network can be chosen to avoid a \fnite number of points\nand to be within an \u000f\u0000neighborhood of the convex hull of the terminal nodes.\nProof. The proof relies on the superposition of networks, i.e., using the fact that\nthe response of two networks that share terminal nodes and only terminal nodes,\nis the sum of the responses of each network (see e.g., [10, x2.4]). To specify the\nbuilding blocks needed to realize the matrix-valued function W(\u0015) in (10), we \frst\nrewrite it as:\nW(\u0015) =W1(\u0015) +W2(\u0015) +W3(\u0015);\nwhere\nW1(\u0015) = (1 +\u000b\u0015)0\n@A\u0000pX\nj=1R(j)\n\u001bj1\nA= (1 +\u000b\u0015)W(0);\nW2(\u0015) = (\f\u0015+\u00152)M;and\nW3(\u0015) =pX\nj=1\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001bj\n\u001bj+\u0015(\u000b\u001bj+\f) +\u00152\u0015R(j)\n\u001bj:\nWe now show how to realize each term W1,W2andW3separately by a network.\nThe superposition principle can then be used to \fnd a network realizing W.\nWe \frst use theorem 3.2 to see that a network realizing W1(\u0015) is the network\nof springs realizing the static response W(0) (existence guaranteed by theorem 3.1\nor [3]), with a damper with damping constant \u000btimes the spring constant added\nin parallel to each spring. To realize W2it su\u000eces to endow each terminal node\nby the mass dictated by Mand surrounding it by a cavity such that the resulting\ndamping is \ftimes the mass.\nWe now show that each term in the sum in the expression of W3is realizable.\nThen the realizability of W3follows from the superposition principle. Let us drop\nthe indexjfor the sake of simplicity and show that there is a Rayleigh damped\nnetwork with response:\n\u0014\n(1 +\u000b\u0015)\u0000(1 +\u000b\u0015)2\u001b\n\u001b+\u0015(\u000b\u001b+\f) +\u00152\u0015R\n\u001b;\nwhen Ris real symmetric positive semide\fnite. By the spectral theorem it is\npossible to \fnd real vectors vksuch that\nR\n\u001b=rX\nk=1vkvT\nk;\nwherer >0 is the rank of R. (The case r= 0 is the trivial R=0case). Hence\nwe have reduced the problem to that of \fnding a network with zero static response,\na rank one response and as resonances the roots of p(\u0015) =\u001b+\u0015(\u000b\u001b+\f) +\u00152.\nLemma 5.1 shows how to build such a network. This completes the construction.\nWe now prove Lemma 5.1.\nProof. The main idea here is to \fnd a massless Rayleigh damped network and add\nappropriate masses to the internal nodes to get the desired resonances. According\nto [10, Lemma 12], it is possible to choose two distinct nodes xn+1andxn+2andRAYLEIGH DAMPED NETWORKS 11\ntwo forces fn+1andfn+2such that the system of forces f1;:::;fn+2is balanced\nwhen supported at the nodes x1;:::;xn+2, regardless of the choice of f. Then\nby theorem 3.2, there exists a massless Rayleigh damped network with rank one\nresponse (1 + \u000b\u0015)[fT;gT]T[fT;gT], where g= [fT\nn+1;fT\nn+2]T. We now endow the\ntwo internal nodes xn+1andxn+2with the same mass m, that is determined later\nto match the desired resonances. We also surround the nodes xn+1andxn+2by\na cavity with a viscous \ruid, designed so that the damping term is \fm. Then\nNewton's second law becomes\n\u0012\u0014f\ng\u0015\u0002\nfTgT\u0003\n(1 +\u000b\u0015) +\u00140\nmI\u0015\n(\f\u0015+\u00152)\u0013\u0014uB\nuI\u0015\n=\u0014W(\u0015)uB\n0\u0015\n;\nwhere uBanduIare the displacements of the boundary nodes x1;:::;xnand\ninterior nodes xn+1andxn+2, respectively, and W(\u0015) is the frequency response of\nthis network.\nNext, we take the Schur complement of the IIblock to \fnd the response W(\u0015):\nW(\u0015) = (1 +\u000b\u0015)\u000bT+ (1 +\u000b\u0015)\u0012\u0000(1 +\u000b\u0015)jgj2\n(1 +\u000b\u0015)jgj2+ (\f\u0015+\u00152)m\u0013\n\u000bT\n= (1 +\u000b\u0015)\u000bT\u0000\u0012(1 +\u000b\u0015)2(jgj2=m)\n\u00152+ (\u000b(jgj2=m) +\f)\u0015+ (jgj2=m)\u0013\n\u000bT:\nLetm=jgj2=\u001b. Then the response is\nW(\u0015) = (1 +\u000b\u0015)\u000bT\u0000\u0012(1 +\u000b\u0015)2\u001b\n\u00152+ (\u000b\u001b+\f)\u0015+\u001b\u0013\n\u000bT;\nwhich is the desired result since W(0) = 0.\n6.Resonances of Rayleigh damped networks. A natural question to ask is\nwhether it is possible to \fnd a Rayleigh network with \u000band\f\fxed with a resonance\nthat is located anywhere in the left half complex plane. Since the conditions of\ntheorem 4.2 are necessary and su\u000ecient for a matrix valued function of frequency\nto be the response of a Rayleigh damped network, this question is equivalent to\n\fnding the set of all possible resonances for a Rayleigh damped network with \u000band\n\f\fxed. Our analysis shows that this is set is not the left hand plane. This means\nthat the Rayleigh damping model is incomplete, since a general damped network\ncan have resonances anywhere in the left half complex plane.\nWe state here the loci of the resonances that can be realized with Rayleigh\ndamped networks. The derivation of these loci is standard and is not included\nhere. According to theorem 4.2, the resonances that can be realized with Rayleigh\ndamped networks are the roots of the quadratic \u00152+ (\u000b\u001b+\f)\u0015+\u001b, i.e.,\n\u0015\u0006=\u0000(\u000b\u001b+\f)\u0006p\n(\u000b\u001b+\f)2\u00004\u001b\n2:\nBecause we only can choose \u001b, the resonances lie in curves in the complex plane.\nThe speci\fc curves depend on \u000band\fand are as follows.\nDamping at the nodes only ( \u000b= 0):Union of the segment Im \u0015= 0;\n\u0000\f\u0014Re\u0015<0 and the line Re \u0015=\u0000\f=2.12 A. GONDOLO AND F. GUEVARA VASQUEZ\nRe\u0015Im\u0015\u0000\u0000\n\u0000\f\u0000\f=2\nDamping with dashpots between the nodes only ( \f= 0):Union of the\nnegative real axis Im \u0015= 0;Re\u0015 < 0 and the circlej\u0015+\u000b\u00001j=\u000b\u00001(the origin\nexcepted).\nRe\u0015Im\u0015\u0000\u0000\u000b\u00001\n\u00001=\u000b\nCase\u000b;\f6= 0and\u000b\f < 1:Union of the negative real axis Im \u0015= 0;Re\u0015<0\nand the circlej\u0015+\u000b\u00001j=\u000b\u00001p1\u0000\u000b\f.\nRe\u0015Im\u0015\u0000\n\u00001=\u000bp1\u0000\u000b\f\n\u000b\nCase\u000b;\f6= 0and\u000b\f\u00151:the negative real axis Im \u0015= 0;Re\u0015<0.\nRe\u0015Im\u0015\n7.Discussion and future work. We have established the characterization and\nsynthesis of the response for mass-spring networks with Rayleigh damping. In\nparticular, our result shows that, for each pair of \u000band\f, there is a class of\nresonances that can be realized when the damping matrix is C=\u000bK+\fM. Clearly,\nfrom section 6, when choosing di\u000berent values of \u000band\f, it is possible to have any\nresonance with negative real part. Hence, if we superimpose Rayleigh damped\nnetworks with di\u000berent values of \u000band\f, it is possible to construct a network\nwith any \fnite number of resonances in the left half complex plane (provided they\nare real or come in complex conjugate pairs). However, it is not clear whether\nthe response of a general damped network can be realized by superposing several\nRayleigh networks with di\u000berent \u000band\f. This is a question that we plan to explore.REFERENCES 13\nAnother extension of this work would be to consider general damping by using\nthe quadratic eigenvalue problem [14, 9]. Speci\fcally, we believe it is possible to use\nthe spectral decomposition for real symmetric quadratic pencils by Chu and Xu [4]\nto characterize the response of general damped networks. Then for the synthesis,\nwe would need to construct networks that realize the form of the response, but this\nis left for future work.\nAppendix A.Schur complement properties. Consider the partition of a ma-\ntrixA2Cn\u0002ninduced by the partition of f1;:::;nginto two sets IandB:\nA=\u0014ABBABI\nAIBAII\u0015\n:\nThe Schur complement of the IIblock in Ais de\fned as\nS=ABB\u0000ABIA\u00001\nIIAIB;\nprovided AIIis invertible. The Schur complement is homogeneous of degree 1, since\nfor nonzero \u00152C,\n\u0015S= (\u0015ABB)\u0000(\u0015ABI)(\u0015AII)\u00001(\u0015AIB):\nA quadratic form of the Schur complement is equivalent to a quadratic form of\nthe original matrix A, indeed by simple manipulations we have\nv\u0003\nBSvB=\u0014\nvB\nvI\u0015\u0003\nA\u0014\nvB\nvI\u0015\n;where vI=\u0000A\u00001\nIIAIBvB: (13)\nA consequence of (13) is that if A2Rn\u0002n,A\u00150 implies S\u00150 (and similarly\nfor the reverse inequality). Here the inequality A\u00150 is understood as Apositive\nsemi-de\fnite.\nThe quadratic form v\u0003AvforA2Cn\u0002nwithAT=A(i.e., complex symmetric),\ncan be written as\nRe(v\u0003Av) = (Re v)T(ReA)(Rev) + (Im v)T(ReA)(Imv);\nIm(v\u0003Av) = (Re v)T(ImA)(Rev) + (Im v)T(ImA)(Imv):\nBy combining this fact with (13) we have that for complex symmetric Aand its\nSchur complement S:\nReA\u00150)ReS\u00150 and Im A\u00150)ImS\u00150; (14)\nand similarly for the reverse inequalities.\nAcknowledgments. The authors are thankful to Graeme W. Milton, Daniel Onofrei\nand Pierre Seppecher for insightful conversations on this subject. The work of AG\nwas supported through the University of Utah Mathematics Department VIGRE-\nREU program. The work of FGV was partially supported by the National Science\nFoundation DMS-0934664.\nREFERENCES.\n[1] R. Bott and R. J. Du\u000en. Impedance synthesis without use of transformers.\nJournal of Applied Physics , 20:804, 1949. doi: 10.1063/1.1698532.\n[2] M. Camar-Eddine and P. Seppecher. Closure of the set of di\u000busion functionals\nwith respect to the Mosco-convergence. Math. Models Methods Appl. Sci. , 12\n(8):1153{1176, 2002. ISSN 0218-2025. doi: 10.1142/S0218202502002069.14 REFERENCES\n[3] M. Camar-Eddine and P. Seppecher. Determination of the closure of the set of\nelasticity functionals. Arch. Ration. Mech. Anal. , 170(3):211{245, 2003. ISSN\n0003-9527. doi: 10.1007/s00205-003-0272-7.\n[4] M. T. Chu and S.-F. Xu. Spectral decomposition of real symmetric quadratic\n\u0015-matrices and its applications. Math. Comp. , 78(265):293{313, 2009. ISSN\n0025-5718. doi: 10.1090/S0025-5718-08-02128-5.\n[5] T. J. Chung. General continuum mechanics . Cambridge University Press,\nCambridge, 2007. ISBN 978-0-521-87406-9; 0-521-87406-8.\n[6] E. B. Curtis, D. Ingerman, and J. A. Morrow. Circular planar graphs and\nresistor networks. Linear Algebra Appl. , 283(1-3):115{150, 1998. ISSN 0024-\n3795. doi: 10.1016/S0024-3795(98)10087-3.\n[7] R. M. Foster. A reactance theorem. The Bell System Technical Journal , 3:\n259{267, 1924. ISSN 0005-8580.\n[8] R. M. Foster. Theorems regarding the driving-point impedance of two-mesh\ncircuits. The Bell System Technical Journal , 3:651{685, 1924. ISSN 0005-8580.\n[9] I. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials . Academic\nPress Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. ISBN\n0-12-287160-X. Computer Science and Applied Mathematics.\n[10] F. Guevara Vasquez, G. W. Milton, and D. Onofrei. Complete characterization\nand synthesis of the response function of elastodynamic networks. J. Elasticity ,\n102(1):31{54, 2011. ISSN 0374-3535. doi: 10.1007/s10659-010-9260-y.\n[11] G. W. Milton and P. Seppecher. Realizable response matrices of multi-terminal\nelectrical, acoustic and elastodynamic networks at a given frequency. Proc. R.\nSoc. Lond. Ser. A Math. Phys. Eng. Sci. , 464(2092):967{986, 2008. ISSN 1364-\n5021. doi: 10.1098/rspa.2007.0345.\n[12] G. W. Milton and P. Seppecher. Electromagnetic circuits. Netw. Heterog.\nMedia , 5(2):335{360, 2010. ISSN 1556-1801. doi: 10.3934/nhm.2010.5.335.\n[13] G. W. Milton and P. Seppecher. Hybrid electromagnetic circuits. Phys-\nica B: Condensed Matter , 405(14):2935 { 2937, 2010. ISSN 0921-4526. doi:\n10.1016/j.physb.2010.01.007. Proceedings of the Eighth International Confer-\nence on Electrical Transport and Optical Properties of Inhomogeneous Media,\nETOPIM-8.\n[14] F. Tisseur and K. Meerbergen. The quadratic eigenvalue problem. SIAM Rev. ,\n43(2):235{286, 2001. ISSN 0036-1445. doi: 10.1137/S0036144500381988.\nE-mail address :agondolo@math.utah.edu\nE-mail address :fguevara@math.utah.edu"    },    {        "title": "1212.0509v3.Inviscid_limit_of_stochastic_damped_2D_Navier_Stokes_equations.pdf",        "content": "arXiv:1212.0509v3  [math.PR]  29 Jul 2013Inviscid limit of stochastic damped 2D\nNavier-Stokes equations\nHakima Bessaih∗& Benedetta Ferrario†\nAbstract\nWe consider the inviscid limit of the stochastic damped 2D Na vier-\nStokes equations. We prove that, when the viscosity vanishe s, the station-\nary solution of the stochastic damped Navier-Stokes equati ons converges\nto a stationary solution of the stochastic damped Euler equa tion and that\nthe rate of dissipation of enstrophy converges to zero. In pa rticular, this\nlimit obeys an enstrophy balance. The rates are computed wit h respect to\na limit measure of the unique invariant measure of the stocha stic damped\nNavier-Stokes equations.\nMSC2010 : 60G10, 60H30, 35Q35.\nKeywords : Inviscid limits, enstrophy balance, stationary processes, invaria nt\nmeasures.\n1 Introduction\nIn this paper, we are interested in the equations of motion of incomp ressible\nfluidsin abounded domainof R2. In particular,weconsidertheEulerorNavier-\nStokes equations damped by a term proportional to the velocity. D amping\nterms in two dimensional turbulence studies have been considered t o model\npumping due to friction with boundaries. Numerical studies of two dim ensional\nturbulence employ devices to remove the energy that piles up at the largescales,\nand damping is the most common such device. We refer to [21, 7] for a physical\nmotivation of the model and to [1, 25, 26] for a mathematical analys is of the\ndeterministic damped Navier-Stokes equations and to [4, 5] for th e stochastic\ndamped Euler equations.\nThese stochastic damped equations are given by\n(1)/braceleftBigg\ndu+[−ν∆u+(u·∇)u+γu+∇p]dt=dw\n∇·u= 0\nThe non negative coefficients νandγare called kinematic viscosity and sticky\nviscosity, respectively. The unknowns are the velocity uand the pressure p.\nSuitable boundary conditions have to be considered; in this paper th e spatial\ndomain is a box and periodic boundary conditions are assumed.\n∗University of Wyoming, Department of Mathematics, Dept. 30 36, 1000 East University\nAvenue, Laramie WY 82071, United States, bessaih@uwyo.edu\n†Universit` a di Pavia, Dipartimento di Matematica, via Ferr ata 1, 27100 Pavia, Italy,\nbenedetta.ferrario@unipv.it\n1For a fixed γ >0, ifν >0 these are called the stochastic damped Navier-\nStokes equations, whereas if ν= 0 they are the stochastic damped Euler equa-\ntions. Ifγ= 0 andν= 0, we refer to [3, 9, 10, 12, 22, 28, 32] for an analysis of\nthe existence and/or uniqueness of solutions and to [15] where som e dissipation\nof enstrophy arguments are discussed in Besov spaces.\nTurbulence theory investigates the behavior of certain quantities as the vis-\ncosityνvanishes. In particular, in the two dimensional setting one is inter-\nested in understanding what happens to the balance equation of en ergy and\nenstrophy (in the stationary regime) as the viscosity vanishes. D. Bernard [2]\nsuggested that there is no anomalous dissipation of enstrophy in da mped and\ndriven Navier-Stokes equations; Constantin and Ramos [11] prove d that there\nis no anomalous dissipation neither of energy nor of enstrophy as ν→0 for the\ndeterministic damped Navier-Stokesequations in the whole plane. So me similar\nquestions were suggested by Kupiainen [30] for the stochastic cas e. Therefore\nwe addressthe same problemwhen the forcingterm is ofwhite noiset ype. Tools\nfrom stochastic analysis are very useful to investigate the same p roblem studied\nin [11], giving a rigorous meaning to the averages of velocity and vortic ity. In-\ndeed, using stochastic PDE’s allows to express the stationary regim e by means\nof an invariant measure, whereas in the deterministic setting the st ationary\nregime is described by taking time averages on the infinite time interva l.\nIn this paper we shall prove that in the stationary regime system (1 ) has no\nanomalous dissipation neither of energy nor of enstrophy as ν→0. However,\nwe shall be working in a finite two dimensional spatial domain and not in t he\nwhole plane; this answers one of the questions posed by Kupiainen in [ 30] about\nthe behaviour of the stochastic damped Navier-Stokes equations on a torus for\nvanishing viscosity.\nAs far as the content of the paper is concerned, in Section 2 we intr oduce\nsome functional spaces, the equations in their vorticity formulatio n and the\nassumptions on the noise term. We also introduce the classical prop erties of the\nnonlinear term associated to these equations. Section 3 is devoted to the well\nposedness of the stochastic 2D damped Navier-Stokes equations , where some\nuniform estimates are computed. Starting from a known result of e xistence and\nuniqueness of the invariant measure, we provide a balance law for th e enstrophy.\nThe vanishing viscosity limit is studied in Section 4 and stationary solutio ns\nare constructed by means of a tightness argument providing a bala nce relation\nfor these stationary solutions. Using these results, we provide a p roof of no\nanomalous of enstrophy and energy for the stochastic damped 2D Navier-Stokes\nequations.\n2 Notations and hypothesis\nLet the spatial domain Dbe the square [ −π,π]2; periodic boundary conditions\nare assumed. A basis of the space L2(D) with periodic boundary conditions\nis{ek}k∈Z2,ek(x) =1\n2πeik·x, whereas a basis for the space of periodic vector\nfields which aresquareintegrableand divergencefree is {k⊥\n|k|ek}k∈Z2, beingk⊥=\n(−k2,k1). Actually we consider k/\\e}atio\\slash= (0,0), since ifuis a solution of system (1)\nthen alsou+cis a solution for any c∈R. Therefore we consider velocity fields\nwith vanishing mean value.\nLetZ2\n0=Z2\\ {(0,0)}, andZ2\n+={k= (k1,k2)∈Z:k1>0} ∪ {k=\n2(0,k2)∈Z2:k2>0}. Givenx= (x1,x2)∈R2we denote by |x|its norm:\n|x|=/radicalbig\n(x1)2+(x2)2. Giveny=ℜy+iℑy∈Cwe denote by |y|its absolute\nvalue and by yits complex conjugate: |y|=/radicalbig\n(ℜy)2+(ℑy)2,y=ℜy−iℑy.\nFor anya∈Rwe define the Hilbert space\nHa={f=/summationdisplay\nk∈Z2\n0fkek(x) :/summationdisplay\nk∈Z2\n0|fk|2|k|2a<∞}\nwith scalar product\n/a\\}b∇acketle{tf,g/a\\}b∇acket∇i}htHa=/summationdisplay\nk∈Z2\n0|k|2afkgk;\nwe set\n/ba∇dblf/ba∇dbl2\nHa=/summationdisplay\nk∈Z2\n0|k|2a|fk|2.\nFor a vector f= (f1,f2) we set\n/ba∇dblf/ba∇dbl2\nHa=/ba∇dblf1/ba∇dbl2\nHa+/ba∇dblf2/ba∇dbl2\nHa.\nIn particular, for scalar functions we have /ba∇dblf/ba∇dbl2\nH0=/ba∇dblf/ba∇dbl2\nL2(D)and/ba∇dblf/ba∇dbl2\nH1=\n/ba∇dbl∇f/ba∇dbl2\nH0.\nThe spaceHais compactly embedded in the space Hbifa>b.\nMoreover, we consider the Banach spaces W1,q(D) (1≤q≤ ∞) endowed\nwith the norm\n/ba∇dblf/ba∇dblq\nW1,q(D)=/ba∇dblf/ba∇dblq\nLq+/ba∇dbl∇f/ba∇dblq\nLq\nwhere/ba∇dbl·/ba∇dblq\nLqis theLq(D)-norm.\nGivena separableHilbert space X, forα>0andp≥1 wedefine the Banach\nspace\nWα,p(0,T;X) =/braceleftBigg\nf∈Lp(0,T;X) :/integraldisplayT\n0/integraldisplayT\n0/ba∇dblf(t)−f(s)/ba∇dblp\nX\n|t−s|1+pαdt ds<∞/bracerightBigg\nand we set\n/ba∇dblf/ba∇dblp\nWα,p(0,T;X)=/integraldisplayT\n0/ba∇dblf(t)/ba∇dblp\nXdt+/integraldisplayT\n0/integraldisplayT\n0/ba∇dblf(t)−f(s)/ba∇dblp\nX\n|t−s|1+pαdt ds.\nLet (Ω,F,P) be a complete probability space, with expectation denoted by\nE. We assume that the stochastic forcing term in (1) is of the form\nw=w(t,x) =/summationdisplay\nk∈Z2\n0√qkβk(t)k⊥\n|k|ek(x).\nHere{βk}k∈Z2\n+is a sequence of independent complex-valued standard Brownian\nmotions on (Ω ,F,P), i.e.βk(t) =ℜβk(t) +iℑβk(t) with{ℜβk} ∪ {ℑβk}k∈Z2\n+\na sequence of independent standard real Brownian motions; more over we set\nβ−k=−βkandqk=q−kfor anyk∈Z2\n+. Therefore\nw(t,x) = 2/summationdisplay\nk∈Z2\n+√qkk⊥\n|k|[ℜβk(t)cos(k·x)−ℑβk(t)sin(k·x)].\n3In the 2D setting it is convenient to introduce the (scalar) vorticity\nξ=∇⊥·u≡∂u2\n∂x1−∂u1\n∂x2.\nSystem (1) corresponds to\n(2)/braceleftBigg\ndξ+[−ν∆ξ+γξ+u·∇ξ]dt=dwcurl\nξ=∇⊥·u\nobtained by taking the curl of both sides of the first equation of (1 ). Periodic\nboundary conditions have to be added to this system. The noise is wcurl(t,x) =\n−2/summationtext\nk∈Z2\n+√qk|k|[ℑβk(t)cos(k·x)+ℜβk(t)sin(k·x)]. Let us define\n(3) Q:=/summationdisplay\nk∈Z2\n0|k|2qk.\nClassical results are\n(4) E/ba∇dblwcurl(t)/ba∇dbl2\nH0= 2tQ∀t≥0\n(5) E/ba∇dblwcurl/ba∇dblp\nWα,p(0,T;H0)≤C(α,p)(T1+p/2+1)(Q)p/2\nfor anyα∈(0,1\n2),p≥2, and the Burkh¨ older-Davies-Gundy inequality\n(6)\nE/parenleftbigg\nsup\n0≤t≤T/integraldisplayt\n0/a\\}b∇acketle{t|ξ(s)|p−2ξ(s),dwcurl(s)/a\\}b∇acket∇i}htL2/parenrightbigg\n≤C(p)/radicalbig\nQE/radicalBigg/integraldisplayT\n0/ba∇dblξ(s)/ba∇dbl2(p−1)\nLpds\nFor this latter inequality we have used that supx∈D|ek(x)|= 1 for allk.\nHereandhenceforth, C(·) denotesapositiveconstantdepending onthespecified\nparameters; it may change from line to line.\nKnowing the vorticity ξ, we recover the velocity uby solving the elliptic\nequation\n(7) −∆u=∇⊥ξ.\nThis means that if ξ(x) =/summationtext\nkξkek(x), thenu(x) =−i/summationtext\nkk⊥\n|k|2ξkek(x).\nWe present basic properties ofthe bilinear term u·ξin the 2D setting. These\nare classical results in the analysis of incompressible fluids (see e.g. [3 3]).\nLemma 2.1 There exists a positive constant Csuch that\n(8)/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nD(u·∇)v·ψ dx/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dblu/ba∇dblL4/ba∇dblv/ba∇dblL4/ba∇dblψ/ba∇dblH1\nfor all divergence free vectors with the regularity specifie d in the r.h.s., and for\nanya>1\n(9)/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDu·∇ξ φ dx/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dblu/ba∇dblH0/ba∇dblξ/ba∇dblH1/ba∇dblφ/ba∇dblHa,\n(10)/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDu·∇ξ φ dx/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dblu/ba∇dblH0/ba∇dblξ/ba∇dblH1+a/ba∇dblφ/ba∇dblH0\nfor all functions with the regularity specified in the r.h.s. .\n4Proof.The key relationship for (8) is\n/integraldisplay\nD[u·∇]v·ψ dx=−/integraldisplay\nD[u·∇]ψ·v dx\nassumingsufficient regularityfor u,v,ψ; this isobtained by integratingby parts.\nThen, we get the estimate by H¨ older inequality and this is extended b y density\nto vectors with the specified regularity. For (9) we use H¨ older ineq uality and\nthe continuous embedding Ha⊂L∞(D) fora >1. Similarly, we obtain the\nlatter estimate. ✷\nLemma 2.2 Letξ=∇⊥·u. We have\n(11)/integraldisplay\nD[u·∇ξ]φ dx=−/integraldisplay\nD[u·∇φ]ξ dx ∀ξ,φ∈H1\nand for any p≥2\n(12)p/integraldisplay\nD[u·∇ξ]ξ|ξ|p−2ψ dx=−/integraldisplay\nD[u·∇ψ]|ξ|pdx∀ξ∈L2p,ψ∈H1.\nMoreover,\n(13)/integraldisplay\nD[u·∇ξ]ξ|ξ|p−2dx= 0∀ξ∈L2p.\nProof.The two first relationships (11)-(12) are easily obtained by integra ting\nby parts, where in (12) the proof is done first with smooth function s and then\nby density it is extended on the spaces specified; notice that for p>1, ifξ∈L2p\nthenu∈W1,2p⊂L∞and the r.h.s. is meaningful (see [27] ). Eventually, (13)\nis the particular case of (12) for ψ= 1. ✷\n3 The stochastic damped Navier-Stokes equa-\ntions\nThe well posedness of the stochastic damped 2D Navier-Stokes eq uations\n(14)/braceleftBigg\ndξν+[−ν∆ξν+uν·∇ξν+γξν]dt=dwcurl\nξν=∇⊥·uν\nis very similar to the case when γ= 0. Here, we assume periodic boundary\nconditions with period box [ −π,π]2.\nThe proof of existence of a unique solution for square summable initia l vor-\nticity is the same as the proof for square summable initial velocity tha t can be\nfound in [17], where the proof is performed for γ= 0. Similar proofs can also\nbe found in [3, 9] with some uniform estimates with respect to the visc osityν.\nHere, we point out the peculiar estimate (16) for γ >0, useful in the analysis\nof the limit as ν→0.\nTheorem 3.1 Letγ,ν >0,p≥2. Assume\nE/ba∇dblξν(0)/ba∇dblp\nLp<∞, Q< ∞.\n5Then, there exists a process ξνwith paths in C([0,∞),Lp)∩L2\nloc(0,∞;H1)P-\na.s., which is a Feller Markov process in Lpand is the unique solution for (14)\nwith initial data ξν(0). Moreover, there exist two positive constants C(p,T)and\nC(p), independent of ν, such that\n(15) Esup\n0≤t≤T/ba∇dblξν(t)/ba∇dblp\nLp≤C(p,T)\nfor any finite T, and\n(16) sup\n0≤t<∞E/ba∇dblξν(t)/ba∇dblp\nLp≤C(p).\nIn particular, the constants depend also on γ,Q,E/ba∇dblξν(0)/ba∇dblp\nLp.\nProof.The proof of the existence of solutions, which is quite classical requ ires\nsome Galerkin approximation of ξν, sayξν,n, for which a priori estimates are\nproved uniformly in n. Using a subsequence of ξν,nwhich converges in the\nweak or weak-star topologies of appropriate spaces, one can the n prove that\nthere exists a solution to (14). The proof of uniqueness and Feller p roperty is\nstandard and hence omitted.\nLetν >0,x∈Dandt∈[0,T]; Itˆ o formula for |ξν(t,x)|pgives\nd|ξν(t,x)|p=p|ξν(t,x)|p−2ξν(t,x)dξν(t,x)+1\n2p(p−1)|ξν(t,x)|p−22Qdt\nhence\nd|ξν(t,x)|p+p|ξν(t,x)|p−2ξν(t,x)[−ν∆ξν(t,x)+uν·∇ξν(t,x)+γξν(t,x)]dt\n−p(p−1)|ξν(t,x)|p−2Qdt=p|ξν(t,x)|p−2ξν(t,x)dwcurl(t,x)\nIntegrating on the spatial domain D, by using (13) and by integrating by parts\nwe get\n(17)d/ba∇dblξν(t)/ba∇dblp\nLp+pν(p−1)/ba∇dbl |ξν(t)|p−2\n2∇ξν(t)/ba∇dbl2\nH0dt+pγ/ba∇dblξν(t)/ba∇dblp\nLpdt\n−Qp(p−1)/ba∇dblξν(t,x)/ba∇dblp−2\nLp−2dt=p/a\\}b∇acketle{t|ξν(t)|p−2ξν(t),dwcurl(t)/a\\}b∇acket∇i}ht.\nIntegrating over the finite time interval (0 ,s) we get that\n(18)/ba∇dblξν(s)/ba∇dblp\nLp+νp(p−1)/integraldisplays\n0/ba∇dbl |ξν(r)|p−2\n2∇ξν(r)/ba∇dbl2\nH0dr+γp/integraldisplays\n0/ba∇dblξν(r)/ba∇dblp\nLpdr\n=/ba∇dblξν(0)/ba∇dblp\nLp+p/integraldisplays\n0/a\\}b∇acketle{t|ξν(r)|p−2ξν(r),dwcurl(r)/a\\}b∇acket∇i}ht\n+Qp(p−1)/integraldisplays\n0/ba∇dblξν(r)/ba∇dblp−2\nLp−2dr.\nTherefore\n(19) sup\n0≤s≤T/ba∇dblξν(s)/ba∇dblp\nLp≤ /ba∇dblξν(0)/ba∇dblp\nLp+psup\n0≤s≤T/integraldisplays\n0/a\\}b∇acketle{t|ξν(r)|p−2ξν(r),dwcurl(r)/a\\}b∇acket∇i}ht\n+Qp(p−1)/integraldisplayT\n0sup\n0≤r≤s/ba∇dblξν(r)/ba∇dblp−2\nLp−2ds.\n6On the other side, using first Burkholder-Davis-Gundy inequality (6 ) and then\nH¨ older inequality, we have that\npE/parenleftBig\nsup\n0≤s≤T/integraldisplays\n0/a\\}b∇acketle{t|ξν(r)|p−2ξν(r),dwcurl(r)/a\\}b∇acket∇i}ht/parenrightBig\n≤pC(p)/radicalbig\nQE/radicalBigg/integraldisplayT\n0/ba∇dblξν(r)/ba∇dbl2p−2\nLpdr\n≤pC(p)/radicalbig\nQE\nsup\n0≤s≤T/ba∇dblξν(s)/ba∇dblp/2\nLp/radicalBigg/integraldisplayT\n0/ba∇dblξν(r)/ba∇dblp−2\nLpdr\n\n≤1\n2Esup\n0≤s≤T/ba∇dblξν(s)/ba∇dblp\nLp+Q\n2C(p)2p2E/integraldisplayT\n0/ba∇dblξν(r)/ba∇dblp−2\nLpdr\n≤1\n2Esup\n0≤s≤T/ba∇dblξν(s)/ba∇dblp\nLp+Q\n2C(p)2p2E/integraldisplayT\n0sup\n0≤r≤s/ba∇dblξν(r)/ba∇dblp−2\nLpds.\nTaking expectation in (19) and collecting all the estimates we get\n1\n2Esup\n0≤s≤T/ba∇dblξν(s)/ba∇dblp\nLp≤E/ba∇dblξν(0)/ba∇dblp\nLp+QC(p)/integraldisplayT\n0Esup\n0≤r≤s/ba∇dblξν(r)/ba∇dblp−2\nLpds\n≤E/ba∇dblξν(0)/ba∇dblp\nLp+ǫ/integraldisplayT\n0Esup\n0≤r≤s/ba∇dblξν(r)/ba∇dblp\nLpds+C(ǫ,p,Q)T(20)\nfor anyǫ >0, by Young inequality. Using Gronwall lemma we obtain (15).\nTaking expectation in (18) and using (15), we also get that\nν(p−1)E/integraldisplayT\n0/ba∇dbl|ξν(s)|p−2\n2∇ξν(s)/ba∇dbl2\nH0ds+γE/integraldisplayT\n0/ba∇dblξν(s)/ba∇dblp\nLpds≤C(p,T,Q,E /ba∇dblξν(0)/ba∇dblp\nLp).\nForp= 2 this gives in particular\nE/integraldisplayT\n0/ba∇dbl∇ξν(s)/ba∇dbl2\nH0ds≤C/parenleftbig\nT,Q,E/ba∇dblξν(0)/ba∇dbl2\nL2/parenrightbig\n.\nGoing back to estimate (18) and taking expectation, we have\nE/ba∇dblξν(s)/ba∇dblp\nLp+γp/integraldisplays\n0E/ba∇dblξν(r)/ba∇dblp\nLpdr\n≤E/ba∇dblξν(0)/ba∇dblp\nLp+Qp(p−1)/integraldisplays\n0E/ba∇dblξν(r)/ba∇dblp−2\nLp−2dr\n≤E/ba∇dblξν(0)/ba∇dblp\nLp+γp\n2/integraldisplays\n0E/ba∇dblξν(r)/ba∇dblp\nLpdr+C(γ,p,Q)s.(21)\nHence\nE/ba∇dblξν(s)/ba∇dblp\nLp≤E/ba∇dblξν(0)/ba∇dblp\nLp−γp\n2/integraldisplays\n0E/ba∇dblξν(r)/ba∇dblp\nLpdr+C(γ,p,Q)s;\nGronwall lemma gives\nE/ba∇dblξν(s)/ba∇dblp\nLp≤E/ba∇dblξν(0)/ba∇dblp\nLpe−γps/2+2C(γ,p,Q)\nγp/parenleftBig\n1−e−γps/2/parenrightBig\nfor anys∈[0,∞). This implies (16). ✷\n7Remark 3.2 The solution ξνis a process whose paths are a.s. in C([0,∞),H0)∩\nL2\nloc(0,∞;H1)at least; therefore it solves system (14)in the following sense:\nfor allt∈[0,∞)andφ∈Hawitha>1, we have\n/integraldisplay\nDξν(t,x)φ(x)dx+ν/integraldisplayt\n0/integraldisplay\nD∇ξν(s,x)·∇φ(x)dx ds\n+/integraldisplayt\n0/integraldisplay\nDuν(s,x)·∇ξν(s,x)φ(x)dx ds+γ/integraldisplayt\n0/integraldisplay\nDξν(s,x)φ(x)dx ds\n=/integraldisplay\nDξν(0,x)φ(x)dx+/integraldisplay\nDwcurl(t,x)φ(x)dx P −a.s.\nThe trilinear term is well defined thanks to (7)and(9).\nMoreover, let us denote by ξν(·;η)the solution with initial data ηand by\nBb(Lp),Cb(Lp)the spaces of Borel bounded functions, respectively contin uous\nand bounded functions, φ:Lp→R. To say that the solution is a Feller process\ninLp(thepdepends on the assumption on the initial vorticity) means th at the\nMarkov semigroup Pν\nt:Bb(Lp)→Bb(Lp), defined as\n(Pν\ntφ)(η) =E[φ(ξν(t;η))],\nactually maps Cb(Lp)into itself.\nWe finally recall what is an invariant measure µν:\n/integraldisplay\nPν\ntφ dµν=/integraldisplay\nφ dµν∀t≥0,φ∈Lp.\nThe Feller property is important to prove the existence of in variant measures by\nmeans of Krylov-Bogoliubov method (see, e.g., [13]).\nFor anyγ >0 one can prove existence and uniqueness of the invariant\nmeasure for system (14), following the lines of the proofs for the 2 D Navier-\nStokes equation (the case γ= 0). Indeed, Krylov-Bogoliubov method provides\na wayto provethe existence ofan invariantmeasure; this applies fo ra wide class\nof noises. On the other side, uniqueness is a more delicate question. We just\nrecall the best result of uniqueness of the invariant measure, pro ved by Hairer\nand Mattingly [23]. They assume that the noise acts on first few mode s, i.e.\n(22)\n\n∃Zfinite :qk/\\e}atio\\slash= 0∀k∈ Z, q k= 0∀k /∈ Z\nwhereZhas to be chosen in such a way that\n•it contains at least two elements with different norms\n•the integer linear combinations of elements of Zgenerates Z2\nActually the kind and the number of forced modes, i.e. the elements o fZ, is\nchosen independently of the viscosity.\nWe summarize the result.\nTheorem 3.3 Letγ >0and2≤p <∞. If(22)holds, then for any ν >0\nsystem(14)has a unique invariant measure µν. Moreover it is ergodic, i.e.\n(23) lim\nT→∞1\nT/integraldisplayT\n0ϕ(ξν(t))dt=/integraldisplay\nϕ dµνinL2(Ω)\n8for anyϕ∈ Cb(Lp)and initial vorticity in Lp. Finally\n(24)\nν(p−1)/integraldisplay\n/ba∇dbl |ξ|p\n2−1∇ξ/ba∇dbl2\nL2dµν(ξ)+γ/integraldisplay\n/ba∇dblξ/ba∇dblp\nLpdµν(ξ) = (p−1)Q/integraldisplay\n/ba∇dblξ/ba∇dblp−2\nLp−2dµν(ξ).\nThe latter equality comes from (17). Notice that this invariant meas ureµνis\nindependent of p, since the assumption on the noise is independent of p.\nRemark 3.4 i) All the previous results hold true when Dis a smooth bounded\ndomain in R2, under the slip boundary condition coupled with a null vorti city\non the boundary. In that case, the assumption on the noise has to be modified\nas/summationtext\nk∈Z2\n0|k|2qk/ba∇dblek/ba∇dbl2\nLp<∞.\nii) For other conditions granting the uniqueness of the inva riant measure see\ne.g. [8, 13, 14, 16, 18, 20, 24, 29, 31]. Anyway, our results ho ld when the noise\nis such that the evolution of system (14)is well defined for initial vorticity in\nLp. In this case, we have that (24)is meaningful.\nIn the following we shall fix p= 4; this allows to choose any kind of finite\ndimensional noise, whereas in the infinite dimensional case (qk/\\e}atio\\slash= 0for allk)\nthis is not a strong restriction.\nNow, we fix the family ofthe unique invariantmeasures, as givenin The orem\n3.3, and consider the limit of vanishing viscosity.\nCorollary 3.5 Letγ >0. Then the family of invariant measures {µν}ν>0is\ntight inH−sfor anys>0; in particular there exists a measure µ0inH−ssuch\nthat\nµν−→µ0weakly in H−s\nasν−→0.\nProof.From (24) with p= 2 we have\n/integraldisplay\n/ba∇dblξ/ba∇dbl2\nH0dµν(ξ)≤Q\nγ\nuniformly in ν∈(0,∞). Then, using that H0is compactly embedded in H−s\nwe get tightness by means of the Chebyshev inequality. ✷\n4 The vanishing viscosity limit\nWhenν= 0, we deal with the stochastic damped Euler equations\n(25)/braceleftBigg\ndξ0+[u0·∇ξ0+γξ0]dt=dwcurl\nξ0=∇⊥·u0\nwith periodic boundary conditions, as before. We always consider γ >0.\nWe are going to prove that this system has a stationary solution who se\nmarginalat fixed time is the measure µ0and that the followingbalanceequation\nholds:\nγ/integraldisplay\n/ba∇dblξ/ba∇dbl2\nH0dµ0(ξ) =Q;\n9moreover,consideringthe limit in the balance equation(24) with p= 2 we prove\nthat\nlim\nν→0ν/integraldisplay\n/ba∇dbl∇ξ/ba∇dbl2\nH0dµν(ξ) = 0.\nThis means that in the limit of vanishing viscosity, the damped stochas tic equa-\ntions (14) have no dissipation of enstrophy.\nHowever, instead of dealing with invariant measures, we deal with st ationary\nprocesses (see next Remark 4.3). Heuristically, we expect that th ere exists a\nstationary solution for the stochastic damped Euler system (25), due to a bal-\nance between the energy injected by the noise term and the dissipa tion of the\ndamping term. More rigorously, in [5] it has been shown that the damp ed Euler\nequation with a multiplicative noise has a stationary solution; there, t he crucial\nestimate (16) wasusedthat holdsfor γ >0(andν≥0). The proofiseven easier\nwith an additive noise; indeed, estimate (16) on the finite dimensional approx-\nimating Galerkin system gives the existence of an invariant measure b y means\nof Krylov-Bogoliubov technique and we recover the existence of a s tationary\nsolution for (25).\nHere, we want to investigate the properties for vanishing viscosity ; in par-\nticular the limit in the balance equation (24) with p= 2, that is\n(26) ν/integraldisplay\n/ba∇dbl∇ξ/ba∇dbl2\nH0dµν(ξ)+γ/integraldisplay\n/ba∇dblξ/ba∇dbl2\nH0dµν(ξ) =Q\nKeeping in mind Corollary3.5, we consider the stationary stochastic p rocess\nξνwhose law at any fixed time is the measure µνof Theorem 3.3, and take the\nlimit of vanishing viscosity. We have\nProposition 4.1 Lets >0. The sequence {ξν}ν>0of stationary processes\nsolving(14)has a subsequence converging, as ν→0, inL2\nloc(0,∞;H−s)∩\nC([0,∞);H−2−2s)(a.s.) to a process, which solves the damped Euler system\n(25). Moreover, for any p≥2the paths of the limit process belong (a.s.) to\nC([0,∞);Lp\nw)∩L∞\nloc(0,∞;Lp), and the limit process is a stationary process in\nLp. The marginal at any fixed time of this limit process is the mea sureµ0.\nProof.The proof is based on two steps: first we show that the sequence o f the\nlaws ofξν,ν >0, is tight; then we pass to the limit in a suitable way and get\nthat the limit process is a weak solution of system (25). Notice that w e find a\nweak solution to system (25) (in the probabilistic sense), whereas s ystem (14)\nhas a unique strong solution.\nActually, the tightness and the convergence of the stationary pr ocesses have\nalready been done in [5] for the damped Navier-Stokes equations wit h a mul-\ntiplicative noise; but there the analysis involved the velocity instead o f the\nvorticity. For the reader’s convenience we recall the basic steps o f the proof; the\ndetails can be found in [3, 5].\nWriting equation (14) in the integral form\nξν(t) =ξν(0)+ν/integraldisplayt\n0∆ξν(s)ds−/integraldisplayt\n0uν(s)·∇ξν(s)ds−γ/integraldisplayt\n0ξν(s)ds+wcurl(t),\nby usual estimations and bearing in mind estimate\nsup\n0≤t<∞E/ba∇dblξν(t)/ba∇dbl4\nL4≤C(4)\n10from Theorem 3.1 (so we estimate sup0≤t<∞E/ba∇dblξν(t)/ba∇dbl4\nH0), one gets that there\nexist constants CandC(p) such that\nE/ba∇dbl/integraldisplay·\n0∆ξν(s)ds/ba∇dbl2\nW1,2(0,T;H−2)≤C\nE/ba∇dbl/integraldisplay·\n0uν(s)·∇ξν(s)ds/ba∇dbl2\nW1,2(0,T;H−2−s)≤Cby (11) and (10)\nE/ba∇dbl/integraldisplay·\n0ξν(s)ds/ba∇dbl2\nW1,2(0,T;H0)≤C\nE/ba∇dblwcurl/ba∇dblp\nWα,p(0,T;H0)≤C(p) by (3)\nfor some (and all) s>0,α∈(0,1\n2) andp≥2. Therefore\nsup\nν∈(0,1)E/ba∇dblξν/ba∇dbl2\nWα,2(0,T;H−2−s)<∞.\nOn the other hand, we already know from Theorem 3.1 that\nsup\nν>0E/ba∇dblξν/ba∇dbl2\nL2(0,T;H0)<∞.\nUsing that the space L2(0,T;H0)∩Wα,2(0,T;H−2−s) is compactly embedded\ninL2(0,T;H−s) (see, e.g., [19]), it follows that the sequence of laws of processes\nξν(0<ν <1) is tight in L2(0,T;H−s). On the other hand, using that both the\nspacesW1,2(0,T;H−2−s) andWα,p(0,T;H−2−s) withαp >1 are compactly\nembedded in C([0,T];H−2−2s), we get tighness in C([0,T];H−2−2s).\nLet us endow L2\nloc(0,∞;H−s) by the distance\nd2(ξ,ζ) =∞/summationdisplay\nn=12−nmin(/ba∇dblξ−ζ/ba∇dblL2(0,n;H−s),1)\nandC([0,∞);H−2−2s) by the distance\nd∞(ξ,ζ) =∞/summationdisplay\nn=12−nmin(/ba∇dblξ−ζ/ba∇dblC([0,n];H−2−2s),1).\nWe have that the sequence {ξν}is tight inL2\nloc(0,∞;H−s)∩C([0,∞);H−2−2s).\nFrom Prokhorov and Skorohod theorems follows that there exists a basis\n(˜Ω,˜F,˜P) and on this basis, L2\nloc(0,∞;H−s)∩C([0,∞);H−2−2s)-valued random\nvariables ˜ξ0,˜ξν, suchthat L(˜ξν) =L(ξν)onL2\nloc(0,∞;H−s)∩C([0,∞);H−2−2s),\nand\n(27) lim\nn→∞˜ξνn=˜ξ0inL2\nloc(0,∞;H−s)∩C([0,∞);H−2−2s),˜P−a.s.\nfor a subsequence with lim n→∞νn= 0.\nThe fact that the process ˜ξ0solves system (25) is classical. Indeed, consid-\nerings=1\n2we have that ˜ξν→˜ξ0inL2\nloc(0,∞;H−1/2); this means, according\nto (7), that ˜ uν→˜u0inL2\nloc(0,∞;H1/2). SinceH1/2(D)⊂L4(D), we get by\nestimates similar to (8) that the quadratic term [˜ uν·∇]˜uνconverges weakly to\n[˜u0·∇]˜u0, i.e.\n/integraldisplay\nD/integraldisplayt\n0[˜uν·∇]˜uν·ψ ds dx−→/integraldisplay\nD/integraldisplayt\n0[˜u0·∇]˜u0·ψ ds dx ˜P−a.s.\n11for alltfinite andψ∈[H1]2. For this it is enough to write\n/integraldisplay\nD{[˜uν·∇]˜uν·ψ−[˜u0·∇]˜u0·ψ}dx\n= +/integraldisplay\nD[(˜uν−˜u0)·∇]˜uν·ψ dx+/integraldisplay\nD[˜u0·∇](˜uν−˜u0)·ψ dx.\nIn addition, ˜ξνandξνhavethe same law; then ˜ξνis a stationaryprocess. By\nthe convergence ˜P-a.s. inC([0,∞);H−2−2s) we get that also ˜ξ0is a stationary\nprocess inH−2−2s.\nFinally, from (15) we have that for 2 ≤p<∞\n˜ξ0∈L∞\nloc(0,∞;Lp)˜P−a.s.\nThen, forT <∞almost each path ˜ξ0∈C([0,T];H−2−2s)∩L∞(0,T;Lp); thus\nit is weakly continuous in Lp, i.e. we have for any φ∈Lp′(1\np+1\np′= 1)\nlim\nt→t0/integraldisplay\nD˜ξ0(t)φ dx=/integraldisplay\nD˜ξ0(t0)φ dx ˜P−a.s.\nand for any t∈[0,T]\n/ba∇dbl˜ξ0(t)/ba∇dblLp≤ /ba∇dbl˜ξ0/ba∇dblL∞(0,T;Lp)˜P−a.s.\n(see [33] p 263).\nHence, for every t≥0, the mapping ˜ ω/ma√sto→˜ξ0(t,˜ω) is well defined from ˜Ω toLp\nand it is weakly measurable. Since Lpis a separable Banach space, it is strongly\nmeasurable (see [34] p 131). Therefore, it is meaningful to speak a bout the law\nof˜ξ0(t) inLp. The stationarity of ˜ξ0inLphas to be understood in this sense.\nBy taking suitable subsequences we have that µ0is the law of ˜ξ0(t) for any\ntimet. ✷\nLet us denote by ˜ξ0the stationary process solving (25), as given in Propo-\nsition 4.1. We have\nProposition 4.2 For any time t\n(28) γ˜E/ba∇dbl˜ξ0(t)/ba∇dbl2\nH0=Q.\nProof.Choosingp= 4 in (16) of Theorem 3.1 we have\n˜E/ba∇dbl˜ξν(t)/ba∇dbl4\nL4≤C(4).\nThis bound implies\n˜ξν(t)−→˜ξ0(t) weakly in L4(˜Ω×D);\nfor the limit we have\n(29) ˜E/ba∇dbl˜ξ0(t)/ba∇dbl4\nL4≤liminf\nν→0˜E/ba∇dbl˜ξν(t)/ba∇dbl4\nL4≤C(4).\nBy working on the first equation of (25), Itˆ o formula for d/ba∇dbl˜ξ0(t)/ba∇dbl2\nH0provides\n(30)/ba∇dbl˜ξ0(t)/ba∇dbl2\nH0+γ/integraldisplayt\n0/ba∇dbl˜ξ0(s)/ba∇dbl2\nH0ds=/ba∇dbl˜ξ0(0)/ba∇dbl2\nH0+tQ+/integraldisplayt\n0/a\\}b∇acketle{t˜ξ0(s),d˜wcurl(s)/a\\}b∇acket∇i}ht,\n12˜P-a.s. For this we have used (13), having that, for any s,˜ξ0(s)∈L4(D) a.s.\nfrom (29).\nTaking expectation and using stationarity we get (28). ✷\nEquation (28) can be rewritten as\nγ/integraldisplay\n/ba∇dblξ/ba∇dbl2\nH0dµ0(ξ) =Q.\nRemark 4.3 At this point, we are not able to prove that µ0is an invariant\nmeasure for the system (25). In fact, the transition semigroup associated to\n(25)can not be defined in H0: existence of a solution holds for initial vorticity\ninH0but uniqueness requires stronger assumptions (see [4] and [ 6]). But to\nget the Feller and Markov properties in a space smaller than H0is not trivial.\nSome work in progress in that direction is being made by the cu rrent authors.\nNow we have our main result\nTheorem 4.4 For anyγ >0, we have\n(31) lim\nν→0ν/integraldisplay\n/ba∇dbl∇ξ/ba∇dbl2\nH0dµν(ξ) = 0.\nProof.Let us write the balance equation (26) in terms ofthe stationarypr ocess\nξν, at any fixed time t:\n(32) νE/ba∇dbl∇ξν(t)/ba∇dbl2\nH0+γE/ba∇dblξν(t)/ba∇dbl2\nH0=Q.\nConsidering the weak limit as in Proposition 4.1 and 4.2 we have\nlimsup\nν→0ν˜E/ba∇dbl∇˜ξν(t)/ba∇dbl2\nH0=Q−γliminf\nν→0˜E/ba∇dbl˜ξν(t)/ba∇dbl2\nH0\n≤Q−γ˜E/ba∇dbl˜ξ0(t)/ba∇dbl2\nH0by (29)\n= 0 by (28) .(33)\nThis gives (31). ✷\nFrom this result we obtain the convergence of the mean enstrophy .\nCorollary 4.5 For anyγ >0, we have\n(34) lim\nν→0/integraldisplay\n/ba∇dblξ/ba∇dbl2\nH0dµν(ξ) =/integraldisplay\n/ba∇dblξ/ba∇dbl2\nH0dµ0(ξ).\nProof.We consider the limit as ν→0 in (26); then use (31) and (28). ✷\nRemark 4.6 All the results proved for the enstrophy ξcan be repeated and\nhence hold for the velocity u; norms of one order less of regularity are involved\nand therefore the proofs are even easier. This means in parti cular that for the\nstochastic damped 2D Navier-Stokes equations, there is no a nomalous dissipa-\ntion of energy as ν→0and energy balance equation holds for ν >0and also\nν= 0.\nAcknowledgment: The work of H. Bessaih was supported in part by the\nGNAMPA-INDAM project ”Professori Visitatori”. We would like to th ank the\nhospitality of the Department of Mathematics of the University of P avia where\npart of this research started and the IMA in Minneapolis where the p aper has\nbeen finalized.\n13References\n[1] V. Barcilon, P. Constantin, E. Titi, Existence of solutions to the Stommel-\nCharney model of the Gulf Stream , SIAM J. Math. Anal. 19(1988), 1355–\n1364.\n[2] D. Bernard, Influence of friction on the direct cascade of 2D forced turbu -\nlence, Europhys. Lett. 50(2000), 333–339.\n[3] H. Bessaih, Martingale solutions for stochastic Euler equations , Stoc. Anal.\nAppl17(1999), no. 5, 713–727.\n[4] H. 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North-\nHolland Publishing Co., Amsterdam-New York, 1979.\n[34] K.Yosida, Functional Analysis , Reprint ofthe sixth (1980)edition. Classics\nin Mathematics. Springer-Verlag, Berlin, 1995.\n16"    },    {        "title": "2203.06902v1.Investigation_of_nonlinear_squeeze_film_damping_involving_rarefied_gas_effect_in_micro_electro_mechanical_systems.pdf",        "content": "arXiv:2203.06902v1  [physics.flu-dyn]  14 Mar 2022Investigation of nonlinear squeeze-film damping\ninvolving rarefied gas effect in\nmicro-electro-mechanical-systems\nYong Wanga, Sha Liua,b, Congshan Zhuoa,b, Chengwen Zhonga,b∗\naSchool of Aeronautics, Northwestern Polytechnical Univer sity, Xi’an, Shaanxi 710072,\nChina\nbNational Key Laboratory of Science and Technology on Aerody namic Design and\nResearch, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China\nAbstract\nIn this paper, the nonlinear squeeze-film damping (SFD) involving rar efied\ngas effect in the micro-electro-mechanical-systems (MEMS) is inves tigated.\nConsidering the motion of structures (beam, cantilever, and memb rane) in\nMEMS, the dynamic response of structure will be influenced largely b y the\nsqueeze-film damping. In the traditional model, a viscous damping as sump-\ntion that damping force is linear with moving velocity is used. As the non lin-\near damping phenomenon is observed for a micro-structure oscillat ing with\na high-velocity, this assumption is invalid and will generates error res ult for\npredicting the response of micro-structure. In addition, due to t he small size\nof device and the low pressure of encapsulation, the gas in MEMS usu ally is\nrarefied gas. Therefore, to correctly predict the damping force , the rarefied\ngas effect must be considered. To study the nonlinear SFD phenome non in-\nvolving the rarefied gas effect, a kinetic method, namely discrete un ified gas\nkinetic scheme (DUGKS), is adopted. And based on DUGKS, two solvin g\nmethods, a traditional decoupled method (Eulerian scheme) and a c oupled\nframework (arbitrary Lagrangian-Eulerian scheme), are adoped . With these\ntwo methods, two basic motion forms, linear (perpendicular) and tilt ing mo-\ntions of a rigid micro-beam, are studied with forced and free oscillatio ns.\n∗Corresponding author\nEmail addresses: wongyung@mail.nwpu.edu.cn (Yong Wanga),\nshaliu@nwpu.edu.cn (Sha Liua,b),zhuocs@nwpu.edu.cn (Congshan Zhuoa,b),\nzhongcw@nwpu.edu.cn (Chengwen Zhonga,b)\nPreprint submitted to Computers & Mathematics with Applica tions March 15, 2022For a forced oscillation, the nonlinear phenomenon of squeeze-film d amping\nis investigated. And for a free oscillation, in the resonance regime, s ome\nnumerical results at different maximum oscillating velocities are prese nted\nand discussed. Besides, the influence of oscillation frequency on th e damp-\ning force or torque is also studied and the cause of the nonlinear dam ping\nphenomenon is investigated.\nKeywords: Squeeze-Film Damping, Discrete Unified Gas Kinetic Scheme,\nRarefied Gas Flow, Micro-Electro-Mechanical-Systems\n1. Introduction\nIn recent years, with the rapid development of fabricating techno logies,\na variety of micro-electro-mechanical-systems (MEMS) devices ha ve more\napplications in everyday life [1]. Among these devices, a moving micro-\nstructure is usually involved; for example, RF switches, high- gacceleration\nsensor, andatomicforcemicroscopy[2,3]. ShowninFig.1, foraRFs witch, a\nmicro-cantilever will oscillate perpendicularly above the substrate; the height\nof gap between the substrate and the micro-cantilever usually is ab out a few\nmicrometers. With the motion of micro-structure, the gas will be pu lled in\nor pushed out from the gap. Usually, as the variation of pressure in the gap\nis more dramatic than that at other areas of the device, the dampin g force or\ntorque acting on the micro-structure will be generated. Further more, due to\nthe small size of structure and the low pressure of encapsulation, these forces\nor torques caused by the gas will increase significantly with a large su rface-\nto-volume ratio of structure. Generally speaking, this type of dam ping is\ncalled the squeeze-film damping (SFD) and is the most important sour ce of\ndamping [4].\nTo study the SFD problem, the traditional numerical methods usua lly\nare based on the Reynolds equation [5], which is simplified from the Navie r-\nStokes equations. Following the definition of the Knudsen ( Kn) number [6],\nin theory, these methods are only available for flows in continuum and near-\ncontinuum regimes. Also due to the small size of micro-structure an d the low\npressure of encapsulation, the gas in MEMS devices usually is the rar efied\ngas. As the response of micro-structure will be influenced largely b y rarefied\ngas [7], the influence of the rarefaction effect must be considered [8 ]. To\nconsider the rarefied gas effect, one way is to improve the Reynolds equation\nwith some techniques. For example, Veijola et al. [9] introduced an effi cient\n2viscosity based on the Knnumber to replace the real physical viscosity. Gal-\nlis et al. [10] modified the coefficients used in the wall boundary conditio n\nof the Reynolds equation based on the Navier–Stokes slip–jump (NS SJ) sim-\nulations for flow at a small Knnumber and the direct simulation Monte\nCarlo (DSMC) molecular gas dynamics simulations for flow at a large Kn\nnumber. Although these methods have lots of applications [11, 12] a nd have\nsatisfactory performance in some benchmark test cases, it can n ot make a\nconclusion that the methods based on the Reynolds equation can ac curately\ndescribe the real physical flow at a high Knnumber [7]. In addition, as sev-\neral assumptions are introduced to derive the Reynolds equation, and very\nsimple micro-structures are considered in these methods, furthe r studies are\nneeded to validate whether these methods can simulate the rarefie d gas flows\naround the complex micro-structures or not. Another way to stu dy the SFD\nproblem is the molecular dynamics (MD) simulations [13], but these meth ods\nare usually used in the free molecular regime. As a consequence, the compu-\ntational cost for this type of method is higher when the Knnumber of the\nflow is moderate. Besides, the DSMC method also has been used to inv esti-\ngate the moving-boundary problem in MEMS devices [14], but it still fa ces\ngreat challenges due to the ultra-low Mach number of the flow. The m ajor\ndisadvantages of the DSMC are slow convergence, large statistica l noise, long\ntime to reach steady state, and extensive number of molecules [8]. U sually,\nthe typical speed of a micro-flow is about 1 mm/sto 1m/s, and the averaging\nmolecular velocities are about 500 m/s, the five to two orders of magnitude\ndifference between those two velocities results in large statistical n oises, and\nrequires 106-108samples to decrease the statistical noise [15]. For exam-\nple, Diab et al. [16] studied the SFD problem with the velocity of a moving\nmicro-beam between 20 m/sand 800m/s, which is a much higher velocity\nfor micro-flow. Although some improved DSMC algorithms [17] have be en\ndeveloped, the restrictions on the time step and mesh size, and the statis-\ntical scatters of density and temperature are not relieved by mos t of those\nmethods [18, 19]. In addition, for unsteady DSMC, the ensemble ave rage at\neach time step replaces the time average used in a steady flow [8], whic h put\nforward a higher requirement for parallel computation of complex fl ows. For\nexample, when studying the oscillating Couette flow, over 5,000 realiz ations\nare implemented to ensemble average this stochastic process at ev ery time\nstep [15]. Consequently, the DSMC method will not has a satisfactor y per-\nformance for solving these problems, as the flow around a moving bo undary\nis an inherently low-speed unsteady flow.\n3Recently, the discrete unified gas kinetic scheme (DUGKS) propose d by\nGuo et al. [20] has become a promising new method to simulate the flows in\nMEMS devices. Due to its kinetic nature, the DUGKS has many advant ages\nover other numerical methods, and has ability to solve the flow prob lems in\nallflowregimes. Forexample, inthecontinuum flowregime, asitcanad opta\nlarger Courant–Friedrichs–Lewy (CFL) number, and is less sensitiv e to mesh\nresolutions, so the DUGKS has a good performance than the tradit ional\nfinite volume lattice Boltzmann method (FVLBM) [21, 22, 23]. And in the\nrarefied flow regime, the computational cost is declined compared t o the\nunified gas kinetic scheme [24] (UGKS). Furthermore, as the DUGKS is an\nunsteady flow simulation method in nature, the ensemble averages a re not\nrequired compared with the DSMC method. Currently, some improve d and\nenhanced schemes based on the DUGKS also have been developed [25 , 26,\n27]. And, Wang et al. [28] proposed an arbitrary Lagrangian-Euleria n-type\nDUGKSforsolvingthemovingboundaryproblemsincontinuumandrar efied\ngas flows. So, the original DUGKS and ALE-DUGKS are the promising\nnumerical schemes for studying the micro-flows and the SFD pheno mena in\nMEMS. In addition, several recently proposed schemes [29, 30, 3 1, 32, 33, 34,\n35, 36] also have advantages or can be further applied to study th e rarefied\ngas flow and SFD problem in MEMS, which can be followed in the future.\nFor the SFD phenomenon in MEMS, in essence, it is a fluid-structure\ninteraction (FSI) problem due to the motion of micro-structure. T o inves-\ntigate this problem, a decoupled method is usually used. Firstly, with a n\nEulerian-framework scheme (based on a stationary mesh), by impo sing dif-\nferent velocities on a stationary micro-structure, the micro-flow s are gener-\nated, then thesqueeze-film damping forcescanbecalculated. And withthese\nresults, a damping force or torque coefficient can be obtained [7]. Ne xt, the\nstructure dynamics equation is solved and the gas damping force or torque\nis considered as an equivalent internal structure damping [12]. In t he above\nprocedure, an assumption that the squeeze-film damping force or torque is\nlinear with moving velocity or angular velocity is used. Usually, this damp -\ning is referred to as a viscous damping [37]. As illustrated in Ref. [38],\nwhen a micro-structure is moved at a high velocity, the downward an d up-\nward motions will generate different values of damping force. There fore, the\nnonlinear phenomenon between the moving velocity and damping forc e is\nobserved. Due to the assumption of viscous damping is not correct in some\nconditions, more accurate result will be obtained if the gas damping f orce\nor torque is treated as an external one. Consequently, introduc ing a new\n4coupled framework that can simulate the moving micro-structure in fluenced\nby the squeeze-film damping for all flow regimes will has a great value in\nengineering applications. According to the above reasons, the main objec-\ntive of this paper is to study the nonlinear SFD phenomenon with a cou pled\nframework based on the ALE-DUGKS scheme. In the past several decades,\na variety of numerical methods in the field of FSI have been propose d [39].\nAnd in these methods, a loosely coupled method is usually adopted, th at is\nthe fluid and structure dynamic solvers are used alternately, and t he data of\nforce or torque are exchanged between those solvers in each iter ation step.\nAs this type of method is easy to implement, it will be adopted in this pap er.\nTo the best of the authors’ knowledge, in corresponding studies, this is the\nfirst attempt to introduce a coupled FSI framework into the DUGKS or dis-\ncrete velocity method (DVM) for solving the SFD phenomenon. More over,\nas the linear (perpendicular) and tilting motions of a rigid micro-beam a re\nthe basic two-dimensional motion forms [40] in MEMS, the FSI problem for\nthese motions will be comprehensively studied in this work.\nThe rest of the paper is organized as follows. In Sec. 2, the ALE-DU GKS\nsolution procedure, and the traditional decoupled method and the proposed\nnew coupled framework for solving SFD problem are introduced. In S ec. 3,\ntwotestcases, themicro-Couetteflowinrarefiedgasandthefre eoscillationof\na square cylinder in continuum flow, are conducted to validate the me thods.\nIn Sec. 4, the SFD coefficient is calculated based on the traditional d ecoupled\nmethod, andthenonlinear squeeze-film damping forceandtorquea restudied\nbased on the new coupled framework. Both the forced and free os cillations\nwith the linear and tilting motions are considered. Finally, a brief conclu sion\nis presented in Sec. 5.\n2. Numerical method\nIn this section, to study the SFD phenomenon which involves the rar efied\ngas effect, the ALE-DUGKS is firstly introduced briefly, then the tr aditional\ndecoupledmethodandthecoupledframeworkforcalculatingthesq ueeze-film\ndamping are presented.\n2.1. Boltzmann-BGK equation\nThe original DUGKS proposed by Guo et al. [20] and the ALE-DUGKS\nproposed by Wang et al. [28] are the numerical schemes based on th e Boltz-\nmann model equation. In this work, the Boltzmann-BGK equation is u sed,\n5(a)\nGap\n(b)\nFigure 1: (a) Microphotograph of a cantilever-type RF-MEMS series ohmic sw itch, and\n(b) 3D cross-sectionof a cantilever-typeRF-MEMS switch (these two figures arepresented\nby Iannacci et al. in Ref. [41]).\nwhich can be expressed as\n∂f\n∂t+ξ·∇f= Ω =−1\nτ[f−feq], (1)\nwheref=f(x,ξ,t) is the velocity distribution function for particles moving\nwith velocity ξat position xand time t,τis the relaxation time depending\non the fluid dynamic viscosity µand pressure pwithτ=µ/p. And,feqis\nthe Maxwell equilibrium function; for the two-dimensional flow, it is giv en\nby\nfeq=ρ\n2πRTexp(−|ξ−u|2\n2RT), (2)\nwhereRis the gas constant, ρis the fluid density, uis the fluid velocity,\nandTis the fluid temperature. Finally, the macro-physical quantities can\nbe calculated as\nρ=/integraldisplay\nfdξ,ρu=/integraldisplay\nξfdξ,p=ρRT, (3)\nwhere the ideal gas law is used for the calculation of pressure.\n2.2. Arbitrary Lagrangian-Eulerian-type discrete unified gas kinetic scheme\nFor the simulation of moving boundary problem, the original Boltzman n-\nBGK equation (Eq. (1)) is extended to the ALE framework; then Eq . (1) can\nbe rewritten as [28, 42]\n∂f\n∂t+(ξ−v)·∇f= Ω =−1\nτ[f−feq]. (4)\n6wherevis the mesh motion velocity. As explained in Ref. [42], the update\nrule of the equilibrium distribution function feqdoes not depend onthe mesh\nmotionvelocity v, so onlythe calculation of the convection term is influenced\nin the ALE-DUGKS.\nFor the discretization of Eq. (4) in the particle velocity–space, a fin ite\nset of discretized particle-velocities is used; and ξirepresents the i-th dis-\ncretized velocity. As shown in Eq. (3), to integrate the macro-qua ntities, the\nmicro-velocities of particle can be set to coincide with the abscissas o f the\nquadrature rule. In this study, for a low-speed continuum flow, th e D2Q9\ndiscretized velocity model, weights and corresponding equilibrium fun ction\ndeveloped in the LBM [43] are used. And for the rarefied gas flows in M EMS,\nthe Gauss-Hermit and Newton-Cotes quadrature rules are used.\nFor the discretization of Eq. (4) in the macro physical-space, an un struc-\ntured mesh finite volume scheme is used. Fig. 2(a) shows the sketch of an\nunstructured mesh, where jis the center of triangular cell ABCand sub-\nscript represents the index number of a cell. If the mid-point rule is u sed for\nthe integration of the convection term, and the trapezoidal rule is used for\nthe calculation of the collision term, Eq. (4) can be discretized as\nfn+1\nj(ξ)/vextendsingle/vextendsingleVn+1,∗\nj/vextendsingle/vextendsingle−fn\nj(ξ)/vextendsingle/vextendsingleVn,∗\nj/vextendsingle/vextendsingle+∆tFn+1/2\nALE(ξ) =∆t\n2[Ωn+1\nj(ξ)/vextendsingle/vextendsingleVn+1,∗\nj/vextendsingle/vextendsingle+Ωn\nj(ξ)/vextendsingle/vextendsingleVn,∗\nj/vextendsingle/vextendsingle],\n(5)\nand the micro-flux of a cell surface Fn+1/2\nALE(ξ) is given as\nFn+1/2\nALE(ξ) =/integraldisplay\n∂Vj(ξ−v)·nf(x,ξ,tn+1/2)dS=/summationdisplay\nk(ξ−vn+1/2\nb,k)·n∗\nb,kfn+1/2(xb,k,ξ)S∗\nk,\n(6)\nwherenis the time level, ∆ t=tn+1−tnis the time step, xbis the center\nof cell interface, and kis the total number of cell interfaces. Under the ALE\nframework, as the geometrical information of a grid cell changes t emporally\nduring the simulation, the cell volumes Vn+1,∗andVn,∗atnandn+ 1\ntime levels, the moving velocity of cell interface vn+1/2\nbatn+ 1/2 time,\nthe outward unit normal vector n∗\nb, and the area of cell interface S∗\nbmust\nbe calculated by the discretized geometric conservation law (DGCL) [28, 44],\nwhere superscript∗means that the values of variables at corresponding times\nmaybe not equal to the real values of variables at those times. In t his study,\nthe DGCL scheme-2 presented in Ref. [28] is used, where\nvn+1/2\nb=xn+1\nb−xn\nb\n∆t, (7)\n7and\nVn,∗\nj=Vn\nj,S∗\nb=Sn\nb,Vn+1,∗\nj=Vn\nj+∆t/summationdisplay\nkvn+1/2\nb,kSn\nb,k. (8)\nTo remove the implicit collision term, two new distribution functions are\nintroduced:\n˜f=f−∆t\n2Ω =2τ+∆t\n2τf−∆t\n2τfeq, (9)\n˜f+=f+∆t\n2Ω =2τ−∆t\n2τ+∆t˜f+2∆t\n2τ+∆tfeq, (10)\nthen Eq. (5) can be rewritten as\n˜fn+1\nj=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleVn,∗\nj\nVn+1,∗\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜f+,n\nj−∆t/vextendsingle/vextendsingleVn+1,∗\nj/vextendsingle/vextendsingleFn+1/2\nALE(ξ). (11)\nAnd with the conservative property of the collision term:\n/integraldisplay\nΩdξ= 0,/integraldisplay\nξΩdξ= 0, (12)\nthe calculation of macro-quantities in Eq. (3) can be replaced by\nρ=/integraldisplay\n˜fdξ,ρu=/integraldisplay\nξ˜fdξ, (13)\nand˜fwill be solved instead of fin the practical computation.\nFor the calculation of the interface micro-flux shown in Fig. 2(b), if in te-\ngrate Eq. (1) along the characteristic line within a half time step s= ∆t/2,\nthe original distribution function f(xb,ξ,tn+s) in Eq. (6) can be updated\nas\nf(xb,ξ,tn+s)−f(xb−ξs,ξ,tn) =s\n2[Ω(xb,ξ,tn+s)+Ω(xb−ξs,ξ,tn)].(14)\nHere, two additional distribution functions are introduced:\n¯f=f−s\n2Ω =2τ+s\n2τf−s\n2τfeq, (15)\n¯f+=f+s\n2Ω =2τ−s\n2τ+s¯f+2s\n2τ+sfeq. (16)\n8A B\nCDE\nFj1j2\nj3j4\nG\n(a)A B\nCFj1\nj4xb\nnbξsxa\n(b)\nFigure 2: Sketches of (a) an unstructured mesh used in the ALE-D UGKS and (b) micro-\nflux calculation for a cell interface.\nThen, Eq. (14) can be rewritten as\n¯f(xb,ξ,tn+s) =¯f+(xb−ξs,ξ,tn), (17)\nand the original distribution function at ( xb,tn+1/2) is given by\nf(xb,ξ,tn+s) =2τ\n2τ+s¯f(xb,ξ,tn+s)+s\n2τ+sfeq(xb,ξ,tn+s),(18)\nwherethemacro-quantitiesatacellinterfacecanbecalculatedif ˜finEq.(13)\nis replaced by ¯f. Besides, the time step ∆ tused in this paper is given by\n∆t=α∆x\n|ξmax|, (19)\nwhere 0< α <1 is theCFLnumber, and ∆ xis the minimum size of grid\ncells. Finally, two relations are used in the practical computation:\n¯f+=2τ−s\n2τ+∆t˜f+3s\n2τ+∆tfeq,˜f+=4\n3¯f+−1\n3˜f, (20)\nand the Taylor expansion and the least-squares method are used t o recon-\nstruct the ¯f+at location xb−ξs. For the details of the implementation of\nthe ALE-DUGKS, it can be found in Ref. [28].\n2.3. Decoupled method and coupled framework for calculatin g the squeeze-film\ndamping force and torque\nShowninFig.1(b),fortheoscillationofamicro-cantilever, atwo-dim ensional\nflow simulation can be adopted with the micro-flow far away from the a n-\nchor. Fig. 3 shows two states of a micro-beam: linear (perpendicula r) motion\n9kL\nWallOutletOutlet\nOutlet\nhD\n(a)KL\nm\nWallOutletOutlet\nOutlet\nD\nh\n(b)\nFigure 3: Configuration for an elastically mounted micro-beam oscillat ing in rarefied gas\n(not drawn to scale) with (a) linear (perpendicular) motion and (b) t ilting motion.\nand tilting motion; LandDare the width and thickness of a micro-beam,\nrespectively, and his the height of gap. For linear motion, the structure\ndynamic equation is given by\nm¨y(t)+c˙y(t)+ky=Fext, (21)\nwhereyis the perpendicular displacement of structure, mis the mass of\nstructure, cisthedampingcoefficient ofstructure, kisthestiffness coefficient\nof structure, and Fextis the external excitation force. And for tilting motion,\nthe corresponding equation is given by [45]\nI¨θ(t)+η˙θ(t)+Kθ=Text, (22)\nwhereθistherotationangleofstructure, Iisthepolarmomentofinertia, ηis\nthe torsional damping coefficient, Kis the torsional stiffness coefficient, and\nTextis the external excitation torque. To predict the response of micr o-beam\nwith the given external force or torque, both the structure intr insic damping\nand the gas squeeze-film damping must be determined. In this paper , only\nthe gas SFD is considered, and the structure intrinsic damping is igno red to\nencourage a larger amplitude oscillation.\n2.3.1. Decoupled method\nSimilar to the traditional method, the DUGKS also can be adopted to\ncalculate the SFD coefficient ( candηin Eqs. (21) and (22)). Fig. 4 shows\nthe sketches of decoupled method based on an Eulerian-framewor k scheme,\nwheretheheightofgap hisconstant, andtheprofileofvelocityonthesurface\nof a stationary beam is given by the physical condition. Then, Eq. (1 ) or\n10WallOutletOutlet\nOutlet\nh     = const(t)V  s\n(a)WallOutletOutlet\nOutlet\nh     = const(t)V  s\n(b)\nFigure 4: Sketches of decoupled method for solving squeeze-film damping at ( a) linear\n(perpendicular) motion and (b) tilting motion.\nEq. (4) with v= 0 can be used to describe the micro-flow. By calculating\nthe force and torque acting on the micro-beam, damping coefficient s,cfand\ncη, are given as\ncf=F\nVsL,cη=T\nL, (23)\nrespectively, where Fis the rarefied gas damping force, and Tis the damping\ntorque. As discussed in Sec. 1, for the decoupled method, when th e damping\ncoefficient isdetermined, it will be treatedasthe structure intrinsic damping;\nthen, Eq. (21) or (22) will be solved to predict the response the mic ro-beam.\nAs the assumption of viscous damping [37] is used, the damping force is\nalways linear with the velocity, no matter what the value of motion velo city.\nConsequently, nonlinearSFDforce[38]cannotbepredictedbythe decoupled\nmethod.\n2.3.2. Coupled framework\nIn this paper, based ona loosely-coupled FSIalgorithm[46], a new fr ame-\nwork for solving SFD is used. As the damping force or torque are tre ated as\nan external one, Eqs. (21) and (22) are rewritten as\nm¨y(t)+ky=Fext+F, (24)\nand\nI¨θ(t)+Kθ=Text+T, (25)\nrespectively, where the structure intrinsic damping is ignored. For the dis-\ncretization of structure dynamic equation, the implicit Newmark sch eme [47]\n11is introduced. And a second-order extrapolation scheme is used to predict\nthe force or torque at n+1 time level:\nFn+1,∗= 2Fn−Fn−1,Tn+1,∗= 2Tn−Tn−1. (26)\nThe advantage of above coupled framework is that the nonlinear da mping\nforce or torque can be calculated dynamically. Finally, the detailed imp le-\nmentation procedure of this FSI framework is as follows:\n(1)predict the damping force For torque Tat new time level with Eq. (26);\n(2)update thestructure displacement yor rotationangle θwith theimplicit\nNewmark scheme [47];\n(3)deformthemeshwithanewstructurelocationbytheLaplacesmoot hing\nequation [48];\n(4)update the distribution function ˜ffromnton+1 time level according\nto Eq. (11);\n(5)calculate the force or torque acting on a micro-structure based o n the\ndistribution function (Eq. (2.18) in Ref. [49]).\nFrom our numerical tests, one inner iteration of the above proced ure in\none time step is enough to obtain a convergent displacement of the s tructure\n(|yn+1,∗−yn+1|<10−6), so the inner iterative cycle used in the traditional\nFSI framework [46] is not required. The reason is that the present ALE-\nDUGKS is an explicit numerical scheme, and the coupled time step is set\nto ∆tCFD= ∆tCSD, where ∆ tCFDis time step used for computational fluid\ndynamics (CFD) simulation and ∆ tCSDis that used for computational struc-\ntural dynamics (CSD) simulation, so the coupled numerical error is s mall\nfor the implicit Newmark scheme at a small time step. The present cou -\npled FSI framework has been coded with the help of Code Saturne [50], an\nopen-source computational fluid dynamics software of Electricite De France\n(EDF), France ( http://www.code-saturne.org/cms/ ). We appreciate the\ndevelopment team of Code Saturne for their great works.\n3.Validation of the numerical framework\nIn this section, to validate the decoupled method and coupled frame work\nbased on the DUGKS, two test cases, namely micro-Couette flow in r arefied\ngasandfreeoscillation of a square cylinder in continuum flow, arecon ducted.\n123.1. Micro-Couette flow in rarefied gas\nThe micro-Couette flow is driven by two parallel moving plates with a\ndistantH. It can be treated as a benchmark test case for the decoupled\nmethod, as the displacement of moving wall is set to zero. In the simu lation,\n400 quadrangular cells are used, with 101 grid nodes are placed in y-direction\nand5 inx-direction. For theboundary conditions, the topand bottomplate s\naresettowallboundarieswithmovingvelocities ±Uw, andleftandrightsides\nof the channel are set to periodic boundaries. The working gas is ar gon (the\nspecific gas constant R= 208J/kg/K), and four Knnumbers, 0 .01, 0.2/√π,\n2.0/√πand20/√π, arecarriedout. Theinitialtemperature T0inthechannel\nandTwat the walls are set to 273 K. For the moving velocities of walls Uw,\ntwo values, ±16.85m/sand±119.15m/s, are used. Besides, the reference\ntemperature and velocity are Tref= 273KandUref=/radicalbig\n2RTref= 337m/s,\nrespectively. The CFLnumber used in Eq. (19) is 0.8 for Kn= 0.01 and\n0.2/√π, and0.7forother Knnumbers. Finally, the28-pointsGauss-Hermite\nquadrature rule is used for flow at Kn= 0.01, and 80 ×80 points of Newton-\nCotes quadrature rules with a range of [ −4,4]×[−4,4] is used for flows\nat other Knnumbers (for flow at 119 .15m/s, the compressible DUGKS is\nused [28]). Fig. 5 shows the comparisons of velocity profiles with thos e of the\nDSMC [15, 17] and the UGKS [51]. Fig. 6 shows the contours of x-direction\nvelocityuand the convergence history eof velocity at two Knnumbers,\nrespectively, where eis given by\ne=/radicalBig/summationtext\ni/bracketleftbig\n(un+1000\ni−un\ni)2+(vn+1000\ni−vn\ni)2/bracketrightbig\n/radicalbig/summationtext\ni[(un\ni)2+(vn\ni)2], (27)\nandiis the index number of grid cells. As shown in the figures, the DUGKS\nobtainssatisfactory results inall flow regimes comparedwith other numerical\nmethods, and also shows good convergence property without sta tistics noise.\nConsequently, the DUGKS demonstrates great potentials in simulat ing the\nlow-speed micro-flows in MEMS.\n3.2. Free vibration of a square cylinder in continuum flow\nForlaminarflowaroundacylinder atReynoldsnumber Re >50(Re∼47\nfor circular cylinder [52]), the flow is unsteady and vortex shedding c an\nbe observed. Then if a cylinder is elastically mounted in a uniform flow,\nthe unsteady aerodynamic force will leads to the vortex-induced v ibrations\n13y / Hu / Uw\n00.10.20.30.40.500.20.40.60.81\nDSMC\nPresent\nK = 0.1\nK = 1\nK = 10\n(a)y / Hu / Uw\n00.10.20.30.40.500.20.40.60.81\nIP-DSMC\nUGKS\nPresent\nKn = 0.01\nK = 0.1\nK = 1\nK = 10\n(b)\nFigure 5: u-velocity profile of micro-Couette flow at (a) Uw= 16.85m/s(DSMC: [15]) and\n(b)Uw= 119.15m/s(IP-DSMC [17], UGKS [51]), where Knis set to Kn= 2K/√π.\n-13-11-9-7-5-3-1135791113\nxy\n-0.0500.05-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5K = 0.1\n-7-5-3-11357\nxy\n-0.0500.05-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5K = 1.0\n(a)iteration stepe\n5000 100001500010-1010-910-810-710-610-510-410-3\nK = 0.1\nK = 1.0\n0 2000 3000\n(b)\nFigure 6: (a)uvelocity contours and (b) convergence history of the micro-Coue tte flow\nat twoKnnumbers, where Knis set to Kn= 2K/√π, and wall moving velocity is\nUw= 16.85m/s.\n14fdU\u0001\ns\nFigure 7: Schematic diagram of flow around an elastically mounted squ are cylinder.\n(VIV) [53]. In this work, flow around an elastically mounted square cy linder\nshown in Fig. 7 is studied, and Reis set to 100. It can be treated as a bench-\nmark test case for the coupled framework in continuum flow region. Fig. 8\nshows the hybrid unstructured mesh used in this case. The total n umber of\ngrid cells is 44992, with 800 points at the surface of cylinder. The reg ion near\nthe surface of cylinder is discretized into quadrangular cells with the mini-\nmum size of grid cells being d/150, where dis the side length of the square\ncylinder. The computational domain is set to [80 d×140d], which is large\nenough to eliminate the influence of the far-field boundary condition . In this\ntest case, two-degree-of-freedom structure dynamics equat ions is used [54]:\n¨x+(2πFs)2x=Cd\n2m∗,¨y+(2πFs)2y=Cl\n2m∗, (28)\nwhereFs=fsd/U∞is the reduced natural frequency relating to the natural\nfrequency of a mass–spring system fs(U∞is velocity of free stream), m∗\nis the non-dimensional mass of square with m∗=ms/ρd2(msis the mass\nof square per unit length and ρis the density of free stream) and is set to\nm∗= 3 in this case, and CdandClare the drag and lift coefficients of a\nsquare, respectively. Fig. 9 shows the maximum vibration amplitude Ayof\nsquareintransverse direction, where Ayisgivenby Ay= (ymax−ymin)/2,and\nU∗is the reduced velocity defined as U∗= 1/Fs=U∞/fsd. In general, our\nresult agrees well with the numerical result of Li et al. [54], and demo nstrates\nthe capability of the present coupled framework to further simulat e the SFD\nproblem in MEMS. Furthermore, although the focus of this paper is r arefied\ngas flow, this continuum flow test also shows the performance of th e DUGKS\nfor simulating the unsteady flow; on the contrary, higher computa tional cost\nis required for the DSMC method to obtain a smooth result of the flow field.\n15xy\n-40-20 020406080100-40-2002040\n(a)xy\n0.5 1 1.5 20.60.91.21.5\n(b)\nFigure 8: Mesh for flow around an elastically mounted square cylinder : (a) full domain\nand (b) near the cylinder surface.\nU*Ay\n2 4 6 8 10 12 1400.10.20.3Li et al.\nPresent\nFigure 9: Transverse vibration amplitude of flow around an elastically mounted square\ncylinder at Re= 100 and m∗= 3.\n164. Results and discussion\nIn this section, two-dimensional forced or free oscillation of a micro -beam\nwith linear (perpendicular) or tilting motion in rarefied gas is fully stud-\nied. Fig. 3 shows the configuration of computational domain and bou ndary\nconditions. The substrate and the surface of micro-beam are set to diffuse-\nscattering wall boundaryconditions, and theleft, right andtopbo undaries of\nthe computational domain are set to outlet boundary conditions. F ollowing\nthe setup described in Ref. [7], the width Land thickness Dof micro-beam\nare set to 18 .0×10−6mand 2.25×10−6m, respectively, and the gap height h\nis 1.0×10−6m. Fig. 10 shows the mesh used in this study, the total number\nof grid cells is 37200, and the minimum size of grid cells near the wall is\n0.025h, with 40 grid cells are located in the gap. The working gas also is\nargon, and the reference temperature Trefand velocity Urefare 273Kand\n307.6m/s(Uref=/radicalbig\nγRTref, andγ= 5/3 is the specific heat ratio), respec-\ntively. The relation between mean-free-path λand viscosity µ[55] is given\nby\nλ=2µ(7−2ω)(5−2ω)\n15ρ√\n2πRT, (29)\nwhereω= 0.5 is the index related to the HS model. The Knudsen number\nis defined as Kn(h)=λ/hin the following section, where his the reference\nlength of flow. For the initial conditions of rarefied gas flow, the ref erence\ndensityρrefand viscosity µrefare set to the corresponding values at Kn(h)=\n3.61×10−4, Mach number Ma= 2.19×10−4and Reynolds number Re= 1.0.\nAnd by keeping a constant value of gas viscosity, the density of rar efied gas\nat other Knudsen numbers can be obtained. Besides, the Gauss-H ermite\nquadrature rule is used for all the considered Knudsen numbers.\n4.1.Decoupled method: squeeze-film damping at different Knudsen numbers\n4.1.1. Squeeze-film damping coefficient and nonlinear dampin g phenomenon\nFirstly, the calculation of the SFD coefficient [7] is considered, which is\nbased on the traditional decoupled method. Shown in Fig. 4(a), dur ing the\nsimulation, by imposing a constant moving velocity on the surface of m icro-\nbeam, the damping force Facting on the micro-beam from continuum to\nfree-molecule flow regimes is calculated. The moving velocity Vsis set to\n−0.0674m/sin this case (according to the setup described in Ref. [7], this\nvalue is about −0.075m/sbased on the sonic speed of air). Fig. 11 shows\na comparison of present result with Guo et al.’s compact model [7] (b ased\n17on the Boltzmann ellipsoidal statistical BGK equation). And this comp act\nmodel is given as\ncf(x1,x2) =F\nVsL=axc\n1\n1+bxe\n1xf\n2D, (30)\nwherex1=L/h,x2=Kn(h)/x1(or equals to the Knudsen number Kn(L)\nbased on the width of micro-beam), and the constant parameters are set to\na= 10.39,b= 1.374,c= 3.100,e= 1.825 and f= 0.9660, respectively.\nClearly, for Kn(h)>0.3, our result agrees well with Guo et al.’s compact\nmodel. The differences between these two results at Kn(h)>100 maybe is\nthe different collision models used in the schemes. Besides, as the Guo et\nal.’s model is constructed based on the rarefied gas flow simulations ( 0.05<\nKn(h)<50), itisdifficult toidentify which result isbetter atnear-continuum\nandcontinuumflowregimes. Consequently, sameasotherdecoupledmethod,\na similar compact model also can be constructed based on the Euleria n-\nframework DUGKS .\nNext, the nonlinear damping phenomenon discussed in Ref. [38] is also\nstudied. As illustrated above, the squeeze-film damping can be trea ted as an\nequivalent structure damping in the decoupled method. To verify th e correc-\ntionofthemodel, aset ofsimulations withvelocities Vsatdifferent directions\n(downward or upward motion) and magnitudes is conducted. In the follow-\ning sections, the damping force is positive with a downward moving velo city,\nand is negative with an upward moving one. Fig. 12 shows the pressur e\ncontours and streamlines near a micro-beam with the downward and upward\nmoving velocities at Kn(h)= 0.1 and|Vs|= 0.0674m/s. Clearly, compared\nwith other regions, the variation of pressure in the gap is obvious. A nd the\ngas in the gap is driven out by the micro-beam with a downward moving\nvelocity and vice versa. Fig. 13 shows the comparison results at two Kn(h)\nnumbers. As shown in Fig. 13(a), at a small value of |Vs|(about 0.0674m/s),\nthe difference of damping force between downward and upward mot ions is\nlittle. When the magnitude of Vsis increased, the differences of those are\nobvious, and these tendencies are more significant at a large Knnumber.\nFig. 13(b) can be used to explain the reason, that is due to the influe nce of\nthe substrate, the variation of pressure on the bottom surface is much more\nsignificant than that on the top surface. Furthermore, a conclus ion can be\nmade from Fig. 13(c), that is the traditional equivalent damping mod el is\nonly available at a low-speed motion ( |Vs|<0.1m/s), as the assumption of\nthe linear relation cannot be maintained at a high-speed motion.\n18xy\n-36-27-18 -9 0918273605101520\nFigure 10: Mesh for an elastically mounted micro-beam oscillating in rar efied gas.\nKn(h)cf\n10-310-210-1100101102103 10-510-410-310-210-1\nGuo et al.\nPresentValid range for Guo’s model\nFigure 11: Comparison of squeeze-film damping coefficient cfwith Guo et al.’s compact\nmodel [7].\nxy\n-20 -10 0 10 200510Pressure\n1.007\n1.00611\n1.00522\n1.00433\n1.00344\n1.00256\n1.00167\n1.00078\n0.999p / p0  \nxy\n-20 -10 0 10 200510Pressure\n1\n0.998889\n0.997778\n0.996667\n0.995556\n0.994444\n0.993333\n0.992222\n0.991111p / p0    \n(a)\nxy\n-20 -10 0 10 200510\nxy\n-20 -10 0 10 200510\n(b)\nFigure 12: (a) Pressure contours and (b) streamlines for flow aro und a micro-beam in\nrarefied gas with downward (left) and upward (right) moving velocit ies atKn(h)= 0.1\nand|Vs|= 0.0674m/s.\n19tVs / h| F | / L (N / m)\n0 0.1 0.2 0.3 0.400.020.040.060.08\nDownward motion\nUpward motion\n0.84%\n| Vs | = 0.0674 m/s3.4%\n| Vs | = 0.27 m/s| Vs | = 0.54 m/s6.94%14.4%\n| Vs | = 1.08 m/s\ntVs / h| F | / L (N / m)\n0 0.2 0.4 0.600.0050.010.015 Downward motion\nUpward motion\n2.195%\n| Vs | = 0.0674 m/s9%\n| Vs | = 0.27 m/s| Vs | = 0.54 m/s19.12%42.24%\n| Vs | = 1.08 m/s\n(a)\nx / Lp / p0\n-0.5 -0.25 0 0.25 0.511.051.11.15Downward motion\nUpward motion\nTop surfaceBottom surface\n| Vs | = 0.0674 m/s| Vs | = 0.54 m/s| Vs | = 1.08 m/s\nx / Lp / p0\n-0.5 -0.25 0 0.25 0.511.11.21.3Downward motion\nUpward motion\nTop surfaceBottom surface\n| Vs | = 0.0674 m/s| Vs | = 0.54 m/s| Vs | = 1.08 m/s\n(b)\nVs (m / s)| F | / L  (N / m)\n00.5 11.5 22.500.040.080.120.16\nDownward motion\nUpward motion\nLinear relation\nF = 0.0485 | Vs |L\nVs (m / s)| F | / L  (N / m)\n00.5 11.5 22.500.010.020.030.04\nDownward motion\nUpward motion\nLinear relation\nF = 0.0099925 | Vs |L\n(c)\nFigure 13: Comparisons of flow around a micro-beam in rarefied gas a tKn(h)= 0.1 (left)\nandKn(h)= 1.0(right): (a)convergenthistoryofdampingforceactingonamicr o-beamat\ndifferentvelocities Vs, (b)pressuredistributionalongthetopandbottomsurfacesofm icro-\nbeam at different velocities (to make a comparison, symmetrical res ults are presented for\nupward motion) and (c) the variation of damping force at different v elocities.\n204.1.2. Nonlinear damping phenomenon at different oscillati on frequencies\nIn this subsection, the influence of frequency on the damping forc e is\nfurther studied, which is not considered in Ref. [38]. By giving a maximu m\nmoving velocity Umax, a motion form of micro-beam is assumed:\ny=Asin(Umax\nAt), (31)\nwhereAis the oscillation amplitude of a moving micro-beam. Then with\nEq. (31), the instantaneous relative height of gap x1in Eq. (30) can be\nobtained. Shown in Fig. 14, with Guo et al.’s compact model [7], moving\nvelocities U, damping coefficients cfand damping forces Fat two oscillation\namplitudes, 0 .02h0and 0.16h0, are compared, where h0= 1.0×10−6mis the\ninitial gap height. Besides, a small value of Umax, 0.0674m/s, is used; similar\nresultswillbeobtainedathighmovingvelocitiesduetothelinearassum ption\nof model. Although the damping coefficient has obvious variation durin g\nthe oscillation at a large amplitude than that at a small one (Fig. 14(b) ),\nthe maximum and minimum values of damping forces are almost the same\n(Fig. 14(c)). So, the oscillation frequency does not influence the d amping\nforce in Guo et al.’s compact model. To further verify this model, a ser ies\nof flows at Umaxof 0.0674m/s, 0.27m/s, and 1.08m/sis simulated, and the\noscillating velocity is given by\nU=−Umaxsin(ft), (32)\nwherefis the oscillation frequency. Shown in Fig. 15, the maximum damp-\ning forces at the initial test frequency f0(f0/(2π) = 1.685MHz) are lower\nthan that obtained by the constant motion velocities (dashed lines s hown\nin figures). And by decreasing the frequency f, the amplitudes of damping\nforcegraduallyconverges tothedashedlines, andthenonlinearph enomenons\nalso can be observed as the absolute values of the maximum and minimu m\nvalues of damping force are not equal to each others at a high oscilla ting\nvelocity. So, Guo et al.’s compact model is only available at a low oscilla-\ntion frequency. Finally, Fig. 16 shows the largest damping forces at different\noscillation frequencies. For the oscillation frequency lower than a th reshold\nvalue, the largest damping forces will converge to a constant numb er, andthe\nnonlinear phenomenon can be observed at a higher oscillating velocity . By\nincreasing the oscillation frequency, the damping force decreases . Besides,\nwhen the oscillation frequency is larger than that threshold value, it seems\n21t (s)U (m / s)\n0 10 20 30 40-0.1-0.0500.050.1\nU ( A = 0.02 h0 )\nU ( A = 0.16 h0 )\n 10-6\n(a)t (s)cf (Ns / m2 )\n0 10 20 30 4068101214\ncf ( A = 0.02 h0 )\ncf ( A = 0.16 h0 ) 10-3\n.\n 10-6\n(b)\nt (s)F / L (N / m)\n0 10 20 30 40-1-0.500.51\nF ( A = 0.02 h0 )\nF ( A = 0.16 h0 ) 10-3\n 10-6\n(c)\nFigure 14: Comparisons of (a) motion velocities U, (b) damping coefficients cfand (c)\ndampingforces Fattwooscillationamplitudes byGuoetal.’scompactmodel [7] ( Kn(h)=\n1.0).\nthat a linear relation can be found between the different oscillating ve locities.\nSo, further work can be continued to construct a modified compac t model\nwhich is to consider the influence of oscillation frequency.\n4.2.Coupled framework: Squeeze-film damping force at the linear motion\nIn this section, with the coupled FSI framework, the rarefied gas d amping\nforce acting on a micro-beam by the forced or free oscillation under the linear\nmotion is studied. Some comparisons of result between decoupled me thod\nand coupled framework also are presented.\n22t (s)F / L (N / m)\n0 2 4 6 8 10-5-2.502.55f = f0\nf = f0 / 2\nf = f0 / 4\nf = f0 / 8\nf = f0 / 16 10-3\n 10-6t (s)F / L (N / m)\n0 2 4 6 8 10-1-0.500.51f = f0\nf = f0 / 2\nf = f0 / 4\nf = f0 / 8\nf = f0 / 16\nf = f0 / 32 10-3\n 10-6\n(a)\nt (s)F / L (N / m)\n0 2 4 6 8 10-2-1012f = f0\nf = f0 / 2\nf = f0 / 4\nf = f0 / 8\nf = f0 / 16\nf = f0 / 32 10-2\n 10-6t (s)F / L (N / m)\n0 2 4 6 8 10-0.5-0.2500.250.5 f = f0\nf = f0 / 2\nf = f0 / 4\nf = f0 / 8\nf = f0 / 16\nf = f0 / 32 10-2\n 10-6\n(b)\nt (s)F / L (N / m)\n0 2 4 6 8 10-10-50510 f = f0\nf = f0 / 2\nf = f0 / 4\nf = f0 / 8\nf = f0 / 16\nf = f0 / 32 10-2\n 10-6t (s)F / L (N / m)\n0 2 4 6 8 10-2-1012 f = f0\nf = f0 / 2\nf = f0 / 4\nf = f0 / 8\nf = f0 / 16\nf = f0 / 32 10-2\n 10-6\n(c)\nFigure 15: Time evolutions of damping forces acting on a micro-beam w ith (a)Umax=\n0.0674m/s, (b)Umax= 0.27m/sand (c)Umax= 1.08m/s. TwoKn(h)numbers, 0.1 (left)\nand 1.0 (right), are considered, f0is the initial test frequency with f0/(2π) = 1.685MHz,\nand the values of dashed lines shown in figures are obtained from Fig. 13(c) at correspond-\ning velocities.\n23f / f0| F | / L (N / m)\n1001011022468\nFmax\nFmin 10-2\nUmax = 1.08 m/s\nUmax = 0.27 m/s\nUmax = 0.0674 m/s1.2\n0.1\n0.02\n(a)f / f0| F | / L (N / m)\n10010110261218\nFmax\nFmin 10-3\nUmax = 1.08 m/s\nUmax = 0.27 m/s\nUmax = 0.0674 m/s2.4\n0.030.2\n(b)\nFigure 16: Comparisons of the largest damping force acting on a micr o-beam at two\nKn(h)numbers of (a) 0.1 and (b) 1.0. f0is the initial test frequency with f0/(2π) =\n0.052656MHz.\n4.2.1. Squeeze-film damping force at forced oscillation\nFor the forced oscillation of a micro-beam, the motion form given by\nEq. (31) is also used here, and the minimum reference amplitude A0is\n0.0025h0. With this equation, a smaller value of oscillation amplitude A\ngenerates a higher oscillation frequency; and for the same oscillatio n am-\nplitude, a larger maximum oscillating velocity Umaxalso generates a higher\noscillation frequency. Fig. 17 shows the time evolutions of damping fo rces\nat different maximum moving velocities and amplitudes. It is clear that f or\nall the considered velocities, a smaller damping force will be generate d by\na high frequency and vice versa. And at a small value of moving velocit y\n(Umax= 0.0674m/s), the maximum and minimum values of damping force\nare almost the same. By increasing the oscillation amplitude, the corr e-\nsponding values gradually converge to the results obtained by the d ecoupled\nmethoddescribedinSec.4.1. Furthermore,thereexistsathresh oldvaluethat\nthe largest damping force does not change when the oscillation freq uency is\nlower than that value. For a higher oscillating velocity, Umax= 0.27m/sor\n1.08m/s, the nonlinear damping phenomenon can be observed as the max-\nimum and minimum values of damping force are not equal to each other s.\nBesides, the convergent values of damping forces will be much highe r than\nthe results obtained by the decoupled method, especially at a more h igher\noscillating velocity, 1 .08m/s. So, it proves againthat thetraditional compact\nmodel [7] is only available at the low velocity and frequency of oscillation .\n24Then, the results obtained by decoupled method (Eq. (32)) and co u-\npled framework (Eq. (31)) are compared at the same oscillation fre quency.\nFig. 18 shows the comparisons of time evolutions of damping force. F or the\noscillation at a small value of amplitude (Figs. 18(a) and 18(b)), the la rgest\ndamping forces are almost the same, so the influences of Umaxon the damp-\ning force are not obvious. As a result, for the high-frequency osc illation\nof a micro-beam, the decoupled method still exhibits a good perform ance\nto predict the damping force. For the oscillation at a high oscillating ve -\nlocityUmax= 1.08m/sand a moderate oscillation amplitude A= 0.08h0\n(Fig. 18(c)), due to the influence of displacement of a micro-beam, the max-\nimum of damping force obtained by the coupled framework is a little high er\nthan that by the decoupled method (downward moving direction) an d this\ntendency is inverse for the minimum one (upward moving direction). S o, the\nadvantage of the coupled framework is more accurate to predict t he damp-\ning force at that computational condition. And for the oscillation at a high\nvelocity and low frequency (Figs. 18(d) and 18(e)), the difference s of results\nobtained by two methods are obvious. The sinusoidal shape of oscilla tion of\ndamping force can not be maintained, and more larger damping force will be\nobtained by the coupled framework. So for the oscillation at a large d isplace-\nment and low frequency, the decoupled method can not predict the damping\nforce correctly. Fig. 19 shows the comparisons of amplitude of damping force\nat different computational conditions. In consideration of the com putational\ncost, a cost-effective framework can be adopted between these two methods\nin the practical application.\n4.2.2. Squeeze-film damping force at free oscillation\nNext, the free oscillation problem of a micro-beam is considered. By\ngiving an external excitation force, Eq. (24) is rewritten as\nm¨y(t)+ky=Fext+F=F0cos(ωnt)+F, (33)\nwhereF0is the amplitude of external excitation force and ωnis the frequency\nofF0. Ifωnis set equal to the natural frequency of structure ω(ω=/radicalbig\nk/m),\nthe resonance phenomenon is excited. Due to the gas damping forc e, the\ndisplacement of the structure is no divergent. So, Eq. (33) is used in the\nsimulation. Furthermore, to make a comparison, a theoretical solu tion can\nbe obtained for a low-speed oscillation. Based on the theory of ordin ary\ndifferentialequation, thetheoreticalsolutionofastructuredyn amicequation:\nm¨y(t)+c˙y(t)+ky=F0cos(ωnt), (34)\n25t (s)F / L (N / m)\n0 2 4 6 8-5-2.502.55A = A0\nA = 2A0\nA = 4A0\nA = 8A0\nA = 16A0\nA = 32A0 10-3\n 10-6t (s)F / L (N / m)\n0 2 4 6 8-1-0.500.51A = A0\nA = 2A0\nA = 4A0\nA = 8A0\nA = 16A0\nA = 32A0\nA = 64A0 10-3\n 10-6\n(a)\nt (s)F / L (N / m)\n0 0.5 1 1.5 2-2-1012A =  A0\nA =  2A0\nA =  4A0\nA =  8A0\nA =  16A0\nA =  32A0\nA =  64A0 10-2\n 10-6t (s)F / L (N / m)\n0 0.5 1 1.5 2-0.5-0.2500.250.5 A = A0\nA = 2A0\nA = 4A0\nA = 8A0\nA = 16A0\nA = 32A0\nA = 64A0 10-2\n 10-6\n(b)\nt (s)F / L (N / m)\n00.125 0.25 0.375 0.5 0.625 0.75-10-50510 A = A0\nA = 2A0\nA = 4A0\nA = 8A0\nA = 16A0\nA = 32A0\nA = 64A0 10-2\n 10-6t (s)F / L (N / m)\n00.125 0.25 0.375 0.5 0.625 0.75-2-1012 A = A0\nA = 2A0\nA = 4A0\nA = 8A0\nA = 16A0\nA = 32A0\nA = 64A0 10-2\n 10-6\n(c)\nFigure 17: Time evolutions of damping forces at (a) Umax= 0.0674m/s, (b)Umax=\n0.27m/sand (c)Umax= 1.08m/s. TwoKn(h0)numbers, 0.1 (left) and 1.0 (right), are\nconsidered; the values of dashed lines shown in figures are obtained from Fig. 13(c) at\ncorresponding velocities and the reference amplitude A0equals to 0 .0025h0.\n26t (s)F / L (N / m)\n0 1 2 3 4 5-4-2024 Decoupled method\nCoupled framework 10-3\n 10-6t (s)F / L (N / m)\n0 1 2 3 4 5-0.5-0.2500.250.5 Decoupled method\nCoupled framework 10-3\n 10-6\n(a)\nt (s)F / L (N / m)\n0 0.2 0.4 0.6 0.8-505Decoupled method\nCoupled framework 10-3\n 10-6t (s)F / L (N / m)\n0 0.2 0.4 0.6 0.8-1-0.500.51Decoupled method\nCoupled framework 10-3\n 10-6\n(b)\nt (s)F / L (N / m)\n0 0.5 1 1.5 2 2.5 3-505Decoupled method\nCoupled framework 10-2\n 10-6t (s)F / L (N / m)\n00.5 11.5 22.5 3-0.8-0.400.40.8 Decoupled method\nCoupled framework 10-2\n 10-6\n(c)\nt (s)F / L (N / m)\n0 2 4 6 8-202Decoupled method\nCoupled framework 10-2\n 10-6t (s)F / L (N / m)\n0 1.5 3 4.5 6 7.5-0.5-0.2500.250.5 Decoupled method\nCoupled framework 10-2\n 10-6\n(d)\nt (s)F / L (N / m)\n0 0.5 1 1.5 2 2.5 3-10-50510 Decoupled method\nCoupled framework 10-2\n 10-6t (s)F / L (N / m)\n00.5 11.5 22.5 3-101Decoupled method\nCoupled framework 10-2\n 10-6\n(e)\nFigure 18: Comparisons of time evolutions of damping forces at two Kn(h0)numbers, 0.1\n(left) and 1.0 (right). The maximum oscillating velocities Umaxand amplitudes Aare\nset to (a) Umax= 0.0674m/sandA= 0.005h0, (b)Umax= 1.08m/sandA= 0.005h0,\n(c)Umax= 1.08m/sandA= 0.08h0, (d)Umax= 0.27m/sandA= 0.16h0, and (e)\nUmax= 1.08m/sandA= 0.16h0, respectively (the dash dot lines shown in figures are\nused for comparison).\n27A / h0Famp / L (N / m)\n0 0.05 0.1 0.15-8-4048\nDecoupled method\nCoupled framework 10-2\nDownward motion\nUpward motionUmax = 1.08 m/s\nUmax = 1.08 m/sUmax = 0.27 m/s\n(a)A / h0Famp / L (N / m)\n0 0.05 0.1 0.15-1.4-0.700.71.4\nDecoupled method\nCoupled framework 10-2\nDownward motion\nUpward motionUmax = 1.08 m/s\nUmax = 1.08 m/sUmax = 0.27 m/s\n(b)\nFigure 19: Comparisons of the maximum and minimum values of damping f orceFampat\nKn(h0)numbers of (a) 0.1 and (b) 1.0.\ncan be obtained in the resonance regime:\ny(t) =F0\ncωn[sin(ωnt)−1/radicalbig\n1−ζ2e−ζωntsin(/radicalbig\n1−ζ2ωnt)],(35)\nwhereζ=c/2mωis the damping ratio. With Eq. (35), the gas damping\nforce can be verified by the present coupled framework. As the ma ximum\noscillation velocity is F0/c, by giving a damping coefficient, F0can be de-\ntermined. Then with the maximum displacement of structure F0/cωn, the\nfrequency of external excitation force ωncan also be determined. Finally, k\nis obtained by assuming a mass of structure m.\nFirstly, the free oscillation at a low-speed Umax= 0.0674m/sand a small\namplitude A= 0.02h0is studied. For flow at Kn(h0)= 0.1, the equivalent\nstructure damping coefficient c(c=cfL) is 0.0485L, and the value of cfis\nobtained from Fig. 11. For flow at Kn(h0)= 1.0,cequals to 0 .01L. Fig. 20\nshows the time evolutions of displacement, moving velocity and dampin g\nforce of a micro-beam. Here, a non-dimensional mass of micro-bea mM∗is\nused with M∗=m/ρLD, wheremthe actual mass of micro-beam and ρis\nthe density of rarefied gas. In our simulations, two values of M∗, 2769.5 and\n1384.7, are considered. For flow at the continuum flow regime, thos e values\nare 10 and 5, respectively. Generally, the results of numerical simu lation\nagree well with the theoretical solution. And the convergence time of dis-\nplacement of a micro-beam developing to its maximum value with a heavie r\nmass is slower than that with a lighter one. Shown in Fig. 17, for the fo rced\n28oscillation at Kn(h0)= 0.1 andA= 0.02h0, the maximum damping force is\na little lower than that obtained by the decoupled method (about 3%) . So\ncused to calculate the theoretical solution in Eq. (35) is slightly larger than\nthe real one in the simulation, and the numerical result is also slightly la rger\nthan that of the theoretical one. For flow at Kn(h0)= 1.0 shown in Fig. 21,\nas that difference increases to about 10%, the numerical results a re much\nlarger than the theoretical solutions. Consequently, it illustrates again that\nthe influence of oscillation frequency must be introduced into the da mping\nmodel.\nSecondly, two freeoscillation cases at higher velocities aresimulated , with\nthe computational conditions are set to Umax= 0.27m/s,A= 0.04h0and\nUmax= 1.08m/s,A= 0.08h0, respectively. And Kn(h0)is set to 0.1. Due to\nthe nonlinear phenomenon of damping force, the damping coefficient cused\nfor a theoretical solution is difficult to construct and is also set to 0 .0485L.\nShown in Figs. 22 and 23, for the displacements and moving velocities o f a\nmicro-beam, as cobtained from a low-frequency simulation can not reflect\nthe real damping at a high-frequency oscillation, the numerical res ults are\nmuch higher than the theoretical solutions. Besides, although the nonlinear\nphenomenon of damping forces also can be observed, the maximum a nd\nminimum values of displacement and moving velocity of a micro-beam are\nalmost the same. For this reason, in the practical computation, an empirical\nparameter cmaybe be constructed and used for high velocity oscillation.\n4.3.Decoupled method and coupled framework: squeeze-film dampi ng torque\nat the tilting motion\nIn this section, with the coupled FSI framework, the rarefied gas d amping\ntorque acting on a micro-beam by the forced or free oscillation unde r the\ntiltingmotionisstudied. Theinitialheightofgap h0alsoissetto1 .0×10−6m.\n4.3.1. Decoupled method: squeeze-film damping torque\nFor the forced tilting oscillation, the motion form is given as\nθ=θ0sin(ft), (36)\nwhereθ0is the maximum tilting angle; and two values, 0 .5◦and 1.0◦, are\nconsidered. With Eq. (36), assuming the maximum damping torque is o b-\ntained at the maximum angular velocity, the tilting angle of a micro-bea m\n29t (s)y / g0\n0 2 4 6 810 12 14-4-2024theory\npresent 10-2\n 10-6t (s)y / g0\n0 2 4 6 810 12 14-4-2024theory\npresent 10-2\n 10-6\n(a)\nt (s)U (m / s)\n0 2 4 6 810 12 14-10010theory\npresent 10-2\n 10-6t (s)U (m / s)\n0 2 4 6 810 12 14-10010theory\npresent 10-2\n 10-6\n(b)\nt (s)F / L (N / m)\n0 2 4 6 810 12 14-505theory\npresent 10-3\n 10-6t (s)F / L (N / m)\n0 2 4 6 810 12 14-505theory\npresent 10-3\n 10-6\n(c)\nFigure 20: Time evolutions of (a) displacement, (b) moving velocity an d (c) damping\nforce of a micro-beam with linear free oscillation at Kn(h0)= 0.1. The maximum moving\nvelocity is 0 .0674m/s, and two non-dimensional mass of micro-beam M∗, 2769.5 (left\nfigures) and 1384.7 (right figures) are considered (the dash lines s hown in the figures are\nthe maximum theoretical values used for comparison).\n30t (s)y / g0\n0 2 4 6 810 12 14-4-2024theory\npresent 10-2\n 10-6\n(a)\nt (s)U (m / s)\n0 2 4 6 810 12 14-10010theory\npresent 10-2\n 10-6\n(b)\nt (s)F / L (N / m)\n0 2 4 6 810 12 14-101theory\npresent 10-3\n 10-6\n(c)\nFigure 21: Time evolutions of (a) displacement, (b) moving velocity an d (c) damping\nforce of a micro-beam with linear free oscillation at Kn(h0)= 1.0. The maximum moving\nvelocity is 0 .0674m/s, and the non-dimensional mass of micro-beam M∗is 2769.5 (the\ndash lines shown in the figures are the maximum theoretical values us ed for comparison).\n31t (s)y / g0\n0 5 10 15-505 10-2\n 10-6\n(a)\nt (s)U (m / s)\n0 3 6 9 12 15-4-2024 10-1\n 10-6\n(b)\nt (s)F / L (N / m)\n0 3 6 9 12 15-2-1012 10-2\n 10-6\n(c)\nFigure 22: Time evolutions of (a) displacement, (b) moving velocity an d (c) damping\nforce of a micro-beam with linear free oscillation at Kn(h0)= 0.1. In the simulation,\nUmax= 0.27m/s,M∗= 2769.5, andA= 0.04h0are used. (the dash lines shown in\nthe figures are the maximum theoretical values, and the dash-dot lines are the maximum\nvalues obtained by numerical simulation).\n32t (s)y / g0\n0 5 10 15-101 10-1\n 10-6\n(a)\nt (s)U (m / s)\n0 3 6 9 12 15-2-1012\n 10-6\n(b)\nt (s)F / L (N / m)\n0 3 6 9 12 15-10-50510 10-2\n 10-6\n(c)\nFigure 23: Time evolutions of (a) displacement, (b) moving velocity an d (c) damping\nforce of a micro-beam with linear free oscillation at Kn(h0)= 0.1. In the simulation,\nUmax= 1.08m/s,M∗= 2769.5, andA= 0.08h0are used. (the dash lines shown in\nthe figures are the maximum theoretical values, and the dash-dot lines are the maximum\nvalues obtained by numerical simulation).\n33used in the simulation is 0◦. Shown in Fig. 4(b), the velocity profile imposed\non the surface of wall is given by\nV=Rθ0fπ\n180, (37)\nwhereVis the magnitude of velocity vector, and Ris the length from the\nsurface of micro-beam to its center. Fig. 24 shows the pressure c ontours and\nstreamlines near the micro-beam at θ0= 0.5◦andf/(2π) = 0.2106MHz. In\nthis study, similar to the definition of the Strouhal number, a param eter of\nfh0/Vmaxis used to nondimensionalize the tilting frequency, where Vmaxis\nthemaximumvelocity atthesurfaceofmicro-beam. Clearly, duetot heeffect\nof tilt, the pressure in one side of the gapincreases, and that at th e other side\ndecreases; then it generates the damping torque. Figs. 25 and 26 show the\npressure distributions along the surface of a micro-beam, and the damping\ntorqueacting onit, respectively. Here, acoefficient T/(0.25ρV2\nmaxh0L)isused\nto nondimensionalize the damping torque T. Similar to the linear motion\ndescribed in Sec. 4.2, due to the influence of the substrate, the va riation of\npressure at the bottom surface is much more significant than that at the\ntop surface, and the pressure distributions show some kind of linea r relation\nbetween the different frequencies. Further, the variation of dam ping torque\nalsoshowsalinearrelationbetweentheoscillationfrequency f, themaximum\ntilting angle θ0and the Knudsen number Kn(h0). Although, the maximum\nmoving velocity on the surface of a micro-beamis about 6 .65m/satθ0= 1.0◦\nandf/(2π) = 6.74MHz, the nonlinear phenomenon can not be observed.\n4.3.2. Coupled method: squeeze-film damping torque at force d oscillation\nFor the forced tilting oscillation, Eq. (36) is used to control the var iation\nof tilting angle. Fig. 27 shows the time evolutions of damping torque at three\ntilting oscillation frequencies. Fig. 28 shows the comparisons of the la rgest\ndampingtorqueactingonamicro-beambytwodifferentmethods. Fo rflowat\nKn(h0)= 0.1, the largest damping torques obtained by two different methods\nare almost the same, so the traditional decoupled method still exhib its a\ngood performance to predict the rarefied gas damping torque coe fficientηin\nEq. (22). And for flow at Kn(h0)= 1.0, due to the rarefaction effect, the\ndifferences of results are obvious at a high tilting oscillation frequenc y. For\nexample, the result obtained by the decoupled method is about twice larger\nthanthatby thecoupled frameworkat f/(2π) = 6.74MHz. So, theinfluence\nof tilting oscillation frequency must be introduced to build the gas dam ping\n34xy\n-10 -5 0 5 1002468Pressure\n1.002\n1.00156\n1.00111\n1.00067\n1.00022\n0.999778\n0.999333\n0.998889\n0.998444\n0.998p / p0    \n(a)\nxy\n-20 -10 0 10 200510\n(b)\nFigure 24: (a) Pressure contours and (b) streamlines for flow aro und a micro-beam in the\nrarefied gas at Kn(h0)= 0.1.θ0andfin Eq. (37) are set to θ0= 0.5◦andf/(2π) =\n0.2106MHz, respectively.\nx / Lp / p0\n-0.5 -0.25 0 0.25 0.50.9960.99811.0021.004\nTop surface\nBottom surface\nf\nf\n(a)x / Lp / p0\n-0.5 -0.25 0 0.25 0.50.9920.99611.0041.008\nTop surface\nBottom surface\nf\nf\n(b)\nFigure 25: Pressure distributions along the top and bottom surfac es of micro-beam with\ndifferent tilting frequencies at (a) Kn(h0)= 0.1 and (b) Kn(h0)= 1.0. Four tilting\nfrequencies ( f/(2π)), 0.05265MHz, 0.1053MHz, 0.2106MHz, and 0.4213MHzatθ0=\n0.5◦are shown in figures.\n35f / f0T (N·m)\n0 50 100-3-2.5-2-1.5-1-0.50\nKn     = 0.1, θ0 = 0.5°\nKn     = 0.1, θ0 = 1.0°\nKn     = 1.0, θ0 = 0.5°\nKn     = 1.0, θ0 = 1.0°× 10-9\n(h0)\n(h0)(h0)\n(h0)\n(a)f / f0| T | (N·m)\n10010110210-1210-1110-1010-9\nKn     = 0.1, θ0 = 0.5°\nKn     = 0.1, θ0 = 1.0°\nKn     = 1.0, θ0 = 0.5°\nKn     = 1.0, θ0 = 1.0°(h0)(h0)\n(h0)(h0)\n(b)\nFigure 26: Comparisons of gas damping torque at different tilting fre quencies with (a)\noriginal coordinates and (b) logarithm coordinates. The initial test frequency f0equals to\nf0/(2π) = 0.052656MHz.\ntorque model at a high tilting oscillation frequency and a high Knudsen\nnumber. Besides, the nonlinear phenomenon also can not be observ ed as\nthe absolute values of maximum and minimum values of damping torque\nare almost the same. Different from the linear motion at a high velocity ,\nthe high moving velocity regime only focuses on the left and right sides of\na tilting micro-beam. So, the cause of the nonlinear phenomenon may be a\nlow oscillation frequency and a large contact area of high velocity bet ween\nrarefied gas and micro-beam (see Fig. 29).\n4.3.3. Coupled method: squeeze-film damping torque at free o scillation\nForthefreetiltingoscillation, similartolinearmotiondescribedinSec.4 .2.2,\nan external excitation torque is also introduced, and Eq. (25) is mo dified as\nI¨θ(t)+Kθ=Text+T=T0cos(ωnt)+T, (38)\nwhereTis the gas damping torque, T0is the amplitude of external excitation\ntorque, and ωnis the frequency of T0. Ifyandcin Eq. (35) are replaced\nbyθandη, respectively, the theoretical solution of the time evolution of\ntilting angle θalso can be obtained. So, Eq. (38) is used in the numerical\nsimulation, and the modified form of Eq. (35) is used to make a compar ison.\nThen assuming the maximum tilting angle θ0equals to 0 .5◦, the tilting oscil-\nlation frequency equals to ωn/(2π) = 1.073MHz, and the torsional damping\ncoefficient ηis calculated from Fig. 26 at the corresponding frequency, T0\n36t (s)T (N·m)\n0 1 2 3 4 5-4-2024θ0 = 1.0°\nθ0 = 0.5°× 10-9\n× 10-6t (s)T (N·m)\n0 1 2 3 4 5-1-0.500.51 θ0 = 1.0°\nθ0 = 0.5°× 10-9\n× 10-6\n(a)\nt (s)T (N·m)\n0 1 2 3 4 5-1-0.500.51 θ0 = 1.0°\nθ0 = 0.5°× 10-9\n× 10-6t (s)T (N·m)\n0 1 2 3 4 5-0.200.2θ0 = 1.0°\nθ0 = 0.5°× 10-9\n× 10-6\n(b)\nt (s)T (N·m)\n0 1 2 3 4 5-0.3-0.1500.150.3θ0 = 1.0°\nθ0 = 0.5°× 10-9\n× 10-6t (s)T (N·m)\n0 1 2 3 4 5-0.0500.05θ0 = 1.0°\nθ0 = 0.5°× 10-9\n× 10-6\n(c)\nFigure 27: Time evolutions of damping torque at two Kn(h0)numbers, 0.1 (left) and\n1.0 (right), and the tilting oscillation frequencies ( f/(2π)) are set to (a) 6 .74MHz, (b)\n1.685MHzand (c) 0 .42125MHz(the values of dash lines and dash dots lines shown in\nfigures are obtained from Fig. 26 for comparison).\n37f / f0| T | (N·m)\n10010110210-1110-1010-9\nθ0 = 0.5°, coupled framework\nθ0 = 0.5°, decoupled method\nθ0 = 1.0°, coupled framework\nθ0 = 1.0°, decoupled method\n(a)f / f0| T | (N·m)\n100101102 10-1210-1110-1010-9\nθ0 = 0.5°, coupled framework\nθ0 = 0.5°, decoupled method\nθ0 = 1.0°, coupled framework\nθ0 = 1.0°, decoupled method\n(b)\nFigure 28: Comparisons of the maximum gas damping torque at differe nt tilting frequen-\nciesfand angles θ0with (a) Kn(h0)= 0.1 and (b) Kn(h0)= 1.0, where f0is the initial\ntest frequency and equals to f0/(2π) = 0.1053MHz.\ncan be obtained. In this case, the non-dimensional mass of micro-b eamM∗\nis 2769.5, and the polar moment of inertia IisM∗(L2+D2)/12 for a rigid\nplate tilted around its center. Fig. 30 shows the numerical results a t two\nKn(h0)numbers, 0.1 and 1.0. Generally, numerical results agree well with\nthe theoretical solution. For flow at Kn(h0)= 1.0, due to the damping torque\ncoefficient used in the theoretical solution is a little different from the real\none in the numerical simulation, a little difference between the two res ults\ncan be observed. Consequently, with Fig. 28, for predicting the response of a\nmicro-beam by free tilting oscillation, the traditional decoupled meth od can\nbe used for low-frequency oscillation, and the coupled framework m ust be\nused for high-frequency oscillation due the rarefied gas effect.\n5. Conclusion\nInthepresentstudy, adecoupledmethodbasedontheDUGKSand acou-\npledframeworkbasedontheALE-DUGKSareusedforstudying the squeeze-\nfilm damping in MEMS. For the decoupled method, based an Eulerian-\nscheme, the damping force is calculated by imposing a velocity profile o n\nthe stationary wall. And for the implementation of coupled framewor k, a\nloosely-coupled algorithm is used, in which the fluid and structure dyn amic\nsolvers are used alternately in each time iteration step. To validate t hese two\nmethods, a micro-Couette flow in rarefied gas and an elastically moun ted\nsquare cylinder oscillating in continuum flow are simulated. Results of b oth\n38t (s)F / L (N / m)\n00.5 11.5 22.5 33.5-101 10-2\n 10-60.825\n-0.62Nonlinear effect\nt (s)T (N·m)\n0 1 2 3 4 5-0.100.1 10-9\n 10-60.072\n-0.072Linear effect\n(a)\nxy\n-15 -10 -5 0 5 10 15048 4.5\n4.0\n3.5\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0Uw, max = 1.08 m/s, t = 3.25  10-6 s U|    |\nxy\n-15 -10 -5 0 5 10 15048 2.5\n2.2\n1.9\n1.7\n1.4\n1.1\n0.8\n0.6\n0.3\n0.0Uw, max = 0.83 m/s, t = 4.15  10-6 s U|    |\n(b)\nWallOutletOutlet\nOutletHigh speed flow region\nWallOutletOutlet\nOutletHigh speed flow region\n(c)\nFigure 29: (a) Time evolutions of force and torque, (b) velocity mag nitude contours |U|\nand (c) sketches of high speed flow region at forced linear oscillation (left) and tilting oscil-\nlation (right) with Kn(h0)= 1.0, where Uw,maxis maximum moving velocity at the surface\nof a micro-beam. For the forced linear oscillation, UmaxandAin Eq. (31) are 1 .08m/s\nand 0.16h0, respectively. So, the corresponding oscillation frequency is 1 .074MHz. And\nfor the forced tilting oscillation, the frequency fin Eq. (36) is f/(2π) = 0.8425MHz, and\nthe maximum tilting angle θ0is 1.0◦.\n39t (s)θ (angle)\n0 2 4 6 8 10-1-0.500.51\ntheory\npresent\n× 10-6\n(a)\nt (s)θ (angle)\n0 2 4 6 8 10-1-0.500.51\ntheory\npresent\n× 10-6\n(b)\nFigure 30: Time evolutions of tilting angle of a micro-beam with tilting osc illation at\n(a)Kn(h0)= 0.1 and (b) Kn(h0)= 1.0. The maximum tilting angle θ0is 0.5◦, and the\nnon-dimensional mass of micro-beam M∗is 2769.5.\n40test cases agree well with existing numerical results. For the SFD p roblems\ninMEMS, two basic motionforms, linear (perpendicular) andtilting mot ions\nof a rigid micro-beam, are fully studied with the forced and free oscilla tions.\nFirstly, based on the decoupled method, the damping coefficients at different\nKnudsen numbers are calculated. A consistent result is obtained co mpared\nwith the compact damping model. In addition, the nonlinear phenomen on\nof damping force at a high moving velocity is reproduced. Next, the f orced\nlinear oscillations are studied. It can be found that the nonlinear dam ping\nis only generated at a low-frequency high-velocity oscillation, so the influ-\nence of oscillation frequency must be introduced to construct the damping\nmodel. Consequently, the advantage of the coupled framework is t o study\nthe large linear displacement problems of a micro-structure, such a s shock\nproblem of a high- gMEMS accelerometer [56]. And for the high-frequency\nsmall-displacement oscillation, the decoupled method still exhibits a go od\nperformance to predict the damping force or torque. Besides, th e influence\nof oscillation frequency also must be considered for tilting oscillation, as the\ndamping torques obtained by the decoupled method are higher than that by\nthe coupled method at a high oscillation frequency, especially for flow at a\nhigh Knudsen number. Finally, the free oscillation in the resonance re gime\nare studied. The maximum perpendicular displacements or tilting angle s cal-\nculated by the numerical method agree well with the theoretical so lutions.\nFurther work such as the improvement of the squeeze-film damping model\nand the prediction of nonlinear damping for the complex micro-struc ture can\nbe continued to enlarge the application range of the DUGKS in MEMS.\nAcknowledgements\nThis work is sponsored by the National Numerical Wind Tunnel Proje ct,\nthe National Natural Science Foundation of China (No. 11902266, 11902264,\n12072283), the Innovation Foundation for Doctor Dissertation o f Northwest-\nernPolytechnical University(CX202015),theNaturalScienceBa sicResearch\nPlan in Shaanxi Province of China (Program No. 2019JQ-315), and t he 111\nProject of China (B17037).\n41References\n[1] S. D. Senturia, N. Azuru, J. 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Xu, Direct modeling for computational fluid dynamics: Constr uction\nand application of unified gas-kinetic schemes, World Scientific, 2015 .\n[50] F. Archambeau, N. M´ echitoua, M. Sakiz, Code Saturne: Afinit e volume\ncode for the computation of turbulent incompressible flows - Indus trial\napplications, International Journal on Finite Volumes 1 (1) (2004) 1–62.\n46[51] Y. Zhu, C. Zhong, K. Xu, Implicit unified gas-kinetic scheme for s teady\nstate solutions inall flow regimes, Journal ofComputational Physic s 315\n(2016) 16–38.\n[52] C. H. K. Williamson, Vortex dynamics in the cylinder wake, Annual\nReview of Fluid Mechanics 28 (1) (1996) 477–539.\n[53] S. P. Singh, S. Mittal, Vortex-induced oscillations at low Reynolds num-\nbers: Hysteresis and vortex-shedding modes, Journal of Fluids & Struc-\ntures 20 (8) (2005) 1085–1104.\n[54] X. Li, Z. Lyu, J. Kou, W. Zhang, Mode competition in galloping of a\nsquare cylinder at low Reynolds number, Journal of Fluid Mechanics\n867 (2019) 516–555.\n[55] Z. Guo, R. Wang, K. Xu, Discrete unified gas kinetic scheme for a ll\nKnudsen number flows: II. Compressible case, Physical Review E 91 (3)\n(2015) 033313.\n[56] D. Parkos, N. Raghunathan, A. Venkattraman, B. Sanborn, Near-\ncontact gas damping and dynamic response of high-g MEMS accelero m-\neter beams, Journal of Microelectromechanical Systems 22 (5) ( 2013)\n1089–1099.\n47"    },    {        "title": "1307.5722v1.Lorentz_Breaking_and_Gravity.pdf",        "content": "arXiv:1307.5722v1  [hep-ph]  22 Jul 2013Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n1\nLORENTZ VIOLATION AND GRAVITY\nR. BLUHM\nDepartment of Physics, Colby College\nWaterville, ME 04901, USA\nE-mail: rtbluhm@colby.edu\nGravitational theories with Lorentz violation must accoun t for a number of\npossible features in order to be consistent theoretically a nd phenomenologi-\ncally. A brief summary of these features is given here. They i nclude evasion of\na no-go theorem, connections between spontaneous Lorentz b reaking and dif-\nfeomorphism breaking, the appearance of massless Nambu-Go ldstone modes\nand massive Higgs modes, and the possibility of a Higgs mecha nism in gravity.\n1. Gravity and the SME\nThe Standard-Model Extension1(SME) consists of the most general\nobserver-independent effective field theory incorporating Loren tz violation.\nIt is routinely used by both theorists and experimentalists to study and ob-\ntain bounds on possible forms of Lorentz violation.2,3As an effective field\ntheory, the SME can accommodate both explicit and spontaneous L orentz\nbreaking. However, there are differences in these two forms of sy mmetry\nbreaking that arise in the context of gravity. This overview looks at these\ndifferences and what their primary consequences are.\nIn a gravitational theory with Lorentz violation it is useful to use a\nvierbein formalism. In this approach, both the local Lorentz frame s and\nspacetimeframesareaccessibleandlinkagebetweenthesymmetrie sinthese\nframes can be examined. The vierbein provides the connection betw een\ntensor components in local Lorentz frames and tensor componen ts in the\nspacetime frame.\nTheLagrangianintheSMEisformedasthemostgeneralscalarfunc tion\n(underbothlocalLorentzanddiffeomorphismtransformations)u singgravi-\ntationalfields, particle fields, and Lorentz-violatingSME coefficient s.When\nthe Lorentz breaking is explicit, the SME coefficients are viewed as fix ed\nbackground fields. However, when the Lorentz breaking is sponta neous, theProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n2\nSME coefficients are vacuum expectation values (vevs) of dynamica l tensor\nfields.\nIn the gravity sector of the SME, a no-go theorem shows that with\nexplicit Lorentz breaking an inconsistency can occur between cond itions\nstemming from the field variations and symmetry considerations with ge-\nometrical constraints that must hold, such as the Bianchi identitie s.4In\ncontrast, the case of spontaneous Lorentz breaking was found to evade the\nno-go theorem. The main difference is that in a theory with explicit bre ak-\ning the SME coefficients are not associated with dynamical fields, while\nwith spontaneous Lorentz breaking they are, which creates a diffe rence\nin the conditions that must hold. An important consequence of the n o-go\ntheorem is that the gravity sector of the SME can only avoid incompa tibil-\nity with conventional geometrical constraints if the symmetry bre aking is\nspontaneous.\n2. Spontaneous symmetry breaking\nThe fact that the SME coefficients must be associated with vevs of d ynam-\nical fields that undergo spontaneous Lorentz violation leads to a nu mber\nof effects that must be accounted for in the gravity sector of the SME.\nFor example, when Lorentz symmetry is spontaneously broken, th ere is\nalso spontaneous breaking of diffeomorphism symmetry. The spont aneous\nLorentz breaking occurs when a nonzero tensor-valued vacuum o ccurs in\nthe local Lorentz frames, which is necessarily accompanied by a vac uum\nvalue for the vierbein. When products of the vierbein vev act on the lo-\ncal tensor vevs, the result is that tensor vevs also appear in the s pacetime\nframe. These tensor vevs spontaneously break local diffeomorph isms in the\nspacetime frame. Conversely, if a vev in the spacetime frame spont aneously\nbreaks diffeomorphisms, then the inverse vierbein acting on it gives r ise to\nvevs in the local frames. Consequently, spontaneous local Loren tz breaking\nimplies spontaneous diffeomorphism breaking and vice versa.5\nWith spontaneous symmetry breaking of both Lorentz and diffeomo r-\nphismsymmetry,therearestandardfeaturesinparticlephysicst hatneedto\nbe investigated. These include the possible appearance of massless Nambu-\nGoldstone (NG) modes and massive Higgs modes, or there is the poss ibility\nof a Higgs mechanism in which the NG modes are reinterpreted as addi-\ntional degrees of freedom in a theory with massive gauge fields.\nIn the absence of a Higgs mechanism, there can be up to as many\nNG modes as there are broken spacetime symmetries. Since the max imal\nsymmetry-breakingcasewould yield six brokenLorentzgenerator sand fourProceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13)\n3\nbroken diffeomorphisms, there can be as many as ten NG modes. A na tural\ngauge choice puts all of the NG modes into the vierbein. However, th is will\nin general lead to the appearance of ghosts, and it is for this reaso n that\nmost models involve breaking fewer than ten of the spacetime symme tries.\nSpontaneous symmetry breaking is usually induced by a potential te rm\nin the Lagrangian that has a degenerate minimum space. The NG mode s\nappear as excitations away from the vacuum that stay in the minimum\nspace, while massive Higgs modes are excitations that go up the pote ntial\nwell away from the minimum. In conventional gauge theory, the pot ential\ninvolves only scalar fields, and the massive Higgs modes are independe nt of\nthe gauge fields. However, with spontaneous Lorentz breaking, t he metric\ntypically appears in the potential along with the tensor fields, and fo r this\nreason massive Higgs modes can occur that include metric excitation s. This\nis an effect that has no analog in the case of conventional gauge the ory.\nIn a Higgs mechanism, the would-be NG modes become additional de-\ngrees of freedom for massive gauge fields. The gauge fields associa ted with\ndiffeomorphisms are the metric excitations. However, a Higgs mecha nism\ninvolving the metric has been shown not to occur.6This is because the\nmass term that is generated by covariant derivatives involves the c onnec-\ntion, which consists of derivatives of the metric and not the metric it self.\nHowever, for the broken Lorentz symmetry, where the relevant gauge fields\nare the spin connection, a conventional Higgs mechanism can occur .5This\nis because the spin connection appears directly in covariant derivat ives act-\ning on local tensor components, and when the local tensors acquir e a vev,\nquadratic mass terms for the spin connection can be generated. N ote, how-\never,aviableHiggsmechanisminvolvingthespinconnectioncanonlyoc cur\nif the spin connection is a dynamical field, which requires nonzero tor sion\nand that the geometry be Riemann-Cartan.\nReferences\n1. V.A. Kosteleck´ y and R. Potting, Phys. Rev. D 51, 3923 (1995); D. Colladay\nand V.A. Kosteleck´ y, Phys. Rev. D 55, 6760 (1997); Phys. Rev. D 58, 116002\n(1998).\n2. R. Bluhm, arXiv:1302.1150.\n3.Data Tables for Lorentz and CPT Violation, V.A. Kosteleck´ y and N. Russell,\n2013 edition, arXiv:0801.0287v6.\n4. V.A. Kosteleck´ y, Phys. Rev. D 69, 105009 (2004).\n5. R. Bluhm and V.A. Kosteleck´ y, Phys. Rev. D 71, 065008 (2005); R. Bluhm,\nS.-H. Fung, and V.A. Kosteleck´ y, Phys. Rev. D 77, 065020 (2008).\n6. V.A. Kosteleck´ y and S. Samuel, Phys. Rev. D 40, 1886 (1989)."    },    {        "title": "1109.5268v1.Proof_of_a_Lorentz_and_Levi_Civita_thesis.pdf",        "content": "arXiv:1109.5268v1  [physics.gen-ph]  24 Sep 2011PROOF OF A LORENTZ AND LEVI-CIVITA THESIS\nANGELO LOINGER\nAbstract. A formal proof of the thesis by Lorentz and Levi-Civita\nthat the left-hand side of Einstein field equations represen ts the real\nenergy-momentum-stress tensor of the gravitational field.\nSummary –1. Introduction. Aim of the paper. – 2. Mathematical preliminaries.\n–3. Proof that the left-hand side of the Einstein field equations gives t he true\nenergy-momentum-stress tensor of the gravitational field. – 4. A fundamental\nconsequence. – Appendix : On the pseudo energy-tensor.\nPACS 04.20 – General relativity.\n1.– As it has been remarked [1], if Iis theactionintegral of any field (of\nany tensorial nature) – say ϕ(x),x≡(x0,x1,x2,x3) – acting in a pseudo-\nRiemannian spacetime, and we perform the variation of I– sayδgI– ge-\nnerated by the variation δgjk, (j,k= 0,1,2,3), of the metric tensor gjk(x)\n(possibly interacting with ϕ(x)),\n(1) δgI=/integraldisplay\nD(...)jkδgjk√−gd4x ,\n– where Dis a fixed spacetime domain – , the expression ( ...)jkis a\nsymmetrical tensor, which represents the energy-momentum -stress tensor of\nϕ(x). This statement has been verifiedfor various fields [1]. And its general\nvalidity can be intuitively understood bearing in mind that Iis an action\nintegral, with the Lagrange density of ϕ(x) as integrand.\nWe shall prove that the above statement holds also if ϕ(x)≡gjk(x), thus\ncorroborating a famous (and debated!) thesis by Lorentz [2] and Levi-Civita\n[3] – see also Pauli [4] (and the references therein).\nThe essential merit of the following demonstration is its independence\nof the Einstein field equations (and of the Bianchi relations).\n2.– Let√−g S[gjk(x),gjk,m(x),gjk,mn(x),...] be a generic scalar density\nwhich is a function of the metric gjk(x) and of a finite number of its ordinary\nderivatives [5]. We do notassume that√−g Sis a Lagrange density, and\ntherefore the integral\n12 ANGELO LOINGER\n(2) J=/integraldisplay\nDS√−gd4x\nisnotan action integral. We have:\n(3) δgJ=/integraldisplay\nDδ(S√−g)\nδgjkδgjkd4x;\nthe variational derivative δ(S√−g)/δgjkis equal to\n(4)∂(S√−g)\n∂gjk−∂\n∂xm/bracketleftbigg∂(S√−g)\n∂gjk,m/bracketrightbigg\n+∂2\n∂xmxn/bracketleftbigg∂(S√−g)\n∂gjk,mn/bracketrightbigg\n−...;\nputtingδ(S√−g)/δgjk:=Pjk√−g, we can write:\n(3′) δgJ=/integraldisplay\nDPjk√−g δgjkd4x .\nLet us now consider the particular δgjk– sayδ∗gjk–, which is generated\nby an infinitesimal change of the co-ordinates x:\n(5) x′j=xj+εj(x) ;\nwe assume that εj(x) is zero on the bounding surface ∂D. The corres-\nponding variation of J– sayδ∗\ngJ– will be equal to zero, because Jis an\ninvariant.\nWe have:\n(6) gmn(x) =∂x′j\n∂xm∂x′k\n∂xng′\njk(x′),\nand we consider the δ∗gjkforfixedvalues of the coordinates, i.e.:\n(6′)δ∗gjk:=g′\njk(x′)−gjk(x′) =g′\njk(x′)−gjk(x)−gjk,s(x)εs.\nIt follows immediately from eqs.(5), (6), (6′) that\n(7) δ∗gmn=−gmn,jεj−gmjεj\n,n−gnjεs\n,m;\nfrom eq.(3′) we get:\nδ∗\ngJ=/integraldisplay\nDPmn√−gδ∗gmnd4x=\n=/integraldisplay\nDPmn/parenleftbig\n−gm;εj\n,n−gnjεj\n,m−gmn,j;εj/parenrightbig√−gd4x=\n=/integraldisplay\nD/bracketleftbig\n2(Pn\nj√−g),n−gmn,jPmn√−g)/bracketrightbig\nεjd4x=\n= 2/integraldisplay\nDPm\nj:mεj√−gd4x= 0, (8)PROOF OF A LORENTZ AND LEVI-CIVITA THESIS 3\nif the colon denotes a covariant derivative; in the last pass age we use the\nfollowing property of any symmetrical tensor Smn:\n(8′) Sm\nj:m√−g=/parenleftbig\nSn\nj√−g/parenrightbig\n,n−1\n2gmn,jSmn√−g .\nAccordingly:\n(9) Pm\nj:m= 0 ; ( j= 0,1,2,3).\n3.– The result (9) has a mere mathematical interest. It becomes physically\nsignificant when Jis the action integral, say A, given by\n(10) A=/integraldisplay\nDR√−gd4x ,\nwhereR=Rjkgjkis the Ricci scalar. We shall not use the fact that the\ngjk’s are (a priori) independent variables, because we do not wish to deduce\nfrom the action Athe Einstein field equations.\nStandard procedures (see, e.g., Hilbert’s method in Appendix ,β)) tell us\nthat\n(11) δgA=/integraldisplay\nD/parenleftbigg\nRjk−1\n2gjkR/parenrightbigg√−g δgjkd4x;\nthe analogue of eq.(8) is:\n(12) δ∗\ngA= 2/integraldisplay\nD/parenleftbigg\nRk\nj−1\n2δj\nkR/parenrightbigg\n:kεj√−gd4x= 0,\nfrom which:\n(13)/parenleftbigg\nRjk−1\n2gjkR/parenrightbigg\n:k= 0,(j= 0,1,2,3).\nThus, quite independently of the field equations, we see that thesym-\nmetrical tensorRjk−(1/2)gjkRsatisfies four conservation equations . Of\ncourse, eqs.(13) are identically satisfied by virtue of Bian chi relations, but\nthe above method – which is essentially due to the conception s of Emmy\nNoether [6] – evidences the conservative property of Rjk−(1/2)gjkR, and\nattributes it the nature of an energy-momentum-stress tens or. Properly\nspeaking, [ Rjk−(1/2)gjkR]/κ, ifκis the Newton-Einstein gravitational\nconstant, represents the Einsteinian energy tensor, as it w as emphasized by\nLorentz [2] and Levi-Civita [3]. And the fact that this tenso r is a function\nonlyof the potential gjkimplies that it is the unique energy-momentum-\nstress tensor of the gravitational field.4 ANGELO LOINGER\n4.– The fact that [ Rjk−(1/2)gjkR]/κisthe true energy-momentum-stress\ntensor of the gravitational field has a very important conseq uence [3]: the\nmathematical undulatory solutions of the equations Rjk−(1/2)gjkR= 0 =\nRjkare quite devoid of physical meaning, because they do not tra nsport\nenergy, momentum, stress. This was the firstdemonstration of the physical\nnon-existence of the gravitational waves. Quite different de monstrations\nhave been given in recent years, see e.g.[7], and references therein.\nIn his fundamental memoir [3], Levi-Civita proved also the n ature of mere\nmathematical fiction (Eddington[8]) ofthewell-known pseudoenergy-tensor\nof the metric field gjk. –\nA useful discussion with Dr. T. Marsico is gratefully acknow ledged.\nAPPENDIX\nα) The full illogicality of the notion of pseudo energy-tenso r can be seen\nalso in the following way. The usual definition of this pseudo tensor is:\n(A.1)√−g tn\nmDEF=∂(L√−g)\n∂gjk,ngjk,m−δn\nmL√−g;\nthe function L:\n(A.2) L≡gmn(Γs\nmnΓr\nsr−Γr\nmsΓs\nnr)\nyields the Lagrangean field equations:\n(A.3)∂(L√−g)\n∂gjk−∂\n∂xn/bracketleftbigg∂(L√−g)\n∂gjk,n/bracketrightbigg\n= 0.\nNow, the left-hand side of (A.3) is notequal to\n(A.4) −/parenleftbigg\nRjk−1\n2gjkR/parenrightbigg√−g\nas it is commonly affirmed. Indeed:\ni) A non tensor entity cannot be equal to a tensor density –\nii) The above affirmed equality has its origin in a “negligence”: in the\ncustomary variational deduction of the Einstein field equat ions the\nvariationof/integraltext\nDR√−gd4xis“reduced”tothevariationof/integraltext\nDL√−gd4x.\nBut in his “reduction” two perfect differentials in the integr and have\nbeen omitted, because on the boundary ∂Dthe variations of the gjk\nand of their first derivatives are zero (by assumption): this omission\ndestroys the tensor-density character of the initial expressions. –\nβ) It is likely that the pseudo energy-tensor would not have be en invented\nif the authors had followed Hilbert’s procedure [9]. This Au thor started\nfrom the fact that (with our previous notations) the explici t evaluation ofPROOF OF A LORENTZ AND LEVI-CIVITA THESIS 5\nthe variational derivative δ(R√−g)/δgmngives the following Lagrangean\nexpressions:\n(A.5)∂(R√−g)\n∂gmn−∂\n∂xk/bracketleftBigg\n∂(R√−g)\n∂gmn\n,k/bracketrightBigg\n+∂2\n∂xkxl/bracketleftBigg\n∂(R√−g)\n∂gmn\n,kl/bracketrightBigg\n;\nHilbert wrote: “ ...specializiere man zun¨ achst das Koordinatensystem\nso, daß f¨ ur den betrachteten Weltpunkt die gmn\n,ss¨ amtlich verschwinden.”.\nI.e., he chose a localcoordinate-system for which the firstderivatives of\ngmnare equal to zero. Thus, only the first term of (A.5) gives a non -zero\ncontribution, and we have that (A.5) is equal to\n(A.6)√−g/parenleftbigg\nRmn−1\n2gmnR/parenrightbigg\n.\nThere is no room in this procedure for false (pseudo) tensor e ntities.\nReferences\n[1] W. Pauli, Teoria della Relativit` a (Boringhieri, Torino) 1958, sect. 55. See also: V.\nFock,The Theory of Space, Time and Gravitation , Second Revised Edition (Pergamon\nPress, Oxford, etc) 1964, sects. 31*, 48, 60; A. Loinger, Nuovo Cimento ,110A(1997)\n341.\n[2] H.A. Lorentz, Amst. Versl. ,25(1916) 468; (this memoir is written in Dutch – an\nEnglish translation would be desirable).\n[3] T. Levi-Civita, Rend. Acc. Lincei ,26(1917) 381; an English translation in\narXiv:physics/9906004 (June 2nd, 1999). See also: Idem,ibid.,11(s.6a) (1930) 3\nand 113.\n[4] See Pauli [1], sects. 23and61.\n[5] E. Schr¨ odinger, Space-Time Structure (Cambridge University Press, Cambridge) 1960,\nChapt. XI; P.A.M. Dirac, General Theory of Relativity (J. Wiley and Sons, New York,\netc) 1975, sect. 30.\n[6] E. Noether, G¨ ott Nachr. , (1918) 235 (“Invariante Variationsprobleme”).\n[7] A. Loinger and T. Marsico, arXiv:1006.3844 [physics.gen-ph] 19 Jun 2010.\n[8] A. S. Eddington, The Mathematical Theory of Relativity , Second Edition (Cambridge\nUniversity Press, Cambridge) 1960, p.148. See also H. Bauer ,Phys. Z.,19(1918) 163.\n[9] D. Hilbert, G¨ ott Nachr. : Erste Mitteilung, vorgelegt am 20. Nov. 1915; zweite Mit-\nteilung, vorgelegt am 23. Dez. 1916 – Math. Annalen ,92(1924) 1.\nA.L. – Dipartimento di Fisica, Universit `a di Milano, Via Celoria, 16 - 20133\nMilano (Italy)\nE-mail address :angelo.loinger@mi.infn.it"    },    {        "title": "2403.13067v1.Weakly_elliptic_damping_gives_sharp_decay.pdf",        "content": "arXiv:2403.13067v1  [math.AP]  19 Mar 2024WEAKLY ELLIPTIC DAMPING GIVES SHARP DECAY\nLASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\nAbstract. We prove that weakly elliptic damping gives sharp energy decay for th e\nabstractdampedwavesemigroup,wherethedampingisnotinthefu nctionalcalculus.\nIn this case, there is no overdamping. We show applications in linearise d water waves\nand Kelvin–Voigt damping.\n1.Introduction\n1.1.Motivation. Let∆≥0betheLaplace–Beltrami operatoronacompactmanifold\nMof dimension d≥1 without boundary. There is recent interest in the study of decay\nof the damped water wave equation, linearised via paradifferential d iagonalisation,\n(∂2\nt+∆1\n2x+∆1\n4xa(x)∆1\n4x∂t)u(t,x) = 0, (1.1)\nu(0,x)∈W1\n2,2(M), ∂tu(0,x)∈L2(M)\ndescribes the evolution of a fluid interface in the gravity-capillary wa ter wave system\nsubject to an external pressure, studied in [ ABZ11,Ala17,Ala18,ABHK18 ,AMW23,\nKW23]. We define the energy of the solution to ( 1.1) by\nE(u,t) =/ba∇dbl∂tu/ba∇dbl2\nL2(M)+/ba∇dbl∆1\n4u/ba∇dbl2\nL2(M).\nWe want to understand the decay of energy, when the dissipation c oefficienta(x)∈\nL∞(M) may vanish on a measure zero set in M. Here the damping term ∆1\n4xa(x)∆1\n4x,\nthough not relatively compact, may still have some weak elliptic prope rties. This\nmotivatesustostudyhowellipticdampinggivessharpenergydecayr atesingeneralised\nsemigroup setting. As a corollary, in Example 1.9we prove that the energy of ( 1.1)\ndecays exponentially when a(x) degenerates fast near its zeros.\nCorollary 1.1. Assume(a(x))−1∈Lp(M)forp∈(d,∞). Then there is C >0that\nE(u,t)≤e−CtE(u,0),\nuniformly in t>0andusatisfying (1.1). Forp∈(1,d], we have\nE(u,t)1\n2≤C/an}b∇acketle{tt/an}b∇acket∇i}ht−p\nd−p(/ba∇dblu(0,x)/ba∇dblW1,2(M)+/ba∇dbl∂tu(0,x)/ba∇dblW1\n2,2(M))\nuniformly in t>0andusolving(1.1)with initial data in W1,2(M)×W1\n2,2(M).\n12 LASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\n1.2.Introduction. LetH=H0be an infinite-dimensional Hilbert space and P:\nH1→Hbe a nonnegative self-adjoint operator with compact resolvent, d efined onH1,\na dense subspace of H. The operator Padmits a spectral resolution and a functional\ncalculus\nPu=/integraldisplay∞\n0ρ2dEρ(u), f(P)u=/integraldisplay∞\n0f(ρ2)dEρ(u),\nwhereEρis a projection-valued measure on Hand suppEρ⊂[0,∞), andfis a Borel\nmeasurable function on [0 ,∞) that formally yields an operator f(P). Fors∈R, define\nthe scaling operators and the interpolation spaces via\nΛsu=/integraldisplay∞\n0(1+ρ2)sdEρ(u), Hs= Λ−s(H0).\nThose operators Λ−s:H→Hsare bounded from above and below, and they commute\nwithP. Fors>0,H−sis isomorphic to the dual space of Hswith respect to H.\nLet the observation space Ybe a Hilbert space. We will consider damping of the\nformQ∗Q, where the control operator Q∗∈ L(Y,H−1\n2) and the observation operator\nQ∈ L(H1\n2,Y). Note that Q∗Qis not necessarily a bounded operator on H. We\nconsider an abstract damped second-order evolution equation:\n(∂2\nt+P+Q∗Q∂t)u= 0.\nIt can be written as a first-order evolution system:\n∂t/parenleftbiggu\n∂tu/parenrightbigg\n=A/parenleftbiggu\n∂tu/parenrightbigg\n,A=/parenleftbigg0 1\n−P−Q∗Q/parenrightbigg\n.\nHereA, defined on {(u,v)∈H1\n2×H1\n2:Pu+Q∗Qv∈H}, generates a strongly\ncontinuous semigroup etAonH=H1\n2×H. See [KW23,CPS+23] for further details.\n1.3.Main results. The goal of this note is to understand the stability, that is the\nnorm of ( A+iλ)−1, and thus decay, of etAon the energy space ˙H=H/KerA, when\nQ∗Qsatisfies some ellipticity conditions but is not by itself in the functional calculus\nofP. We define the weak ellipticity and boundedness:\nDefinition 1.2 (m-ellipticity) .Letm(λ)be a positive continuous function on (0,∞)λ.\nWe say the observation operator Q:H1\n2→Yism(λ)-elliptic, if for some χ∈\nC0([0,∞))withχ(1)>0, there exist and C,λ0,N >0that\nm(λ)/ba∇dblχ(λ−2P)u/ba∇dbl2\nH≤C/ba∇dblQu/ba∇dbl2\nY+o(min{m(λ)λ4N,λ})/ba∇dblΛ−Nu/ba∇dbl2\nH(1.2)\nuniformly for all u∈H1\n2and allλ∈Rwithλ≥λ0.WEAKLY ELLIPTIC DAMPING 3\nDefinition 1.3 (m-boundedness) .Letm(λ)be a positivecontinuous functionon (0,∞)λ.\nWe say the observation operator Q:H1\n2→Yism(λ)-bounded , if for some χ∈\nC0([0,∞))withχ(1)>0, there isC >0such that\n/ba∇dbl(1+λ−2P)−1\n2Q∗Qχ(λ−2P)u/ba∇dbl2\nH≤Cm(λ)/ba∇dbl(1+λ−2P)1\n2u/ba∇dbl2\nH,\nfor allu∈H1\n2.\nRemark 1.4. (1) For readers familiar with semiclassical analysis, m-ellipticity (or\nm-boundedness) heuristically means ( m(h−1))−1\n2Q(or (m(h−1))−1Q∗Q) being semi-\nclassically elliptic (or bounded) over {h2ρ2= 1}, the characteristic variety of h2P−1.\n(2) Consider two positive continuous functions m−(λ)≤m+(λ) on (0,∞)λ. IfQis\nm+-elliptic, then Qisalsom−-elliptic. IfQism−-bounded, then Qisalsom+-bounded.\n(3) Consider two observation operators Q±:H1\n2→Y±such that /ba∇dblQ−u/ba∇dbl2\nY−≤\nC/ba∇dblQ+u/ba∇dbl2\nY+uniformly for u∈H1\n2. IfQ−ism-elliptic, then Q+is alsom-elliptic.\n(4) Any observation operator Qbounded from H1\n2toYis a prioriλ2-bounded.\nFor classically elliptic and bounded operators, the m-ellipticity and m-boundedness\nare easy to verify via Theorem 3proved at the end of this note:\nExample 1.5 (Linearised water waves) .In the setting of §1.1,P= ∆,H=L2(M),\nHs=Ws,2(M) are the Sobolev spaces of order sfors∈R,Q=/radicalbig\na(x)∆1\n4:H1\n2→\nY=H. Whena(x)∈L∞(M) is bounded from above and below by positive constants,\nQis classically elliptic with respect to Λ1\n2:\n/ba∇dbl/radicalbig\na(x)∆1\n4u/ba∇dblL2(M)≥C−1/ba∇dblΛ1\n2u/ba∇dblL2(M)−C/ba∇dblu/ba∇dblL2(M)\nand Theorem 3impliesQisλ2-elliptic and λ2-bounded.\nOur result is bifold. The first result is that m-elliptic damping gives an upper bound\nfor semigroup stability.\nTheorem 1 (Weak ellipticity gives stability) .Letm(λ)be positive continuous function\nand letQbem(λ)-elliptic. Then there are C,λ0>0such that\n/ba∇dbl(A+iλ)−1/ba∇dblL(H)≤Cmax/braceleftbigg1\nm(|λ|),1/bracerightbigg\nuniformly for all |λ| ≥λ0,λ∈R.\nWhen the unique continuation hypothesis holds (that is, ( P−λ2)u= 0 implies\nQu/ne}ationslash= 0 forλ >0), andm(λ) is chosen to be 1, or the reciprocal of a function of\npositive increase, for example, λ−sore−sλfor somes≥0, one can use [ KW23, Lemma\n3.10] (based on semigroup equivalence results in [ BT10,RSS19]), to turn the stability\nresults into exponential, polynomial and logarithmic energy decay fo retArespectively.4 LASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\nThe second result is that m-bounded damping gives a lower bound for the semigroup\nstability, which is asymptotic to the upper bound found in Theorem 1.\nTheorem 2 (Weak boundedness gives sharpness) .Letm(λ)be positive continuous\nfunction and let Qbem(λ)-bounded. Then there exist a sequence of λk→ ∞and\nC >0,\n/ba∇dbl(A+iλk)−1/ba∇dblL(H)≥1\nCm(|λk|).\nIt implies that when Qism-bounded, it is not possible to improve the bound in\nTheorem 1too(1/M(|λ|)). Thus, we have proved that a m-elliptic and m-bounded\ndamping gives sharp semigroup stability. We have a handy corollary be low to show\nenergy decay:\nCorollary 1.6 (Weak ellipticity gives decay) .LetQ:H1\n2→Ybe bounded from below\nbyΛsfor somes∈(−∞,0), that is, there exists C >0such that\n/ba∇dblΛsu/ba∇dblH≤C/ba∇dblQu/ba∇dblY (1.3)\nuniformly for u∈H1\n2/KerP. Then there is C >0such that for all t≥0,\n/ba∇dbletAA−1/ba∇dblH→˙H≤C/an}b∇acketle{tt/an}b∇acket∇i}ht1\n4s.\nWhen(1.3)holds withs= 0, then there is C >0such that\n/ba∇dbletA/ba∇dbl˙H→˙H≤e−Ct.\nRemark 1.7 (Strong monotonicity) .Corollary 1.6is interpreted that damping given\nbyQlarger than Λsmust give at least the same decay rate as that given by Λs. In\nparticular, in this case there is no overdamping, contrary to the ge neral case of weak\nmonotonicitydiscussedin[ KW23,§2.2](seealso[ ALN14,Sta17,Kle19,DK20,KW22]).\nTheorems 1,2and Corollary 1.6constitute improvements over [ DP21, Theorems\n12.1 and 12.2], where Q∗Q=f(P) for some nonnegative continuous function f. Such\nan assumption is often too strong and restrictive. Our results do n ot requireQ∗Qto be\nin the functional calculus of Pand also accommodates compact errors. The compact\nerrors are natural in the studies of elliptic estimates, and accommo dating them in our\ntheorems allows us to obtain new stability results: see Example 1.12. Corollary 1.6\nalso generalises the result of [ LZ15], the authors of which studied the case Ker P= 0\nwithout allowing compact errors in ( 1.2). The separation of Ker PfromP≥0 is\nnot trivial, when Q∗Qis no longer relatively compact. Furthermore, in contrast to\n[DP21,LZ15], we are able to assess sharpness of the decay rates even for dam ping\noutside of the functional calculus.\nHere we present concrete examples where we get new results.WEAKLY ELLIPTIC DAMPING 5\nExample 1.8 (Linearised water waves, bounded from below) .In the setting of §1.1\nand Example 1.8, assumea(x) is bounded from above and below by positive constants.\nNote uniformly for all u∈W1\n2,2(M)/Span{1},\n/ba∇dblΛ1\n2u/ba∇dblL2(M)≤C/ba∇dbl/radicalbig\na(x)∆1\n4u/ba∇dblL2(M).\nCorollary 1.6implies for some C >0,\nE(u,t)≤e−CtE(u,0),\nfor allt>0 and for all uthat solves ( 1.1).\nExample 1.9 (Linearised water waves, degenerate) .We now consider that in the\nsetting of §1.1thata(x)≥0 may vanish on a measure zero set. In order to control\nits degree of degeneracy, we assume ( a(x))−1∈Lp(M) forp∈(1,∞): the larger p\nis, the faster a(x) vanishes near its zeros. Sobolev embedding implies that ( a(x))−1\n2:\nL2(M)→W−d\n2p,2(M) is bounded (see for example, [ KW23, Lemma 2.17]). Thus\n/ba∇dbla1\n2∆1\n4u/ba∇dblL2≥C−1/ba∇dbl(1+∆)−d\n4p∆1\n4u/ba∇dblL2≥C−1/ba∇dblΛ−1\n4(d\np−1)u/ba∇dblL2−C/ba∇dblΛ−1\n4(d\np−1)−1u/ba∇dblL2.\nThusQ=a1\n2∆1\n4is classically elliptic with respect to Λ−1\n4(d\np−1). Apply Theorem 3\nto seeQisλ−(d\np−1)-elliptic. Furthermore, since aonly vanishes on a measure zero\nset,a1\n2∆1\n4u= 0 implies ∆1\n4u= 0 onM, and thus ∆ u= 0. This implies the unique\ncontinuation holds. Apply Theorem 1with [KW23, Lemma 3.10] to see\n(1) Whenp∈[d,∞) andp/ne}ationslash= 1, there is C >0 such that uniformly for all t>0,\nE(u,t)≤e−CtE(u,0), (1.4)\nfor allusolving (1.1).\n(2) Whenp∈(1,d), there isC >0 such that uniformly for all t>0,\nE(u,t)1\n2≤C/an}b∇acketle{tt/an}b∇acket∇i}ht−p\nd−p(/ba∇dblu(0,x)/ba∇dblW1,2(M)+/ba∇dbl∂tu(0,x)/ba∇dblW1\n2,2(M))\nfor allusolving (1.1) with initial data in W1,2(M)×W1\n2,2(M).\n(3) Whenp= 1 andd≥2, theSobolevembedding works with( a(x))−1\n2:L2(M)→\nW−d\n2p−0,2(M). For each ǫ>0,there isCǫ>0 such that uniformly for all t>0,\nE(u,t)1\n2≤Cǫ/an}b∇acketle{tt/an}b∇acket∇i}ht−p\nd−p+ǫ(/ba∇dblu(0,x)/ba∇dblW1,2(M)+/ba∇dbl∂tu(0,x)/ba∇dblW1\n2,2(M))\nfor allusolving (1.1) with initial data in W1,2(M)×W1\n2,2(M).\n(4) Whenp= 1 andd= 1, for any N >0, there isCN>0 such that uniformly for\nallt>0,\nE(u,t)1\n2≤CN/an}b∇acketle{tt/an}b∇acket∇i}ht−N(/ba∇dblu(0,x)/ba∇dblW1,2(M)+/ba∇dbl∂tu(0,x)/ba∇dblW1\n2,2(M))\nfor allusolving (1.1) with initial data in W1,2(M)×W1\n2,2(M).6 LASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\nAs an example, consider M=S1= [−1,1]xwith endpoints identified. Consider the\ndampinga(x) =x2sfors∈[0,1\n2), then(a(x))−1∈L1\n2s−0(M)⊂L1+0(M)andwealways\nhave exponential decay ( 1.4). Contextually, in [ AMW23], only polynomial decay has\nbeen shown with a smaller damping Q=a1\n2that degenerates slowly near its zeros.\nThe next few examples are devoted to the damped wave equations w ith unbounded,\nKelvin–Voigt, and pseudodifferential damping. Let Mbe a compact smooth manifold,\nH=L2(M),P= ∆≥0 andQ:W1,2(M)→Y. The damped wave equation is\n(∂2\nt+∆+Q∗Q∂t)u(t,x) = 0, (1.5)\nu(0,x)∈W1,2(M),∂tu(0,x)∈L2(M),\nThe energy of the solution is\nE(u,t) =/ba∇dbl∂tu/ba∇dbl2\nL2(M)+/ba∇dbl∇u/ba∇dbl2\nL2(M).\nExample 1.10 (Damped wave equation) .Consider a function a(x)∈L∞(M) with\ninfx∈M|a(x)|>0, and letQ=/radicalbig\na(x)/an}b∇acketle{t∆/an}b∇acket∇i}hts:W1,2(M)→L2(M) fors∈(−∞,1\n2].\nHere/an}b∇acketle{t·/an}b∇acket∇i}ht= (1+|·|2)1\n2. Note that [ Q∗Q,P]/ne}ationslash= 0 and hence Q∗Qis not in the functional\ncalculus of P. HereQis a classically elliptic operator, and Theorem 3impliesQis\nλ4s-elliptic. Corollary 1.6further gives the energy decay rates:\n(1) Whens∈[0,1\n2], there isC >0 such that uniformly for all t>0,\nE(u,t)≤e−CtE(u,0),\nfor allusolving (1.5). This rate still holds when we replace /an}b∇acketle{t∆/an}b∇acket∇i}htsby ∆s.\n(2) Whens∈(−∞,0), there is C >0 such that uniformly for all t>0,\nE(u,t)1\n2≤C/an}b∇acketle{tt/an}b∇acket∇i}ht1\n4s(/ba∇dblu(0,x)/ba∇dblW2,2(M)+/ba∇dbl∂tu(0,x)/ba∇dblW1,2(M))\nfor allusolving (1.5) with initial data in W2,2(M)×W1,2(M).\nWhens∈[0,1\n2] anda(x)∈L∞(M), Theorem 3impliesQisλ4s-bounded, andTheorem\n2implies the rate in (1) are optimal. When s∈(−∞,0), if we further impose the\nregularity assumption to a(x)∈W−2s+0,∞(M) being ( −2s+ 0)-H¨ older, Theorem 3\nimpliesQisλ4s-bounded and the rate in (2) is optimal.\nExample 1.11 (Kelvin–Voigt damping) .LetAbe a bundle isomorphism on TMsuch\nthatAx∈Iso(TxM), not necessarily continuous in x∈M. Assume\nesssup\nx∈M/ba∇dblAx/ba∇dblL(TxM)<∞,esssup\nx∈M/ba∇dblA−1\nx/ba∇dblL(TxM)<∞.\nThenQ=A∇:W1,2(M)→L2(M,TM) isλ2-bounded and λ2-elliptic, where ∇is the\ngradient. Apply Corollary 1.6to see there is C >0 such that for all t>0,\nE(u,t)≤e−CtE(u,0),WEAKLY ELLIPTIC DAMPING 7\nfor allusolving (1.5). In the special case that Ax(x,v) = (x,a(x)v) fora(x)∈L∞(M)\nwith essinf x∈M|a(x)|>0, we recovered the exponential decay for the Kelvin–Voigt\ndamping of L∞-regularity. Contextually, under additional C1-regularity, it was shown\nin [Bur20,BS22b] that the same exponential rate holds when a(x) may vanish on some\nopen sets but still satisfies the geometric control condition. L∞-regularity in this case\nonly gives optimal polynomial decay: see [ BS22a].\nExample 1.12 (Pseudodifferential damping) .Fors∈(−∞,1\n2], considerQ∈Ψ2s(M),\nan(one-step polyhomogeneous) pseudodifferential operator of order 2s. Then Theorem\n3impliesQisλ4s-bounded. If Qis classically elliptic, that is, its principal symbol\n|σ2s(Q)(x,ξ)|is uniformly bounded from below by C−1/an}b∇acketle{tξ/an}b∇acket∇i}ht2son{|ξ| ≥C}for some\nC >0, then for any N >0, we have the elliptic estimate\n/ba∇dblQu/ba∇dblL2(M)≤C/ba∇dblu/ba∇dblW2s,2(M)+CN/ba∇dblu/ba∇dblW−N,2(M),\nuniformly for all u∈W1,2(M). Theorem 3impliesQisλ4s-elliptic. Our Theorems\n1and2below imply the sharp estimate /ba∇dbl(A+iλ)−1/ba∇dblL(H)≤Cmax{λ−4s,1}for large\nλ. With extra unique continuation hypotheses on Qexplicitly, one can obtain energy\ndecay results. Pseudodifferential damping models dissipation in aniso tropic materials:\nthe energy of waves is dissipated at different rates depending on th e direction of prop-\nagation. The case s= 0 was studied in [ KK23], and cases/ne}ationslash= 0 in [KW23]. See further\nin Theorem 3and references [ Zwo12,DZ19].\n1.4.Acknowledgement. The authorsthank Jeffrey Galkowski andJared Wunsch for\ndiscussions around the results. LP and NV are supported by the Re search Council of\nFinland grant 349002. RPTW is supported by EPSRC grant EP/V0017 60/1.\n2.Proof\nConsider the semiclassical operator Ph=h2P−ihQ∗Q−1:H1\n2→H−1\n2, whereh∈\n(0,h0) for some small h0>0. It is Fredholm: see [ KW23, Lemma 3.4]. In this section,\nwe prove the upper bounds and lower bounds for P−1\nhin Propositions 2.1and2.6,\nrespectively. We state and prove Theorem 3at the end of this section. Throughout\nthis section, we use the abbreviation m=m(λ) =m(h−1). We start with an upper\nbound forP−1\nh.\nProposition 2.1 (Semiclassical resolvent estimate) .Letmbe a positive continuous\nfunction on (0,∞)and letQbem-elliptic, that is, for some χ∈C0([0,∞))with\nχ(1)>0, there exist C,N >0such that\n/ba∇dblχ(h2P)u/ba∇dbl2≤C(m(h−1))−1/ba∇dblQu/ba∇dbl2\nY+o(min{h−4N,m(h−1)−1h−1})/ba∇dblΛ−Nu/ba∇dbl2,(2.1)8 LASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\nuniformly for all u∈H1\n2and all small h>0. Then there is C >0such that\n/ba∇dblP−1\nh/ba∇dblL(H)≤Cmax/braceleftbiggh−1\nm(h−1),1/bracerightbigg\nuniformly for all h>0small.\nTo prove Proposition 2.1, we need to estimate the compact error in ( 2.1).\nLemma 2.2 (Compact error estimates) .ForN >0, there exists C >0such that\n/ba∇dblΛ−Nu/ba∇dbl2≤Ch4N/ba∇dblu/ba∇dbl2+C/vextendsingle/vextendsingle/vextendsingle/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2/vextendsingle/vextendsingle/vextendsingle.\nuniformly for all u∈H1\n2and all small h>0.\nProof.Consider that\n/ba∇dblΛ−Nu/ba∇dbl2=/integraldisplay\n|h2ρ2−1|<1\n2(1+ρ2)−2N/an}b∇acketle{tdEρu,u/an}b∇acket∇i}ht+/integraldisplay\n|h2ρ2−1|≥1\n2(1+ρ2)−2N/an}b∇acketle{tdEρu,u/an}b∇acket∇i}ht.\nWe can estimate the second term/integraldisplay\n|h2ρ2−1|≥1\n2(1+ρ2)−2N/an}b∇acketle{tdEρu,u/an}b∇acket∇i}ht ≤2/integraldisplay\n|h2ρ2−1|≥1\n2/vextendsingle/vextendsingleh2ρ2−1/vextendsingle/vextendsingle/an}b∇acketle{tdEρu,u/an}b∇acket∇i}ht\nand therefore bound it by C/vextendsingle/vextendsingle/vextendsingle/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2/vextendsingle/vextendsingle/vextendsingle. To bound the first term, we note that\non{|h2ρ2−1|<1\n2}, we have 1+ ρ2>1+1\n2h−2and thus (1+ ρ2)−2N<(1+1\n2h−2)−2N≤\nCh4Nfor smallh. Thus we can estimate the first term/integraldisplay\n|h2ρ2−1|<1\n2(1+ρ2)−2N/an}b∇acketle{tdEρu,u/an}b∇acket∇i}ht ≤Ch4N/integraldisplay\n|h2ρ2−1|<1\n2/an}b∇acketle{tdEρu,u/an}b∇acket∇i}ht\nand bound it by Ch4N/ba∇dblu/ba∇dbl2as desired. /square\nThefollowinglemma allows ustoobtainahigh-frequency unique continu ationresult.\nLemma 2.3 (High-frequency unique continuation) .AssumeQism-elliptic. Then\n(ImPh)⊥= KerP∗\nh={0}\nandPh:H1\n2→H−1\n2is invertible for all hsmall.\nProof.AssumeP∗\nhu= (h2P+ihQ∗Q−1)u= 0 for some u. This implies\n/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2= 0,/ba∇dblQu/ba∇dbl2\nY= 0, Qu= 0,(h2P−1)u= 0.\nWe then have\nEh−1u=Eh−1h2Pu=u,\nand thus\nχ(h2P)u=/integraldisplay∞\n0χ(h2ρ2)dEρu=χ(1)u.WEAKLY ELLIPTIC DAMPING 9\nNow apply Lemma 2.2to see (2.1) reduces to\nχ(1)2/ba∇dblu/ba∇dbl2≤o(1)/ba∇dblu/ba∇dbl2.\nThus uniformly for small h >0, this implies u= 0 and Ker P∗\nh={0}. From [KW23,\nLemma 3.4], we know Ph:H1\n2→H−1\n2is Fredholm for all hsmall. Thus (Im Ph)⊥=\nKerP∗\nh={0}for allhsmall. /square\nProof of Proposition 2.1.PairPhuwithuto observe that\n/an}b∇acketle{tPhu,u/an}b∇acket∇i}ht=/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2−ih/ba∇dblQu/ba∇dbl2,\nwhose real and imaginary parts satisfy\n/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2= Re/an}b∇acketle{tPhu,u/an}b∇acket∇i}ht, h/ba∇dblQu/ba∇dbl2\nY=−Im/an}b∇acketle{tPhu,u/an}b∇acket∇i}ht.\nWe can estimate them by/vextendsingle/vextendsingle/vextendsingle/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2/vextendsingle/vextendsingle/vextendsingle≤4ǫ−2/ba∇dblPhu/ba∇dbl2+ǫ2/ba∇dblu/ba∇dbl2, (2.2)\n/ba∇dblQu/ba∇dbl2\nY≤4ǫ−2h−2/ba∇dblPhu/ba∇dbl2+ǫ2/ba∇dblu/ba∇dbl2.\nThem-ellipticity ( 2.1) implies\n/ba∇dblχ(h2P)u/ba∇dbl2≤Cm−1/ba∇dblQu/ba∇dbl2\nY+e(h)/ba∇dblΛ−Nu/ba∇dbl2\n≤Cǫ−2m−2h−2/ba∇dblPhu/ba∇dbl2+ǫ2/ba∇dblu/ba∇dbl2+e(h)/ba∇dblΛ−Nu/ba∇dbl2,\nwheree(h) =o(min{h−4N,m(h−1)−1h−1})asin(2.1). WeapplyLemma 2.2toestimate\nthe compact error:\ne(h)/ba∇dblΛ−Nu/ba∇dbl2≤o(1)/ba∇dblu/ba∇dbl2+e(h)/vextendsingle/vextendsingle/vextendsingle/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2/vextendsingle/vextendsingle/vextendsingle≤Cǫ−2e(h)2/ba∇dblPhu/ba∇dbl2+ǫ2/ba∇dblu/ba∇dbl,\nfor the last inequality of which we used a variant of ( 2.2). Notinge(h)2=o(m−2h−2)\nby assumption, we have\n/ba∇dblχ(h2P)u/ba∇dbl2≤Cǫ−2m−2h−2/ba∇dblPhu/ba∇dbl2+ǫ2/ba∇dblu/ba∇dbl2.\nNow since |χ(s)|is uniformly bounded from below near s= 1, the algebraic inequality\nC−1≤/vextendsingle/vextendsingleh2ρ2−1/vextendsingle/vextendsingle+/vextendsingle/vextendsingleχ(h2ρ2)/vextendsingle/vextendsingle2\nholds uniformly for all h>0,ρ∈R. Thus\n/ba∇dblu/ba∇dbl2=/integraldisplay∞\n0d/an}b∇acketle{tEρu,u/an}b∇acket∇i}ht ≤C/integraldisplay∞\n0/vextendsingle/vextendsingleh2ρ2−1/vextendsingle/vextendsingle+/vextendsingle/vextendsingleχ(h2ρ2)/vextendsingle/vextendsingle2d/an}b∇acketle{tEρu,u/an}b∇acket∇i}ht\ncan be estimated by\n/ba∇dblu/ba∇dbl2≤C/parenleftBig/vextendsingle/vextendsingle/vextendsingle/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2/vextendsingle/vextendsingle/vextendsingle+/ba∇dblχ(h2P)u/ba∇dbl2/parenrightBig\n≤C(1+m−2h−2)/ba∇dblPhu/ba∇dbl2+ǫ2/ba∇dblu/ba∇dbl2.\nWe absorb the ǫ-term to observe\n/ba∇dblP−1\nh/ba∇dblL(H)≤Cmax{m−1h−1,1},10 LASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\nas desired. /square\nWe now move on to prove the lower bound for P−1\nh. To do so, it is convenient to\nintroduce the semiclassical interpolation spaces Hh\ns. Define the semiclassical scaling\noperators\nΛs\nhu=/integraldisplay∞\n0(1+h2ρ2)sdEρ(u),\nand interpolation spaces Hh\ns= Λ−s\nh(H) equipped with the norm /ba∇dbl·/ba∇dblHhs=/ba∇dblΛs\nh·/ba∇dblH.\nLemma 2.4 (Norm equivalence) .Fors≥0and anyχBorel measurable on (0,∞)\nwhose support is away from 0, there exists C >0such that\nC−1h2s/ba∇dblΛsu/ba∇dbl ≤ /ba∇dblΛs\nhu/ba∇dbl ≤C/ba∇dblΛsu/ba∇dbl,\n/ba∇dblΛs\nhχ(h2P)u/ba∇dbl ≤Ch2s/ba∇dblΛsχ(h2P)u/ba∇dbl,\nh−2s/ba∇dblΛ−sχ(h2P)u/ba∇dbl ≤C/ba∇dblΛ−s\nhχ(h2P)u/ba∇dbl,\nuniformly for all u∈Hsand allh>0small.\nProof.It suffices to note the algebraic inequalities\nC−1h2s(1+ρ2)s≤(1+h2ρ2)s≤C(1+ρ2)s,\n(1+h2ρ2)sχ(h2ρ2)≤C(h2ρ2)sχ(h2ρ2)≤Ch2s(1+ρ2)sχ(h2ρ2),\nh−2s(1+ρ2)−sχ(h2ρ2)≤C(h2+h2ρ2)−sχ(h2ρ2)≤C(1+h2ρ2)−sχ(h2ρ2),\nfor the last two lines of which we used that χis supported away from 0. /square\nWe will use the following lemma to compare different norms of P−1\nh: it is a semiclas-\nsical version of [ KW23, Lemma 3.9].\nLemma 2.5 (Operator norm estimate) .Assume that Ph:H1\n2→H−1\n2is invertible for\nfixedh>0. Then\n/ba∇dblP−1\nh/ba∇dblH→Hh\n1\n2≤C(1+/ba∇dblP−1\nh/ba∇dblL(H)).\nHereCdoes not depend on h.\nProof.Given anyf∈H−1\n2, there exists a unique u∈H1\n2such thatPhu=f. Pair\n/an}b∇acketle{tPhu,u/an}b∇acket∇i}ht=/ba∇dblhP1\n2u/ba∇dbl2−/ba∇dblu/ba∇dbl2−ih/ba∇dblQu/ba∇dbl2\nY,\nwhose real part implies\n/ba∇dblhP1\n2u/ba∇dbl2≤Cǫ−1/ba∇dblPhu/ba∇dbl2+/ba∇dblu/ba∇dbl2+ǫ/ba∇dblu/ba∇dbl2.\nNote that, after absorption of ǫ, we have\n/ba∇dblu/ba∇dbl2\nHh\n1\n2=/ba∇dblu/ba∇dbl2+/ba∇dblhP1\n2u/ba∇dbl2≤C/ba∇dblPhu/ba∇dbl2+C/ba∇dblu/ba∇dbl2,WEAKLY ELLIPTIC DAMPING 11\nthat is\n/ba∇dblP−1\nhf/ba∇dblHh\n1\n2≤C(/ba∇dblf/ba∇dblH+/ba∇dblP−1\nhf/ba∇dblH),\nyielding the desired operator norm bound. /square\nWe now prove the lower bound for P−1\nh.\nProposition 2.6 (Resolvent lower bound) .Letmbe a positive continuous function\nand letQism-bounded, that is, for some χ∈C0([0,∞))withχ(1)>0, there isC >0\nsuch that\n/ba∇dblQ∗Qχ(h2P)u/ba∇dblHh\n−1\n2≤Cm(h−1)/ba∇dblu/ba∇dblHh\n1\n2. (2.3)\nuniformly for all u∈H1\n2and all real h≤C−1. Then there is C >0such that\n/ba∇dblP−1\nh/ba∇dblL(H)≥C−1(m(h−1))−1h−1, (2.4)\nalong some sequence of h→0.\nProof.IfPh:H1\n2→H−1\n2fails to be invertible on a sequence of h→0, then ( 2.4)\ntrivially holds due to [ KW23, Lemmata 3.6 and 3.9]. Without loss of generality we\nassumePhis invertible for all hsmall. Since Phas compact resolvent, Pis unbounded\nand has discrete spectrum. There then exists a sequence of nont rivialuh∈H1\n2as\nh→0, such that ( h2P−1)uh= 0. Letχ∈C∞\nc((0,∞)) be a cutoff with χ(1) = 1.\nNow we have uh=χ(h2P)uhand\n/ba∇dblP∗\nhuh/ba∇dblHh\n−1\n2=h/ba∇dblQ∗Qχ(h2P)uh/ba∇dblHh\n−1\n2≤Cmh/ba∇dbluh/ba∇dblHh\n1\n2≤Cmh/ba∇dbluh/ba∇dbl.\nRecalling that Hh\n1\n2andHh\n−1\n2are in duality, we then have\n/ba∇dblP−1\nh/ba∇dblH→Hh\n1\n2=/ba∇dbl(P∗\nh)−1/ba∇dblHh\n−1\n2→H≥C−1m−1h−1.\nApply the norm inequality in Lemma 2.5while noting mis bounded from below to see\n/ba∇dblP−1\nh/ba∇dblL(H)≥C−1m−1h−1\nas desired. /square\nProof of Theorems 1and2.It remains to convert those resolvent estimates to upper\nandlowerboundsfor /ba∇dbl(A+iλ)−1/ba∇dbl. Wehavethefollowingcharacterisationfrom[ KW23,\nLemma 3.7]: for any mbounded from above,\n/ba∇dblP−1\nh/ba∇dblL(H)≤C(m(h−1))−1h−1\nuniformly for hsmall is equivalent to\n/ba∇dbl(A+iλ)−1/ba∇dblL(H)≤C\nm(λ),\nas desired. Apply [ KW23, Lemma 3.9] to obtain the lower bound of ( A+iλ)−1./square12 LASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\nProof of Corollary 1.6.Theorem 1already gives bounds on ( A+iλ)−1. With [KW23,\nLemma 3.10], it remains to show that the unique continuation propert y holds to con-\nclude the energy decay. That is, we want to show for any λ>0, (P−λ2)u= 0 implies\nQu/ne}ationslash= 0. When ( P−λ2)u= 0, [KW23, Lemma 3.3(4)] implies Λsu/ne}ationslash= 0. Now ( 1.3)\nimpliesQu/ne}ationslash= 0 as desired. /square\nWe conclude the note with a lemma to conveniently turn classical elliptic ity and\nboundedness of Qinto weak ellipticity ( 2.1) and weak boundedness ( 2.3).\nTheorem 3 (Classical estimates) .Leth=λ−1. The following are true:\n(1) IfQis classically elliptic with respect to Λsfors∈(−∞,1\n2], that is, there are\nC,N >0such that −N <sand\n/ba∇dblΛsu/ba∇dbl2≤C/ba∇dblQu/ba∇dbl2\nY+C/ba∇dblΛ−Nu/ba∇dbl2,\nfor allu∈H1\n2. ThenQisλ4s-elliptic: for some χ∈C∞([0,∞))withχ(1)>0,\nh−4s/ba∇dblχ(h2P)u/ba∇dbl2≤C/ba∇dblQu/ba∇dbl2\nY+C/ba∇dblΛ−Nu/ba∇dbl2.\nNote here the compact error is of desired size C=o(h−4s−4N) =o(h−1).\n(2) IfQis bounded by Λsfors∈[0,1\n2], that is, there is C >0such that\n/ba∇dblQu/ba∇dbl2\nY≤ /ba∇dblΛsu/ba∇dbl2\nH\nfor allu∈Hs. ThenQisλ4s-bounded, that is,\n/ba∇dblΛ−1\n2\nhQ∗Qχ(h2P)u/ba∇dbl ≤h−4sC/ba∇dblΛ1\n2\nhu/ba∇dbl.\n(3) IfQis bounded by Λsfors∈(−∞,0), thenQisλ2s-bounded. If additionally\nQ∗Qismicrolocal , in the sense that there are cutoffs φ,ψ∈C∞[0,∞)withφ(s)≡1\nnears= 0,φ(s)≡1nears= 1,φψ= 0such that for all hsmall and all u∈H,\n/ba∇dblφ(h2P)Q∗Qψ(h2P)u/ba∇dblH≤Ch−4s/ba∇dblu/ba∇dblH. (2.5)\nThenQisλ4s-bounded.\n(4) Whens∈(−∞,0), ifQ∗Qis bounded by Λ2sonH2s, that is\n/ba∇dblQ∗Qu/ba∇dblH≤C/ba∇dblΛ2su/ba∇dblH\nuniformly for u∈H2s. ThenQ∗Qis microlocal as in (3), and Qisλ4s-bounded.\nProof.We again use the semiclassicalisation h=λ−1.\n1. Assume Qis classically elliptic with respect to Λs. When 0 ≤s≤1\n2, we have for\nlargeλ, that\n/ba∇dbl(h2P)s/ba∇dbl2≤Ch4s/ba∇dblΛsu/ba∇dbl2≤Ch4s/ba∇dblQu/ba∇dbl2\nY+Ch4s/ba∇dblΛ−Nu/ba∇dbl2WEAKLY ELLIPTIC DAMPING 13\nas desired. When s<0, consider a nonnegative cutoff ψ∈C∞\nc((0,∞)) withψ(1) = 1.\nThen from Lemma 2.4\n/ba∇dblψ(h2P)Λs\nhu/ba∇dbl2≤Ch4s/ba∇dblΛsu/ba∇dbl2≤Ch4s/ba∇dblQu/ba∇dbl2\nY+Ch4s/ba∇dblΛ−Nu/ba∇dbl2\nas desired.\n2. Assume Qis bounded by Λs. Note then Λ−sQ∗:Y→His bounded and we have\n/ba∇dblΛ−sQ∗Qu/ba∇dbl ≤C/ba∇dblΛsu/ba∇dbl.\n2a. When 0 ≤s≤1\n2, with Lemma 2.4, we have\n/ba∇dblΛ−1\n2\nhQ∗Qu/ba∇dbl ≤C/ba∇dblΛ−s\nhQ∗Qu/ba∇dbl ≤Ch−2s/ba∇dblΛ−sQ∗Qu/ba∇dbl ≤Ch−2s/ba∇dblΛsu/ba∇dbl ≤Ch−4s/ba∇dblΛs\nhu/ba∇dbl\nas desired.\n2b. Whens<0, consider a nonnegative cutoff ψ∈C∞\nc((0,∞)) withψ(1) = 1. Note\nQ∗:Y→His bounded and we have with Lemma 2.4,\n/ba∇dblΛ−1\n2\nhQ∗Qψ(h2P)u/ba∇dbl ≤C/ba∇dblQ∗Qψ(h2P)u/ba∇dbl ≤C/ba∇dblΛsψ(h2P)u/ba∇dbl ≤Ch−2s/ba∇dblΛs\nhu/ba∇dbl\nshowing that Qisλ2s-bounded as desired.\n2c. When s <0, assume additionally there are cutoffs φ,ψ∈C0([0,∞)) with\nφ(s)≡1 nears= 0,ψ(s)≡1 nears= 1,φψ= 0 such that for all hsmall and all\nu∈H,\n/ba∇dblφ(h2P)Q∗Qψ(h2P)u/ba∇dblH≤Ch−4s/ba∇dblu/ba∇dblH.\nNow consider\nΛ−1\n2\nhQ∗Qψ(h2P)u= Λ−1\n2\nh(1−φ)(h2P)Q∗Qψ(h2P)u+Λ−1\n2\nhφ(h2P)Q∗Qψ(h2P)u,\nthe second term has the size Ch−4s/ba∇dblu/ba∇dblfrom the assumption. We can estimate the first\nterm. Note 1 −φis supported away from 0. Apply Lemma 2.4to see\n/ba∇dblΛ−1\n2\nh(1−φ)(h2P)Q∗Qψ(h2P)u/ba∇dbl ≤ /ba∇dblΛ−s\nh(1−φ)(h2P)Q∗Qψ(h2P)u/ba∇dbl\n≤Ch−2s/ba∇dblΛ−s(1−φ)(h2P)Q∗Qψ(h2P)u/ba∇dbl ≤Ch−2s/ba∇dblQψ(h2P)u/ba∇dblY\n≤Ch−2s/ba∇dblΛsψ(h2P)u/ba∇dbl ≤Ch−4s/ba∇dblu/ba∇dbl.\nWe then have\n/ba∇dblΛ−1\n2\nhQ∗Qψ(h2P)u/ba∇dbl ≤Ch−4s/ba∇dblu/ba∇dbl\nyielding the desired λ4s-boundedness.\n3. Assume s<0 andQ∗Qis bounded by Λ2s. Pick any cutoffs φ,ψas described in\nStep 2c. Then\n/ba∇dblφ(h2P)Q∗Qψ(h2P)u/ba∇dbl ≤ /ba∇dblQ∗Qψ(h2P)u/ba∇dbl ≤C/ba∇dblΛ2sψ(h2P)u/ba∇dbl ≤Ch−4s/ba∇dblu/ba∇dbl\nuniformly for all Has desired. Thus Q∗Qis microlocal and we apply (3) to see Qis\nλ4s-bounded. /square14 LASSI PAUNONEN, NICOLAS VANSPRANGHE, AND RUOYU P. T. WANG\nRemark 2.7. In the case s <0, Theorem 3(3) may be suboptimal without the mi-\ncrolocality of Q∗Q. The microlocality ( 2.5) forbids the communication between zero\nsections{ρ= 0}and semiclassical characteristics {ρ=h−1}. Note that if Q∗Q=f(P)\nis indeed within the functional calculus of P, thenQ∗Qis automatically microlocal:\nφ(h2P)Q∗Qψ(h2P)u=/integraldisplay∞\n0φ(ρ2)f(ρ2)ψ(ρ2)dEρu= 0, (2.6)\nwheneverφψ= 0. When Pis a self-adjoint nonnegative pseudodifferential operator\nof positive order on a compact smooth manifold without boundary, a ndQ∗Qis the\nmultiplication by a smooth function, the microlocality also holds with the right of\n(2.6) replaced by h∞: see [DZ19, equation (E.2.5)]. In practice, it is easier to check the\nclassical boundedness of Q∗Qand use Theorem 3(4) instead: see Examples 1.10,1.12.\nReferences\n[ABHK18] T. Alazard, P. Baldi, and D. Han-Kwan. Control of water w aves.J. Eur. Math. Soc.\n(JEMS), 20(3):657–745, 2018.\n[ABZ11] T. Alazard, N. Burq, and C. Zuily. On the water-wave equat ions with surface tension.\nDuke Math. J. , 158(3):413–499, 2011.\n[Ala17] T. Alazard. Stabilization of the water-wave equations with su rface tension. Ann. PDE ,\n3(2):41, 2017.\n[Ala18] T. Alazard.Stabilization of gravitywaterwaves. J. Math. Pures Appl. , 114(9):51–84,2018.\n[ALN14] N. Anantharaman, M. L´ eautaud, and S. Nonnenmacher. Sharp polynomial decay rates for\nthe damped wave equation on the torus. Anal. PDE , 7(1):159–214, 2014.\n[AMW23] T. Alazard, J. L. Marzuola, and J. Wang. Damping for fract ional wave equations and\napplications to water waves. Preprint, arXiv:2308.09288, 2023.\n[BS22a] N. Burq and C. Sun. Decay for the Kelvin-Voigt damped wave equation: piecewise smooth\ndamping. J. Lond. Math. Soc. (2) , 106(1):446–483, 2022.\n[BS22b] N.BurqandC.Sun.DecayratesforKelvin-Voigtdampedwa veequationsII:thegeometric\ncontrol condition. Proc. Amer. Math. Soc. , 150(3):1021–1039, 2022.\n[BT10] A. Borichev and Y. Tomilov. Optimal polynomial decay of funct ions and operator semi-\ngroups.Math. Ann. , 347:455–478, 2010.\n[Bur20] N. Burq. Decays for Kelvin-Voigt damped wave equations I: The black box perturbative\nmethod. SIAM J. Control Optim. , 58(4):1893–1905, 2020.\n[CPS+23] R. Chill, L. Paunonen, D. Seifert, R. Stahn, and Y. Tomilov. Non- uniform stability of\ndamped contraction semigroups. Anal. PDE , 16(5):1089–1132, 2023.\n[DK20] K. Datchev and P. Kleinhenz. Sharp polynomial decayrates f or the damped waveequation\nwith H¨ older-like damping. Proc. Amer. Math. Soc. , 148(8):3417–3425, 2020.\n[DP21] F. Dell’Oro and V. Pata. Second order linear evolution equation s with general dissipation.\nAppl. Math. Optim. , 83(3):1877–1917, 2021.\n[DZ19] S. Dyatlov and M. Zworski. Mathematical Theory of Scattering Resonances , volume 200\nofGraduate Studies in Mathematics . American Mathematical Society, 2019.\n[KK23] B. Keeler and P. Kleinhenz. Sharp exponential decay rates f or anisotropically damped\nwaves.Ann. Henri Poincar´ e , 24(5):1561–1595, 2023.\n[Kle19] P. Kleinhenz. Stabilization rates for the damped wave equatio n with H¨ older-regulardamp-\ning.Comm. Math. Phys. , 369(3):1187–1205, 2019.WEAKLY ELLIPTIC DAMPING 15\n[KW22] P. Kleinhenz and R. P. T. Wang. Sharp polynomial decay for po lynomially singular damp-\ning on the torus. Preprint, arXiv:2210.15697, 2022.\n[KW23] P. Kleinhenz and R. P. T. Wang. Optimal backward uniqueness and polynomial stability\nof second order equations with unbounded damping. Preprint, arX iv:2310.19911, 2023.\n[LZ15] Z. Liu and Q. Zhang. A note on the polynomial stability of a weak ly damped elastic\nabstract system. Z. Angew. Math. Phys. , 66:1799–1804, 2015.\n[RSS19] J. Rozendaal, D. Seifert, and R. Stahn. Optimal rates of d ecay for operator semigroups on\nHilbert spaces. Adv. Math. , 346:359–388, 2019.\n[Sta17] R. Stahn. Optimal decay rate for the wave equation on a sq uare with constant damping\non a strip. Z. Angew. Math. Phys. , 68(2), 2017.\n[Zwo12] M.Zworski. Semiclassical analysis , volume138of Graduate Studies in Mathematics . Amer-\nican Mathematical Society, Providence, RI, 2012.\nMathematics and Statistics, Faculty of Information Techno logy and Communica-\ntion Sciences, Tampere University, PO Box 692, 33101 Tamper e, Finland\nEmail address , Lassi Paunonen: lassi.paunonen@tuni.fi\nMathematics and Statistics, Faculty of Information Techno logy and Communica-\ntion Sciences, Tampere University, PO Box 692, 33101 Tamper e, Finland\nEmail address , Nicolas Vanspranghe: nicolas.vanspranghe@tuni.fi\nDepartment of Mathematics, University College London, Lon don, WC1H 0AY,United\nKingdom\nEmail address , Ruoyu P. T. Wang: ruoyu.wang@ucl.ac.uk"    },    {        "title": "1206.6915v3.Tests_of_Lorentz_and_CPT_violation_with_MiniBooNE_neutrino_oscillation_excesses.pdf",        "content": "arXiv:1206.6915v3  [hep-ex]  28 Apr 2014October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nModern Physics Letters A\nc/circlecopyrtWorld Scientific Publishing Company\nTESTS OF LORENTZ AND CPT VIOLATION WITH MINIBOONE\nNEUTRINO OSCILLATION EXCESSES\nTEPPEI KATORI\nLaboratory for Nuclear Science, Massachusetts Institute o f Technology,\nCambridge, MA 02139, U.S.A.\nkatori@mit.edu\nReceived (Day Month Year)\nRevised (Day Month Year)\nViolation of Lorentz invariance and CPT symmetry is a predic ted phenomenon of\nPlanck-scale physics. Various types of data are analyzed to search for Lorentz violation\nunder the Standard-Model Extension (SME) framework, inclu ding neutrino oscillation\ndata. MiniBooNE is a short-baseline neutrino oscillation e xperiment at Fermilab. The\nmeasured excesses from MiniBooNE cannot be reconciled with in the neutrino Standard\nModel (νSM); thus it might be a signal of new physics, such as Lorentz v iolation. We\nhave analyzed the sidereal time dependence of MiniBooNE dat a for signals of the possible\nbreakdown of Lorentz invariance in neutrinos. In this brief review, we introduce Lorentz\nviolation, the neutrino sector of the SME, and the analysis o f short-baseline neutrino\noscillation experiments. We then present the results of the search for Lorentz violation\nin MiniBooNE data. This review is based on the published resu lt1.\nKeywords : MiniBooNE; neutrino oscillation; SME; Lorentz violation .\nPACS: 11.30.Cp, 14.60.Pq, 14.60.St\n1. Spontaneous Lorentz symmetry breaking (SLSB)\nEvery fundamental symmetry needs to be tested, including Loren tz symmetry. The\nbreakdown of Lorentz invariance naturally arises in different scena rios of physics at\nthe Planck scale. For this reason the expected scale of Lorentz-v iolating phenomena\nis more than the Planck mass mP≃1019GeV, or in other words, Lorentz viola-\ntion is expected to be suppressed until at least ≃10−19GeV in our energy scale.\nLorentz symmetry is a fundamental symmetry both in quantum field theory and\ngeneral relativity; the consequence of its violation would be enormo us, and it seems\nit is impossible to establish a self-consistent theory with Lorentz viola tion. However,\nLorentz violation can be incorporated into existing theories by spon taneous break-\ning. In this way, Lorentz-violating terms in the Lagrangian do not co nflict with the\nStandard Model (SM).\nThere are a number of models for spontaneous Lorentz symmetry breaking\n(SLSB),2although the basic idea is the same for all. In the SM, the mass terms\n1October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n2T. Katori\narise from the spontaneous symmetry breaking (SSB) triggered b y a Higgs field\nwith a nonzero vacuum expectation value. In this way, mass terms a re dynamically\ngenerated and they do not exist before SSB. Consequently, the S M does not have\nto violate gauge symmetry where mass terms would necessarily do so .\nFigure 1 illustrates this situation.3The theory starts from perfect symmetry and\nthe vacuum is true null space (Fig. 1a). After the SSB, the vacuum is saturated by a\nscalar Higgs field φ, and the particles obtain mass terms (Fig. 1b). This idea can be\nextended to a vector field (Fig. 1c). The ultra-high-energy theor ies, such as Planck-\nscale physics theories, have many Lorentz vector fields (or, more generally, Lorentz\ntensor fields). When the universe cools, if any of them acquire nonz ero vacuum\nexpectation values, then the vacuum can be saturated with vecto r fields. (Fig. 1d).\nNote that, theoretically, SLSB is conceived to occur earlier than th e SSB of the SM,\nunlike this cartoon. Such vector fields are the background fields of the universe, and\nthey are fixed in space; couplings with the SM fields generate interac tion terms in\nthe vacuum.\n/suppress L =iψγµ∂µ¯ψ+mψ¯ψ+ψγµaµ¯ψ+ψγµcµν∂ν¯ψ+···.\nIn this expression, the coefficients aµandcµνrepresent vacuum expectation\nvalues of vector and tensor fields, and they correspond to backg round fields that fill\nthe universe. The crucial observation is that, since these Lorent z tensors are fixed\nin space and time, they cause direction-dependent physics . In particular, rotation\nof the Earth (period 86164 .1 sec) causes sidereal time dependent physics for any\nterrestrial measurement if the SM fields couple with Lorentz-violat ing background\nfields. Therefore, the smoking gun of Lorentz violation is to find a sid ereal time\ndependence in any physics observable.\n2. What is Lorentz and CPT violation?\nWe introduce Lorentz violation as coupling terms between ordinary S M fields and\nbackground fields in the universe. They are Lorentz scalars of the coordinate trans-\nformation; however since background fields are fixed in space, mot ion of the SM\nparticles generates coordinate-dependent physics.\nThe situation is illustrated in Figure 2. The top cartoon (Fig. 2a) show s our\nsetting: a SM particle is moving in two-dimensional space, from botto m to top, as\nseen by the local observer (Einstein). The space is filled with a hypot hetical Lorentz-\nviolating background field, aµ(depicted by arrows). There are two ways to move\nthis particle from left to right for the local observer: Particle Lore ntz transformation\nand Observer Lorentz transformation.\n2.1.Particle Lorentz transformation\nThe first one is Particle Lorentz transformation, where the motion of a SM particle\nis actively transformed in the fixed coordinate system (Fig. 2b). Sin ce the back-October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 3\nFig. 1. An illustration of spontaneous symmetry breaking (S SB).\nground field is unchanged, as a consequence, a coupling between th e SM particle\nand the background field is not preserved; therefore, one can se e Lorentz violation.\nIn other words, Lorentz violation generally means the Particle Lore ntz violation,\nand it implies direction-dependent physics of SM particles in the fixed c oordinate\nspace.\n2.2.Observer Lorentz transformation\nThe second way is Observer Lorentz transformation, where the c oordinate is in-\nversely transformed (Fig. 2c). In this cartoon, if Einstein (the loc al observer) turns\nhis neck 90◦counterclockwise, the SM particle moves from left to right for the\nlocal observer. However, the background field is also transforme d to the new coor-\ndinate system leaving the coupling with the SM particle unchanged. On e cannot\ngenerate Lorentz violation by Observer Lorentz transformation , because Observer\nLorentz transformation only corresponds to a coordinate trans formation. In other\nwords, coordinate transformations preserve the Lorentz-viola ting effect, and every\nobservers agree with the same Lorentz-violating effect.October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n4T. Katori\nFig. 2. An illustration of Particle Lorentz transformation and Observer Lorentz transformation.\n2.3.CPT violation\nThere is a close link between Lorentz symmetry and CPT symmetry. H ere, “CPT”\nrepresents the combination of charge transformation (C), parit y transformation (P),\nand time reversal (T). It is known that none of these, taken individ ually, is a sym-\nmetry of the SM, but from the CPT theorem4we expect that their combination\nis a perfect symmetry. Since the CPT theorem is based on Lorentz s ymmetry, one\ncan expect CPT violation when Lorentz invariance is broken. This is ma nifest in\nthe appearance of CPT-odd terms in the Lagrangian, which appear as a subset of\nthe terms that break Lorentz invariance. The phase of CPT trans formation is re-\nlated to the number of Lorentz indices, i, transforming under the Particle Lorentz\ntransformation.\nCPT phase = (−1)i.\nIn the SM any Lagrangian is the linear sum of CPT-even, or phase= +1 terms.\nThis is the main consequence of the CPT theorem, and this is why any L agrangian\nis CPT invariant. However, if the theory includes Lorentz violation, it is possible\nthatican be odd. When iis odd number, Lorentz violation causes CPT violation,October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 5\nand this is called “CPT-odd” Lorentz violation. On the other hand, if iis an even\nnumber, the phase of CPT transformation is even, and the theory does not violate\nCPT, even though it contains Lorentz violation. Such Lorentz violat ion is called\n“CPT-even” Lorentz violation. CPT-odd and CPT-even Lorentz vio lation differ\nonly in the number of their Lorentz indices.\n•coefficients of CPT-odd Lorentz-violating terms ( aµ,gµνλ,···)\n•coefficients of CPT-even Lorentz-violating terms ( cµν,κµνλσ,···)\nNote that the interactive quantum field theory necessarily violates Lorentz sym-\nmetry if CPT symmetry is not preserved.5This general theory is consistent with\nthe argument here and Standard-Model Extension (SME), which w e discuss in the\nnext section.\n3. Analysis of Lorentz violation\nLorentz violation is realized as a coupling of SM particles and backgrou nd fields.\nTo specify the components of the Lorentz-violating vector or ten sor fields, the coor-\ndinate system must be specified. Then the general Lagrangian, inc luding all possi-\nble Lorentz-violating terms, is prepared. Finally, using this Lagrang ian, observable\nphysical quantities can be identified.\n3.1.The Sun-centered coordinate system\nThe choice of the coordinate system is arbitrary, since Particle Lor entz violation in\none coordinate system is preserved in another coordinate system through Observer\nLorentz transformation. Nevertheless, in order to compare exp erimental results from\ndifferent experiments in a physically meaningful way, a common frame should be\nused. For this purpose, we need a universal coordinate system (F ig. 3).6The uni-\nversal coordinate system is required to be reasonably inertial in ou r timescale. The\nSun-centered coordinated system is just such a coordinate syst em (Fig. 3a). Here,\nthe rotation axis of the Earth aligns with the orbital axis by tilting the orbital plane\nby 23.4◦, defining the Z-axis. The X-axis points towards the vernal equinox , and\nthe Y-axis completes the right-handed triad. Obviously, we assume Lorentz-violating\nfield is uniform at least the scale of the solar system. This can be just ified in many\nways, for example, we know the weak and the electromagnetic laws a re same in far\nstars through the observation, indicating Lorentz-violating fields are also uniform\nin these scales if they are arisen through the spontaneous breakin g process. Note,\nSun-centered is more suitable than galaxy-centered, because alt hough the galac-\ntic rotation is faster, the galactic rotation takes too long for huma n observation to\nchange the direction, and consequently it cannot help test the viola tion of rotational\nsymmetry. Then the location of the experiment is specified by the Ea rth-centered\ncoordinate system (Fig. 3b). Here, the x-axis points south, the y -axis points east,\nand the z-axis points to the sky from the site of the experiment. Fin ally, local polarOctober 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n6T. Katori\nFig. 3. The coordinate system used by this analysis; (a) first , the motion of the Earth is described\nin Sun-centered coordinates, (b) then the local coordinate s of the experiment site are described in\nEarth-centered coordinates, (c) finally, the direction of t he neutrino beam is described in the local\npolar coordinate system. (d) The time zero is defined when the experiment site is at midnight near\nthe autumnal equinox, in other words, when the large “Y” and s mall “y” axes almost align.\ncoordinates specify the direction of the beam (Fig. 3c). The time ze ro of the side-\nreal time is defined as being the position of the experiment at midnight near the\nautumnal equinox (Fig. 3d).\n3.2.Standard-Model Extension (SME)\nFor the general search for Lorentz violation, the Standard-Mod el Extension\n(SME)7,8is widely used by the community. Various types of data are analyzed\nunder SME,10,11including neutrino oscillation data.6,12,13,14,15The SME is a self-\nconsistent effective field theory including Particle Lorentz violation. In principle,\nSME is an infinite series of all types of interactions, but many analyse s are lim-October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 7\nited to the renormalizable sector, called the minimal SME. For example , under the\nminimal SME, the effective Lagrangian for neutrinos can be written a s,9\nL=1\n2i¯ψAΓµ\nAB↔\nDµψB−¯ψAMABψB+h.c., (1)\nΓν\nAB≡γνδAB+cµν\nABγµ+dµν\nABγ5γµ+eν\nAB+ifν\nABγ5+1\n2gλµν\nABσλµ, (2)\nMAB≡mAB+im5ABγ5+aµ\nABγµ+bµ\nAB+1\n2Hµν\nABσµν. (3)\nHere, theABsubscripts represent Majorana basis flavor space (6 ×6 for con-\nvention). The first term of Eq. 2 and the first and second terms of Eq. 3 are the\nonly nonzero terms in the SM, and the rest of the terms are from th e SME. As we\nsee, these SME coefficients can be classified into two groups (Sec. 2 .3), namely eµ\nAB,\nfµ\nAB,gµνλ\nAB,aµ\nAB, andbµ\nABwhich are CPT-odd SME coefficients, and cµν\nAB,dµν\nAB, and\nHµν\nABwhich are CPT-even SME coefficients.\n3.3.Lorentz-violating neutrino oscillations\nOnce we have a suitable formalism, such as the SME, we are ready to w rite down\nphysical observables. The effective Hamiltonian relevant for the ν−νoscillations\ncan be written,9\n(heff)ab∼1\n|/vector p|[(aL)µpµ−(cL)µνpµpν]ab (4)\nHere, (aL)µ\nab≡(a+b)µ\naband (cL)µν\nab≡(c+d)µν\nab. Note that we drop the neutrino\nmass term since the standard neutrino mass term is negligible for sho rt baseline\nneutrino oscillation experiments, such as MiniBooNE.\nSolutions of this Hamiltonian have very rich physics, but for our purp ose, we\nrestrict ourselves to short-baseline νµ−νe(¯νµ−¯νe) oscillation phenomena. By\nassuming the baseline is short enough compared to the oscillation leng th, theνµ−νe\noscillation probability can be written as follows,16\nPνµ→νe≃L2\n(/planckover2pi1c)2|(C)eµ+ (As)eµsinω⊕T⊕+ (Ac)eµcosω⊕T⊕\n+(Bs)eµsin 2ω⊕T⊕+ (Bc)eµcos 2ω⊕T⊕|2. (5)\nHere,ω⊕stands for the sidereal time angular frequency ( ω⊕=2π\n86164.1rad/s),\nas opposed to the solar time angular frequency ( ω⊙=2π\n86400.0rad/s). The neutrino\noscillation is described by the function of the sidereal time T⊕, with the sidereal time\nindependent amplitude ( C)eµ, and the sidereal time dependent amplitudes, ( As)eµ,\n(Ac)eµ, (Bs)eµ, and (Bc)eµ. Therefore, an analysis of Lorentz and CPT violation in\nneutrino oscillation data involves fitting the data with Eq. 5 to find non zero sidereal\ntime dependent amplitudes.October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n8T. Katori\nIn terms of the coefficients for Lorentz violation, these amplitudes are explicitly\ngiven by,\n(C)eµ= (C(0))eµ+E(C(1))eµ\n(As)eµ= (A(0)\ns)eµ+E(A(1)\ns)eµ\n(Ac)eµ= (A(0)\nc)eµ+E(A(1)\nc)eµ (6)\n(Bs)eµ=E(B(1)\ns)eµ\n(Bc)eµ=E(B(1)\nc)eµ\n(C(0))eµ= (aL)T\neµ+ˆNZ(aL)Z\neµ\n(C(1))eµ=−1\n2(3−ˆNZˆNZ)(cL)TT\neµ+ 2ˆNZ(cL)TZ\neµ+1\n2(1−3ˆNZˆNZ)(cL)ZZ\neµ\n(A(0)\ns)eµ=ˆNY(aL)X\neµ+ˆNX(aL)Y\neµ\n(A(1)\ns)eµ=−2ˆNY(cL)TX\neµ+ 2ˆNX(cL)TY\neµ+ 2ˆNYˆNZ(cL)XZ\neµ−2ˆNXˆNZ(cL)Y Z\neµ\n(A(0)\nc)eµ=−ˆNX(aL)X\neµ+ˆNY(aL)Y\neµ (7)\n(A(1)\nc)eµ= 2ˆNX(cL)TX\neµ+ 2ˆNY(cL)TY\neµ−2ˆNXˆNZ(cL)XZ\neµ−2ˆNYˆNZ(cL)YZ\neµ\n(B(1)\ns)eµ=ˆNXˆNY((cL)XX\neµ−(cL)Y Y\neµ)−(ˆNXˆNX−ˆNYˆNY)(cL)XY\neµ\n(B(1)\nc)eµ=−1\n2(ˆNXˆNX−ˆNYˆNY)((cL)XX\neµ−(cL)Y Y\neµ)−2ˆNXˆNY(cL)XY\neµ\nHere, the ˆNX,ˆNY, and ˆNZare the direction vectors of the neutrino beam\nin the Sun-centered coordinates (Sec. 3.1). The components are described with a\nco-latitude χof detector location in the Earth-centered system (Fig. 3b), and the\nzenith and azimuthal angles θandφof the local beam system. (Fig. 3c):\n\nˆNX\nˆNY\nˆNZ\n=\ncosχsinθcosφ+ sinχcosθ\nsinθsinφ\n−sinχsinθcosφ+ cosχcosθ\n (8)\nFor the antineutrino oscillation analysis, one needs to switch the sign ofaL\naccording to CPT-odd nature of CPT-odd coefficients ( aL→ −aL).\nIn the reality of the analysis, fitting five parameters using Eq. 5 is no t easy.\nTherefore, we also consider the following three-parameter model, by setting ( Bs)eµ\nand (Bc)eµto be zero by hand. This model, Eq. 9, can be motivated, for example ,\nby assuming nature only has CPT-odd SME coefficients.\nPνµ→νe≃L2\n(/planckover2pi1c)2|(C)eµ+ (As)eµsinω⊕T⊕+ (Ac)eµcosω⊕T⊕|2. (9)October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 9\nFig. 4. L-E diagram with the νSM (two straight dotted lines) and the Puma model (two dashed\nand solid curves).\n3.4.Lorentz violation as an alternative neutrino oscillation m odel\nBecause of the unconventional energy dependence of Lorentz- violating terms in the\nHamiltonian ( E0andE1), naively, its energy dependence on neutrino oscillations\nis different from one expected from the three massive neutrino mod el (so-called\nνSM). However, it is also possible to “mimic” neutrino mass-like energy d ependence\n(E−1) using Lorentz violating terms only.17,18,19,20There is a chance that such\ntypes of models could be correct, because we currently have some tensions in the\nworld neutrino oscillation data. For this purpose, it would be helpful t o show the\nphase space of neutrino oscillations in a model-independent way. The L-E diagram\n(Fig. 4) shows world’s neutrino oscillation experiments mapped with th eir energy\nand baseline.19\nThe curves in Figure 4 represent the oscillation length. For example, massive\nneutrino oscillation solutions (= L/E oscillatory dependence) are represented by\nthe lineL∝E. Here, data are consistent with two L/E neutrino oscillations, the\n¯νedisappearance measurement at the KamLAND experiment (2 to 8 Me V), and\ntheνµand ¯νµdisappearance measurements at the long-baseline and atmospher ic\nneutrino experiments (0.3 to 10 GeV). Therefore, we know there a reat least two\nsegments with L ∝E on the L-E diagram . Nevertheless, our knowledge outside of\nthese segments is limited. There are proposed models, such as the P uma model,19,20\nwhich have L/E oscillatory dependencies in these energy ranges. So, here, the mo d-October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n10T. Katori\nFig. 5. An overview of the MiniBooNE experiment. The top left picture shows the Fermilab site,\nincluding the Booster. The top right picture is the tank of th e MiniBooNE detector. The bottom\ncartoon shows the sketch of the Booster Neutrino Beamline (B NB).\nels are consistent with current data,abut outside of these energy ranges they have\ncompletely different dependencies. These alternative models are int eresting because\nthey have a chance to reproduce short-baseline anomalies, such a s the MiniBooNE\noscillation signals,24,25which we discuss in the next sections.\n4. The MiniBooNE experiment\nThe MiniBooNE experiment is a short-baseline neutrino oscillation expe riment at\nFermilab, USA (2002-2012). Its primary goal is to find νµ→νe(¯νµ→¯νe) oscil-\nlations with an ∼800 (600) MeV neutrino (antineutrino) beam with an ∼500 m\nbaseline. Figure 5 shows the overview of the MiniBooNE experiment.26\n4.1.Booster neutrino beamline (BNB)\nMiniBooNE uses neutrinos (antineutrinos) from the Booster neutr ino beamline\n(BNB),27which is illustrated in Figure 5, bottom. The 8 GeV protons, the “pri-\naRecent reactor neutrino disappearance oscillation result s do not support Puma model21,22,23October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 11\nmary” beam, are extracted from the Booster and steered to collid e with the beryl-\nlium target in the magnetic focusing horn. The collision of protons and the target\nmakes a shower of mesons, the “secondary” beam; and the toroid al field created by\nthe horn focuses π+(π−) for neutrino (antineutrino) mode. At the same time it\ndefocusesπ−(π+), which create backgrounds. The decay-in-flight of π+(π−) create\nνµ(¯νµ), the “tertiary beam”. The consequent muon neutrinos (muon an tineutrinos)\nare a wideband beams peaked in around 800 (600) MeV.\n4.2.MiniBooNE detector\nThe MiniBooNE detector is located 541 m to the north of the target.28The detec-\ntor is a 12.2 m diameter spherical Cherenkov detector, filled with mine ral oil. An\noptically separated inner black region is covered with 1,280 8-inch PMT s, and an\nouter white region has 240 8-inch PMTs which act as a veto (Fig. 5, to p right). The\nblack inner cover helps to reduce reflections, so that one can reco nstruct particles\nfrom the Cherenkov light more precisely; the outer white cover help s to enhance\nreflections, so that a smaller number of veto PMTs can cover a large r area.\n4.3.Events in detector\nThe time and charge information of the Cherenkov ring from the cha rged particle is\nused to estimate particle type, energy, and direction.29For example, an electron-like\ntrack is characterized by a fuzzy-edged Cherenkov ring, compar ed with a sharp-\nedged muon-like Cherenkov ring. Based on particle type hypothesis , the track fit-\nter estimates a particle energy and direction. Figure 6 shows typica l particles and\ntheir characteristic tracks, Cherenkov rings, and event candida tes from the event\ndisplay.26\nA variety of track fitters are developed to measure the kinematics of specific types\nof interactions.30,31,32,33Among them, the most important reaction for the oscilla-\ntion analysis is the charged current quasielastic (CCQE) interaction ,34,35which is\ncharacterized by one outgoing charged lepton (at the BNB energy , protons seldom\nexceed a Cherenkov threshold of ∼350 MeV kinetic energy). If a charged lepton\nis detected from the CCQE interaction, one can reconstruct the n eutrino energy,\nEQE\nν, by assuming the target nucleon is at rest and the interaction type is truly\nCCQE34(QE assumption).\nEQE\nν=2(Mn−B)Eµ−((Mn−B)2+m2\nµ−M2\np)\n2·[(Mn−B)−Eµ+/radicalig\nE2µ−m2µcosθµ]. (10)\nHere,Mn,Mp, andmµare the neutron, proton, and muon masses, Eµis the\ntotal muon energy, θµis the muon scattering angle, and Bis the binding energy of\ncarbon. Ability to reconstruct neutrino energy is essential for ne utrino oscillation\nphysics, since neutrino oscillations are function of neutrino energy .October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n12T. Katori\nFig. 6. The particle type and characteristics. From left to r ight, interaction types, characteristics\nof tracks and Cherenkov rings, and event displays of candida te events.\nIt is interesting to note that many of the neutrino interaction cros s sections\nmeasured by MiniBooNE are at a higher rate and harder spectrum th an historically\nknown values and disagree with interaction models tuned with old bubb le chamber\ndata.36This fact triggered the development of a new class of neutrino inter action\nmodels,37,38,39,40,41,42,43mostly by including nucleon correlations. These new models\neven question how to reconstruct neutrino energy44,45with the QE assumption,\ntraditionally done in all Cherenkov-type detectors. Therefore, s imilar to other fields\n(e.g., cosmology), the further study of neutrino physics just incr eases the number\nof mysteries!\n4.4.Oscillation analysis\nThe signature of the νµ→νe(¯νµ→¯νe) oscillation is the single, isolated electron-like\nCherenkov ring produced by the CCQE interaction.\nνµoscillation−→νe+n→µ++p ,\n¯νµoscillation−→ ¯νe+p→µ−+n .\nNote, since MiniBooNE was not magnetized, electrons and positrons were notOctober 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 13\ndistinguished. Thus, the analysis of neutrino mode and antineutrino mode is rea-\nsonably parallel, except for some differences in handling background s.\nThere are two backgrounds which contribute equally to our signals. The first\nclass is the “misID”, and this is dominated by a single gamma ray from th e neutral\ncurrent channels, such as radiative ∆ decay and π◦production, where only one\ngamma ray is detected. It is essential to constrain our misID backg round predictions\non those channels using measurements of controlled samples. For t his purpose, we\nmeasured the neutral current π◦production rate in situ , and the result is used to\ntuneπ◦kinematics in our simulation. We also used the measured π◦production\nuncertainty in our simulation.46\nAnother major background is the intrinsic background, namely νe(¯νe) as beam\ncontamination. Although they can be predicted by the beamline simula tion, and\nare expected to be <0.5%, this is a critical background for the ∼0.5% appearance\noscillation search carried out by experiments, such as MiniBooNE. Ag ain,in situ\nmeasurements largely help to reduce errors in the simulation. For ex ample, the\nmajority of νe(¯νe) are from µ-decay in the beamline but one can constrain their\nvariations from the measured νµ(¯νµ) rate, where both νeandνµ(¯νeand ¯νµ) are\nrelated through the π+(π−) decay (for π+,π+→νµµ+,µ+→¯νµνee+). Another\nmajor source of νe(¯νe) is kaon decay. MiniBooNE utilizes SciBooNE experiment\ndata to constrain it.47SciBooNE is a tracker for the neutrino cross section mea-\nsurement, located upstream of MiniBooNE, and their precise track measurement\nis sensitive to primary mesons in the beamline. More specifically, K-dec ay origin\nneutrinos are higher energy, and tend to make multiple tracks in the SciBooNE\ndetector. This information provides the constraint on the errors on predictions of\nνe(¯νe) from K-decay in MiniBooNE.\nAfter the evaluation of all backgrounds, MiniBooNE finds a signal-to -background\nratio of roughly one to three, with expected oscillation parameters .\n4.5.MiniBooNE neutrino mode oscillation result\nFor neutrino mode data analysis,24we use 6.46×1020protons on target (POT) data.\nAfter all cuts, an excess of νecandidate events in the “low-energy” region (200 <\nEQE\nν(MeV)<475) was observed (Fig. 7). A total of 544 events are observed in this\nregion, as compared to the predicted 409 .8±23.3(stat.)±38.3(syst.). Interestingly,\nthis excess does not show the expected L/E energy dependence of a simple two\nmassive neutrino oscillation model. Therefore, this excess might be n ew physics.\n4.6.MiniBooNE antineutrino mode oscillation result\nFor the antineutrino mode analysis,25we use 5.66×1020POT data. Here, Mini-\nBooNE not only observed an excess in the low energy region, an exce ss in the\n“high-energy” region (475 < EQE\nν(MeV)<1300) was also observed (Fig. 8). There-\nfore, in the “combined” region (200 <EQE\nν(MeV)<1300), MiniBooNE observed 241\n¯νecandidate events as compared to the predicted 200 .7±15.5(stat.)±14.3(syst.).October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n14T. Katori\nFig. 7. MiniBooNE neutrino mode νeappearance search result. The data points with errors are\nshown together with predicted backgrounds.\nFig. 8. MiniBooNE antineutrino mode ¯ νeappearance search result. The data points with errors\nare shown together with predicted backgrounds.\nAgain, the νSM does not predict this excess, which therefore has the potentia l to\nbe new physics.\n5. Lorentz-violating neutrino oscillation analysis in Min iBooNE\nIn this section we follow the procedure described in Sec. 3.\n5.1.The coordinate system\nFirst, we make the time distribution of neutrino events from a stand ard GPS time\nstamp. The analysis is based on the sidereal time distribution, but we also use\nthe local solar time to check time-dependent systematics. The coo rdinate system is\ndescribed in Fig. 3. The local coordinates of the BNB are specified by three angles,October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 15\nrun number4000 6000 8000 10000 12000 14000 16000 18000day\n050100150200250300350\n2003\n2004\n20052007\n2008\nlocal solar time (sec) 0 10000 20000300004000050000600007000080000Events/bin\n05101520253035\nlocal solar time (sec)0 1000020000300004000050000600007000080000sidereal time (sec)\n01000020000300004000050000600007000080000\nsideal time (sec)0 1000020000300004000050000600007000080000Events/bin\n05101520253035\nFig. 9. Time distribution of the MiniBooNE low energy excess in neutrino mode. The top left\nfigure shows the day and run number of each event. The bottom le ft plot is the scatter plot of each\nevent with sidereal and local solar time; projections of the distribution onto each axes are show to\nthe right.\nthe co-latitude χ= 48.2◦, the polar angle θ= 89.8◦, and the azimuthal angle\nφ= 180.0◦.\n5.2.Time dependent systematics\nThe signature of the Lorentz violation in this analysis is the sidereal t ime-dependent\nneutrino oscillations. However, any solar time distribution may show u p in the\nsidereal time distribution, if data taking is not completely continuous . This is the\ncase for MiniBooNE, as you see from top left plots of Figure 9 and 10. There were\noccasional shutdowns of the accelerator, and data taking was no t continuous over\nany of the years. Therefore, we need to check time-dependent s ystematics, which\ninclude any day-night effects of the detector and the beam. For ex ample, electronics\nmay have a higher efficiency in cold nighttime than hot daytime, etc. Th e quick way\nto evaluate all of these effects together is to utilize the neutrino da ta itself. The high-\nstatisticsνµCCQE35data are used to evaluate day-night variations of the neutrino\ninteraction rate. Figure 11, above, shows the variation of νµCCQE events in local\nsolar time.1As you see, the νµCCQE sample shows day-night variations. These\noriginate from the variation of the beam. The number of POT shows s inusoidal\ncurve, namely maximum POT is at midnight and decreases towards the daytime:\npresumably the beamline is occasionally accessed in the daytime, and c umulativeOctober 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n16T. Katori\nrun number13000140001500016000170001800019000200002100022000day\n050100150200250300350\n2006\n2007 2008\n2009\n2010\nlocal solar time (sec) 0 10000 20000300004000050000600007000080000Events/bin\n0246810121416\nlocal solar time (sec)0 1000020000300004000050000600007000080000sidereal time (sec)\n01000020000300004000050000600007000080000\nsideal time (sec)0 1000020000300004000050000600007000080000Events/bin\n024681012141618\nFig. 10. Time distribution of the MiniBooNE excess in antine utrino mode. The keys are same\nwith Figure 9.\nhuman behavior over several years make a nice sinusoidal curve in P OT, and in\nneutrino rate! After correcting POT, νµCCQE event shows flat (Fig. 11, bottom).\nThen, the question is how much the solar time variation of the POT act ually\nshows up in the oscillation candidate sidereal variation. We found tha t a small\nvariation persists in the νµCCQE sidereal distribution. We evaluated the impact of\nthis on our analysis by correcting the POT variation event by event in νecandidate\ndata. It turned out that, because of the low statistics of oscillatio n candidate events\nand the smearing effect from solar time distribution to sidereal time d istribution,\nthe correction only had a negligible effect. Thus, we decided to use un corrected\nevents in later analysis.\nSimilar study is needed for the antineutrino mode, but the conclusion is likely\nto be the same due to lower statistics of ¯ νesample.\n5.3.Unbinned likelihood fit\nTo find best-fit (BF) parameters to describe MiniBooNE νe(¯νe) candidate data with\nLorentz violation, we employed an unbinned likelihood fit. The likelihood f unction\nΛ has following expression with two probability density functions (PDF s).\nΛ =e−(µs+µb)\nN!N/productdisplay\ni=1(µsFi\ns+µbFi\nb)×1√2πσb2exp/parenleftigg\n−(¯µb−µb)2\n2σb2/parenrightigg\n(11)October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 17\nlocal solar time (sec) 0 10000 20000 30000 40000 50000 60000 70000 80000CCQE eventsµν\n260027002800290030003100\nlocal solar time (sec) 0 10000 20000 30000 40000 50000 60000 70000 80000CCQEµν POT corrected \n0.940.960.9811.021.041.06\n/dof=53.9/492χ\nFig. 11. The top histogram shows the νµCCQE local solar time distribution, with the arbitrary\nnormalized fit curve extracted from the POT local solar time d istribution in the same time period\n(solid curve). The bottom plot shows the same events after co rrecting for variations in POT. The\nχ2of this distribution with a flat hypothesis is 53 .9/49 (29.3% compatibility).\nN, the number of observed candidate events\nµs, the predicted number of signal events, the function of fitting pa rameters\nµb, the predicted number of background events, floating within 1 σrange\nFs, the PDF for the signal, the function of sidereal time and fitting par ameters\nFb, the PDF for the background, not the function of the sidereal tim e\nσb, the 1σerror on the predicted background\n¯µb, the central value of the predicted total background events\nThis method is suitable because it has the highest statistical power f or a low-\nstatistics sample. The computation is performed to maximize this fun ction. But in\nthe reality, we maximize the log likelihood function. The maximum log likeliho od\n(MLL) point provide the best fit (BF) parameter set. Then the con stant surface of\nlog likelihood function provide the errors. Neither the neutrino nor t he antineutrino\nmode data allow us to extract errors if we fit all five parameters at o nce (Eq. 5),\ndue to the high correlation of parameters. Therefore, we focus o n three-parameter\nfit (Eq. 9) to discuss errors and limits. Since the five-parameter fit is quantitatively\nsimilar to the three-parameter fit, we will focus the discussion of th ese results on\nthe three-parameter fit.\n6. Results\nThis section describes the results of the fits.October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n18T. Katori\nsidereal time (sec)0 10000 20000 30000 40000 50000 60000 70000 80000-osc candidate eventsν1020304050607080 data\nflat solution\nbackground\n3 parameter fit\n5 parameter fit\nFig. 12. The fit results for the neutrino mode low energy regio n. The plot shows the curves\ncorresponding to the flatsolution (dotted), three-paramet er fit(solid), and five-parameter fit(dash-\ndotted), together with binned data (solid marker). Here the fitted background is shown as a dashed\nline, however the best fit for the background is 1.00 (the defa ult prediction).\n6.1.Fit result of neutrino mode data\nFigure 12 shows the neutrino mode low energy region fit results.1The solid and\ndash-dotted lines are the best fit curves from three- and five-pa rameter fit. The\ndotted line is the flat solution. Since the fit is dominated by the ( C)eµparameter\n(sidereal time-independent amplitude), both three- and five-par ameter fit solutions\nhave small time-dependent amplitudes and look like the flat solution. T he details of\nthe fit results were tested by using a fake data set. We construct ed a fake data set\nwith signal to evaluate errors of fit parameters of three-parame ter fit ((C)eµ, (As)eµ,\nand (Ac)eµ). Then we constructed a fake data set without signal, to evaluate the\ncompatibility with flat solution over three parameter fit solution by ∆ χ2method.\nIt turns out data is compatible with flat solution over a 26 .9%, and it concludes νe\ncandidate data is consistent with flat.\n6.2.Fit result of antineutrino mode data\nFigure 13 shows the fit result for combined energy region in antineut rino mode,\nwhich is analogous with Figure 12.1For antineutrino mode, the combined region is\nmore interesting due to lower statistics. The fit result is more curiou s in antineutrino\nmode, because now the ( C)eµparameter no longer significantly deviates from zero,\nbut the fit favors a nonzero solution for the ( As)eµand (Ac)eµparameters. The\nfit solutions look more different from the flat distribution. We again co nstructed\na fake data set to find the significance of this solution, and it turns o ut that the\ncompatibility with the flat solution is now only 3 .0%. Although this is interesting,\nhowever the significance is not high enough to claim discovery.October 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\nTests of Lorentz and CPT violation with MiniBooNE neutrino o scillation excesses 19\nsidereal time (sec)0 10000 20000 30000 40000 50000 60000 70000 80000-osc candidate eventsν510152025303540data\nflat solution\nbackground\n3 parameter fit\n5 parameter fit\nFig. 13. The fit results for the antineutrino mode combined re gion. Notations are the same as\nFig. 12. The fitted background is shown as a dashed line, and th e best fit for the background is\n0.97 (3% lower than the default prediction).\n6.3.Summary of the fit\nIn this analysis, three parameters, ( C)eµ, (As)eµ, and (Ac)eµfor neutrino and an-\ntineutrino mode are obtained by fits. Fits provide the BF values of ab ove three\nparameters, as well as 1 σerrors and 2 σlimits. The expressions of these parameters\nare found in Eqs. 6, 7, and 8. From the 2 σlimits we obtain we estimate the limit of\neach SME coefficient by setting all but one of SME coefficient as nonze ro. Table 1\nis the result. As you see, the limits of the SME coefficients from the Min iBooNE\ndata are of the order of 10−20GeV (CPT-odd), and 10−20to 10−19(CPT-even).\nSimilar analysis have been done for the LSND data.49However, these limits exclude\nany SME coefficients needed to explain the LSND data. Therefore, a simple picture\nusing Lorentz violation to explain both LSND and MiniBooNE leaves some tension,\nand a mechanism to cancel the Lorentz-violating effect at high ener gy17,18,19,20is\nneeded.\nConclusion\nLorentz and CPT violation is a predicted signal at the Planck scale, an d there\nare worldwide efforts to search for it. Neutrino oscillation is a natura l interfer-\nometer, and the sensitivity to Lorentz violation is comparable with pr ecise opti-\ncal experiments. The MiniBooNE short-baseline neutrino oscillation e xperiment at\nFermilab observed a νe(¯νe) candidate event excess from a νµ(¯νµ) beam. These\nexcesses are not understood by νSM; therefore, they might be the first signal of\nnew physics. Since Lorentz-violation-motivated phenomenological neutrino oscilla-\ntion models, such as the Puma model, predict an oscillation signal for t he Mini-\nBooNE, it is interesting to check the sidereal variation of MiniBooNE o scillation\ncandidate data. The analysis found that neutrino mode data prefe r a sidereal time\nindependent solution, and the data is compatible with a flat distributio n over 26.9%.\nThe antineutrino mode prefers a sidereal time dependent solution, and the data isOctober 26, 2018 10:27 WSPC/INSTRUCTION FILE LV˙review˙v5\n20T. Katori\nTable 1. List of SME coefficient limits, derived from 2 σlimits of fitting parameters, setting\nall but one of the SME coefficients to be zero.\nCoefficient e µ(νmode low energy region) e µ(¯νmode combined region)\nRe(aL)Tor Im(aL)T4.2×10−20GeV 2.6 ×10−20GeV\nRe(aL)Xor Im(aL)X6.0×10−20GeV 5.6 ×10−20GeV\nRe(aL)Yor Im(aL)Y5.0×10−20GeV 5.9 ×10−20GeV\nRe(aL)Zor Im(aL)Z5.6×10−20GeV 3.5 ×10−20GeV\nRe(cL)XYor Im(cL)XY— —\nRe(cL)XZor Im(cL)XZ1.1×10−196.2×10−20\nRe(cL)Y Zor Im(cL)Y Z9.2×10−206.5×10−20\nRe(cL)XXor Im(cL)XX— —\nRe(cL)Y Yor Im(cL)Y Y— —\nRe(cL)ZZor Im(cL)ZZ3.4×10−191.3×10−19\nRe(cL)TTor Im(cL)TT9.6×10−203.6×10−20\nRe(cL)TXor Im(cL)TX8.4×10−204.6×10−20\nRe(cL)TYor Im(cL)TY6.9×10−204.9×10−20\nRe(cL)TZor Im(cL)TZ7.8×10−202.9×10−20\ncompatible with a flat solution only 3 .0%, making this solution very interesting;\nhowever, the statistical significance is not high enough to claim as ev idence. Since\nthe data set we used for this analysis is about ∼50% of the total antineutrino mode\ndata set, reanalysis including full data set may increase the significa nce. Finally,\nfrom the fits, we extract limits of each minimal SME coefficient. The re sults from\nMiniBooNE leave tension with LSND under the simple Lorentz violation mo tivated\nmodel.\nAcknowledgments\nI thank Jorge D´ ıaz for valuable comments, I also thank Janet Conr ad, Josh Spitz,\nBen Jones, and Clemmie Jones for their careful reading of this manu script. This\nwork is supported by NSF PHYS-084784.\nReferences\n1. A. A. Aguilar-Arevalo et al., arXiv:1109.3480v2 [hep-ex].\n2. For example, see, V. A. Kosteleck´ y and S. Samuel, Phys. Re v. D39, 683 (1989).\n3. Ralf Rehnert, [arXiv:hep-ph/0611177].\n4. R..F. StreaterandA.S.Wightman, PCT, Spin and Statistics, and All That ,Princeton\nUniversity Press(2000).\n5. O. W. Greenberg, Phys. Rev. Lett, 89, 231602 (2002).\n6. L. B. Auerbach et al., Phys. Rev. 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The pathological s olutions of the 1-particle theory may be removed\nby expanding the latter equation in powers of τ:=q2/6πm. The radiation-induced change in entropy is\nexplored, and its physical origin is discussed. As a simple d emonstration of the theory, the radiative damping\nrate of longitudinal plasma waves is calculated.\n1 Introduction\nThe question of how a charged particle interacts with its own radiatio n field remains unclear, despite intensive\ntheoreticalinvestigationsoverthelastcentury. Until recently, thisissuehasbeen largelyoftheoreticalinterestas\nin current facilities the self-force is small in comparison to the Loren tz force due to the applied fields. However,\nthe advent of new ultra-high intensity laser facilities requires that r adiation reaction effects be taken seriously.\nFor instance, the Extreme Light Infrastructure (ELI) [1] is exp ected to operate with intensities exceeding 1023\nW cm−2, and electron energiesin the GeV range, at which level the radiation reactionforcebecomes comparable\nto and can even exceed the Lorentz force.\nThe problem was first addressed by Lorentz [2] and Abraham [3], who introduced a third order differential\nequation to describe the trajectory of a radiating particle. This wo rk was later generalized to the relativistic\nregime by Dirac [4], leading to what is now known as the Abraham-Loren tz-Dirac equation or, more simply, the\nLorentz-Dirac equation,\n¨Ca=−q\nmFab˙Cb+τ∆ab...\nCb, (1)\nfor the components {Ca}of the worldline C:λ/ma√sto→xa=Ca(λ) of an electron of charge qand mass min an\nelectromagnetic field Fabexpressed in inertial coordinates ( xa) on Minkowski spacetime Mwith metric tensor\nη,\nη=−dx0⊗dx0+dx1⊗dx1+dx2⊗dx2+dx3⊗dx3. (2)\nτ:=q2/6πm≃10−23s is the characteristic time of the electron1, ∆ab:=δa\nb+˙Ca˙Cbis the˙C-orthogonal\nprojection, and an overdot indicates differentiation with respect t o proper time λ,\nηab˙Ca˙Cb=−1, (3)\nwhere [ηab] = diag( −1,1,1,1). We work in Heaviside-Lorentz units with c= 1, and raise and lower indices with\nηab. Latin indices run from 0 to 3, and the Einstein summation convention is used.\n∗Department of Physics, SUPA and University of Strathclyde, Glasgow, G4 0NG, UK\n†Department of Physics, Lancaster University, Lancaster, L A1 4YB, UK and Cockcroft Institute, Daresbury, WA4 4AD, UK\n1For definiteness we consider electrons, though the formalis m could be applied to other charged particles.\n1Equation (1) has proved highly controversial. The difficulties stem fr om the fact that it contains derivatives\nof the acceleration, so specifying the initial position and velocity is no t sufficient to determine the solution.\nOn the other hand, a generic specification of the initial acceleration leads to exponentially growing proper\nacceleration,\n/radicalBig\n¨Ca¨Ca∼eλ/τ, (4)\neven in the absence of applied forces. Such ‘runaway solutions’ are clearly at odds with our observations, and\nso must be eliminated.\nThe solution proposed by Dirac was to use the elimination of the nonph ysical solutions, rather than the\ninitial acceleration, as part of the initial data, effectively convertin g the Lorentz-Dirac equation into a second\norder integro-differential equation,\n¨Ca(λ) =/integraldisplay∞\n0Ka(λ+ατ)e−αdα,\nKa:=−q\nmFab˙Cb−τ¨Cb¨Cb˙Ca. (5)\nEquation (5) requires the acceleration at a given time to depend on t he force applied at all subsequent times,\nand so is acausal. However, the acausal effects are exponentially d amped for times /greaterorsimilarτinto the future, and so\nin practice are unobservable.\nDespitethemanydistinctderivationsoftheLorentz-Diracequatio n[4–11], theproblemsofrunawaysolutions\nand acausality have led many researchers to propose alternative d escriptions of radiation reaction [12–15]. The\nmost widely adopted of these is the Landau-Lifshitz equation [16],\n¨Ca=−q\nmFa\nb˙Cb−q\nmτ/bracketleftbig\n∂dFa\nb˙Cb−q\nm∆a\nbFb\ncFc\nd/bracketrightbig˙Cd, (6)\nobtained by perturbing (1) about the Lorentz force and neglectin g terms higher than first order in τ. It has\nbeen argued by Spohn [17] that the solutions approximated by (6) a re precisely those corresponding to Dirac’s\nasymptotic condition (5). Further, it is often claimed [18] that effec ts neglected in the derivation of the LD\nequation – such as spin and quantum effects – are of order τ2, so that (6) is as accurate as (1). Though\nplausible in the case of a single radiating electron, this claim remains unp roven, and it is more reasonable to\nconsider (6) an approximation to (1). There has been some interes t recently in exploring the validity of this\napproximation [19,20].\nIn practice, radiation reaction is unlikely to ever be observed in the c ontext of a single radiating particle.\nHowever, modern laser facilities accelerate electron bunches with c harge of the order of 10pC, containing 108\nparticles, and it follows that an appropriate description of radiation reaction is not the one-particle equation\nof motion, but rather a kinetic theory. A kinetic theory based on (6 ) has appeared in the literature [21–23].\nHowever, it is not clear a priori that the approximations leading to (6) are appropriate for describ ing a large\nnumber of interacting electrons.\nThe Vlasov equation describes a large collection of particles as a cont inuum. Charged continua typically do\nnot suffer from the pathologies associated with radiating point part icles, and since the Maxwell-Vlasov system\nsatisfiesthe energyand momentum conservationlawsthat led to (1 ) it mayseem that it doesnot need modifying\nto describe radiation reaction. However, the continuum comprises ‘particles’ with infinitesimal mass and charge,\n(q,m)→0 withq/mfixed, in which limit (1) becomes the usual Lorentz force equation. T o describe a collection\nof real electrons with finite qandm, the Vlasov equation must be modified so that elements of the contin uum\nfollow trajectories of the Lorentz-Dirac equation (1). Such a kine tic equation has been introduced in [24];\nhowever, in that work the physical phase space was not identified, so the 1-particle distribution is required to\nbe distributional (in the sense of Schwartz). As well as impeding the physical interpretation, this has led to a\nnumber of errors, for example in calculating the rate of change of e ntropy.\nIn this article, we introduce a kinetic theory for electrons obeying ( 1), with the 1-particle distribution a\nregular function on the physical phase space, and explore some of its consequences.\n22 Kinetic theory with radiation reaction\nThe usual approach to obtaining an equation for a large collection of particles from the equation of motion for\na single particle is to consider a 1-particle distribution on a subspace o fTM. This distribution is then taken to\nbe constant along the lifts of particle orbits to TM, which may be obtained as integral curves of a vector field\nonTM. This prescription is not quite general enough to incorporate radia tion reaction, since the Lorentz-Dirac\nequation is third order. The approach adopted in the present artic le is to consider a 1-particle distribution on\na subspace of the bundle with total space\nTM⊕TM=/uniondisplay\np∈MTpM⊕TpM (7)\nand whose fibres are two copies of the tangent space, one repres enting 4-velocity and the other representing\n4-acceleration, with (˙ xa) and (¨xa) the induced coordinates on the first and second copies of the tan gent space.\nWe are interested in the physical subspace Q ⊂TM⊕TMgiven by\nQ={(x,˙x,¨x)∈TM⊕TM|ϕ1= 0, ϕ2= 0,˙x0>0} (8)\nwhere\nϕ1=1\n2(ηab˙xa˙xb+1), (9)\nϕ2=ηab˙xa¨xb. (10)\nWorldlines parametrized by proper time satisfy ϕ1= 0, from which it follows that their 4-velocity and 4-\nacceleration should be orthogonal as encoded in ϕ2= 0. It is straightforward to show that solutions to the\nLorentz-Dirac equation are integral curves of the vector field L,\nL:= ˙xa∂\n∂xa+ ¨xa∂\n∂˙xa+/bracketleftbig\n¨xb¨xb˙xa+τ−1(¨xa+q\nmFab˙xb)/bracketrightbig∂\n∂¨xa, (11)\nonTM⊕TM.\nSinceLis tangent to Q(Lϕ1=ϕ2andLϕ2= 2ηab¨xa¨xbϕ1+τ−1ϕ2) it can be expressed as the push-forward\nof a vector field LQonQ,L=ι∗LQ, whereι:Q֒→TM⊕TMis the inclusion map. The Liouville vector field\nLQis calculated explicitly below.\nThe 1-particle distribution must satisfy the Vlasov equation, which s tates that the 1-particle distribution\nis preserved under the flow of the Liouville vector field. However, it is important to note that it is not the\nparticle density fwhich must be preserved along the flow, but rather the particle dist ribution fω, whereωis\na non-vanishing top-dimensional form. Thus, the Vlasov equation m ay be cast as\nLLQ(fω) = 0 (12)\nwhereLXis the Lie derivative with respect to X,fis a 0-form on Q, andωis a non-vanishing 10-form on Q.\nIn principle, ωmay beanymeasure on Q; however, in practice, it is convenient to use the Leray measure of\nQ ⊂TM⊕TMderived from the natural measure ˆ ωonTM⊕TM,\nˆω=dx0123∧d˙x0123∧d¨x0123, (13)\nwheredx0123≡dx0∧dx1∧dx2∧dx3, etc. The Leray measure ωis defined by\nω:=ι∗˜ω, (14)\nwhere ˜ωis any 10-form on TM⊕TMsatisfying\nˆω= ˜ω∧dϕ1∧dϕ2. (15)\n3We choose\n˜ω= (˙x1¨x0−˙x0¨x1)−1dx0123∧d¨x0123∧d˙x23, (16)\nwhich can readily be shown to satisfy (15).\nTo calculate ι∗˜ω, we use coordinates ( xa,vµ,aν) onQsuch that\nι∗xa=xa, (17)\nι∗˙x0=/radicalbig\n1+v2, ι∗˙xµ=vµ, (18)\nι∗¨x0=a·v√\n1+v2, ι∗¨xµ=aµ, (19)\nwhere Greek indices run from 1 to 3 and a·v≡aµvµ, etc.\nUsing (16), (17), (18), (19) it follows that the Leray measure ω=ι∗˜ωonQis\nω=1\n1+v2dx0123∧da123∧dv123. (20)\nSimilarly, on the fibre Qp=π−1(p) of the bundle ( Q,π,M) overp∈ M, the induced measure Ω is identified as\nΩ :=da123∧dv123\n1+v2(21)\nbecause we adopt dx0123as the measure on M2.\nIt may be shown that\nι∗∂\n∂xa=∂\n∂xa, (22)\nι∗∂\n∂vµ=∂\n∂˙xµ+vµ√\n1+v2∂\n∂˙x0+aν√\n1+v2/parenleftbigg\nδν\nµ−vνvµ\n1+v2/parenrightbigg∂\n∂¨x0, (23)\nι∗∂\n∂aµ=∂\n∂¨xµ+vµ√\n1+v2∂\n∂¨x0(24)\nand it follows ι∗LQ=Lwhere\nLQ= ˙xa∂\n∂xa+aµ∂\n∂vµ+/parenleftbigg\n¨xa¨xavµ+τ−1(aµ+q\nmFµa˙xa)/parenrightbigg∂\n∂aµ. (25)\nIn (25), and from now on, it is to be understood that ˙ xaand ¨xaare shorthand for the values of ι∗˙xaandι∗¨xa\nrespectively.\nIn the absence of radiation reaction, the Leray measure on the to tal space Eis preserved by the flow of the\nLiouville vector field induced from the Lorentz force equation. This w ell-known result is commonly described\nas “conservation of phase space volume”. By contrast, the Lera y measure ωonQis not preserved by the flow\nofLQ:\nLLQω=3\nτω. (26)\nThis may be understood physically as a consequence of losses due to radiation. It follows from (26) that (12)\ncan be written\nLQf+3\nτf= 0. (27)\n2The same procedure leads to dx0123∧dv123/√\n1+v2for the Leray measure on the bundle with total space E:={(x,˙x)∈\nTM|ϕ1= 0,˙x0>0}, where vµis the pull-back of ˙ xµtoEwith respect to the inclusion E֒→TM. It follows that the induced\nmeasure on a fibre of that bundle is dv123/√\n1+v2as expected.\n4The electrons couple to the electromagnetic field through the final term in (25), and through the Maxwell\nequations\n∂Fbc\n∂xa+∂Fca\n∂xb+∂Fab\n∂xc= 0,\n∂Fab\n∂xa=Jb+Jb\next, (28)\nwhere the electron current is given by\nJa=q/integraldisplay\nf˙xaΩ (29)\nandJa\nextis the current of any source other than the electrons.\n3 Physical solutions\nEquation (27) does not in general reduce to the usual Vlasov equa tion in the limit τ→0. This is because, for\na regular solution to (27), fwill be nonzero over a range of afor given values of ( x,v), and so will necessarily\ninclude runaway solutions as well as physical ones. To avoid this, look for solutions of the form\nf(x,v,a) =/radicalbig\n1+v2g(x,v)δ(3)/parenleftbig\na−A(x,v)/parenrightbig\n, (30)\nwhere the (as yet undetermined) functions Aµdescribe the submanifold of phase space containing physical\ntrajectories. The factor√\n1+v2in (30) ensures that gcan be interpreted as a particle distribution function on\nthe space of unit timelike 4-vectors, from which it follows that the cu rrent (29) becomes\nJa=q/integraldisplay\ng˙xad3v√\n1+v2. (31)\nUsing (30) in (27) and integrating out the acceleration variables lead s to the coupled equations\n˙xa∂Aµ\n∂xa+Aν∂Aµ\n∂vν=AaAavµ+1\nτ(Aµ+q\nmFµa˙xa), (32)\n˙xa∂g\n∂xa+/radicalbig\n1+v2∂\n∂vµ/parenleftBiggAµ\n√\n1+v2/parenrightBig\n= 0, (33)\nwithA0:=vµAµ/√\n1+v2. Equation (32) describes the evolution of the physical submanifold , as governed\nby (1), while (33) is a generalized Vlasov equation for the 1-particle d istribution g. It is the reduced system\n(32)–(33), rather than (27), that reduces to the usual Vlasov equation in the limit τ→0.\nSo far we have placed no restrictions on the submanifold represent ed byAµ, other than that it satisfy (32).\nIn order that it represent the physical solutions to (1), it must be regular in the limit τ→0.Aµmay then be\nexpanded in powers of τ:\nAµ=∞/summationdisplay\nn=0τnAµ\n(n), (34)\nwith\nAµ\n(0)=−q\nmFµ\na(0)˙xa, (35)\nAµ\n(n+1)=−q\nmFµ\na(n+1)˙xa+ ˙xa∂Aµ\n(n)\n∂xa+n/summationdisplay\nj=0Aν\n(n−j)∂Aµ\n(j)\n∂vν−vµn/summationdisplay\nj=0Aa\n(n−j)Aa(j). (36)\n5Although τappears explicitly only in the accelerationequation (32), the distribu tiongand electromagneticfield\nFabacquire a τdependence via their couplings to Aµin the Vlasov equation (33) and the Maxwell equations\n(28), respectively, and may be similarly expanded:\ng=∞/summationdisplay\nn=0τng(n), Fab=∞/summationdisplay\nn=0τnFab\n(n). (37)\nTruncating at n= 0 yields the usual Vlasov equation without radiation reaction, while t runcating at n= 1\nyields the kinetic theory derived from the Landau-Lifshitz equation (6). This supports the large body of\nwork [21–23] utilizing the latter approach.\n4 Entropy\nIn the absence of radiation reaction, the entropy of the electron s is conserved. This is not generally true if\nradiation reaction is included, as the velocity-divergence of the phy sical acceleration acts as a source of entropy.\nConsider the entropy density s=−glngon phase space, which from the Vlasov equation (33) satisfies\n˙xa∂s\n∂xa+/radicalbig\n1+v2∂\n∂vµ/parenleftbiggsAµ\n√\n1+v2/parenrightbigg\n=g/radicalbig\n1+v2∂\n∂vµ/parenleftbiggAµ\n√\n1+v2/parenrightbigg\n. (38)\nThen the entropy current sa=/integraltext\n˙xasd3v/√\n1+v2on spacetime satisfies\n∂sa\n∂xa=/integraldisplay\ng∂\n∂vµ/parenleftbiggAµ\n√\n1+v2/parenrightbigg\nd3v, (39)\nand the change in the total entropy S=/integraltext\ns0dxof the electrons is\ndS\ndt=/integraldisplay\ng∂\n∂vµ/parenleftbiggAµ\n√\n1+v2/parenrightbigg\nd3vd3x. (40)\nAlthough the entropy density sis defined with respect to a reference density, conservation of pa rticle number\nensures that changing this reference density merely shifts the to tal entropy by a constant, so the rate of change\nof entropy (40) is well-defined.\nRestricting to the physical solutions of (32), we can expand S=/summationtext∞\nn=0τnS(n), and it follows that\ndS(n)\ndt=n/summationdisplay\nj=0/integraldisplay\ng(n−j)∂\n∂vµ/parenleftbiggAµ\n(j)√\n1+v2/parenrightbigg\nd3vd3x. (41)\nThe contribution from Aµ\n(0)vanishes, so the leading order entropy change comes from n= 1,\ndS(1)\ndt=−1\nm/integraldisplay/parenleftBig\nJa(Ja+Ja\next)+4q2\nm2TabSab/parenrightBig\nd3x, (42)\nwhere\nTab=FacFbc−1\n4ηabFcdFcd(43)\nis the stress-energy-momentum tensor of the electromagnetic fi eld and\nSab=m/integraldisplay\ng˙xa˙xbd3v√\n1+v2(44)\nis that of the electrons. Note that JaandSabin (42) are calculated in the limit τ→0.\nThe rate of change of entropy (42) has previously been derived in [2 1] from a kinetic theory based on\nthe Landau-Lifshitz equation, though the physical interpretatio n was there less apparent. The self-interaction\nterm,JaJa, increases the entropy, while the interaction with the field, TabSab, leads to an entropy decrease.\nThe effect of the remaining term, JaJa\next, depends on the nature of any external currents. In [24], only t heJaJa\ncontribution was found, leading to the incorrect statement that t he entropy of the electrons always increases.\nIn fact, it is quite possible for the TabSabterm to dominate, leading to radiation cooling of the electrons.\n65 Plasma oscillations\nA simple demonstration of the theory comes from exploring the effec t of radiation reaction on plasma waves.\nLinearize (28), (32)–(33) about the background\ng=/hatwideg(v), Aµ= 0, F ab= 0, (45)\nwithJa\next=−qn0δa\n0andn0=/integraltext\n/hatwidegd3v. Assuming longitudinal perturbations of the form exp i(kz−ωt) yields\nthe dispersion relation\n1 =q2\nm/integraldisplay/parenleftbig\n1+v2\n1+v2\n2/parenrightbig\n/hatwideg\n∆(ω√\n1+v2−kv3)2d3v√\n1+v2, (46)\nwhere ∆ = 1+ iτ(ω√\n1+v2−kv3) represents the modification of the plasma waves due to radiation r eaction.\nThe dispersion relation (46) possesses a more complicated mode str ucture than its counterpart in the\nMaxwell-Vlasov system. As well as small modifications to the usual so lutions, there exist entirely new roots.\nHowever, the latter do not exist in the limit τ→0, and so must be rejected as unphysical.\nTaking the cold equilibrium /hatwideg(v) =n0δ(3)(v) yields\nω2\np=ω2(1+iτω), (47)\nwhereωp=/radicalbig\nq2n0/mis the plasma frequency. Though rather simplistic, this nicely illustrat es the effects of\nincorporating radiation reaction into the Vlasov equation. Two (phy sically equivalent) roots of (47),\nω≈ ±ωp/parenleftbig\n1−5\n8(ωpτ)2/parenrightbig\n−i\n2ω2\npτ, (48)\nindicate a damping rate of order τand an order τ2frequency downshift. There also exists a third (purely\nimaginary) root,\nω≈i\nτ, (49)\nwhichrepresentsextremelyrapidgrowth,withoutoscillation. Thism ode, whichdoesnotexistinthelimit τ→0,\ncorresponds to the runaway solutions of the Lorentz-Dirac equa tion, and must be discarded as nonphysical.\nResults equivalent to (48) may be obtained by truncating the expan sion inτatn= 2, while the root (49) is\nexcluded in this approach. Similar results may be found for other typ es of waves in plasmas.\n6 Conclusions\nIn summary, with the advent of ultra-high intensity laser facilities, s uch as ELI, it is important to have a reliable\nkinetic theory of radiating particles, incorporating radiation react ion. We have developed such a theory based\non the full Lorentz-Dirac equation, and found that it reduces to t he usual Vlasov theory and to a kinetic theory\nbased on the Landau-Lifshitz equation in appropriate limits. As simple demonstrations of the theory, we have\nexplored the effects of radiation reaction on entropy and on longitu dinal plasma waves.\n7 Acknowledgments\nWe would like to thank Robin Tucker for useful discussions. This work was supported by UK EPSRC, the\nLaserlab-Europe consortium and the FP7–Extreme Light Infrast ructure (ELI) project.\n7References\n[1] http://www.extreme-light-infrastructure.eu/\n[2] Lorentz HA 1916 The theory of electrons and its applications to the phenomen a of light and radiant heat (Stechert,\nNew York)\n[3] Abraham M 1932 The classical theory of electricity and magnetism (Blackie, London)\n[4] Dirac PAM 1938 Proc. Roy. Soc. A167148\n[5] Bhabha HJ 1939 Proc. Roy. Soc. A172384\n[6] Wheeler JA and Feynman RP 1945 Rev. Mod. Phys. 17157\n[7] Rohrlich F 1964 Phys. Rev. Lett. 12375\n[8] Teitelboim C 1970 Phys. Rev. D11572\n[9] Barut AO 1974 Phys. Rev. D103335\n[10] Burton DA, Gratus J and Tucker RW 2006 Ann. Phys. 322599\n[11] Ferris MR and Gratus J 2011 J. Math. Phys. 52092902\n[12] Eliezer CJ 1948 Proc. Roy. Soc. A194543\n[13] Ford GW and O’Connell RF 1993 Phys. Lett. A174182\n[14] Sokolov IV et al.2009Phys. Plasmas 16093115\n[15] Hammond RT 2010 Phys. Rev. A81062104\n[16] Landau LD and Lifshitz EM 1962 The Classical Theory of Fields (Pergamon, London)\n[17] Spohn H 2000 Europhys. Lett. 50(3) 287\n[18] Rohrlich F 2007 Classical charged particles (World Scientific, Singapore)\n[19] Griffiths DJ, Proctor TC and Schroeter DF 2010 Am. J. Phys. 78(4) 391\n[20] Bulanov SV et al.2011Phys. Rev. E84056605\n[21] Tamburini M et al.2011Nucl. Instrum. Methods A653(1) 181\n[22] Berezhiani VI, Hazeltine RD and Mahajan SM 2004 Phys. Rev. E69056406\n[23] Hazeltine RD and Mahajan SM 2004 Phys. Rev. E70046407\n[24] Hakim R and Mangeney A 1968 J. Math. Phys. 9116\n8"    },    {        "title": "1703.02132v1.Differences_between_Doppler_velocities_of_ions_and_neutral_atoms_in_a_solar_prominence.pdf",        "content": "arXiv:1703.02132v1  [astro-ph.SR]  6 Mar 2017Astronomy&Astrophysics manuscript no. submit05 c∝circleco√yrtESO 2018\nSeptember 7, 2018\nDifferences between Doppler velocities of ions and neutral atoms\nin a solar prominence\nT. Anan1, K. Ichimoto1,2, and A. Hillier3\n1Kwasan and Hida Observatories, Kyoto University, Gifu, Jap an\n2SOLAR-C Project O ffice, National Astronomical Observatory of Japan, Tokyo, Jap an\n3CEMPS, University of Exeter, Exeter EX4 4QF U.K.\ne-mail:anan@kwasan.kyoto-u.ac.jp\nReceived XXX, 2015; accepted xxx, 2015\nABSTRACT\nContext. In astrophysical systems with partially ionized plasma the motion of ions is governed by the magnetic field while the neut ral\nparticles can only feel the magnetic field’s Lorentz force in directly through collisions with ions. The drift in the velo city between\nionized and neutral species plays a key role in modifying imp ortant physical processes like magnetic reconnection, dam ping of\nmagnetohydrodynamic waves, transport of angular momentum in plasma through the magnetic field, and heating.\nAims. This paper investigates the di fferences between Doppler velocities of calcium ions and neut ral hydrogen in a solar prominence\nto look for velocity di fferences between the neutral and ionized species.\nMethods. We simultaneously observed spectra of a prominence over an a ctive region in H I 397 nm, H I 434 nm, Ca II 397 nm, and\nCa II 854 nm using a high dispersion spectrograph of the Domel ess Solar Telescope at Hida observatory, and compared the Do ppler\nvelocities, derived from the shift of the peak of the spectra l lines presumably emitted from optically-thin plasma.\nResults. There are instances when the di fference in velocities between neutral atoms and ions is signi ficant, e.g. 1433 events ( ∼3 %\nof sets of compared profiles) with a di fference in velocity between neutral hydrogen atoms and calci um ions greater than 3 σof the\nmeasurement error. However, we also found significant di fferences between the Doppler velocities of two spectral line s emitted from\nthe same species, and the probability density functions of v elocity difference between the same species is not significantly di fferent\nfrom those between neutral atoms and ions.\nConclusions. We interpreted the di fference of Doppler velocities as a result of motions of di fferent components in the prominence\nalong the line of sight, rather than the decoupling of neutra l atoms from plasma.\nKey words. Sun: filaments, prominences – Magnetohydrodynamics (MHD) – Methods: observational – Techniques: spectroscopic\n1. Introduction\nThe plasma in the solar photosphere, the solar chromosphere , the interstellar medium, and protoplanetary disks, to giv e just a few\nexamples, is partially ionized. In many of these systems, ma gnetic fields play an crucial role in the plasma dynamics, but they cannot\ndirectly exert a force on the neutral particles. Therefore, the Lorentz force is indirectly exerted on the neutrals thro ugh collisional\nfriction between the neutral and charged particles. Howeve r, this coupling of the two fluids, and by extension the neutra ls to the\nmagnetic field, is not perfect causing the neutral particles to diffuse across the magnetic field in a process called ambipolar di ffusion\n(e.g. Brandenburg & Zweibel 1994).\nThis diffusion of neutrals across the magnetic field is a crucial proce ss in astrophysical systems. It increases the rate at which\nmagnetic fields reconnect (e.g. Zweibel 1989; Leake et al. 20 12), damping rates of propagating magnetohydrodynamic wav es (e.g.\nOsterbrock 1961; Khodachenko et al. 2004), heating rates in the solar chromosphere (e.g. Osterbrock 1961; Khomenko & Co llados\n2012), and rate at which magnetic field emerges from the conve ction zone into the solar corona (e.g. Arber et al. 2007; Leak e & Linton\n2013a). The effect also changes the energy flux of Alfv´ en wave (Vranjes et al . 2008), the structure of slow-mode MHD shocks\n(Hillier et al. 2016), the thermodynamic structure of quiet -Sun magnetic features (Cheung & Cameron 2012), and the stru cture of\nthe solar prominences (Hillier et al. 2010). Furthermore, i t has significant impact on the angular momentum transport by magnetic\nfields in the formation and evolution of circumstellar disks and stars (e.g. Mestel & Spitzer 1956; Tomida et al. 2015). Ho wever,\ndirect observation of ion-neutral drift is di fficult. This is partially due to the fact that the appreciable d ecoupling of neutral atoms\nfrom plasma is too small to be measured on observable scales b ecause the expected collisional coupling in many partial-i onized\nastrophysical plasmas is strong.\nThe object of our observation is a solar prominence. Solar pr ominences are relatively cool ( ∼104K) and dense (3 - 6 ×1011cm−3)\nstructures observed in chromospheric lines sustained in th e hotter and the sparser corona above the solar limb (Hirayam a 1986;\nArticle number, page 1 of 13A&A proofs: manuscript no. submit05\nFig. 1. Prominence in (a) the AIA 304 Å imager onboard SDO, (b) H αslit-jaw image, (c) spectrum including Ca+H, and Hǫ, (d) spectrum of H γ,\nand (e) spectrum of Ca+IR. The oblique white line in the slit-jaw image shows the spe ctral slit. The temporal evolution is available online1.\nTandberg-Hanssen 1995). Because of their low temperature, prominences are made of partially-ionized plasma, with the ionization\nfraction of hydrogen characteristically 0.2 (Ruzdjak & Tan dberg-Hanssen 1989; Engvold et al. 1990; Labrosse et al. 201 0) in the\ncentre of the dense prominence material. In prominences, it has been suggested that plasma is supported against gravity by the\nLorentz force (Kippenhahn & Schlüter 1957; Kuperus & Raadu 1 974), and neutral atoms are supported by the frictional forc e\nbetween them and the plasma (Low et al. 2012). Prominences ar e dynamic structures, displaying motions of various kinds, such as\nturbulence (e.g. Hillier et al. 2017), oscillations (e.g. O kamoto et al. 2007), and convection (Berger et al. 2008, 2011 ).\nIn a partially ionized system, when neglecting the electron and ion inertia terms and assuming that magnetic forces domi nate,\nwe obtain the relative velocity between ions, vi, and neutrals, vnas,\nvi−vn∝ξ\nαn(J×B),\nwhereξis the neutral fraction, αnis the sum of collisional frequencies between the neutral an d ionized species multiplied by the\ncorresponding mass densities, Jis the current and Bis the magnetic field vector.\nAn analytical calculation (Gilbert et al. 2002) using a simp le prominence model in which the Lorentz force is balanced wi th the\nfrictional force shows the relative flow of the neutral hydro gen and ionized components of 3 .7×10−3km s−1. The relative flow of\nneutrals to ions was modeled using numerical simulations in a 2D prominence model (Terradas et al. 2015). Simulations al so show\ndecoupling of neutral atoms from plasma as a result of the Ray leigh-Taylor instability in prominences (Khomenko et al. 2 014b).\nCharge exchange is another process through which ionized an d neutral fluids can couple and it e ffectively increases the cross sec-\ntion of momentum transfer between neutral and ionized hydro gen by approximately a factor 2 (Krstic & Schultz 1999; Vranj es & Krstic\nArticle number, page 2 of 13T. Anan , K. Ichimoto , and A. Hillier : Di fferences between Doppler velocities of ions and neutral atom s in a solar prominence\nFig. 2. Example of the observed profiles and fits with a parabola. Blac k symbols indicate the observed profiles in Ca+H, Ca+IR, Hǫ, and Hγ\nnormalized by the maximum intensity of each profile. The wave length position of each peak (blue solid lines) is determine d from fitting (red lines)\nwith a parabola using interval at which the intensity is larg er than 75% of the peak intensity. The inferred Doppler veloc ities of Ca+H, Ca+IR, Hǫ,\nand Hγare 14.5±0.2 km s−1, 18.5±1.7 km s−1, 19.1±0.5 km s−1, and 16.2±0.4 km s−1, respectively. The measurement error is indicated in each\ntitle for the 1-sigma uncertainty, and in each plot as the blu e dotted lines for the 3-sigma uncertainty. The minimum and m aximum wavelength\nranges displayed in the individual panels are −70 and+70 km s−1, respectively. The black vertical solid lines denote the wa velength where the\nDoppler velocities are equal to zero.\n2013). Charge exchange is able to increases the momentum cou pling between a range of neutral atoms and ions (e.g. Leake & L inton\n2013b; Vranjes et al. 2016), however as the process requires resonance it is naturally most common between two atoms of th e same\nspecies. Terradas et al. (2015) described that the relative flow of neutral hydrogen to protons in a prominence is reduced by the\ncharge exchange interactions.\nKhomenko et al. (2016) detected di fferences between ion and neutral velocities in a prominence, as shown by the di fference\nin Doppler velocities of He I 1083 nm and Ca II 854 nm of the orde r of 0.1 km s−1. They discussed coherency of the di fferent\nvelocity in time and space, but they didn’t compare between t he same neutral species to confirm that the di fferences in Doppler shift\nobserved were actually a result of ion-neutral drift. In thi s paper, we present an observation of another solar prominen ces in two\nspectral lines of the neutral hydrogen and two lines of the ca lcium ions, in order to confirm the decoupling of neutral atom s from\nthe plasma. In the following sections, we describe the detai ls of the observations (Section 2), the inference of the Dopp ler velocities\nArticle number, page 3 of 13A&A proofs: manuscript no. submit05\nFig. 3. Maximum intensity of Ca+H vs Hǫ, Hγvs Hǫ, and Ca+IR vs Hǫ. The white dotted line indicates a threshold in the maximum i ntensity of\nCa+H to determine the Ca+H intensities that are emitted by optically-thin plasma.\n(Section 3), the results of di fferences between the Doppler velocity of neutral atoms and io ns (Section 4), the discussions (Section\n5), and finally we summarize (Section 6).\n2. Observation and image processing\nA prominence over an active region NOAA12339 at the east sola r limb was observed in Ca II 397 nm (Ca+H), H I 397 nm (Hǫ), H I\n434 nm (Hγ), and Ca II 854 nm (Ca+IR) using the horizontal spectrograph of the Domeless Solar Telescope (Nakai & Hattori 1985)\nat Hida observatory, Japan (Fig. 1). The observation ran fro m 10:53 to 11:37 local standard time on 2015 May 5. The helioce ntric\ncoordinates of the prominence at the time of observation was (N15◦, E90◦). The prominence was dynamic, which is demonstrated\nby the accompanying movie1. The mass drained to the chromosphere from the prominence al ong the loop-like structures with the\nplane-of-the-sky velocity of ∼50 km s−1, which appear in 304 Å images from Atmospheric Imaging Assem bly (AIA) (Lemen et al.\n2012) on Solar Dynamic Observatory (SDO) (Pesnell et al. 201 2). In the spectra, the intensity profiles were changing quit e rapidly.\nThe horizontal spectrograph can image the entire visible an d near infrared solar spectrum for all the spatial points alo ng the\nspectrograph slit. Four spectral regions including 396 nm, 397 nm, 434 nm, and 854 nm are taken with three CCD cameras (Pro silica\nGE1650), with an exposure time of 0 .6 s, and with a time cadence of 1 s. One of the cameras took the sp ectral region which includes\n396 nm and 397 nm. The spectral samplings in 396 nm, 434 nm, and 854 nm are 17 mÅ pixel−1, 21 mÅ pixel−1, and 32 mÅ pixel−1,\nrespectively and the spatial sampling of spectra in 396 nm, 4 34 nm, and 854 nm are 0 .28 arcsec pixel−1, 0.36 arcsec pixel−1and\n0.28 arcsec pixel−1, respectively. The linear dispersion of spectra was determ ined by using neighboring solar lines in the background\nsky spectrum using the solar atlas (Moore et al. 1966).\nWhen one camera starts its exposure, the camera produces a tr igger signal for the other cameras to start their exposure. T he\ntime lag of the start of an exposure between the three cameras is less than 8µs. Assuming the Fried’s parameter of 40 mm (Fried\n1966; Kawate et al. 2011) and wind velocity at the turbulent l atitude of 40 m s−1, the time scale of the seeing is approximately equal\nto 1 ms, that is it is much larger than the synchronization acc uracy, and thus the exposures of all three cameras can be rega rded as\n1Movie, http://www.kwasan.kyoto-u.ac.jp /∼anan/shed/2016submit _prominence_fig1.gif\n(a) the AIA 304 Å imager onboard SDO, (b) H αslit-jaw image, (c) spectrum including Ca+H, and Hǫ, (d) spectrum of H γ, and (e) spectrum of\nCa+IR. The oblique white line in the slit-jaw image shows the spe ctral slit.\nArticle number, page 4 of 13T. Anan , K. Ichimoto , and A. Hillier : Di fferences between Doppler velocities of ions and neutral atom s in a solar prominence\nexactly simultaneous. Motion of the solar image on the slit c aused by the local turbulence of air (seeing) was approximat ely 3 arcsec\nin amplitude during the run of the observation. The typical s patial resolution during the observation is also about 4 arc sec, which\ncorresponds to the Fried’s parameter of about 40 mm at 600 nm.\nDuring the observations, the zenith angle of the sun was appr oximately equal to 25◦. The solar rays pass through the atmosphere\nof the earth and are refracted before reaching the entrance w indow of the telescope. Since the refraction angle can chang e with the\nwavelength, there may be a slight shift of images in three wav elengths. We place the spectrograph slit on the prominence w ith an\norientation parallel to the line connecting the prominence and the zenith, to sample exactly the same region of prominen ce on the\nslit in 396 nm, 434 nm, and 854 nm.\nAfter dark-frame and flat-field corrections, the observed sp ectra are de-convolved using the point spread function of th e hori-\nzontal spectrograph. Then we subtracted the sky spectrum, w hich was made from the average of 80 spectral profiles near by t he\nprominence. In order to compare the three spectral lines, we reduce the spatial sampling of 396 nm and 854 nm to that of 434 n m,\nand align the two hair lines of the spectra in 396 nm and 854 nm t o those in 434 nm. We also checked the spatial alignment of the\nthree spectral lines using the cross-correlation between p rofiles of Doppler velocities of the three lines along the sli t and confirmed\nthat the differential refraction between the three lines is negligible d uring the observations. Even though there was motion of the s o-\nlar image on the slit caused by the seeing, the high synchroni zation accuracy and the negligible displacement due to the a tmospheric\nrefraction allow us to sample the same ensemble of plasma in t he four spectral lines.\nThe point spread functions of the telescope also depend on th e wavelength,λ, i.e., the theoretical size of the point spread function\nof the telescope is proportional to the wavelength. However , under seeing, it is proportional to λ−1/5(Roddier 1981), if the exposure\ntime is long enough compared to the seeing time scale as in our observation. Thus the di fference of the point spread function among\nthe wavelengths is much smaller than the ideal case, and we su spect that the difference of the point spread function among the\nwavelengths is not a serious problem in our analysis, though we cannot absolutely exclude its small e ffect on our results.\nThe photometric accuracy of spectral measurement was evalu ated from the random variation of intensity in continuum of s ky\ncomponent of each spectrum and it is multiplied by square roo t of the intensity to obtain the random error in spectral line s by\nassuming that the photon noise is the dominant source. The ro ot-mean-square photometric accuracy depends on the intens ity in\nCa+H, Hǫ, Hγ, and Ca+IR and varies in the range of 1 - 10%, 1 - 20%, 1 - 20%, and 1 - 25% ac ross the respective emission lines.\n3. Doppler velocity of optically-thin plasma\nFrom the 44 min of observation, we obtained 2626 sets of spect ral images of the prominence and 347677 independent sets of l ine\nprofiles of the prominence. We fit the peak of spectral profiles with a parabola using a wavelength interval at which the inte nsity is\nlarger than 75% of the peak intensity to determine the wavele ngth position of the line peak by applying the IDL routine poly_fit.pro\n(Fig. 2). We confirmed that results with di fferent thresholds, i.e. 65 %, 70 %, 80 %, and 85 % of the peak inte nsity, are the\nqualitatively the same as those with the threshold of 75 %. Th en, we derived the Doppler velocity of each spectral line fro m the\nshift of its peak. The measurement error of the Doppler veloc ity is determined from the uncertainty of the center positio n of the\nparabola, i.e, the keyword SIGMA of the poly_fit.pro , by which the residual error of the fitting is calculated from the random error\nin the spectral data.\nThe observed line profiles used for the present analysis have to satisfy the following conditions: (1) The maximum intens ity of\nthe spectral line is more than five times larger than the root- mean-square of the random noise in the background scattered spectrum;\n(2) The fitted parabola is concave downward; (3) The center po sition of the parabola is within the wavelength interval for fitting,\ni.e. the wavelength range where the intensity is larger than 75% of the peak intensity, with a 99 .7% confidence level (3 σ); and (4)\nThe spectral line is emitted by optically-thin plasma as is judged by the following method. Condition (4) is to ensure that we are\nable to deduce information integrated along the full depth o f the emitting plasma. In the optically thick case the radiat ion emitted by\nthe plasma carries information which is dependent on the dep th of the formation of the specific spectral line used for obse rvations.\nBecause different optically thick spectral lines can form at very di fferent depths, we cannot interpret di fference of Doppler shifts\nbetween lines of neutral and ionized atoms as evidence of rel ative velocity of the di fferent species. Such analysis can be performed\nonly in the optically thin case when the intensity of emissio n is proportional to the optical depth. Figure 3 shows a scatt er plot of the\nmaximum intensity of H ǫvs. that of the other lines. The Ca+H intensity saturates due to the increase of the optical thic kness while\nthe other lines remain optically thin. Thus, we adopt a thres hold in the maximum intensity of Ca+H as 35000 digital number (DN)\nto select a data point satisfying condition (4), and to safel y remove all data points in which the spectral line of Ca+H was possibly\nemitted by optically thick plasma. However, we don’t adopt a threshold for the other lines, because they remain opticall y thin. We\nnote that the observational data used in this paper were not a bsolutely calibrated but instead the DN units are used as a me asure of\nthe intensity throughout the present paper.\nFigure 4 shows time-space diagram of the inferred Doppler ve locities of Ca+H, Ca+IR, Hǫ, and Hγin the prominence. The\nDoppler velocity of the prominence varies between −60 km s−1and 40 km s−1. We set the zero LOS velocity as the mean position\nof each line over the final sample. There are approximately 74 759 pixels of the four profiles ( ∼20 % of the original data set), which\nsatisfy the condition (1), (2), and (3) in all lines, and 43 % o f them are excluded by the (4) criterion. Finally, the number of the\npixels of the line profiles that are subject to our further ana lysis is 42585. The pixels that satisfy all conditions are sh own in Fig. 4\nwith the color scale.\n4. Results\nDifferences between the Doppler velocities of two spectral line s are shown as a function of space and time in Fig. 5. By compari ng\nthe Doppler velocities that are derived from the shift of the peak of the spectral lines emitted by optically-thin plasma , we find\nArticle number, page 5 of 13A&A proofs: manuscript no. submit05\nFig. 4. Doppler velocity of the prominence in Ca+H, Ca+IR, Hǫ, and Hγas functions of position along the slit and time. Pixels whic h satisfy the\ncriteria 1), 2), 3), and 4) are shown by color scale, while the others by gray scale. The solar surface is located at the bott om.\nthat there are instances when the di fference in velocities of di fferent lines is significant. For example 1433 cases ( ∼3 % of the\nsets of compared profiles) show a di fference of velocities between neutral hydrogen atoms (H γand Hǫ) and calcium ions (Ca+H\nand Ca+IR) greater than 3σ. Pixels with large velocity di fference are located coherently in time and in space. However, velocity\ndifferences are also significant ( >3σ) for another 37561 pairs of spectral lines of the same specie s, i.e., between Ca+H and Ca+IR\nand between Hγand Hǫ.\nFigure 6 shows the probability density functions (PDFs) of t he velocity difference,∆V, between two of the observed spectral\nlines (a) and those of velocity di fference normalized by measurement error (b). Here, the PDFs a re normalized so that its integral\nalong∆V(a) and∆V/error (b) equals one. If there is a decoupling of neutral atom s from plasma, we can expect to obtain PDFs\nof velocity difference between neutral atoms and ions to be wider than those o f velocity differences between lines of the same\nspecies. However, some PDFs of the velocity di fference between neutral atoms and ions are narrower than eith er PDF of two\nspectral lines emitted from the same species. The probabili ty density function of the di fferences in the Doppler velocity, normalized\nto the measurement errors (Fig. 6 b), display departures fro m the standard normal distribution (black dotted line). If t he differences\nwere only determined by measurement errors, the distributi on of the probability density function would be the same as a s tandard\nnormal distribution. This means that we have found significa nt Doppler velocity di fferences among Ca+H, Hǫ, Hγ, and Ca+IR in\nthe prominence.\nArticle number, page 6 of 13T. Anan , K. Ichimoto , and A. Hillier : Di fferences between Doppler velocities of ions and neutral atom s in a solar prominence\nFig. 5. Doppler velocity di fferences in the prominence between pairs of spectral lines Ca+H, Ca+IR, Hǫ, and Hγas functions of position along the\nslit and time. Pixels which satisfy criteria 1), 2), 3), and 4 ) are shown by color scale, while the others by gray scale. The solar surface is located at\nthe bottom.\n5. Discussion\nIn order to investigate the decoupling of neutral atoms from plasma in a solar prominence, we measured the Doppler veloci ties of\nCa+H, Hǫ, Hγ, and Ca+IR observed simultaneously. By comparing the Doppler veloc ity of neutral hydrogen and calcium ions (Fig.\n5 and 6), derived from the shift of the peak of the spectral lin es emitted by optically-thin plasma, we find that there is sig nificant\ndifference between the Doppler velocity of neutral hydrogen and calcium ions. However, significant velocity di fferences are also\nfound even between di fferent spectral lines of the same species, for example betwee n Ca+H and Ca+IR. In this section, we discuss\nArticle number, page 7 of 13A&A proofs: manuscript no. submit05\nFig. 6. Probability density functions of di fferences between the Doppler velocity of Ca+H and Hǫ(red solid line), Ca+H and Hγ(red dashed line),\nCa+IR and Hǫ(blue solid line), Ca+IR and Hγ(blue dashed line), Ca+H and Ca+IR (black solid line), and H ǫand Hγ(black dashed line). For the\nfunctions (b), the velocity di fferences were normalized to the measurement errors, σ. The black dotted line indicates the normal distribution wi th\na standard deviation of 1.\npossible errors in the measurement and whether they cause th e observed velocity di fferences. Finally, we propose an interpretation\nof the velocity differences between spectral lines of the same species.\n5.1. Parabola fitting\nThe measurement error of the Doppler velocity is determined from the uncertainty of the center position of the parabola, which\ncorrespond to the residual error of the fitting equal to the ra ndom error in spectral data. Figure 7 is the same as Fig. 6, but obtained\nfrom spectral profiles, in which the random error is reduced b y averaging three adjacent profiles in the slit direction. If the differences\nwere only determined by the parabola fitting errors, the widt h of the PDFs would be 1 /√\n3∼0.6 times smaller than those of Fig.\n6 a. However, the standard deviation is almost the same as tha t before the reduction of the random errors (Fig. 6 a and Fig. 7 a),\nand the departures of the probability density function of th e velocity differences normalized to the measurement errors from the\nstandard normal distribution (black dotted line) are even l arger than those from the single pixel profiles (compare Fig. 6 b and Fig.\n7 b). Therefore, we can conclude that the velocity di fferences do not originate from the uncertainty of the center p osition of the\nparabola in the line fitting.\n5.2. Optical depth\nIf some data sets that include spectral lines emitted from op tically-thick plasma remain in our selection, we are compar ing the\nDoppler velocities of the plasma at di fferent depth dependent on spectral lines, and the velocity di fference among the four lines may\nincrease. Figure 8 shows the maximum of absolute values of th e Doppler velocity di fferences of a spectral line against the others as\na function of the maximum intensity of a line. Because the int ensity of emission is proportional to the optical depth in th e optically\nArticle number, page 8 of 13T. Anan , K. Ichimoto , and A. Hillier : Di fferences between Doppler velocities of ions and neutral atom s in a solar prominence\nFig. 7. The same as Fig. 6, but obtained from spectral profiles, in whi ch the measurement error is reduced by averaging over three a djacent profiles\nalong the slit.\nFig. 8. Scatter plots of the maximum of absolute values of the Dopple r velocity differences of a spectral line against the other spectral lines v s.\nthe maximum intensity of the spectral line, and fits with a lin ear function (white solid lines).\nArticle number, page 9 of 13A&A proofs: manuscript no. submit05\nFig. 9. Scatter plot of the di fferences between the wavelength of the gravity center ( λg) and the peak (λp) of Ca+IR vs. the differences of Doppler\nvelocity between Ca+H and Ca+IR in the pixels that satisfy all the conditions described in Section 3. The linear Pearson correlation coe fficient, r,\nis 0.65. The white vertical lines indicate the threshold of λg−λpof Ca+IR to exclude the highly asymmetric profiles to give the absol ute value of\nthe linear Pearson correlation coe fficient of all combinations of the four lines to be less than 0.3 .\nthin case, the higher intensity the higher possibility that we sample data sets including spectral lines emitted by opti cally-thick\nplasma. But, as is shown in Fig.8, the absolute value of the Do ppler velocity difference among spectral lines does not tend to be\nlarger. Therefore, we conclude that the optical-thickness of the plasma is not the cause of the velocity di fference.\n5.3. Asymmetry of the line profile\nFor some pairs of the analyzed spectral lines, we found that d ifferences between the wavelength of the gravity center, λg, and the\npeak,λp, of a spectral-line profile, which characterize asymmetry o f the line profile, correlate with Doppler velocity di fferences\nbetween the spectral lines. Figure 9 shows an example of this strong correlation, where λg−λpof Ca+IR strongly correlates (linear\nPearson correlation coe fficient is 0.65) with Doppler velocity di fferences between Ca+H and Ca+IR,VCa+H−VCa+IR, in the pixels\nthat satisfy all the conditions described in Section 3. The a symmetric profiles can be formed if there are multiple compon ents, which\nhave different velocities, along the line of sight or in a resolution e lement. The peak wavelength may be di fferent for different spectral\nlines, because sensitivities to physical parameters (e.g. density, temperature) is di fferent for different spectral lines. Incomplete sky\nsubtractions can also result in asymmetric profiles. Theref ore, there is the possibility that asymmetry produces spuri ous Doppler\nvelocity differences caused by the contamination of scattered light spec trum or by sampling of di fferent plasmas.\nWe excluded the profiles where |λg−λp|of a spectral-line profile is larger than a threshold. We adop t the thresholds for the\ndifferences as 1.5×10−3nm, 1.0×10−2nm, and 2.0×10−3nm in Hǫ, Hγ, and Ca+IR, respectively, for the absolute value of the\nlinear Pearson correlation coe fficient of all combinations of the four lines to be less than 0.3 . For example, the correlation coe fficient\nbetweenλg−λpof Ca+IR and VCa+H−VCa+IRbecomes 0.14 after the exclusion (Fig. 9). We also excluded t he profiles that have two\npeaks within the wavelength interval for the parabola fittin g. The number of the sets of the line profiles that pass this fur ther criteria\nare 1865.\nFigure 10 is the same as Fig. 6, but obtained from the remainin g spectral profiles. Probability density functions of di fferences\nbetween the Doppler velocity of neutral hydrogen and calciu m ions remarkably resemble those of the same species. All PDF s of\nvelocity difference are fitted with a Gaussian distribution with a standar d deviation of 1.4 km s−1. In section 4, we described that the\nPDFs of velocity di fference between neutral atoms and ions are not wider than thos e of two spectral lines emitted from the same\nspecies. Here, we can conclude that the asymmetry of the line profile, which affects on the measurement of the Doppler velocity in\nsome cases, does not change the result.\nWe evaluated the similarity of the PDFs by using the Kolmogor ov-Smirnov test, which is a method for comparing two PDFs.\nFrom differences of their cumulative distributions, the test diagno ses the probability that a single parent distribution draws two\nPDFs. We applied the test to the PDFs of velocity di fference in the prominence. The results from this test show tha t our PDFs are so\nsimilar that we cannot ignore the possibility of the sharing the same parent distribution for Ca+H & Ca+IR, Ca+H & Hγ, and Ca+IR\n& Hǫ(13%).\nArticle number, page 10 of 13T. Anan , K. Ichimoto , and A. Hillier : Di fferences between Doppler velocities of ions and neutral atom s in a solar prominence\nFig. 10. The same as Fig. 6, but obtained from spectral profiles, which do not have large asymmetry of the line profile or two clear pea ks within the\nwavelength interval for the parabola fitting. For the functi ons (a), the black dotted line indicates a Gaussian distribu tion with a standard deviation\nof 1.4 km s−1. Note that the x-axes and the y-axes over smaller ranges than those of Fig. 6.\n5.4. Interpretation\nIn case that we observed the same plasma in di fferent spectral lines we could expect that the Doppler veloci ty derived from the\nspectral lines of the same ion are the same. However, the obse rvations analyzed in the present paper show that significant differences\nexist even between spectral lines of the same species, i.e., between Ca+H and Ca+IR, or between Hγand Hǫ. Moreover, the PDFs\nof velocity difference between neutral atoms and ions are not wider than thos e of two spectral lines emitted from the same species.\nWe concluded in the above discussions that none of the possib le candidates of the error explain the distribution of the ve locity\ndifferences.\nAnalyzing the spectral line emitted by optically-thin part s of the prominence, we measured the Doppler velocity of the e mitting\nplasma integrated along the full depth of the prominence. Mo reover, we assumed that only the dominant component is sampl ed\nalong the line-of-sight for each line, because the Doppler v elocity is determined from the peak of the spectral line. How ever,\nthe prominence is not necessarily optically thin for the inc ident radiation that excites the atoms or the ions.Non-LTE m odels of\nprominences show that the complex absorption and emission o f the Lyman lines have to be taken into account to explain the\nobserved emission lines from prominences (Gunár et al. 2008 , 2010; Schwartz et al. 2015), and it is shown that Ly βis the key\nfor the formation of H αfor example (Gunár et al. 2012). In our case, Ly δand Lyζare the main spectral lines that excite neutral\nhydrogen to the upper state of H γand Hǫ, respectively. The optical depths of prominences in Ly δand Lyζare generally larger than\nthose in Hγand Hǫ, which we observed. The path of the incident radiation is als o different from that of the observed radiation, as\nthe incident radiation is emitted mainly from the solar surf ace. Therefore, the distribution of the neutral excited hyd rogen atoms\nemitting Hγdepends on the radiation field of Ly δand the distribution of atoms emitting H ǫdepends on the radiation field of Ly ζ.\nThis means that the ratio between the number density of neutr al hydrogen atoms emitting H γand Hǫis not constant throughout the\nobserved prominence (Gun´ ar, private communication). Suc h difference may result in a di fferent rate of emission of the H γand Hǫ\nlines at different positions along a line of sight. The same arguments as a re presented here for the hydrogen lines can be expected\nArticle number, page 11 of 13A&A proofs: manuscript no. submit05\nto be valid also for calcium lines. Hence, the dominant compo nent in a emission line is not necessarily the same as that of a nother\nline in the same species.\nHigh-resolution images and a radiative transfer visualiza tion technique demonstrate that thin threads constitute th e fundamental\nstructure of the prominences (Lin et al. 2008; Gunár & Mackay 2015a,b). The inferred significant velocity di fferences between\ntwo spectral lines of the same species may be attributed to ve locity differences between the threads along the line-of-sight (e.g.\nGunár et al. 2008). Because the PDFs of the velocity di fferences between spectral lines of calcium ions and neutral h ydrogen\nremarkably resemble the velocity di fferences for the same species, these di fferences also may show the velocity di fferences among\nthreads, rather than the decoupling of neutral atoms from pl asma. Charge exchange may contribute to the reduction of the relative\nflow velocity of neutrals and ions in prominences, but as with collisional coupling it would not provide an explanation fo r the\nvelocity difference between lines from the same species.\n6. Summary\nThe drift in the velocity between ionized and neutral specie s plays a key role in modifying important physical processes like\nmagnetic reconnection, damping of magnetohydrodynamic wa ves, transport of angular momenta in the formation and evolu tion of\nstars and disks, and heating in the solar chromosphere. Khom enko et al. (2016) detected the di fferences in the Doppler velocity of\nHe I 1083 nm and Ca II 854 nm in prominences, and interpreted as the drift of neutral atoms from ions.\nThis paper presents an analysis of the di fference between the Doppler velocities of ions and neutral at oms in an active region\nprominence. We use observations of the spectral lines of H ǫ, Hγ, Ca+H, and Ca+IR obtained by a high dispersion spectrograph of\nthe Domeless Solar Telescope at Hida observatory. We compar ed the Doppler velocities, derived from the shift of the peak of the\nspectral lines emitted by optically-thin plasma. There are instances when the di fference between the velocities of neutral atoms and\nions is significant, e.g. 1433 events ( ∼3 % of sets of compared profiles) show a di fference in the velocity between neutral hydrogen\nand calcium ions greater than 3 σ. However, velocity di fferences are also significant between di fferent spectral lines of the same\nspecies, and their PDFs remarkably resemble those of the vel ocity difference between neutral atoms and ions. We interpreted the\ndifference of Doppler velocities observed in di fferent spectral lines as a result of motions of di fferent components in the prominence\nin a resolution element, rather than the decoupling of neutr al atoms from plasma. In our interpretation, di fferent spectral lines sample\ndifferent components in the prominence because the optical dept h and the path of the incident radiation, which excites the at oms\ninto the upper levels that emit the observed spectral lines, are different among the observed spectral lines.\nElectric fields act on neutral atoms that decouple from plasm a and move across the magnetic field. If we assume a magnetic fie ld\nstrength of 200 G in the prominence over the active region, th e electric field experienced by neutral atoms moving across t he mag-\nnetic field is 0.56 V cm−1with a speed of 2.8 km s−1, which corresponds to twice of the standard deviation of the Gauss distribution\nin Fig. 10 (a). Anan et al. (2014) observed the full Stokes spe ctra of the Paschen series of neutral hydrogen in chromosphe ric jets,\nand obtained upper limits for possible electric fields of 0.3 V cm−1using magnetic field strength of 200 G. Thus, there is possibi lity\nthat the spectro-polarimetric observations allow us to inv estigate directly whether or not the decoupling of neutral a toms from the\nplasma cause the inferred Doppler velocity di fferences between calcium ions and neutral hydrogen. In contr ast, we cannot determine\nthe decoupling of neutral atoms from the observational data of this study.\nAcknowledgements. This work was supported by a Grant-in-Aid for Scientific Rese arch (No. 22244013, P.I. K. Ichimoto; No. 15K17609, P.I. T. A nan; No.\n16H01177, P.I. T. Anan) from the Ministry of Education, Cult ure, Sports, Science and Technology of Japan. A.H. is suppor ted by his STFC Emest Rutherford\nFellowship grant number ST /L00397X/2. 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Pop/suppress lawski\nConservation laws for a general Lorentz connection\nthe date of receipt and acceptance should be inserted later\nAbstract We derive conservation laws for energy–momentum (canonical and dynamical) and angular\nmomentum for a general Lorentz connection.\nKeywords Conservation law ·Canonical energy–momentum ·Dynamical energy–momentum ·\nAngular momentum ·Lorentz connection ·Spin density ·Metric compatibility ·Nonmetricity ·Tetrad\nrotation\nPACS04.20.Fy ·04.50.Kd\n1 Introduction\nGaugeformulation of gravitation [1] attempts to derive a unified picture of known interactions. This\nformulationis metric–affine : both themetric(ortetrad)andaffine(orLorentz)connectiona reregarded\nas gravitational potentials [2,3,4]. Explicit dynamical variables in met ric–affine theories can be taken\nas: metric and symmetric connection (Palatini formulation) [5], met ric and torsion (Einstein–Cartan\ntheory) [6], metric and asymmetric connection [7,8], metric, torsion and nonmetricity [9], tetrad and\ntorsion [10], and tetrad and Lorentz connection (Kibble–Sciama the ory) [2].\nThe principle of general covariance imposes the invariance of the to tal action under general co-\nordinate transformations. Since the metric and tetrad are relate d by the orthonormality condition,\nthe group of tetrad rotations is the Lorentz group [11,12]. The loc al Poincar´ e invariance, i.e. the in-\nvariance of a Lagrangian density for matter under coordinate tra nsformations and tetrad rotations,\nleads to identities ( conservation laws ) satisfied by matter sources [13]. An energy–momentum conser-\nvation (4 equations) results from the coordinate invariance and an angular momentum conservation\n(6 equations) results from the invariance under tetrad rotations [11,14]. The same invariance of the\ntotal Lagrangian for matter and gravitational field gives analogou sBianchi identities satisfied by the\nfield equations. Since there are 80 gravitational equations for 16 + 64 = 80 gravitational potentials\n(tetrad and connection) with 4+6 = 10 identities, 80 −10 = 70 potentials are independent dynamical\nvariables [11].\nIn this paper, which is a sequel of [15], we present a brief derivation o f conservation laws for the\ncanonical and dynamical energy–momentum tensors and angular m omentum (spin) density for a gen-\neralLorentz connection. Such a connection corresponds to an affine c onnection that is not restricted\nto be metric compatible and torsionless. Examples of a physical theo ry with a general affine connec-\ntion are Weyl’s conformal geometry [12,16] and the generalized Eins tein–Maxwell theory with the\nN. J. Pop/suppress lawski\nDepartment of Physics, Indiana University, Swain Hall West 117, 727 East Third Street, Bloomington, Indiana\n47405, USA\nE-mail: nipoplaw@indiana.edu2\nelectromagnetic field tensor represented by the homothetic curv ature tensor [17]. We use the notation\nof [15].\n2 Infinitesimal coordinate transformations\nUnder an infinitesimal coordinate transformation\nxµ→x′µ=xµ+ξµ, (1)\nwhereξµis an infinitesimal vector, the transformation law for any tensor or tensor density Φis given\nby\nδΦ=Φ′(x′)−Φ(x) =ξα\n,βCβ\nαΦ, (2)\nwhere the constant linear operators ˆCβ\nαare determined by the covariant derivative ofΦwith respect\nto the affine connection [12]:\nΦ;µ=Φ,µ+Γα\nβµˆCβ\nαΦ. (3)\nFor example, the operator ˆCacting on a scalar φ, contravariant vector Vν, covariant vector Vνand\nscalar density Vof weightwreturns, respectively:\nˆCβ\nαφ= 0, (4)\nˆCβ\nαVν=δν\nαVβ, (5)\nˆCβ\nαVν=−δβ\nνVα, (6)\nˆCβ\nαV=−wδβ\nαV. (7)\nThe transformation law for a Lagrangian density /C4, which is a scalar density (of weight 1) since /C4d4x\nis a scalar, is thus\nδ /C4=−ξµ\n,µ\n/C4. (8)\nALie derivative with respect to ξµof a quantity Φis defined as\nLξΦ=¯δΦ=Φ′(x)−Φ(x) =δΦ−ξνΦ,ν. (9)\nIt can be shown that ¯δΦ, unlikeδΦ, transforms under general coordinate transformations the sa me\nway asΦ. For example, the Lie derivative of the contravariant metric tenso rgµνis also a tensor:\nLξgµν=ξµ\n,αgαν+ξν\n,αgµα−ξαgµν\n,α= 2ξ(µ;ν)+(4S(µν)\nα+Nµν\nα)ξα. (10)\nAKilling vector is defined as a vector ξµthat preserves the metric: Lξgµν= 0.1\n3 Canonical energy–momentum tensors\nThe change δ /C4of a Lagrangiandensity for matter under an infinitesimal coordinat e transformation (1)\nis given by Eq. (8). If we assume that /C4depends, in addition to the coordinates xµ, on matter fields\nφand their first derivatives φ,µthen\nδ /C4=∂ /C4\n∂φδφ+∂ /C4\n∂φ,µδ(φ,µ)+¯∂ /C4\n∂xµξµ, (11)\nwhere the changes δφandδ(φ,µ) are brought by the transformation (1) and ¯∂denotes partial differ-\nentiation with respect to xµregardingφandφ,µas constant. Using the Lagrange field equations,\n∂ /C4\n∂φ−/parenleftBig∂ /C4\n∂φ,µ/parenrightBig\n,µ= 0, (12)\n1The invariance of the matter action under a coordinate trans lation generated by a Killing vector yields\nthe covariant conservation of the corresponding dynamical energy–momentum tensor [18].3\nand the identities /C4,µ=¯∂ /C4\n∂xµ+∂ /C4\n∂φφ,µ+∂ /C4\n∂φ,νφ,νµandδ(φ,µ) = (δφ),µ−ξν\n,µφ,ν, we bring Eq. (11) to\nδ /C4=ξµ/C4,µ+/parenleftBig∂ /C4\n∂φ,µ(δφ−ξνφ,ν)/parenrightBig\n,µ. (13)\nCombining Eqs. (8) and (13) gives a conservation law [12]:/C2µ\n,µ= /C2µ\n;µ−2Sµ\n/C2µ= 0, (14)\nwith the current:/C2µ=ξµ/C4+∂ /C4\n∂φ,µ(δφ−ξνφ,ν) =ξµ/C4+∂ /C4\n∂φ,µ¯δφ. (15)\nEquations (14) and (15) represent Noether’s theorem : the correspondence between continuous symme-\ntries of a Lagrangian and conservation laws.2\nIf we assume that the matter fields φare purely tensorial then applying Eqs. (2) and (3) to the\ndefinition of the conserved current (15) gives/C2µ=ξµ/C4+∂ /C4\n∂φ,µ/parenleftBig\n(ξα\n;β+2Sα\nβνξν)ˆCβ\nαφ−ξνφ;ν/parenrightBig\n. (16)\nIn order to obtain a local conservation law that does not contain th e vectorξµwe must impose a\ncovariant restriction on this vector at a particular point in spacetim e:\nξα\n;β+2Sα\nβνξν= 0, (17)\nwhich brings Eqs. (14) and (15) to\n/parenleftBig\nξµ/C4−∂ /C4\n∂φ,µξνφ;ν/parenrightBig\n;µ−2Sµ/parenleftBig\nξµ/C4−∂ /C4\n∂φ,µξνφ;ν/parenrightBig\n= 0. (18)\nUsing again Eq. (17) allows to eliminate ξµ, leading to/C4;ν−/parenleftBig∂ /C4\n∂φ,µφ;ν/parenrightBig\n;µ+2Sρ\nµν∂ /C4\n∂φ,µφ;ρ+2Sµ∂ /C4\n∂φ,µφ;ν= 0, (19)\nwhich represents a conservation law:\nHν\nµ;ν−2SνHν\nµ+2Sν\nµρHρ\nν= 0, (20)\nfor thecanonical energy–momentum tensor density:3\nHµ\nν=∂ /C4\n∂φ,µφ;ν−δµ\nν\n/C4. (21)\n2Ifxµare Cartesian coordinates then for Lorentz translations, ξµ= const and δφ= 0, we obtain a\nconservation of energy–momentum :Θµ\nν,µ= 0, where Θµ\nν=∂ /C4\n∂φ,µφ,ν−δµ\nν\n/C4is theenergy–momentum tensor\ndensity. This conservation also follows from the Lagrange fi eld equations (12). For Lorentz rotations, ξµ=ǫµ\nνxν\nandφ=1\n2ǫµνGµνφ, where Gµνare the generators of the Lorentz group, the conservation la w (14) yields a\nconservation of angular momentum : (Λµ\nαβ+Σµ\nαβ),µ= 0, where Λµ\nαβ=xαΘµ\nβ−xβΘµ\nαis theorbital angular\nmomentum density and Σµ\nαβ=∂ /C4\n∂φ,µGαβφis thespin density .\n3The canonical energy–momentum tensor density (21) general izes the Cartesian energy–momentum tensor\ndensityΘµ\nν(cf. footnote 2) to a general affine connection, replacing the ordinary derivative φ,νby the covariant\nderivative φ;ν. The corresponding canonical energy–momentum tensor can b e symmetrized using the generalized\nBelinfante–Rosenfeld formula [8,19].4\n4 Dynamical energy–momentum tensors\nAdynamical energy–momentum tensor density in the tetrad formulation of gravity, /CCa\nµ, is defined via\nthe variation of a matter Lagrangian density /C4with respect to a tetrad:4\nδ /C4= /CCa\nµδeµ\na. (22)\nIf /C4depends only on tensor matter fields then the tetrad enters /C4only where there is the metric tensor,\nin a combination gµν=ηabeµ\naeν\nb, yielding\nδeµ\na=1\n2eaνδgµν. (23)\nSubstituting Eq. (23) to (22) gives the standard general-relativis tic formδ /C4=1\n2\n/CCµνδgµν, where/CCµν=eaµ\n/CCa\nν. (24)\nTherefore, for purely tensorial matter fields, the symmetric par t /CC(µν)of the dynamical energy–\nmomentum tensor density in the tetrad formulation coincides with th e dynamical energy–momentum\ntensor density Tµνin the metric formulation [12,18]. If /C4depends also on spinor matter fields then\nEq. (22) becomes the sum of two parts, tensorial and spinorial:\nδ /C4=1\n2Tµνδgµν+ /CCa(spin)\nµ˜δeµ\na, (25)\nwhere˜δeµ\nadenotes the variation of the tetrad that cannot be related to the variation of the metric\ntensor.\nLet us assume that the matter Lagrangian density /C4depends on matter fields φ(and their first\nderivatives φ,µ) that can be expressed in terms of Lorentz and spinor indices only. Consequently,\nthe tetrad field appears in /C4only where there is a derivative of φ, in a covariant combination\neµ\naφ|µ. For example, the Dirac Lagrangian density for a massless particle isi\n2\n/CT(¯ψγaeµ\naψ|µ−eµ\na¯ψ|µγaψ)\nand the Maxwell Lagrangian density −1\n4√− /CVFµνFµν, whereFµν=Aν,µ−Aµ,ν, can be written as\n1\n2\n/CT(−eµ\naAb|µFab+Sc\nabAcFab).5Since /C4= /CTL, whereLis a scalar, we obtain6\nδ /C4= /CTδL− /CTea\nµLδeµ\na= /CT∂L\n∂φ|aφ|µδeµ\na− /C4ea\nµδeµ\na=/parenleftBig∂ /C4\n∂φ|aφ|µ− /C4ea\nµ/parenrightBig\nδeµ\na. (26)\nComparing Eq. (26) with (22) shows that the dynamical energy–mo mentum tensor density /CCa\nµis a\ngeneralized canonical energy–momentum tensor density [12]:/CCa\nµ=∂ /C4\n∂φ|aφ|µ−ea\nµ\n/C4, (27)\nor equivalently/CCµ\nν=∂ /C4\n∂φ,µφ|ν−δµ\nν\n/C4. (28)\nThe canonical energy–momentum tensor density (28) generalizes the density (21), replacing the\nderivativeφ;νwithφ|ν. The difference between the tensor densities (28) and (21) is/CCµ\nν−Hµ\nν=−∂ /C4\n∂φ,µΓνφ, (29)\nwhere the connection Γν=−1\n2ωabνGabdepends on the generators Gabof the representation of the\nLorentz group governing a transformation law of φ[12,15]. If the matter fields φin the Lagrangian\ndensity /C4are expressed in terms of coordinate indices only, they must be pur ely tensorial and the\ndynamical energy–momentum tensor density (27) corresponds t o the dynamical matter density Tµνin\nthe metric formulation.\n4The energy–momentum tensor, canonical or dynamical, is obt ained from the corresponding energy–\nmomentum tensor density by dividing the latter by /CT.\n5The definition Fµν=Aν,µ−Aµ,νdoes not yield Fab=Ab,a−Aa,b.\n6For the Maxwell Lagrangian density the variation δLdue to the variation of the tetrad is: δL=\n−Ab|µFabδeµ\na.5\n5 Spin density\nFor a general connection, not restricted to be metric compatible, the Lorentz connection ωab\nµis not\nantisymmetric in the indices a,b[3] and its symmetric part is related to the nonmetricity tensor:\nω(ab)\nµ=−1\n2Nab\nµ[15]. The variation of a matter Lagrangian density /C4with respect to a Lorentz\nconnection:\nδ /C4=1\n2\n/CBµ\nabδωab\nµ, (30)\ndefines the Lorentz connection-conjugate density /CBµ\nab.7The variation of /C4with respect to the anti-\nsymmetric part of the Lorentz connection defines the spin density /C5µ\nabin the tetrad formulation of\ngravity:\nδ /C4=1\n2\n/C5µ\nabδω[ab]\nµ. (31)\nThe variation of /C4with respect to the symmetric part of the Lorentz connection, i.e. the nonmetricity\ntensor, defines the nonmetricity-conjugate density /C6µ\nab:\nδ /C4=1\n2\n/C6µ\nabδω(ab)\nµ. (32)\nThe densities /C5µ\naband /C6µ\nabare the symmetric and antisymmetric (in the indices a,b) parts of the\ndensity /CBµ\nab, respectively:/CBµ\nab= /C5µ\nab+ /C6µ\nab. (33)\nThe Lorentz connection ωab\nµenters /C4only where there is a derivative of φ, in a combination\n−∂ /C4\n∂φ,µΓµφ. Consequently, the spin density is equal to/C5µ\nab=∂ /C4\n∂φ,µGabφ, (34)\nand generalizes the spin density Σµ\nαβin the Cartesian coordinates (cf. footnote 2). The difference (29 )\nbetween the tensor densities (28) and (21) is then/CCµ\nν−Hµ\nν=1\n2ωab\nν\n/C5µ\nab. (35)\n6 Conservation of angular momentum\nThe Lorentz group is the group of tetrad rotations [3,12]. Since a physical matter Lagrangian density/C4is invariant under local, proper Lorentz transformations, it is invar iant under tetrad rotations:\nδ /C4=∂ /C4\n∂φδφ+∂ /C4\n∂φ,µδ(φ,µ)+ /CCa\nµδeµ\na+1\n2\n/CBµ\nabδωab\nµ= 0, (36)\nwhere the changes δcorrespond to a tetrad rotation. Under integration of Eq. (36) o ver spacetime the\nfirst two terms vanish because of the field equation for φ(12):\n/integraldisplay/parenleftBig/CCa\nµδeµ\na+1\n2\n/CBµ\nabδωab\nµ/parenrightBig\nd4x= 0. (37)\nFor an infinitesimal Lorentz transformation:\nΛa\nb=δa\nb+ǫa\nb, (38)\n7The analogous affine connection-conjugate density Πµν\nρ, defined via the variation of /C4with respect to a\ngeneral affine connection: δ /C4=Πµν\nρδΓρ\nµν, is called hypermomentum [8].6\nwhereǫab=−ǫba, the tetrad ea\nµchanges according to\nδea\nµ= ˜ea\nµ−ea\nµ=Λa\nbeb\nµ−ea\nµ=ǫa\nµ, (39)\nand the tetrad eµ\na, because of the identity δ(ea\nµeν\na) = 0, according to\nδeµ\na=−ǫµ\na. (40)\nThe Lorentz connection changes according to\nδωab\nµ=δ(ea\nνωνb\nµ) =ǫa\nνωνb\nµ−ea\nνǫνb\n;µ=ǫa\ncωcb\nµ−ea\nνǫνb\n|µ+ǫa\ncωbc\nµ=−ǫab\n|µ−ǫa\ncNbc\nµ.(41)\nSubstituting Eqs. (24), (40) and (41) to (37) gives8\n0 =−/integraldisplay/parenleftBig/CCa\nµǫµ\na+1\n2\n/CBµ\nabǫab\n|µ+1\n2\n/CBµ\nabǫa\ncNbc\nµ/parenrightBig\nd4x\n=/integraldisplay/parenleftBig/CCµνǫµν−1\n2\n/CBρ\nµνǫµν\n|ρ−1\n2\n/CBσ\nµρǫµνNρ\nνσ/parenrightBig\nd4x\n=/integraldisplay/parenleftBig/CC[µν]−Sρ\n/CBρ\n[µν]+1\n2\n/CBρ\n[µν] ;ρ+1\n2Nρσ\n[µ\n/CBν]ρσ/parenrightBig\nǫµνd4x. (42)\nSince the infinitesimal Lorentz rotation ǫµνis arbitrary, we obtain a generalized conservation law for\nangular momentum (spin density):9/C5ρ\nµν;ρ=− /CCµν+ /CCνµ+2Sρ\n/C5ρ\nµν−Nρσ\n[µ\n/C5ν]ρσ−Nρσ\n[µ\n/C6ν]ρσ. (43)\nEq. (43) corresponds to the conservation law for the hypermome ntum density in the orthonormal\ngauge [11].\n7 Conservation of energy–momentum\nA matter Lagrangian density /C4is also invariant under infinitesimal translations of the coordinate sy s-\ntem (1). The corresponding changes of the tetrad and Lorentz c onnection are given by Lie derivatives:\n¯δeµ\na=Lξeµ\na=ξµ\n,νeν\na−ξνeµ\na,ν, (44)\n¯δωab\nµ=Lξωab\nµ=−ξν\n,µωab\nν−ξνωab\nµ,ν. (45)\nEq. (37) becomes now/integraldisplay/parenleftBig/CCa\nµ¯δeµ\na+1\n2\n/CBµ\nab¯δωab\nµ/parenrightBig\nd4x= 0, (46)\nand holds for an arbitrary vector ξµ:\n0 =/integraldisplay/parenleftBig/CCa\nµξµ\n,νeν\na− /CCa\nµξνeµ\na,ν−1\n2\n/CBµ\nabξν\n,µωab\nν−1\n2\n/CBµ\nabξνωab\nµ,ν/parenrightBig\nd4x\n=/integraldisplay/parenleftBig\n− /CCν\nµ,ν− /CCa\nνeν\na,µ+1\n2( /CBν\nabωab\nµ),ν−1\n2\n/CBν\nabωab\nν,µ/parenrightBig\nξµd4x. (47)\nConsequently we can write\n0 = /CBν\nab ,νωab\nµ+ /CBν\nab(ωab\nµ,ν−ωab\nν,µ)−2 /CCν\nµ,ν−2 /CCa\nνeν\na,µ\n= ( /CBν\nab|ν−2Sρ\n/CBρ\nab+ /CBν\ncbωc\naν+ /CBν\nacωc\nbν)ωab\nµ+ /CBν\nab(−Rab\nµν+ωa\ncµωcb\nν−ωa\ncνωcb\nµ)\n−2 /CCν\nµ,ν−2 /CCa\nνeν\na,µ, (48)\n8We also use the partial-integration identity/integraltext\nd4x( /CEµ)|µ= 2/integraltext\nd4xSµ\n/CEµ, valid for an arbitrary vector\ndensity /CEµ.\n9If we use the affine connection Γρ\nµν, which is invariant under tetrad rotations, instead of the L orentz\nconnection ωab\nµas a variable in a Lagrangian density /C4then we must replace the term with δωab\nµin Eq. (36)\nby a term with δ(eµ\na,ν). The resulting equation is equivalent to Eq. (43), cf. [17] .7\nwhich reduces to\n0 = ( /CBν\nab|ν−2Sρ\n/CBρ\nab− /CBν\nacNc\nb ν)ωab\nµ−Rab\nµν\n/CBν\nab\n−2 /CCν\nµ;ν+4Sν\n/CCν\nµ−2 /CCρνωνρ\nµ+4Sνρ\nµ\n/CCρν\n= ( /C5ν\nαβ;ν+ /C6ν\nαβ;ν−2Sρ\n/CBρ\nαβ− /CBν\nασNσ\nβ ν)ωαβ\nµ−Rαβ\nµν\n/CBν\nαβ\n−2 /CCν\nµ;ν+4Sν\n/CCν\nµ−2 /CCρνωνρ\nµ+4Sνρ\nµ\n/CCρν. (49)\nEquations (33), (43) and the identity10R(αβ)\nµν=Nαβ\n[µ;ν]+Sρ\nµνNαβ\nρbring Eq. (49) to the form\nof a generalized conservation law for energy–momentum :/CCν\nµ;ν= 2Sν\n/CCν\nµ−2Sν\nµρ\n/CCρ\nν+1\n2\n/CCνρNνρ\nµ−1\n4\n/C6σ\nνρ;σNνρ\nµ+1\n4\n/C5αρσNρσ\nβNαβ\nµ\n+1\n4\n/C6αρσNρσ\nβNαβ\nµ+1\n2\n/C6ρ\nαβSρNαβ\nµ−1\n2\n/C5ν\nαβRαβ\nµν−1\n2\n/C6ν\nαβ(Nαβ\n[µ;ν]+Sρ\nµνNαβ\nρ).(50)\nEq. (50) corresponds to the conservation law for the canonical e nergy–momentum density in the or-\nthonormal gauge [11], cf. also [17,20].\nThe conservation law (50) for the dynamical energy–momentum te nsor density /CCν\nµcoincides with\nthe conservation law (20) for the canonical energy–momentum de nsityHν\nµif the nonmetricity tensor\nNµνρand the spin density /C5ρ\nµνvanish. In order to derive Eq. (20) we assumed that the matter fie lds\nφare purely tensorial and only considered a coordinate translation in the variation of φ. Therefore the\nspin density does not contribute to the conservation of the tenso r density Hν\nµand does not appear in\nEq. (20). In fact, the difference (35) between the tensor densit ies (28) and (21) is linear in the spin\ndensity. The absence of the nonmetricity tensor in Eq. (20) is relat ed to the fact that we imposed the\nconstraint (17) on the vector ξµto derive a covariant conservation law independent of ξµ, while there\nis no restriction on this vector in the derivation of the conservation law (50). The full conservation law\nforHν\nµ(corresponding to unrestricted ξµ) can be derived by combining Eqs. (35), (43) and (50).\nReferences\n1. R. Utiyama, Phys. Rev. 101, 1597 (1956).\n2. T. W. B. Kibble, J. Math. Phys. 2, 212 (1961); D. W. Sciama, in: Recent Developments in General Relativity\n(Pergamon, Oxford, 1962), p. 415; D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964).\n3. F. W. Hehl, P. von der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976).\n4. F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’eman, Phys. Rep. 258, 1 (1995).\n5. A. Palatini, Rend. Circ. Mat. (Palermo) 43, 203 (1919); A. Einstein, Sitzungsber. Preuss. Akad. Wiss.\n(Berlin), 32 (1923).\n6.´E. Cartan, Compt. Rend. Acad. Sci. (Paris)174, 593 (1922).\n7. F. W. Hehl and G. D. Kerlick, Gen. Relativ. Gravit. 9, 691 (1978).\n8. F. W. Hehl, E. A. Lord and L. L. Smalley, Gen. Relativ. Gravit. 13, 1037 (1981).\n9. L. L. Smalley, Phys. Lett. A 61, 436 (1977).\n10. F. W. Hehl and B. K. Datta, J. Math. Phys. 12, 1334 (1971).\n11. E. A. Lord, Phys. Lett. A 65, 1 (1978).\n12. E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, New Delhi, 1976).\n13. K. Hayashi and A. Bregman, Ann. Phys. (N.Y.)75, 562 (1973).\n14. P. von der Heyde, Phys. Lett. A 58, 141 (1976).\n15. N. J. Pop/suppress lawski, ArXiv: 0710.3982 [gr-qc].\n16. H. Weyl, Space, Time, Matter (Methuen, London, 1922); R. Carroll, ArXiv: 0705.3921 [gr- qc].\n17. V. N. Ponomariov and Ju. Obuchov, Gen. Relativ. Gravit. 14, 309 (1982).\n18. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1975).\n19. F. J. Belinfante, Physica7, 449 (1940); L. Rosenfeld, M´ em. Acad. R. Belg. 18, 6 (1940).\n20. R. Hecht, F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’em an,Phys. Lett. A 172, 13 (1992); G.\nSardanashvily, Class. Quantum Grav. 14, 1357 (1997).\n10This identity follows from the formula for the commutator of the covariant derivatives of the metric tensor."    },    {        "title": "2302.11968v1.Buckling_Metamaterials_for_Extreme_Vibration_Damping.pdf",        "content": "Buckling Metamaterials for Extreme Vibration Damping\nDavid M.J. Dykstra,\u0003Coen Lenting, Alexandre Masurier, and Corentin Coulaisy\nInstitute of Physics, University of Amsterdam, Science Park 904,\n1098 XH, Amsterdam, the Netherlands\nDamping mechanical resonances is a formidable challenge in an increasing number of applications.\nMany of the passive damping methods rely on using low sti\u000bness dissipative elements, complex\nmechanical structures or electrical systems, while active vibration damping systems typically add\nan additional layer of complexity. However, in many cases, the reduced sti\u000bness or additional\ncomplexity and mass render these vibration damping methods unfeasible. This article introduces\na new method for passive vibration damping by allowing buckling of the primary load path, which\nsets an upper limit for vibration transmission: the transmitted acceleration saturates at a maximum\nvalue, no matter what the input acceleration is. This nonlinear mechanism leads to an extreme\ndamping coe\u000ecient tan \u000e\u00190:23 in a metal metamaterial|orders of magnitude larger than the linear\ndamping of traditional lightweight structural materials. This article demonstrates this principle\nexperimentally and numerically in free-standing rubber and metal mechanical metamaterials over a\nrange of accelerations, and shows that bi-directional buckling can further improve its performance.\nBuckling metamaterials pave the way towards extreme vibration damping without mass or sti\u000bness\npenalty, and as such could be applicable in a multitude of high-tech applications, including aerospace\nstructures, vehicles and sensitive instruments.\nAny mechanical system will exhibit a resonance. At\nthis resonance, the transmission of force and accelera-\ntion is maximal. Limiting the ampli\fcation of accel-\neration is paramount in a wide range of applications\nwhere vibrations can cause unwanted noise and failure.\nA paradigmatic example is that of a mass and spring\ndamper shaken from the bottom (Fig. 1A): at reso-\nnance, the mass will vibrate with a much higher acceler-\nation than the input acceleration provided by the shaker.\nWhile myriad strategies using highly viscoelastic materi-\nals [1, 2], negative sti\u000bness components [3{8], band-gap\nmetamaterials [9{15] and active control [2, 16, 17] have\nbeen proposed, they typically su\u000ber from added mass or\nloss in sti\u000bness.\nHere, we propose to use Euler buckling as a functional\nmechanism to create vibrations absorbers (Fig. 1B):\nbuckling structures are simultaneously sti\u000b thanks to\ntheir high sti\u000bness prior to buckling, yet limit the trans-\nmission of acceleration under post-buckling since they\nexhibit a force plateau under compression. Although the\nidea is best illustrated with a single column under axial\nvibrations (Fig. 1B), such a structure would collapse in\npost-buckling and would not be suitable for applications\nthat require freestanding unsupported structures. This\nis precisely where metamaterials could come to the res-\ncue, as they can combine buckling in one direction and\nstructural sti\u000bness in the other directions [18{24]. While\ntheir shock response has been well studied, their nonlin-\near vibration response remains poorly understood.\nIn this article, we demonstrate that free-standing load-\ncarrying metamaterials that undergo a buckling insta-\n\u0003dmj.dykstra@gmail.com\nycoulais@uva.nlbility provide an upper limit for transmission of vibra-\ntions. We show that this e\u000ecient vibration absorption\nstems from elastic and damping nonlinearities that are in-\nduced by buckling. We further generalize the concept to\nmetallic metamaterials and to metamaterials that buckle\nboth under compression and tension. We obtain extreme\ndamping with respect to ordinary lightweight structural\nmaterials. Our work demonstrates that buckling meta-\nmaterials are a competitive solution for lightweight struc-\ntures combining high damping and high speci\fc sti\u000bness\nin high-tech applications.\nTo create a structure that can maintain its own lat-\neral stability in post-buckling, we \frst turn to one of the\nmost common designs in \rexible mechanical metamate-\nrials [25{27]: a polymeric slab that is patterned with a\nsquare array of circular holes (Fig. 1CD). This metama-\nterial exhibits a global buckling mode, where the pattern\nof pores becomes an array of ellipses with alternating\norientations (Fig. 1D, centre). To determine its nonlin-\near vibration characteristics, we mount a mass on top\nand subject the sample to a base excitation around the\neigenfrequency at 2 di\u000berent levels: a low level of 0.26 G\n(Fig. 1CE) and a high level of 0.89 G (Fig. 1df), where\nG=9.81 m/s2is the acceleration of gravity.\nAt the lower excitation level, we observe that the holes\nremain close to circular (Fig. 1C, see also Supplemen-\ntary Video 1) and we observe both a sinusoidal output\nresponse, which has a \u0019=2 phase lag with respect to the\ninput excitation (Fig. 1E). This was to be expected based\non linear vibrations [28]. At the higher excitation level,\nthe sample buckles as seen in Fig. 1D, see also Supple-\nmentary Video 1. As a result, the output response in\nFig. 1F is no longer perfectly sinusoidal. Importantly,\ndespite the input level increasing by more than a factor\nthree (from 0.26 G to 0.89 G), the output level in com-\npressive direction merely changes by a third (from 4.3 GarXiv:2302.11968v1  [cond-mat.soft]  23 Feb 20232\nB A\nMM\nMMM\nD C\nF EMMMMM\n1 cm\nFIG. 1: Damping vibrations with buckling. (A) A mass ( M) spring damper system, with base excitation\n(blue) can show a large ampli\fed response (orange) around resonance. (B) When the spring is a slender beam,\nwhich can buckle when subjected to a su\u000ecient compressive load from the base excitation, the ampli\fed response\nmay be lower. (C,D) show the deformation of a holar sample with mass mounted on top when subjected to a base\nexcitation from the bottom around the eigenfrequency. (C) corresponds to a base excitation acceleration of 0.26 G\nat 33.8 Hz, while (D) corresponds to a base excitation acceleration of 0.89 G at 33.0 Hz. The ellipticity of the holes,\n\n, is tracked with red and blue ellipses (Section IV D, color bar). (E,F) Base excitations (blue) of 0.26 G (E) and\n0.89 G (F) induce output accelerations (orange) of 4.3 G (E) and 5.7 G (F) respectively.\nto 5.7 G). More surprisingly, the maximum acceleration\nin tensile direction also only increases by a factor of two\ninstead of three (from 4.3 G to 8.9 G). This suggests that\ncompressive buckling also dampens vibrations in tensile\ndirection.\nThis e\u000ecient vibration damping stems from nonlineari-\nties induced by buckling. To quantify such nonlinearities,\nwe perform compression and tension mechanical tests at\nvarious strain rates (Fig. 2A, see Methods for details).\nAs expected [26, 29], while the response is nearly linear\nin tension and for compression less than 2 mm, a buck-\nling instability occurs at a compressive displacement of\n2 mm. This instability induces a force plateau, a key\nnonlinearity that explains the saturation of acceleration\nunder compression seen in Fig. 1F. However, this nonlin-\nearity alone does not su\u000ece to explain the reduction of\nacceleration in the tensile direction.\nThe missing ingredient is an additional damping non-\nlinearity that is also rooted in buckling. Indeed, when we\nvary the loading rate, we observe that the amount of hys-\nteresis increases signi\fcantly around the point of buck-\nling when we increase the loading rate from 1 mm/min\nto 1000 mm/min (Fig. 2A-inset). To better quantify thise\u000bect, we compress and extend the sample at a loading\nrate of 1000 mm/min up to di\u000berent compression levels\nand measure the average hysteretic force (di\u000berence be-\ntween loading and unloading) across the loading regime\nas function of the compression. We see that the hysteresis\nis non-monotonic with a maximum at a compression of 4\nmm, which corresponds to the buckling point. This non-\nmonotonic damping di\u000bers from that of linear viscoelastic\nmaterials (e.g. the Voigt damper in Fig. 1A), where the\naverage hysteretic force is constant. One concludes that\nbuckling ampli\fes viscoelastic e\u000bects. This can be inter-\npreted by the fact that the material that makes up the\nslender parts of the metamaterial undergoes much larger\nstrain rates than the full structure does. Moreover, an-\nother key component is present speci\fcally in vibrations\nwith a base excitation. E\u000bectively speaking, nonlinear-\nities break resonance. As the peak force levels do not\nincrease linearly with acceleration exciting level, the en-\nergy injected in the system does not increase linearly with\nacceleration exciting level either.\nThese combined elastic and damping nonlinearities\nboth team up to e\u000eciently dissipate vibrations. To quan-\ntify such dissipation, we measure the ampli\fcation fac-3\nC D A B\nH IE F\nG\nFIG. 2: Vibration damping performance of buckling metamaterials. (A) Force ( F) displacement ( d)\ncurves during compression-tension tests of the sample of Fig. 1DE. The four curves in D and E correspond to\nloading rates of 1, 10, 100 and 1000 mm/min from yellow to red respectively. (E) Equivalent force-displacement\ncurve of simpli\fed model (see Section IV E). (B,F) Average hysteretic force over compression range when performing\ntension-compression experiments at 1000 mm/min: (B) experimental, (F) numerical, normalized with the average\nhysteretic force for 0.1mm compression-tension. (G) Numerical Voigt damper strength as function of the\ncompressive displacement, normalized by the linear Voigt damper strength (Methods). (C,H) Maximum acceleration\nampli\fcation factor Aas function of frequency fduring a frequency sweep with rising and (inset) dropping\nfrequency: (C) experimental and (H) numerical. Solid lines correspond to tension while dashed lines correspond to\ncompression. The color of the curve indicates the input acceleration ain. (D,I) Maximum output acceleration aout\nacross the frequency range: (E) experimental and (I) numerical. Blue (orange) curves correspond to rising\n(dropping) frequencies. Circles and solid lines (squares and dashed lines) correspond to compression (tension) in E\nand I respectively. Grey lines show the linearized trend.\ntor (ratio between output and input acceleration) as a\nfunction of frequency: we perform frequency sweeps at\nvarious input levels with both rising (Fig. 2C) and drop-\nping (Fig. 2C-inset) frequency levels, passing the reso-\nnance. For rising frequencies, buckling is very e\u000ecient\nat limiting the vibration transmission. For small excita-\ntions (blue in Fig. 2C), ampli\fcation factors up to 19\nare found. During post-buckling the peak ampli\fcation\nfactors across the frequency domain become as low as 6\nin compression (bordeaux solid lines) and 9 in tension\n(bordeaux dashed lines). This corresponds to more than\ndoubling and tripling the amount of damping for tension\nand compression respectively. The peak ampli\fcation\nfactors also shift to lower frequencies. For dropping fre-\nquencies, we experience a di\u000berent trend. We observe\nlarger ampli\fcation factors (Fig. 2C-inset) at smaller\nfrequencies compared to a frequency sweep with a rising\nfrequency. This di\u000berence between rising and dropping\nfrequency sweeps demonstrates bistability post-buckling\nat frequency ranges below that of the linear resonance.Equivalently, this reduction of the ampli\fcation factor\ncorresponds to a saturation of the maximum output ac-\nceleration, aout, across the frequency range as function\nof the input acceleration, ain(Fig. 2D). In the com-\npressive direction (circles), we observe a hard limit on\nthe maximum vibration transmission at 5.9 G. In tension\n(squares), we observe that the trend becomes markedly\nlower than the linear trendline (grey) post-buckling. This\nbehavior is consistent with analytical examples of vibra-\ntions of mass-spring dampers with softening springs such\nas quadratic or cubic examples [30{32]. However, while\nanalytical examples of vibrations of mass-spring dampers\nwith weakly nonlinear softening springs typically demon-\nstrate increased ampli\fcation factors at larger excita-\ntions [30, 31], Fig. 2C demonstrates the opposite: the\ndrastic nonlinearities spawned by buckling shave o\u000b the\nresonance peak and e\u000eciently limit vibration transmis-\nsion.\nHowever, the fact that buckling dampens e\u000bectively\nsimple frequency sweeps does not yet guarantee that it4\ndampens more complex vibrations. After all, nonlinear\nvibration responses can not be linearly combined like lin-\near vibration responses can be combined. Therefore, we\nalso subject the sample of Fig. 1 to random vibrations\n(See Section IV G) and we \fnd again that the maxi-\nmum transmission of acceleration saturates at 6.0 G.\nThis demonstrates that buckling based vibration damp-\ning works e\u000bectively regardless of the type of vibration:\ncontrolled or random.\nIn order to design buckling-based vibration damping, it\nis also necessary to be able to predict it. In theory, \fnite\nelement methods could be used to model the response\nof buckling based vibration damping. However, nonlin-\near dynamic \fnite element methods are notoriously ex-\npensive computationally, especially for a very large num-\nber of cycles [33]. For this purpose, we develop a sim-\nple numerical model, based on a nonlinear mass spring\ndamper, similar to the simpli\fed representation of Fig.\n1A. However, instead of a linear model, we use nonlin-\near force-displacement (Fig. 2E) and dashpot strength-\ndisplacement (Fig. 2FG) curves. We tune these param-\neters to \ft the experimental results of Fig. 2AB and\nthe low excitation eigenfrequency of Fig. 2C (Methods).\nWe then subject the model numerically to the same fre-\nquency sweeps as Fig. 2CD (Methods) and obtain the\nequivalent numerical results in Fig. 2HI.\nWe observe that the results in Fig. 2HI match the re-\nsults of Fig. 2CD well: not only qualitatively but also\nquantitatively. The only signi\fcant di\u000berence found is\nthat the numerical model predicts slightly higher output\naccelerations post-buckling than the experiments. This\ncould suggest that the bilinear approximations of Fig.\n2EG oversimplify the nonlinear dissipation of the exper-\niments. However, the fact that the numerical predic-\ntions are slightly higher than the experimental results\nalso show that the numerical results slightly underesti-\nmate the damping performance. Moreover, while the \ft-\nting parameters of Fig. 2EG have been obtained based on\nthe experiments (See Section IV E), these could also be\ngenerated using nonlinear \fnite element methods, with-\nout modeling the entire frequency sweep tests with these\nsame \fnite element methods. Together, this implies that\nbuckling metamaterials with target sti\u000bness and target\ndamping could conservatively be designed and predicted\nusing a very simple single element numerical model.\nI. HIGH STIFFNESS MATERIALS\nSo far, we have demonstrated that vibration damping\nusing buckling metamaterials made out of an elastomer\nis e\u000ecient. However, our results su\u000ber from three main\nshortcomings. First of all, elastomers inherently su\u000ber\nfrom a low speci\fc sti\u000bness, which makes them inher-\nently unsuited for load bearing structures. Second, it is\nfar from obvious that buckling metamaterials can be gen-\neralized to sti\u000b materials, such as metals, \fbre reinforced\ncomposites or ceramics, which have a low yield or failurestrain [34]. This implies that sti\u000b metamaterials cannot\nundergo repeated thick-walled buckling. Finally, while\nviscoelastic buckling will increase the amount of dissi-\npation (Fig. 1D (inset)), the resulting extra amount of\ndissipated energy can still be relatively small compared\nto the total strain energy in the system for base materials\nwith low damping coe\u000ecients.\nTo overcome the \frst two problems, we can use thin-\nwalled metal metamaterials. We can prevent yielding\nwhile buckling by letting the entire mechanism bend, as\nis demonstrated in Fig. 3EF, as opposed to localized\nmechanism deformation in Fig. 1D. Furthermore, while\nstraight thin-walled beams inherently exhibit a low buck-\nling load, the buckling load can be increased and tailored\nusing curvature about the longitudinal axis of each beam.\nIn turn, by using the thin-walled design of Fig. 3EF, the\ndesign becomes inherently sensitive to shear. We can re-\nmove this compliance to shear by creating a 3D structure\nmade out of two copies of the 2D mechanism, wherein the\ntwo 2D mechanism stabilize one another.\nWe construct the metamaterial sample of Fig. 3AEF.\nThis sample consists of 0.15 mm thin laser cut steel sheets\n(AISI 301 Full Hard) of 90 \u000250 mm, bolted to 3D printed\naluminium connectors (AlSi10Mg, selective laser sinter-\ning). The steel sheets are pre-curved by 1.5 mm. The\ntop and bottom crosses are made from 5 mm thick CNC\nmilled aluminium, while the cross sheets in the middle\nare made from 0.25 mm thin laser cut steel sheets (AISI\n301 Full Hard). The geometry has been selected with\nthe help of static nonlinear \fnite element methods in\nAbaqus. The sample measures 333 \u0002333\u0002328 mm\nand has a mass of 1.9 kg.\nWhen we compress the sample slowly (1 mm/min) and\nplot the force-displacement curve in Fig. 3B, we ob-\nserve a buckling force plateau at 78 N. While buckling,\nwe observe snap-through instabilities (Fig. 3B inset).\nWhen the pre-curved members buckle, they snap to an\nuncurved state about their longitudinal axis. In particu-\nlar, we observe that snap-through releases strain energy\nand excites local vibration modes of the metal sheets.\nThis immediately helps to solve the third problem: insuf-\n\fcient dissipation for low damping base materials, which\nwas identi\fed in the beginning of this section. The snap-\nthrough instabilities show negative sti\u000bness and dissipate\nenergy [3, 5, 35].\nIn turn, this dissipated energy as function of the dis-\nplacement is plotted in Fig. 3C. It is also noteworthy\nthat we observe that the force-displacement curve before\nbuckling is not entirely linear due to imperfections in the\nsample.\nSimilar to what we did for the elastomeric sample in\nFig. 1 and Fig. 2, we add a mass of 1.5 kg on the top of\nthe sample and subject the metal sample to a vibrational\nbase excitation, see also Supplementary Video 2. We sub-\nject the sample to frequency sweeps from low to high fre-\nquencies and back at several acceleration excitation levels\n(See Section IV F). We track the acceleration response at\nthe base and top of the sample, similar to what we did5\nE\nF\nA B\nG\nC D\nH I\n333\n328\nFIG. 3: Metallic buckling metamaterial for vibration damping. (A) Metallic metamaterial consisting of\ntwo crossing 3\u00023 unit cells, allowing buckling, with side view: (E) unbuckled and uncompressed, (F) buckled and\ncompressed by 1.2mm. (B) Force displacement curve of metal metamaterial with zoomed insert. (C) Cumulative\ndissipated energy, H, corresponding to (B). (D) When subjected to a base vibration, with a 1.4kg mounted on top:\nas function of the average input acceleration ain, the peak output acceleration aoutacross the frequency domain.\nOrange (blue) curves correspond to rising (dropping) frequencies. Circles (squares) correspond to compression\n(tension). Grey lines show the average linearized trend. (G-I) Numerical results for simulations, equivalent to the\nexperimental results of B-D (See Methods).\nfor the elastomeric sample in Fig. 2. We also compute\nthe maximum output acceleration across the frequency\ndomain as function of the input acceleration level in Fig.\n3D. Again, we \fnd that buckling based vibration damp-\ning works. More speci\fcally, the metal sample features\na loss coe\u000ecient tan \u000e\u00191=Amax\u00190:23 atain= 1 G,\na tripling of the damping coe\u000ecient with respect to the\nlow excitation response. . This loss coe\u000ecient greatly\nsurpasses those of traditional light-weight structural ma-\nterials such as aluminium alloys (1 \u000110\u00004\u00002\u000110\u00003), steels\n(2\u000110\u00004\u00003\u000110\u00003) or carbon \fbre reinforced polymers\n(1\u000110\u00003\u00003\u000110\u00003) [34]. This shows that buckling based\nvibration damping can be used to surpass the Ashby lim-\nits of loss coe\u000ecient versus speci\fc modulus [34].\nMoreover, it is noteworthy that the metal sample did\nnot su\u000ber from fatigue damage during the performed vi-\nbration tests. Over the course of the test campaign, the\nmetal sample has been subjected to around \u0019105vibra-\ntion cycles, across a variety of frequency and accelera-\ntion levels. At no point during this testing campaign has\nany visual damage been identi\fed. This is all the more\nimpressive considering that snap-through induces local\nmodes, which may even vibrate at a higher frequency.\nThis shows that it is also possible to produce buckling\nmetamaterials for vibration damping with a high speci\fcsti\u000bness without being fatigue sensitive.\nSimilar to what we did for the elastomeric sample, we\ncan also model the performance of the metal sample using\na numerical model in Fig. 3GHI (See Section IV E). We\ncan approximate the force-displacement curve of Fig. 3B\nusing the bilinear force-displacement curve of Fig. 3G\n(blue). Here, we omit the imperfections in the force-\ndisplacement curve of Fig. 3B to see how a sample with\nlittle imperfections would respond. However, unlike what\nwe considered for the numerical model of the elastomeric\nsample in Fig. 2EFG, we will omit nonlinear viscos-\nity and instead only consider the dissipation due to the\nsnap-through instabilities. To do this, we smear the hys-\nteresis of Fig. 3B linearly over the range between the\nstart of buckling in Fig. 3G and the distance of the last\nsnap-through instability in Fig. 3B. As such, we obtain\nthe unbuckling force-displacement curve of Fig. 3G (or-\nange), with corresponding hysteresis-displacement curve\nin Fig. 3H. We then subject the model numerically to\nthe same back-and-forth frequency sweeps as the exper-\nimental experimental of Fig. 3A and obtain the equiva-\nlent numerical results of Fig. 3D in Fig. 3I. Again, we\nalso \fnd that buckling based vibration damping works\nnumerically, particularly in compression.\nWe do however obtain some signi\fcant di\u000berences be-6\ntween the experiments and numerics. Namely, the nu-\nmerics predict lower peak accelerations in compression\nthan the experiments and higher peak accelerations in\ntension, especially for dropping frequencies. The lower\npeak acceleration in compression could be because the\nmetal sample also has additional local modes, which the\nnumerical model does not take into account. Further-\nmore, viscoelasticity can delay buckling, which can in-\ncrease the peak load [23]. A similar e\u000bect in turn explains\nwhy the numerical mode predicts higher accelerations\nin tensile direction: delayed buckling and unbuckling\ninduced by viscoelasticity can increase dissipation [23],\nwhich can lead to lower ampli\fcation factors, as discussed\nabove in the case of the elastomeric sample. Despite these\ndiscrepancies, even without considering nonlinear damp-\ning, but when considering snap-through induced dissi-\npation, the numerical model shows valid trends. This\ndemonstrates that such a simple numerical model can\nstill be used to predict and design the performance of\nbuckling based vibration damping metamaterials, even\nwhen they are made from thin-walled structures with\nhigh speci\fc moduli and snap-through instabilities.\nII. BI-DIRECTIONAL BUCKLING\nSo far, we have demonstrated that buckling metama-\nterials dampen vibrations for both soft and sti\u000b materi-\nals. However, all of the cases analyzed so far consider\nbuckling exclusively in the compressive direction. Yet, it\nis also possible to realize geometries that can buckle in\nmultiple directions, such as the geometries considered in\nFig. 4. In Fig. 4A, we have created a sample, which can\nbuckle in both the compressive (Fig. 4B) and the tensile\n(Fig. 4C) direction(See section IV A 2 for manufacturing\ndetails).\nThis sample has in fact been optimized to buckle in\ntension and compression at the same strain level using\nBayesian optimisation and nonlinear \fnite element meth-\nods with Python and Abaqus [36]. The sample can even\nbuckle simultaneously in tension and compression when\nsubjected to more complex loading cases, such as bending\nin Fig. 4D.\nWhile this sample demonstrates that buckling can be\nachieved simultaneously in both tension and compres-\nsion, it does not o\u000ber su\u000ecient sti\u000bness by itself to\nbe used as a load bearing vibration damping structure.\nHowever, we can generalize our numerical model to sim-\nulate the vibration response of structures that o\u000ber snap-\nthrough buckling in both tension and compression. To do\nso, we adjust the model we used to simulate Fig. 3G-I,\nand apply the force-displacement curve of Fig. 4E in-\nstead. This force-displacement curve is in fact the sym-\nmetrically buckling version of Fig. 3G, with the numeri-\ncally considered mass subtracted (See Section IV E). We\nthen use our model to subject the sample numerically to\nthe same up and down frequency sweeps at di\u000berent base\naccelerations as we did for Fig. 3I. When we track themaximum output acceleration across the frequency do-\nmain,aoutas function of the input acceleration, ain, we\nobtain the results of Fig. 4F. Here, we \fnd that buckling\nbased vibration damping is very e\u000bective to set an up-\nper limit for vibration transmission. In compression, it is\nalready more e\u000bective than what was the case when buck-\nling only occurred in compressive direction, as in Fig. 3I.\nThis is because the snap-through induced hysteresis per\ncycle is twice as large, as snap-through occurs in both\ncompression and tension. Furthermore, Fig. 4F demon-\nstrates that the same upper level for vibration transmis-\nsion can be set in tensile direction, making it much more\ne\u000bective in tensile direction than what was the case in\nFig. 3I. This shows that buckling metamaterials o\u000bering\nbuckling in both tension and compression can o\u000ber in-\ncreased vibration damping over those that only buckle in\ncompression.\nIII. OUTLOOK\nOf course, the geometry of Fig. 4A-D is not the only\nmetamaterial design, which can o\u000ber buckling in both\ntensile and compressive direction. For instance, kirigami\nforms a common alternative [37{39]. However, if we look\ncarefully at the structure of Fig. 4A, we can see that it\nis very similar to lightweight lattice structures [40, 41].\nSuch \fbre-reinforced lattice structures are speci\fcally de-\nsigned for their high strength and sti\u000bness to weight ratio\nand can be made in both 2D or 3D geometries. Typically\nin such structures, the members have been sized such\nthat they do not buckle under load [40, 41]. However,\ntheir members could also be sized instead to allow for\nelastic buckling, possibly accompanied by snap-through,\nsuch as the sample shown in Fig. 3. A speci\fc advantage\nof such lattice structures, but also of the metal sample\nof Fig. 3, is that they are stretching dominated pre-\nbuckling. This implies that a much higher speci\fc sti\u000b-\nness can be achieved than in many bending dominated\nstructures [19{21, 24, 42]. As such, it may be possible to\nproduce buckling based vibration damping metamateri-\nals with a very high sti\u000bness to weight ratio.\nFurthermore, while the current research demonstrates\na threefold improvement in damping for both samples, we\nexpect that much larger improvements in damping coef-\n\fcient could be obtained in systems with large ampli\fca-\ntion factors at small excitations. More speci\fcally, if the\nsample of Fig. 3, or another similar structure produced\nout of high sti\u000bness materials, had less imperfections,\nwe would expect a much larger ampli\fcation at small\nexcitation. Meanwhile, we would not expect the ampli-\n\fcation factors of excitations that induce post-buckling\nto increase signi\fcantly. As such, the increase in damp-\ning performance of buckling metamaterials for vibration\ndamping could even be much larger than reported in this\nmanuscript. Finally, while this manuscript has focused\non limiting the transmission of a speci\fc resonance, the\nmethod itself does not depend on a frequency. In fact,7\nCB E\nD FA\n24 cm\nFIG. 4: Buckling based vibration damping in tension and compression . (A) 3D printed TPU sample\nallowing buckling in two directions, buckling respectively in (B) compression, (C) tension and (D) bending. (E)\nForce-displacement for featuring tensile and compressive snap-through buckling, based on the force-displacement\ncurve of Fig. 3G. (F) For numerical up and down frequency sweeps at various acceleration levels, the maximum\noutput acceleration aoutacross the frequency range. Blue (orange) curves correspond to rising (dropping)\nfrequencies. Solid (dashed) lines correspond to compression (tension). Grey lines show the linearized trend.\nthis method should work to restrict transmissions of any\nvibration. This could be a big advantage in cases with\nmultiple modes.\nOpen challenges remain however. Speci\fcally, it still\nhas to be demonstrated that buckling based vibration\ndamping metamaterials can be produced with an even\nhigher speci\fc sti\u000bness, without signi\fcant imperfec-\ntions. Similarly, the possible ceiling limits of performance\nstill need to be identi\fed.\nWhen these points are addressed, we anticipate ap-\nplications in any situation, where a high speci\fc sti\u000b-\nness is required along with high damping, including\naerospace, sensitive instruments and high-tech machin-\nery [1, 2, 8, 43{45].8\nIV. MATERIALS AND METHODS\nA. Sample design and fabrication\n1. Elastomeric holar sample fabrication\nTo create the sample of Fig. 1CD, we pour a two\ncomponent silicone rubber (Zhermack Elite Double 32) in\na 3D printed mold, manufactured with a Stratasys Objet\nConnex 500 3D printer. The mold contains a pattern of\n5\u00025 circular holes in which we place steel rods. The\nholes have a diameter of 10mm and wall thickness of 1.5\nmm, implying a sample width and height of 59 mm. The\ndepth of the sample is 50 mm. The sample is connected\nto two perspex plates using silicone rubber glue. The\nperspex plates can be used to mount the sample to a test\nrig at the bottom and to mount additional weight and an\naccelerometer at the top.\n2. Multi-directional buckling sample fabrication\nThe sample of Fig. 4A-D has been manufactured from\na 2-component slow curing silicone rubber using an En-\nvisionTEC 3D Bioplotter. The sample measures 240 \u0002\n115\u000222 mm.\nB. Experimental methods\n1. Uniaxial testing\nTo perform the tests of Fig. 2AB and Fig. 3B, we\ncompress and extend the samples using a uniaxial testing\ndevice (Instron 5943 with a 500 N load cell). We do so\nat constant rates of 1 mm/min (Fig. 3B) and 1, 10, 100\nand 1000 mm/min (Fig. 2AB).\n2. Vibration testing\nTo perform the vibration tests of Fig. 1, Fig. 2 and\nFig. 6, we vibrate the sample using an Instron Elec-\ntropuls 3000. We suspend a rectangular steel frame from\nthe machine and mount the sample on the bottom of\nthis frame, such that we e\u000bectively obtain a base exciting\nfrom the bottom. A mass is mounted on top of the frame\nto e\u000bectively obtain a mass-spring system with base ex-\ncitation (Fig. 1CD.\nFor the vibration tests with the metal sample of Fig.\n3, we vibrate the sample using a Tira Vibration Test Sys-\ntem TV 5220-120 instead, where we control the frequency\nwith an Aim-TTi TG5011 function generator. We add a\nlarge mass of 4.5 kg on the shaker next to the sample to\nkeep the vibration input acceleration level more constant\nduring a frequency sweep. The set-up is shown in Fig. 5.\nShakerStabilising massAccelerometer 2\nAccelerometer 1Top mass\n328333FIG. 5: Metal buckling based vibration damping\nsample mounted on shaker table.\nIn both cases, we measure the accelerations at 5000 Hz\nwith two PCB Piezotronics 352C33 accelerometers: one\nat the bottom, capturing the base excitation and one\non top, capturing the output accelerations. The tests of\nFig. 1, Fig. 2 are also recorded using a Phantom VEO\n640 monochrome high-speed CMOS camera with Nikon\n200mm f/4 Macro lens, at 250 frames per second at a\nresolution of 1024 \u00021024, inducing a spatial resolution of\n0:10 mm.\nC. Processing accelerometer data\nTo process the accelerometer data, we \frst apply a\nfourth order Butterworth bandpass \flter from 4 to 60\nHz for the elastomeric sample and from 5 to 100 Hz for\nthe metal sample. This way, we \flter out low frequency\nsensor drift and high frequency noise generated by the\ntest set-up.\nWe then use the Python scipy.signal.\fnd peaks algo-\nrithm to \fnd the trends of the peak acceleration and as\nfunction of time and frequency. We also apply a Savitzky-\nGolay \flter to smooth the resulting peak acceleration\ntrends.\nD. Image analysis\nTo get a quantitative understanding of the degree of\nbuckling of the sample used in Fig. 1 and Fig. 2, we use\nparticle tracking (OPENCV and Python) and custom-\nmade tracking algorithms to quantify the \rattening f\nand orientation \u001ew.r.t. the horizontal of each pore and\ncalculate the polarisation [18, 23, 46]:\n\nnxny:= (\u00001)nx+nyfcos 2\u001e; (1)\nwherenx(ny) is the hole's column (row). We then plot9\nthese tracked ellipses along with their polarisation in red\nand blue in Fig. 1CD.\nE. Numerical model\nThe numerical model consists of a nonlinear mass\nspring damper system (Fig. 1A). The system contains\na massM, output displacement y(t) and acceleration __y\n(orange in Fig. 1A),coupled through a spring to a base\nexcitationu(t),__u(blue in Fig. 1A). The spring extension\ns=y\u0000u. The spring sti\u000bness Kis multilinear, while\nthe damper Cis multilinear only in the spring extension\ns.KandCcan be normalized with the mass to obtain\nthekandc. These can then be coupled to the linear\nsystem response as follows, where !n=fn\n2\u0019is the angu-\nlar eigenfrequency, \u0010is the damping coe\u000ecient, Gis the\nampli\fcation factor and fnis the eigenfrequency:\nk=K\nm=!2\nn (2)\nc=C\nm= 2\u0010!n (3)\n\u0010=1\n2G(4)\nWe derive the linearized spring sti\u000bness K, eigenfre-\nquencyfnand ampli\fcation factor Gfrom the experi-\nmental data of Fig. 2A,C for the elastomeric sample and\nFig. 3B as well as the underlying data at small excita-\ntions of Fig. 3D for the metal sample. We derive for small\nexcitations, K= 11:38 N/mm, fn= 32 andG= 19 for\nthe elastomeric sample and K= 368:8 N/mm,fn= 48\nHz andG= 13:2 for the metal sample. We use these\nvalues to derive M= 0:28 kg andC= 2:83 Ns/m for the\nelastomeric sample and M= 4:0 kg andC= 92:6 Ns/m\nfor the metal sample. These masses are higher than the\nactual masses mounted on top as the sample mass also\na\u000bects the eigenfrequency. We also derive the buckling\nforce plateaus to be 25 N and 78 N, for the elastomeric\n(Fig. 2E) and metal (Fig. 3) samples respectively. To\naccount for gravity, we subtract the gravitational accel-\neration,gM, whereg= 9:81 m/s2in the model to end\nup with the base state at s= 0 without excitation in the\nmodel. The force-displacement curve for the model with\nbuckling in two directions, seen in Fig. 4E presents the\nsame force plateau as the one for the metal sample in\nFig. 3G, with gMsubtracted.\nThe system is governed by the following di\u000berential\nequation.\n__y+c( _y\u0000_u) +k(y\u0000u) = 0 (5)When the system is subjected to a base excitation u(t),\n__u, the system can be solved numerically in the time do-\nmain using a forward Taylor series, which we do at a\nfrequency of 2000 Hz:\n__y=\u0000c( _y\u0000_u)\u0000k(y\u0000u) (6)\n___y=\u0000c\u0000__y\u0000__u\u0001\n\u0000k( _y\u0000_u) (7)\nCompared to some alternative ways to numerically\nsolve the system, a forward Taylor series has a number\nof distinct advantages. Compared to closed form solu-\ntions, such as piecewise solutions [47, 48], a forward Tay-\nlor series has the distinct advantage that it can easily\nbe modi\fed to allow for more complicated sti\u000bness and\ndamping characteristics. Compared to Euler's method,\nwhich is a \frst order Taylor series, a higher order forward\nTaylor series o\u000bers more numerical stability and o\u000bers a\nmuch higher speed of convergence. Finally, compared to\nexplicit nonlinear \fnite element methods, a simple single\ndegree of freedom system o\u000bers large advantages in sim-\nplicity, computational needs and numerical stability, at\nthe cost of a lower accuracy.\nF. De\fning the vibration signal\nTo obtain the frequency response, we subject the nu-\nmerical and experimental sample to a sinusoidal vibra-\ntion with constant acceleration amplitude, where the sine\nslightly changes frequency after every cycle. At each start\nof a full cycle, implying the acceleration and displacement\nare 0, the frequency of the sine cycle is updated. This\nimplies that the input acceleration, velocity and displace-\nment are de\fned as follow, with Bthe input acceleration\namplitude, fthe frequency and tcthe time in the cycle\nbetweentc= 0s totc=1\nf:\n__u=Bsin (2\u0019ftc) (8)\n_u=\u0000B\n2\u0019fcos (2\u0019ftc) (9)\nu=\u0000B\n(2\u0019f)2sin (2\u0019ftc) (10)\nG. Random vibration\nWe \frst de\fne a random vibration spectrum. We start\nfrom the acceleration Power Spectral Density Function of\nFig. 6A, calculated at 1 standard deviation (1 \u001b).PSDn10\nEC D\nF\nA B\nFIG. 6: Random vibration. (A) Power Spectral\nDensity (PSD) pro\fle of the input acceleration,\nnormalized with the peak PSD level. (B) Corresponding\ntime-displacement history applied in the tests. (C,D)\nExperimental time-acceleration history for input\naccelerations of 0.06 gRMS(c) and 0.30 gRMS(d).\nOrange and blue curves correspond to the input and\nout accelerations respectively. (E) Output acceleration\n(gRMSout) as function of input acceleration (gRMS in).\n(F) Maximum acceleration, amax, as function of the\ninput acceleration. Blue and orange correspond to\ncompression and tension respectively. All RMS values\nare shown at 1 \u001b.\nis the PSD value, normalized with the maximum PSD\nvalue. This PSD shows a constant PSD value between\n12.5 Hz and 50 Hz, which also includes the area surround-\ning the eigenfrequency of the sample of Fig. 1. This im-\nplies that vibration with this PSD will only excite the\nfrequencies around the eigenfrequency.\nThe 1\u001broot-mean-square of any PSD spectrum can be\ncalculated as [49]:RMS =sZ1\n0PSDdf: (11)\nWe can convert the acceleration PSD to a time sig-\nnal using an inverse Fourier transform [49], where we\nassume random values for the starting phase of the\nequivalent wave at each frequency. We can integrate\nthe time-acceleration signal to a time-velocity and time-\ndisplacement signal and use a high-pass Butterworth \fl-\nter to remove low-frequency drift. As such, we can ob-\ntain the time-displacement signal of Fig. 1B. We can\nalso convert time signals to PSD signals using a Fourier\ntransform.\nWe can then shake the sample, as described in the\nMain Text and in Section IV B 2, using this time signal\nas an input. If we subject the sample to a low random\nbase excitation of 0.06 gRMS in Fig. 6D, we observe that\nthe output acceleration in blue is much larger than the\ninput acceleration in orange and that acceleration levels\nare similar in tension and compression. However, when\nwe do the same at a higher base excitation of 0.30 gRMS\nin Fig. 6D, we observe a clear upper limit in compression,\ndue to buckling.\nWe can also track how the output accelerations change\nwith the applied input acceleration. In Fig. 6E, we ob-\nserve forgRMSin>0:15 thatgRMSoutstarts to go\nlower than the linear trend, which suggests that buckling\nbased vibration damping also works in random vibra-\ntions. However, to convert a time signal to a frequency\nbased signal, including a gRMS value, the implicit as-\nsumption is made that the signal is a sum of sinusoidal\nsignals, which is approximately symmetric about 0. That\nis not the case here, as seen in Fig. 6D. As such, it is more\nrepresentative to look at the peak accelerations instead.\nThese are plotted in Fig. 6F. Here we observe a clear up-\nper limit in compression (blue) for gRMSin>0:1. How-\never, forgRMSin>0:2, we observe a similar plateau\nin tension (orange). We presume that the reason why\nit requires a much larger base excitation to set an upper\nlimit in tension, is because only a few cycles induce buck-\nling and therefore induce additional dissipation. This is\ndi\u000berent from vibrations at constant frequency and ac-\nceleration, where buckling occurs every cycle, along with\nthe accompanying increased dissipation. However, both\nFig. 6E and Fig. 6F demonstrate that buckling based\nvibration damping also works in tension.\nH. Design & Finite Element Analysis\nFor the sample of Fig. 4, we designed the sample us-\ning a form of Bayesian optimisation combined with \fnite\nelement simulations in Abaqus [50].\nTo predict the response of the metamaterial samples\nof Fig. 3 and Fig. 4, we use nonlinear \fnite element\nsimulations in Abaqus 2021 (Dassault Syst\u0012 emes).11\nB A C\nD E F G\na/2tbeam/2 thinge/2thinge/2\nthinge\nα\nFIG. 7: Simulating and optimising bi-directional buckling metamaterials . (A) Finite element mesh. (B)\nTensile response. (C) Compressive response. (D) Quarter unit cell. (E) Sti\u000bness ratios for training points before and\nafter buckling: minimum of tension of compression. (F) Sti\u000bness ratios interpolated with Gaussian Processes\nRegression before and after buckling: minimum of tension of compression. (G) Normalized stress ( \u001b\u0003)-strain (\u000f)\ncurve in tension (blue) and compression (orange) of sample in A.\n1. Metal sample\nWe designed the sample of Fig. 3 using a combination\nof \fnite element analyses and experiments. We started\ndesigning the sample using 3D analyses with 2D plane\nstress plate elements (CPS8R). We used a combination\nof single strip analyses, 3 \u00023 and 5\u00025 metamaterial anal-\nyses. We modelled the entire sample using aluminium\nwith a Young's modulus of 70 GPa, Poisson's ratio of 0.3\nand density of 2800 kg/m3. We performed various sets of\nanalyses, varying the aspect ratio, thickness and out-of-\nplane curvature. We compressed and uncompressed the\nsample numerically using a nonlinear quasi-static analy-\nsis. In some cases, snap-through instabilities made the\nanalyses unstable, in which cases we opted for dynamic\nexplicit analyses instead. We sized the design using two\nmain criteria:\n1. At \fves times the post-buckling strain, the force-\ndrop from peak is less than 50% and a 5 \u00025 alu-\nminium sample still has at least a 40 N force. This\nprevents unstable collapse post-buckling.\n2. At \fves times the post-buckling strain, the max-\nimum Von Mises Stress is less than 100 MPa in\naluminium. This prevents fatigue damage in high\nstrength metals.\nAfter analysis, we opted to make the sample out of steel\nsheets instead due to the higher load carrying capac-\nity and better availability of high-strength variants with\nsmall thicknesses (AISI 301 Full Hard). After producingthe \frst sample, we opted to produce the \fnal sample of\nFig. 3 out of a higher thickness than originally analysed\nas the produced load carrying capacity was lower than\npredicted due to imperfections.\n2. Bi-directional buckling sample\nThe \fnal design and mesh of the sample of Fig. 4 is\ngiven in Fig. 7A. We modelled the sample in 2D using\nquadratic quad-dominated CPE8R plane strain elements.\nFor the material, we assumed a Neo-Hookean material\nmodel with a Poisson's of 0.48. We fully constrained\nthe left and right side of Fig. 7A and compressed and\nextended the sample numerically using a nonlinear quasi-\nstatic analysis.\nTo maximise the e\u000eciency of bi-directional buckling\nbased vibration damping, the sample had to buckle in\nboth directions at an equal strain. To obtain a suitable\ndesign, we used Bayesian optimisation [36, 50]. First,\nwe speci\fed the design using the parameters as shown in\nthe quarter unit cell of Fig. 7D. We optimized the design\nfor three normalized design parameters: t\u0003\nbeam =tbeam\na,\nt\u0003\nhinge =thinge\naandangle =\u000b. We ran 500 analyses with\na variety of combinations of these three variables. We\npopulated the design space using a Sobol sequence, an\ne\u000ecient method to populate a multivariable design space\nwith an arbitrary number of data points [51]. We tracked\ntwo outputs: (i) the ratio in sti\u000bness before and after\nbuckling,K1=K2: minimum of tension and compression,\nand (ii) the ratio of the buckling force in tension and12\ncompressive direction, Fb;t=Fb;c. ForK1=K2, we plotted\nthe results in Fig. 7E. We then used Bayesian machine\nlearning (Gaussian Processes Regression) to more densely\ninterpolate the design space as seen in Fig. 7F for K1=K2.\nIn doing so, we also obtained the standard deviation, \u001b, of\nthe uncertainty of the interpolation. We then de\fned the\nfollowing criteria to de\fne designs of su\u000ecient quality:\n1.K1=K2>5 + 1:96\u001bK1=K2\n2.jFb;t=Fb;cj\u00001<0:15\u00001:96\u001bFb;t=Fb;c\nThis presents us with several designs which adhere to\nour requirements. Around these locations, we then re-\n\fned our data by running 100 additional analyses using\na Sobol sequence, which then accurately provided us with\na variety of designs adhering to our requirements. One\nof those is the design of Fig. 7A-C, which we produced\nin Fig. 4. The nonlinear normalized stress-strain curve\nas calculated by Abaqus, is presented in Fig. 7G, wherethe stress has been normalized by the cross-section and\nYoung's modulus.\nV. DATA AND CODE AVAILABILITY\nThe data and codes that support the \fgures within this\npaper are publicly available on a Zenodo repository [52].\nTwo videos that support this article can also be found in\nthe supplementary information. Supplementary Videos\n1 and 2 show buckling of samples with base excitations\naround resonance for the rubber sample of Fig. 1 and for\nthe metal sample of Fig. 3 respectively.\nVI. 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Kirillova,b, Diego Misseronia, Davide Bigonia\naUniversit\u0013 a di Trento, DICAM, via Mesiano 77, I-38123 Trento, Italy\nbRussian Academy of Sciences, Steklov Mathematical Institute, Gubkina st. 8, 119991 Moscow, Russia\nAbstract\nElastic structures loaded by nonconservative positional forces are prone to instabilities in-\nduced by dissipation: it is well-known in fact that internal viscous damping destabilizes the\nmarginally stable Ziegler's pendulum and P\r uger column (of which the Beck's column is a\nspecial case), two structures loaded by a tangential follower force. The result is the so-called\n`destabilization paradox', where the critical force for \rutter instability decreases by an order\nof magnitude when the coe\u000ecient of internal damping becomes in\fnitesimally small. Until\nnow external damping, such as that related to air drag, is believed to provide only a stabi-\nlizing e\u000bect, as one would intuitively expect. Contrary to this belief, it will be shown that\nthe e\u000bect of external damping is qualitatively the same as the e\u000bect of internal damping,\nyielding a pronounced destabilization paradox. Previous results relative to destabilization by\nexternal damping of the Ziegler's and P\r uger's elastic structures are corrected in a de\fnitive\nway leading to a new understanding of the destabilizating role played by viscous terms.\nKeywords: P\r uger column, Beck column, Ziegler destabilization paradox, external\ndamping, follower force, mass distribution\n1. Introduction\n1.1. A premise: the Ziegler destabilization paradox\nIn his pioneering work Ziegler (1952) considered asymptotic stability of a two-linked\npendulum loaded by a tangential follower force P, as a function of the internal damping in\nthe viscoelastic joints connecting the two rigid and weightless bars (both of length l, Fig.\n1(c)). The pendulum carries two point masses: the mass m1at the central joint and the\nEmail addresses: mirko.tommasini@unitn.it (Mirko Tommasini), kirillov@mi.ras.ru (Oleg N.\nKirillov), diego.misseroni@unitn.it (Diego Misseroni), davide.bigoni@unitn.it (Davide Bigoni)\nCorresponding author: Davide Bigoni, davide.bigoni@unitn.it; +39 0461 282507\nPreprint submitted to Elsevier October 19, 2021arXiv:1611.03886v1  [physics.class-ph]  1 Oct 2016Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nmassm2mounted at the loaded end of the pendulum. The follower force Pis always aligned\nwith the second bar of the pendulum, so that its work is non-zero along a closed path, which\nprovides a canonical example of a nonconservative positional force.\nFor two non-equal masses ( m1= 2m2) and null damping, Ziegler found that the pendulum\nis marginally stable and all the eigenvalues of the 2 \u00022 matrix governing the dynamics are\npurely imaginary and simple, if the load falls within the interval 0 \u0014P <P\u0000\nu, where\nP\u0000\nu=\u00127\n2\u0000p\n2\u0013k\nl\u00192:086k\nl; (1)\nandkis the sti\u000bness coe\u000ecient, equal for both joints. When the load Preaches the value\nP\u0000\nu, two imaginary eigenvalues merge into a double one and the matrix governing dynamics\nbecomes a Jordan block. With the further increase of Pthis double eigenvalue splits into\ntwo complex conjugate. The eigenvalue with the positive real part corresponds to a mode\nwith an oscillating and exponentially growing amplitude, which is called \rutter, or oscilla-\ntory, instability. Therefore, P=P\u0000\numarks the onset of \rutter in the undamped Ziegler's\npendulum.\nWhen the internal linear viscous damping in the joints is taken into account, Ziegler\nfound another expression for the onset of \rutter: P=Pi, where\nPi=41\n28k\nl+1\n2c2\ni\nm2l3; (2)\nandciis the damping coe\u000ecient, assumed to be equal for both joints. The peculiarity of\nEq. (2) is that in the limit of vanishing damping, ci\u0000! 0, the \rutter load Pitends to\nthe value 41 =28k=l\u00191:464k=l, considerably lower than that calculated when damping is\nabsent from the beginning, namely, the P\u0000\nugiven by Eq. (1). This is the so-called `Ziegler's\ndestabilization paradox' (Ziegler, 1952; Bolotin, 1963).\nThe reason for the paradox is the existence of the Whitney umbrella singularity on\nthe boundary of the asymptotic stability domain of the dissipative system (Bottema, 1956;\nKrechetnikov and Marsden, 2007; Kirillov and Verhulst, 2010)3.\nIn structural mechanics, two types of viscous dampings are considered: (i.) one, called\n`internal', is related to the viscosity of the structural material, and (ii.) another one, called\n`external', is connected to the presence of external actions, such as air drag resistance during\n3In the vicinity of this singularity, the boundary of the asymptotic stability domain is a ruled surface\nwith a self-intersection, which corresponds to a set of marginally stable undamped systems. For a \fxed\ndamping distribution, the convergence to the vanishing damping case occurs along a ruler that meets the\nset of marginally stable undamped systems at a point located far from the undamped instability threshold,\nyielding the singular \rutter onset limit for almost all damping distributions. Nevertheless, there exist\nparticular damping distributions that, if \fxed, allow for a smooth convergence to the \rutter threshold of\nthe undamped system in case of vanishing dissipation (Bottema, 1956; Bolotin, 1963; Banichuk et al., 1989;\nKirillov and Verhulst, 2010; Kirillov, 2013).\n2Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\noscillations. These two terms enter the equations of motion of an elastic rod as proportional\nrespectively to the fourth spatial derivative of the velocity and to the velocity of the points\nof the elastic line.\nOf the two dissipative terms only the internal viscous damping is believed to yield the\nZiegler destabilization paradox (Bolotin, 1963; Bolotin and Zhinzher, 1969; Andreichikov\nand Yudovich, 1974).\n1.2. A new, destabilizing role for external damping\nDi\u000berently from internal damping, the role of external damping is commonly believed to\nbe a stabilizing factor, in an analogy with the role of stationary damping in rotor dynamics\n(Bolotin, 1963; Crandall, 1995). A full account of this statement together with a review of\nthe existing results is provided in Appendix A.\nSince internal and external damping are inevitably present in any experimental realization\nof the follower force (Saw and Wood, 1975; Sugiyama et al., 1995; Bigoni and Noselli, 2011), it\nbecomes imperative to know how these factors a\u000bect the \rutter boundary of both the P\r uger\ncolumn and of the Ziegler pendulum with arbitrary mass distribution. These structures are\nfully analyzed in the present article, with the purpose of showing: (i.) that external damping\nis a destabilizing factor, which leads to the destabilization paradox for all mass distributions;\n(ii.) that surprisingly, for a \fnite number of particular mass distributions, the \rutter loads\nof the externally damped structures converge to the \rutter load of the undamped case (so\nthat only in these exceptional cases the destabilizing e\u000bect is not present); and (iii.) that\nthe destabilization paradox is more pronounced in the case when the mass of the column or\npendulum is smaller then the end mass.\nTaking into account also the destabilizing role of internal damping, the results presented\nin this article demonstrate a completely new role of external damping as a destabilizing\ne\u000bect and suggest that the Ziegler destabilization paradox has a much better chance of being\nobserved in the experiments with both discrete and continuous nonconservative systems than\nwas previously believed.\n2. Ziegler's paradox due to vanishing external damping\nThe linearized equations of motion for the Ziegler pendulum (Fig. 1(c)), made up of two\nrigid bars of length l, loaded by a follower force P, when both internal and external damping\nare present, have the form (Plaut and Infante, 1970; Plaut, 1971)\nMx+ciDi_x+ceDe_x+Kx= 0; (3)\nwhere a superscript dot denotes time derivative and ciandceare the coe\u000ecients of internal\nand external damping, respectively, in front of the corresponding matrices DiandDe\nDi=\u00122\u00001\n\u00001 1\u0013\n;De=l3\n6\u00128 3\n3 2\u0013\n; (4)\n3Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nandMandKare respectively the mass and the sti\u000bness matrices, de\fned as\nM=\u0012m1l2+m2l2m2l2\nm2l2m2l2\u0013\n;K=\u0012\u0000Pl+ 2k Pl\u0000k\n\u0000k k\u0013\n; (5)\nin whichkis the elastic sti\u000bness of both viscoelastic springs acting at the hinges.\n16 16 16 163/c112 /c112/c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 420.00\n-0.05-0.10\n-0.15\n-0.20\n-0.25\n-0.30/c68F\nDA\nBC\nP m2l\nm1l\nk, c , ciek, c , ciea)\nc)b)\n16 16 16 163/c112 /c112/c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 421.52.02.53.0\nFA\nBCD\nm =2m12\ninternalexternalidealFLUTTER\n1.2502.086\n1.464\nFigure 1: (a) The (dimensionless) tangential force F, shown as a function of the (transformed via cot \u000b=\nm1=m2) mass ratio \u000b, represents the \rutter domain of (dashed/red line) the undamped, or `ideal', Ziegler\npendulum and the \rutter boundary of the dissipative system in the limit of vanishing (dot-dashed/green\nline) internal and (continuous/blue line) external damping. (b) Discrepancy \u0001 Fbetween the critical \rutter\nload for the ideal Ziegler pendulum and for the same structure calculated in the limit of vanishing external\ndamping. The discrepancy quanti\fes the Ziegler's paradox.\nAssuming a time-harmonic solution to the Eq. (3) in the form x=ue\u001btand introducing\nthe non-dimensional parameters\n\u0015=\u001bl\nkp\nkm2; E =cel2\npkm2; B =ci\nlpkm2; F =Pl\nk; \u0016 =m2\nm1; (6)\nan eigenvalue problem is obtained, which eigenvalues \u0015are the roots of the characteristic\npolynomial\np(\u0015) = 36\u00154+ 12(15B\u0016+ 2E\u0016+ 3B+E)\u00153+\n(36B2\u0016+ 108BE\u0016 + 7E2\u0016\u000072F\u0016+ 180\u0016+ 36)\u00152+\n6\u0016(\u00005EF+ 12B+ 18E)\u0015+ 36\u0016: (7)\n4Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nIn the undamped case, when B= 0 andE= 0, the pendulum is stable, if 0 \u0014F <F\u0000\nu,\nunstable by \rutter, if F\u0000\nu\u0014F\u0014F+\nu, and unstable by divergence, if F >F+\nu, where\nF\u0006\nu(\u0016) =5\n2+1\n2\u0016\u00061p\u0016: (8)\nIn order to plot the stability map for all mass distributions 0 \u0014\u0016 <1, a parameter\n\u000b2[0;\u0019=2] is introduced, so that cot \u000b=\u0016\u00001and hence\nF\u0006\nu(\u000b) =5\n2+1\n2cot\u000b\u0006p\ncot\u000b: (9)\nThe curves (9) form the boundary of the \rutter domain of the undamped, or `ideal',\nZiegler's pendulum shown in Fig. 1(a) (red/dashed line) in the load versus mass distribution\nplane (Oran, 1972; Kirillov, 2011). The smallest \rutter load F\u0000\nu= 2 corresponds to m1=m2,\ni.e. to\u000b=\u0019=4. When\u000bequals\u0019=2, the mass at the central joint vanishes ( m1= 0) and\nF\u0000\nu=F+\nu= 5=2. When\u000bequals arctan (0 :5)\u00190:464, the two masses are related as\nm1= 2m2andF\u0000\nu= 7=2\u0000p\n2.\nIn the case when only internal damping is present ( E= 0) the Routh-Hurwitz criterion\nyields the \rutter threshold as (Kirillov, 2011)\nFi(\u0016;B) =25\u00162+ 6\u0016+ 1\n4\u0016(5\u0016+ 1)+1\n2B2: (10)\nFor\u0016= 0:5 Eq. (10) reduces to Ziegler's formula (2). The limit for vanishing internal\ndamping is\nlim\nB!0Fi(\u0016;B) =F0\ni(\u0016) =25\u00162+ 6\u0016+ 1\n4\u0016(5\u0016+ 1): (11)\nThe limitF0\ni(\u0016) of the \rutter boundary at vanishing internal damping is shown in green in\nFig. 1(a). Note that F0\ni(0:5) = 41=28 andF0\ni(1) = 5=4. For 0\u0014\u0016<1the limiting curve\nF0\ni(\u0016) has no common points with the \rutter threshold F\u0000\nu(\u0016) of the ideal system, which\nindicates that the internal damping causes the Ziegler destabilization paradox for every mass\ndistribution.\nIn a route similar to the above, by employing the Routh-Hurwitz criterion, the critical\n\rutter load of the Ziegler pendulum with the external damping Fe(\u0016;E) can be found\nFe(\u0016;E) =122\u00162\u000019\u0016+ 5\n5\u0016(8\u0016\u00001)+7(2\u0016+ 1)\n36(8\u0016\u00001)E2\n\u0000(2\u0016+ 1)p\n35E2\u0016(35E2\u0016\u0000792\u0016+ 360) + 1296(281 \u00162\u0000130\u0016+ 25)\n180\u0016(8\u0016\u00001)\nand its limit calculated when E!0, which provides the result\nF0\ne(\u0016) =122\u00162\u000019\u0016+ 5\u0000(2\u0016+ 1)p\n281\u00162\u0000130\u0016+ 25\n5\u0016(8\u0016\u00001): (12)\n5Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nThe limiting curve (12) is shown in blue in Fig. 1(a). It has a minimum min \u0016F0\ne(\u0016) =\n\u000028 + 8p\n14\u00191:933 at\u0016= (31 + 7p\n14)=75\u00190:763.\nRemarkably, for almost all mass ratios, except two (marked as A and C in Fig. 1(a)),\nthe limit of the \rutter load F0\ne(\u0016) isbelow the critical \rutter load F\u0000\nu(\u0016) of the undamped\nsystem. It is therefore concluded that external damping causes the discontinuous decrease in\nthe critical \rutter load exactly as it happens when internal damping vanishes. Qualitatively ,\nthe e\u000bect of vanishing internal and external damping is the same . The only di\u000berence is\nthe magnitude of the discrepancy: the vanishing internal damping limit is larger than the\nvanishing external damping limit, see Fig. 1(b), where \u0001 F(\u0016) =Fe(\u0016)\u0000F\u0000\nu(\u0016) is plotted.\n0.00B\n0.010.020.030.040.050.060.070.080.090.10\n0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9\nEb) a)\nE0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.02.002.102.202.302.40\n2.052.152.252.35\nFFLUTTER\nFLUTTER\n2.086\nFigure 2: Analysis of the Ziegler pendulum with \fxed mass ratio, \u0016=m2=m1= 1=2: (a) contours of the\n\rutter boundary in the internal/external damping plane, ( B;E), and (b) critical \rutter load as a function\nof the external damping E(continuous/blue curve) along the null internal damping line, B= 0, and (dot-\ndashed/orange curve) along the line B=\u0000\n8=123 + 5p\n2=164\u0001\nE.\nFor example, \u0001 F\u0019\u00000:091 at the local minimum for the discrepancy, occurring at the\npoint B with \u000b\u00190:523. The largest \fnite drop in the \rutter load due to external damping\noccurs at\u000b=\u0019=2, marked as point D in Fig. 1(a,b):\n\u0001F=11\n20\u00001\n20p\n281\u0019\u00000:288: (13)\nFor comparison, at the same value of \u000b, the \rutter load drops due to internal damping of\nexactly 50%, namely, from 2 :5 to 1:25, see Fig. 1(a,b).\nAs a particular case, for the mass ratio \u0016= 1=2, considered by Plaut and Infante (1970)\nand Plaut (1971), the following limit \rutter load is found\nF0\ne(1=2) = 2; (14)\n6Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\n16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 42 16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 42b) a)\n-1.0/c98\n-0.8-0.6-0.4-0.20.00.2\n1.52.02.53.0\nFA C0.111\n0.524-2/15FLUTTERideal\nFigure 3: Analysis of the Ziegler pendulum. (a) Stabilizing damping ratios \f(\u0016) according to Eq. (19) with\nthe points A and C corresponding to the tangent points A and C in Fig. 1(a) and to the points A and C of\nvanishing discrepancy \u0001 F= 0 in Fig. 1(b). (b) The limits of the \rutter boundary for di\u000berent damping ratios\n\fhave: two or one or none common points with the \rutter boundary (dashed/red line) of the undamped\nZiegler pendulum, respectively when \f < 0:111 (continuous/blue curves), \f\u00190:111 (continuous/black\ncurve), and \f >0:111 (dot-dashed/green curves).\nonly slightly inferior to the value for the undamped system, F\u0000\nu(1=2) = 7=2\u0000p\n2\u00192:086.\nThis discrepancy passed unnoticed in (Plaut and Infante, 1970; Plaut, 1971) but gives evi-\ndence to the destabilizing e\u000bect of external damping. To appreciate this e\u000bect, the contours\nof the \rutter boundary in the ( B;E) - plane are plotted in Fig. 2(a) for three di\u000berent values\nofF. The contours are typical of a surface with a Whitney umbrella singularity at the origin\n(Kirillov and Verhulst, 2010). At F= 7=2\u0000p\n2 the stability domain assumes the form of a\ncusp with a unique tangent line, B=\fE, at the origin, where\n\f=8\n123+5\n164p\n2\u00190:108: (15)\nFor higher values of Fthe \rutter boundary is displaced from the origin, Fig. 2(a), which\nindicates the possibility of a continuous increase in the \rutter load with damping. Indeed,\nalong the direction in the ( B;E) - plane with the slope (15) the \rutter load increases as\nF(E) =7\n2\u0000p\n2 +\u001247887\n242064+1925\n40344p\n2\u0013\nE2+o(E2); (16)\nsee Fig. 2(b), and monotonously tends to the undamped value as E!0. On the other\nhand, along the direction in the ( B;E) - plane speci\fed by the equation B= 0, the following\n7Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\ncondition is obtained\nF(E) = 2 +14\n99E2+o(E2); (17)\nsee Fig. 2(b), with the convergence to a lower value F= 2 asE!0.\nIn general, the limit of the \rutter load along the line B=\fEwhenE!0 is\nF(\f) =504\f2+ 1467\f+ 104\u0000(4 + 21\f)p\n576\f2+ 1728\f+ 121\n30(1 + 14\f)\u00147\n2\u0000p\n2; (18)\nan equation showing that for almost all directions the limit is lower than the ideal \rutter\nload. The limits only coincide in the sole direction speci\fed by Eq. (15), which is di\u000berent\nfrom theE-axis, characterized by \f= 0. As a conclusion, pure external damping yields the\ndestabilization paradox even at \u0016= 1=2, which was unnoticed in (Plaut and Infante, 1970;\nPlaut, 1971).\nIn the limit of vanishing external ( E) and internal ( B) damping, a ratio of the two\n\f=B=E exists for which the critical load of the undamped system is attained, so that the\nZiegler's paradox does not occur. This ratio can therefore be called `stabilizing', it exists for\nevery mass ratio \u0016=m2=m1, and is given by the expression\n\f(\u0016) =\u00001\n3(10\u0016\u00001)(\u0016\u00001)\n25\u00162+ 6\u0016+ 1+1\n12(13\u0016\u00005)(3\u0016+ 1)\n25\u00162+ 6\u0016+ 1\u0016\u00001=2: (19)\nEq. (19) reduces for \u0016= 1=2 to Eq. (15) and gives \f=\u00002=15 in the limit \u0016!1 . With\nthe damping ratio speci\fed by Eq. (19) the critical \rutter load has the following Taylor\nexpansion near E= 0:\nF(E;\u0016) =F\u0000\nu(\u0016) +\f(\u0016)(5\u0016+ 1)(41\u0016+ 7)\n6(25\u00162+ 6\u0016+ 1)E2\n+636\u00163+ 385\u00162\u0000118\u0016+ 25\n288(25\u00162+ 6\u0016+ 1)\u0016E2+o(E2); (20)\nyielding Eq. (16) when \u0016= 1=2. Eq. (20) shows that the \rutter load reduces to the\nundamped case when E= 0 (called `ideal' in the \fgure).\nWhen the stabilizing damping ratio is null, \f= 0, convergence to the critical \rutter load\nof the undamped system occurs by approaching the origin in the ( B;E) - plane along the E\n- axis. The corresponding mass ratio can be obtained \fnding the roots of the function \f(\u0016)\nde\fned by Eq. (19). This function has only two roots for 0 \u0014\u0016<1, one at\u0016\u00190:273 (or\n\u000b\u00190:267, marked as point A in Fig. 3(a)) and another at \u0016\u00192:559 (or\u000b\u00191:198, marked\nas point C in Fig. 3(a)).\nTherefore, if \f= 0 is kept in the limit when the damping tends to zero, the limit of the\n\rutter boundary in the load versus mass ratio plane will be obtained as a curve showing\ntwo common points with the \rutter boundary of the undamped system, exactly at the mass\n8Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nE Eb) a)\n0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.02.052.072.09\n2.062.08F\n0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.0B\n0.000\n-0.025-0.0500.0250.0500.0750.100\n2.102.112.122.132.142.15\n2.07F=2.10F=2.07 F=2.04FLUTTER FLUTTER\nB=0\nFigure 4: Analysis of the Ziegler pendulum with \fxed mass ratio, \u0016\u00192:559: (a) contours of the \rutter\nboundary in the internal/external damping plane, ( B;E), and (b) critical \rutter load as a function of external\ndampingE(continuous/blue curve) along the null internal damping line, B= 0.\nratios corresponding to the points denoted as A and C in Fig. 1(a), respectively characterized\nbyF\u00192:417 andF\u00192:070.\nIf for instance the mass ratio at the point C is considered and the contour plots are\nanalyzed of the \rutter boundary in the ( B;E) - plane, it can be noted that at the critical\n\rutter load of the undamped system, F\u00192:07, the boundary evidences a cusp with only\none tangent coinciding with the Eaxis, Fig. 4(a). It can be therefore concluded that at the\nmass ratio\u0016\u00192:559 the external damping alone has a stabilizing e\u000bect and the system does\nnot demonstrate the Ziegler paradox due to small external damping, see Fig. 4(b), where\nthe the \rutter load F(E) is shown.\nLooking back at the damping matrices (4) one may ask, what is the property of the\ndamping operator which determines its stabilizing or destabilizing character. The answer to\nthis question (provided by (Kirillov and Seyranian, 2005b; Kirillov, 2013) via perturbation\nof multiple eigenvalues) involves all the three matrices M(mass), D(damping), and K\n(sti\u000bness). In fact, the distributions of mass, sti\u000bness, and damping should be related in a\nspeci\fc manner in order that the three matrices ( M,D,K) have a stabilizing e\u000bect (see\nAppendix B for details).\n3. Ziegler's paradox for the P\r uger column with external damping\nThe Ziegler's pendulum is usually considered as the two-dimensional analog of the Beck\ncolumn, which is a cantilevered (visco)elastic rod loaded by a tangential follower force (Beck,\n9Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\n1952). Strictly speaking, this analogy is not correct because the Beck column has a di\u000berent\nmass distribution (the usual mass distribution of the Ziegler pendulum is m1= 2m2) and this\nmass distribution yields di\u000berent limiting behavior of the stability threshold (Section 2). For\nthis reason, in order to judge the stabilizing or destabilizing in\ruence of external damping in\nthe continuous case and to compare it with the case of the Ziegler pendulum, it is correct to\nconsider the Beck column with the point mass at the loaded end, in other words the so-called\n`P\r uger column' (P\r uger, 1955).\nA viscoelastic column of length l, made up of a Kelvin-Voigt material with Young modulus\nEand viscosity modulus E\u0003, and mass per unit length mis considered, clamped at one end\nand loaded by a tangential follower force Pat the other end (Fig. 5(c)), where a point mass\nMis mounted.\nThe moment of inertia of a cross-section of the column is denoted by Iand a distributed\nexternal damping is assumed, characterized by the coe\u000ecient K.\nSmall lateral vibrations of the viscoelastic P\r uger column near the undeformed equilib-\nrium state is described by the linear partial di\u000berential equation (Detinko, 2003)\nEI@4y\n@x4+E\u0003I@5y\n@t@x4+P@2y\n@x2+K@y\n@t+m@2y\n@t2= 0; (21)\nwherey(x;t) is the amplitude of the vibrations and x2[0;l] is a coordinate along the\ncolumn. At the clamped end ( x= 0) Eq. (21) is equipped with the boundary conditions\ny=@y\n@x= 0; (22)\nwhile at the loaded end ( x=l), the boundary conditions are\nEI@2y\n@x2+E\u0003I@3y\n@t@x2= 0; EI@3y\n@x3+E\u0003I@4y\n@t@x3=M@2y\n@t2: (23)\nIntroducing the dimensionless quantities\n\u0018=x\nl; \u001c =t\nl2q\nEI\nm; p =Pl2\nEI; \u0016 =M\nml;\n\r=E\u0003\nEl2q\nEI\nm; k =Kl2p\nmEI(24)\nand separating the time variable through y(\u0018;\u001c) =lf(\u0018) exp(\u0015\u001c), the dimensionless bound-\nary eigenvalue problem is obtained\n(1 +\r\u0015)@4\n\u0018f+p@2\n\u0018f+ (k\u0015+\u00152)f= 0;\n(1 +\r\u0015)@2\n\u0018f(1) = 0;\n(1 +\r\u0015)@3\n\u0018f(1) =\u0016\u00152f(1);\nf(0) =@\u0018f(0) = 0; (25)\n10Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nde\fned on the interval \u00182[0;1].\nA solution to the boundary eigenvalue problem (25) was found by Pedersen (1977) and\nDetinko (2003) to be\nf(\u0018) =A(cosh(g2\u0018)\u0000cos(g1\u0018)) +B(g1sinh(g2\u0018)\u0000g2sin(g1\u0018)) (26)\nwith\ng2\n1;2=p\np2\u00004\u0015(\u0015+k)(1 +\r\u0015)\u0006p\n2(1 +\r\u0015): (27)\nImposing the boundary conditions (25) on the solution (26) yields the characteristic equation\n\u0001(\u0015) = 0 needed for the determination of the eigenvalues \u0015, where\n\u0001(\u0015) = (1 +\r\u0015)2A1\u0000(1 +\r\u0015)A2\u0016\u00152(28)\nand\nA1=g1g2\u0000\ng4\n1+g4\n2+ 2g2\n1g2\n2coshg2cosg1+g1g2(g2\n1\u0000g2\n2) sinhg2sing1\u0001\n;\nA2= (g2\n1+g2\n2) (g1sinhg2cosg1\u0000g2coshg2sing1): (29)\n16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c1120/c112\n88 42pa)\nexternal\nintenalr8101214161820\nMml\nPb)\n20.0\n19.518.017.517.0\n16/c112/c112\n80external\n/c97FLUTTER\nc)ideal\nidealA\nB\nFigure 5: Analysis of the P\r uger column [scheme reported in (c)]. (a) Stability map for the P\r uger's column\nin the load-mass ratio plane. The dashed/red curve corresponds to the stability boundary in the undamped\ncase, the dot-dashed/green curve to the case of vanishing internal dissipation ( \r= 10\u000010andk= 0 ) and\nthe continuous/blue curve to the case of vanishing external damping ( k= 10\u000010and\r= 0). (b) detail of\nthe curve reported in (a) showing the destabilization e\u000bect of external damping: small, but not null.\n11Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nTransforming the mass ratio parameter in Eq. (28) as \u0016= tan\u000bwith\u000b2[0;\u0019=2] allows\nthe exploration of all possible ratios between the end mass and the mass of the column\ncovering the mass ratios \u0016from zero (\u000b= 0) to in\fnity ( \u000b=\u0019=2). The former case, without\nend mass, corresponds to the Beck column, whereas the latter corresponds to a weightless\nrod with an end mass, which is known as the `Dzhanelidze column' (Bolotin, 1963).\nIt is well-known that the undamped Beck column loses its stability via \rutter at p\u0019\n20:05 (Beck, 1952). In contrast, the undamped Dzhanelidze's column loses its stability via\ndivergence at p\u001920:19, which is the root of the equation tanpp=pp(Bolotin, 1963).\nThese values, corresponding to two extreme situations, are connected by a marginal stability\ncurve in the ( p;\u000b)-plane that was numerically evaluated in (P\r uger, 1955; Bolotin, 1963;\nOran, 1972; Sugiyama et al., 1976; Pedersen, 1977; Ryu and Sugiyama, 2003). The instability\nthreshold of the undamped P\r uger column is shown in Fig. 5 as a dashed/red curve.\n16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c1120/c112\n88 428.0\n7.010.0\n9.011.012.013.014.015.016.017.018.019.020.021.022.023.024.025.0\nSTABILITY\nSTABILITY\n/c103=10 , k=0/UNI207b¹⁰/c103=0.050, k=0\n/c103=0.100, k=0k=5, =0 /c103k=10 , =0 /UNI207b¹⁰/c103\nk=10, =0 /c103\nk1 0/c61/c103=/UNI207b¹⁰\n/c97p\nk 0.010/c61/c49/c44 /c103 =\nFigure 6: Evolution of the marginal stability curve for the P\r uger column in the ( \u000b;p) - plane in the case\nofk= 0 and\rtending to zero (green curves in the lower part of the graph) and in the case of \r= 0 andk\ntending to zero (blue curves in the upper part of the graph). The cases of k=\r= 10\u000010and ofk= 1 and\n\r= 0:01 are reported with continuous/red lines.\n12Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nFor every \fxed value \u000b2[0;\u0019=2), the undamped column loses stability via \rutter when\nan increase in pcauses the imaginary eigenvalues of two di\u000berent modes to approach each\nother and merge into a double eigenvalue with one eigenfunction. When plies above the\ndashed/red curve, the double eigenvalue splits into two complex eigenvalues, one with the\npositive real part, which determines a \rutter unstable mode.\nAt\u000b=\u0019=2 the stability boundary of the undamped P\r uger column has a vertical tangent\nand the type of instability becomes divergence (Bolotin, 1963; Oran, 1972; Sugiyama et al.,\n1976).\nSettingk= 0 in Eq. (28) the location in the ( \u000b;p)-plane of the marginal stability curves\ncan be numerically found for the viscoelastic P\r uger column without external damping, but\nfor di\u000berent values of the coe\u000ecient of internal damping \r, Fig. 6(a). The thresholds tend to\na limit which does not share common points with the stability boundary of the ideal column,\nas shown in Fig. 5(a), where this limit is set by the dot-dashed/green curve.\nThe limiting curve calculated for \r= 10\u000010agrees well with that obtained for \r= 10\u00003\nin (Sugiyama et al., 1995; Ryu and Sugiyama, 2003). At the point \u000b= 0, the limit value of\nthe critical \rutter load when the internal damping is approaching zero equals the well-known\nvalue for the Beck's column, p\u001910:94. At\u000b=\u0019=4 the limiting value becomes p\u00197:91,\nwhile for the case of the Dzhanelidze column ( \u000b=\u0019=2) it becomes p\u00197:49.\nAn interesting question is what is the limit of the stability diagram for the P\r uger column\nin the (\u000b;p)-plane when the coe\u000ecient of internal damping is kept null ( \r= 0), while the\ncoe\u000ecient of external damping ktends to zero.\nThe answer to this question was previously known only for the Beck column ( \u000b= 0), for\nwhich it was established, both numerically (Bolotin and Zhinzher, 1969; Plaut and Infante,\n1970) and analytically (Kirillov and Seyranian, 2005a), that the \rutter threshold of the\nexternally damped Beck's column is higher than that obtained for the undamped Beck's\ncolumn (tending to the ideal value p\u001920:05, when the external damping tends to zero). This\nvery particular example was at the basis of the common and incorrect opinion (maintained\nfor decades until now) that the external damping is only a stabilizing factor, even for non-\nconservative loadings. Perhaps for this reason the e\u000bect of the external damping in the\nP\r uger column has, so far, simply been ignored.\nThe evolution of the \rutter boundary for \r= 0 andktending to zero is illustrated by the\nblue curves in Fig. 6. It can be noted that the marginal stability boundary tends to a limiting\ncurve which has two common tangent points with the stability boundary of the undamped\nP\r uger column, Fig. 5(b). One of the common points, at \u000b= 0 andp\u001920:05, marked as\npoint A, corresponds to the case of the Beck column. The other corresponds to \u000b\u00190:516 and\np\u001916:05, marked as point B. Only for these two `exceptional' mass ratios the critical \rutter\nload of the externally damped P\r uger column coincides with the ideal value when k!0.\nRemarkably, for all other mass ratios the limit of the critical \rutter load for the vanishing\nexternal damping is located below the ideal value, which means that the P\r uger column fully\ndemonstrates the Ziegler destabilization paradox due to vanishing external damping , exactly\nas it does in the case of the vanishing internal damping, see Fig. 5(a), where the two limiting\n13Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\ncurves are compared.\nNote that the discrepancy in case of vanishing external damping is smaller than in case\nof vanishing internal damping, in accordance with the analogous result that was established\nin Section 2 for the Ziegler pendulum with arbitrary mass distribution. As for the discrete\ncase, also for the P\r uger column the \rutter instability threshold calculated in the limit when\nthe external damping tends to zero has only two common points with the ideal marginal\nstability curve. The discrepancy is the most pronounced for the case of Dzhanelidze column\nat\u000b=\u0019=2, where the critical load drops from p\u001920:19 in the ideal case to p\u001916:55 in\nthe case of vanishing external damping.\n4. Conclusions\nSince the \fnding of the Ziegler's paradox for structures loaded by nonconservative follower\nforces, internal damping (due to material viscosity) was considered a destabilizing factor,\nwhile external damping (due for instance to air drag resistance) was believed to merely\nprovide a stabilization. This belief originates from results obtained only for the case of Beck's\ncolumn, which does not carry an end mass. This mass is present in the case of the P\r uger's\ncolumn, which was never analyzed before from the point of view of the Ziegler paradox. A\nrevisitation of the Ziegler's pendulum and the analysis of the P\r uger column has revealed\nthat the Ziegler destabilization paradox occurs as related to the vanishing of the external\ndamping, no matter what is the ratio between the end mass and the mass of the structure.\nResults presented in this article clearly show that the destabilizing role of external damping\nwas until now misunderstood, and that experimental proof of the destabilization paradox\nin a mechanical laboratory is now more plausible than previously thought. Moreover, the\nfact that external damping plays a destabilizing role may have important consequences in\nstructural design and this opens new perspectives for energy harvesting devices.\nAcknowledgements\nThe authors gratefully acknowledge \fnancial support from the ERC Advanced Grant In-\nstabilities and nonlocal multiscale modelling of materials FP7-PEOPLE-IDEAS-ERC-2013-\nAdG (2014-2019).\nReferences\nReferences\nAndreichikov, I. P., Yudovich, V. I., 1974. Stability of viscoelastic bars. Izv. AN SSSR. Mekh.\nTv. Tela 9(2), 78{87.\nBanichuk, N. V., Bratus, A. S., Myshkis, A. D. 1989. Stabilizing and destabilizing e\u000bects in\nnonconservative systems. PMM USSR 53(2), 158{164.\n14Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nBeck, M. 1952. Die Knicklast des einseitig eingespannten, tangential gedr uckten Stabes. Z.\nangew. Math. Phys. 3, 225.\nBigoni, D., Noselli, G., 2011. Experimental evidence of \rutter and divergence instabilities\ninduced by dry friction. J. Mech. Phys. Sol. 59, 2208{2226.\nBolotin, V. V., 1963. Nonconservative Problems of the Theory of Elastic Stability. Pergamon\nPress, Oxford.\nBolotin, V. V., Zhinzher, N. I., 1969. E\u000bects of damping on stability of elastic systems\nsubjected to nonconservative forces. Int. J. Solids Struct. 5, 965{989.\nBottema, O., 1956. The Routh-Hurwitz condition for the biquadratic equation. Indag. Math.\n18, 403{406.\nChen, L. W., Ku, D. M., 1992. Eigenvalue sensitivity in the stability analysis of Beck's\ncolumn with a concentrated mass at the free end. J. Sound Vibr. 153(3), 403{411.\nCrandall, S. H., 1995. The e\u000bect of damping on the stability of gyroscopic pendulums. Z.\nAngew. Math. Phys. 46, S761{S780.\nDetinko, F. M., 2003. Lumped damping and stability of Beck column with a tip mass. Int.\nJ. Solids Struct. 40, 4479{4486.\nDone, G. T. S., 1973. Damping con\fgurations that have a stabilizing in\ruence on non-\nconservative systems. Int. J. Solids Struct. 9, 203{215.\nKirillov, O. N., 2011. Singularities in structural optimization of the Ziegler pendulum. Acta\nPolytechn. 51(4), 32{43.\nKirillov, O. N., 2013. Nonconservative Stability Problems of Modern Physics. De Gruyter\nStudies in Mathematical Physics 14. De Gruyter, Berlin.\nKirillov, O. N., Seyranian, A. P., 2005. The e\u000bect of small internal and external damping on\nthe stability of distributed nonconservative systems. J. Appl. Math. Mech. 69(4), 529{552.\nKirillov, O. N., Seyranian, A. P., 2005. Stabilization and destabilization of a circulatory\nsystem by small velocity-dependent forces, J. Sound Vibr. 283(35), 781{800.\nKirillov, O. N., Seyranian, A. P., 2005. Dissipation induced instabilities in continuous non-\nconservative systems, Proc. Appl. Math. Mech. 5, 97{98.\nKirillov, O. N., Verhulst, F., 2010. Paradoxes of dissipation-induced destabilization or who\nopened Whitney's umbrella? Z. Angew. Math. Mech. 90(6), 462{488.\n15Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nKrechetnikov, R., Marsden, J. E., 2007. Dissipation-induced instabilities in \fnite dimensions.\nRev. Mod. Phys. 79, 519{553.\nOran, C., 1972. On the signi\fcance of a type of divergence. J. Appl. Mech. 39, 263{265.\nPanovko, Ya. G., Sorokin, S. V., 1987. On quasi-stability of viscoelastic systems with the\nfollower forces, Izv. Acad. Nauk SSSR. Mekh. Tverd. Tela. 5, 135{139.\nPedersen, P. 1977. In\ruence of boundary conditions on the stability of a column under\nnon-conservative load. Int. J. Solids Struct. 13, 445{455.\nP\r uger, A., 1955. Zur Stabilit at des tangential gedr uckten Stabes. Z. Angew. Math. Mech.\n35(5), 191.\nPlaut, R. H., 1971. A new destabilization phenomenon in nonconservative systems. Z. Angew.\nMath. Mech. 51(4), 319{321.\nPlaut, R. H., Infante, E. F., 1970. The e\u000bect of external damping on the stability of Beck's\ncolumn. Int. J. Solids Struct. 6(5), 491{496.\nRyu, S., Sugiyama, Y., 2003. Computational dynamics approach to the e\u000bect of damping on\nstability of a cantilevered column subjected to a follower force. Comp. Struct. 81, 265{271.\nSaw, S. S., Wood, W. G., 1975. The stability of a damped elastic system with a follower\nforce. J. Mech. Eng. Sci. 17(3), 163{176.\nSugiyama, Y., Kashima, K., Kawagoe, H., 1976. On an unduly simpli\fed model in the\nnon-conservative problems of elastic stability. J. Sound Vibr. 45(2), 237{247.\nSugiyama, Y., Katayama, K., Kinoi, S. 1995. Flutter of cantilevered column under rocket\nthrust. J. Aerospace Eng. 8(1), 9{15.\nWalker, J. A. 1973. A note on stabilizing damping con\fgurations for linear non-conservative\nsystems. Int. J. Solids Struct. 9, 1543{1545.\nWang, G., Lin, Y. 1993. A new extension of Leverrier's algorithm. Lin. Alg. 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Mech. 20, 49{56.\n16Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nAppendix A. - The stabilizing role of external damping and the destabilizing\nrole of internal damping\nA critical review of the relevant literature is given in this Appendix, with the purpose of\nexplaining the historical origin of the misconception that the external damping introduces a\nmere stabilizing e\u000bect for structures subject to \rutter instability.\nPlaut and Infante (1970) considered the Ziegler pendulum with m1= 2m2, without\ninternal damping (in the joints), but subjected to an external damping proportional to the\nvelocity along the rigid rods of the double pendulum4. In this system the critical \rutter load\nincreases with an increase in the external damping, so that they presented a plot showing\nthat the \rutter load converges to a value which is very close to P\u0000\nu. However, they did not\ncalculate the critical value in the limit of vanishing external damping, which would have\nrevealed a value slightly smaller than the value corresponding to the undamped system5.\nIn a subsequent work, Plaut (1971) con\frmed his previous result and demonstrated that\ninternal damping with equal damping coe\u000ecients destabilizes the Ziegler pendulum, whereas\nexternal damping has a stabilizing e\u000bect, so that it does not lead to the destabilization\nparadox. Plaut (1971) reports a stability diagram (in the external versus internal damping\nplane) that implicitly indicates the existence of the Whitney umbrella singularity on the\nboundary of the asymptotic stability domain. These conclusions agreed with other studies\non the viscoelastic cantilevered Beck's column (Beck, 1952), loaded by a follower force which\ndisplays the paradox only for internal Kelvin-Voigt damping (Bolotin and Zhinzher, 1969;\nPlaut and Infante, 1970; Andreichikov and Yudovich, 1974; Kirillov and Seyranian, 2005a)\nand were supported by studies on the abstract settings (Done, 1973; Walker, 1973; Kirillov\nand Seyranian, 2005b), which have proven the stabilizing character of external damping,\nassumed to be proportional to the mass (Bolotin, 1963; Zhinzher, 1994).\nThe P\r uger column [a generalization of the Beck problem in which a concentrated mass\nis added to the loaded end, P\r uger (1955), see also Sugiyama et al. (1976), Pedersen (1977),\nand Chen and Ku (1992)] was analyzed by Sugiyama et al. (1995) and Ryu and Sugiyama\n(2003), who numerically found that the internal damping leads to the destabilization paradox\nfor all ratios of the end mass to the mass of the column. The role of external damping was\ninvestigated only by Detinko (2003) who concludes that large external damping provides a\nstabilizing e\u000bect.\nThe stabilizing role of external damping was questioned only in the work by Panovko\nand Sorokin (1987), in which the Ziegler pendulum and the Beck column were considered\nwith a dash-pot damper attached to the loaded end (a setting in which the external damper\ncan be seen as something di\u000berent than an air drag, but as merely an additional structural\n4Note that di\u000berent mass distributions were never analyzed in view of external damping e\u000bect. In the\nabsence of damping, stability investigations were carried out by Oran (1972) and Kirillov (2011).\n5In fact, the \rutter load of the externally damped Ziegler pendulum with m1= 2m2, considered by Plaut\nand Infante (1970) and Plaut (1971) tends to the value P= 2 which is smaller than P\u0000\nu\u00192:086, therefore\nrevealing the paradox.\n17Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nelement, as suggested by Zhinzher (1994)). In fact the dash-pot was shown to always yield\nthe destabilization paradox, even in the presence of internal damping, no matter what the\nratio is between the coe\u000ecients of internal and external damping (Kirillov and Seyranian,\n2005c; Kirillov, 2013).\nIn summary, there is a well-established opinion that external damping stabilizes struc-\ntures loaded by nonconservative positional forces.\nAppendix B. - A necessary condition for stabilization of a general 2 d.o.f.\nsystem\nKirillov and Seyranian (2005b) considered the stability of the system\nMx+\"D_x+Kx= 0; (A.1)\nwhere\">0 is a small parameter and M=MT,D=DT, and K6=KTare real matrices of\nordern. In the case n= 2, the characteristic polynomial of the system (A.1),\nq(\u001b;\") = det( M\u001b2+\"D\u001b+K);\ncan be written by means of the Leverrier algorithm (adopted for matrix polynomials by\nWang and Lin (1993)) in a compact form:\nq(\u001b;\") = det M\u001b4+\"tr(D\u0003M)\u001b3+ (tr( K\u0003M) +\"2detD)\u001b2+\"tr(K\u0003D)\u001b+ det K;(A.2)\nwhere D\u0003=D\u00001detDandK\u0003=K\u00001detKare adjugate matrices and tr denotes the trace\noperator.\nLet us assume that at \"= 0 the undamped system (A.1) with n= 2 degrees of freedom\nbe on the \rutter boundary, so that its eigenvalues are imaginary and form a double complex-\nconjugate pair \u001b=\u0006i!0of a Jordan block. In these conditions, the real critical frequency\n!0at the onset of \rutter follows from q(\u001b;0) in the closed form (Kirillov, 2013)\n!2\n0=r\ndetK\ndetM: (A.3)\nA dissipative perturbation \"Dcauses splitting of the double eigenvalue i!0, which is\ndescribed by the Newton-Puiseux series \u001b(\") =i!0\u0006ip\nh\"+o(\"), where the coe\u000ecient his\ndetermined in terms of the derivatives of the polynomial q(\u001b;\") as\nh:=dq\nd\"\u00121\n2@2q\n@\u001b2\u0013\u00001\f\f\f\f\f\n\"=0;\u001b=i!0=tr(K\u0003D)\u0000!2\n0tr(D\u0003M)\n4i!0detM: (A.4)\nSince the coe\u000ecient his imaginary, the double eigenvalue i!0splits generically into two com-\nplex eigenvalues, one of them with the positive real part yielding \rutter instability (Kirillov\n18Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nand Seyranian, 2005b). Consequently, h= 0 represents a necessary condition for\"Dto be\nastabilizing perturbation (Kirillov and Seyranian, 2005b).\nIn the case of the system (3), with matrices (5), it is readily obtained\n!2\n0=k\nl2pm1m2: (A.5)\nAssuming D=Di, eq. (A.4) and the representations (5) and (A.5) yield\nh=hi:=i\nm1l25\u0016\u00002p\u0016+ 1\n4\u0016; (A.6)\nso that the equation hi= 0 has as solution the complex-conjugate pair \u0016= (\u00003\u00064i)=25.\nTherefore, for every real mass distribution \u0016\u00150 the dissipative perturbation with the matrix\nD=Diof internal damping results to be destabilizing.\nSimilarly, eq. (A.4) with D=Deand representations (A.5), (5), and F=F\u0000\nu(\u0016) yield\nh=he:=il\n48m18\u00162\u000011p\n\u00163\u00006\u0016+ 5p\u0016\n\u00162; (A.7)\nso that the constraint he= 0 is satis\fed only by the two following real values of \u0016\n\u0016A\u00190:273; \u0016C\u00192:559: (A.8)\nThe mass distributions (A.8) correspond exactly to the points A and C in Fig. 1, which are\ncommon for the \rutter boundary of the undamped system and for that of the dissipative\nsystem in the limit of vanishing external damping. Consequently, the dissipative perturbation\nwith the matrix D=Deof external damping can have a stabilizing e\u000bect for only two\nparticular mass distributions (A.8). Indeed, as it is shown in the present article, the external\ndamping is destabilizing for every \u0016\u00150, except for \u0016=\u0016Aand\u0016=\u0016C.\nConsequently, the stabilizing or destabilizing e\u000bect of damping with the given matrix D\nis determined not only by its spectral properties, but also by how it `interacts' with the mass\nand sti\u000bness distributions. The condition which selects possibly stabilizing triples ( M,D,\nK) in the general case of n= 2 degrees of freedom is therefore the following\ntr(K\u0003D) =!2\n0tr(D\u0003M): (A.9)\n19"    },    {        "title": "1111.7152v2.Local_phase_damping_of_single_qubits_sets_an_upper_bound_on_the_phase_damping_rate_of_entangled_states.pdf",        "content": "arXiv:1111.7152v2  [quant-ph]  27 Jan 2012Local phase damping of single qubits sets an upper bound on th e phase damping rate\nof entangled states\nStephan D¨ urr\nMax-Planck-Institut f¨ ur Quantenoptik, Hans-Kopfermann -Straße 1, 85748 Garching, Germany\nI derive an inequality in which the phase damping rates of sin gle qubits set an upper bound for the\nphase damping rate of entangled states of many qubits. The de rivation is based on two assumptions,\nfirst, that the phase damping can be described by a dissipator in Lindblad form and, second, that\nthe phase damping preserves the population of qubit states i n a given basis.\nPACS numbers: 03.65.Yz, 03.65.Ud\nI. INTRODUCTION\nQuantum information processing [1] offers a perspec-\ntive for a tremendous reduction of the computation time\nin solving certain problems, such as factorization of large\nnumbers [2], simulations of quantum systems [3], and\ndatabase searches [4]. However, interactions of the quan-\ntum system with its environment induce decoherence [5–\n8]. This is a major limiting factor on the way toward\nlarge-scale experimental implementations. Naively, one\nmight expect that the decoherence rate of an entangled\nmany-qubit state should equal the sum of the decoher-\nence rates of the individual qubits. While this is true if\nthe decoherence processes are local, i.e. if they act inde-\npendently on individual qubits, the situation can change\nif the decoherence processes act on some or all qubits in\na correlated manner. Such correlated decoherence can\ngive rise to decoherence-free subspaces which have been\nstudied theoretically [9–13] and experimentally [14–17].\nThis shows that certain entangled states can decohere\nconsiderably slower than their single-qubit constituents.\nIn other words, a single-qubit decoherence rate is not a\nlowerbound for the decoherence rate of entangled states.\nHere I show that the single-qubit decoherence rates\nset a rigorous upperbound for the decoherence rate of\nentangled states of nqubits. The existence of such an\nupper bound is somewhat surprising because entangle-\nment is difficult to prepare and maintain experimentally.\nThe upper bound derived here is experimentally relevant\nbecause in many experiments where one aims at gener-\nating entangled states one has the capability to measure\nsingle-qubit decoherence rates already during the build-\nup phase of the apparatus, often much earlier than the\ntimewhereonemanagestogenerateanddetectentangled\nstates. In addition, one often uses experimental tech-\nniques such as a magnetic hold field to suppress spin-flip\ntransitions of the qubits. But these techniques cannot\nprotect the qubit against loss of phase coherence. The\nexperimenter thus selects a preferred basis in which the\npopulations are preserved, whereas phase coherence may\ndecay.II. MODEL\nThe derivation of the upper bound rests on two as-\nsumptions. The first assumption is that there is an or-\nthonormal basis Bin which the population of the basis\nstates is time independent whereas the relative phases\nbetween the basis states may rotate and decay. This\ncondition is easy to verify experimentally. I will show\nbelow that this requires the Hamiltonian Hto be diago-\nnal in the basis B. This implies that the final results will\nbe relevant for quantum memories, but they are not nec-\nessarily applicable during quantum gate operations that\nact on the qubits of interest. The second assumption is\nthat the decoherence can be described by a dissipator in\nLindbladform. Thisis areasonableassumptionformany\nexperiments, as discussed now.\nConsider a quantum system coupled to an environ-\nment. If one assumes, first, that the system is initially\nnot entangled with the environment and, second, that\na Born-Markov approximation is appropriate (because\nthe coupling between system and environment is weak\nenough and because the environment is much larger than\nthe system), then the environmentally-induced decoher-\nence can be described by\nd\ndtρ=1\ni¯h[H,ρ]+Dρ (1a)\nwith a dissipator Din Lindblad form [18–20]\nDρ=/summationdisplay\nmγ(m)\n2/parenleftBig\n2A(m)ρ(A(m))†\n−(A(m))†A(m)ρ−ρ(A(m))†A(m)/parenrightBig\n.(1b)\nHere,ρis the density matrix, the γ(m)>0 are deco-\nherence rates, and the dimensionless operators A(m)are\ncalled Lindblad operators. In the following, it is always\nassumed that the time evolution is described by Eq. (1).\nDefinition : I call a time evolution generated by Eq.\n(1)population preserving with respect to an orthonormal\nbasisBifalldiagonalelementsof ρinthebasis Baretime\nindependent for all initial states, i.e. ( d/dt)ρii= 0.2\nIII. RESULTS\nI now formulate three theorems, which are proven in\nappendix A.\nTheorem 1 : If all the A(m)are diagonal in the same\nbasisBwith eigenvalues λ(m)\ni, then\nDρij= (i∆ij−Γij)ρij (2)\nwith decay coefficients Γ ijand angular frequencies ∆ ij\ngiven by\nΓij=1\n2/summationdisplay\nmγ(m)/vextendsingle/vextendsingle/vextendsingleλ(m)\ni−λ(m)\nj/vextendsingle/vextendsingle/vextendsingle2\n, (3)\n∆ij=/summationdisplay\nmγ(m)Im/parenleftBig\nλ(m)\ni(λ(m)\nj)∗/parenrightBig\n.(4)\nIfadditionally, Hisdiagonalinthesamebasiswitheigen-\nvaluesEi, then\nd\ndtρij= (iωij−Γij)ρij (5)\nwith angular frequencies\nωij=Ej−Ei\n¯h+∆ij. (6)\nNote that Γ ij≥0 and Γii= ∆ii=ωii= 0. Hence, only\noff-diagonal elements of ρchange over time. These ele-\nments do not mix. Instead, each off-diagonal element ex-\nperiences two effects. The first effect, described by Γ ij, is\nan exponentialdecayof |ρij|, which is calledphase damp-\ning [1]. The second effect, described by ωij, is a phase\nrotation [21]. This paper focuses on phase damping.\nTheorem 2 : A time evolution is population preserving\nwith respect to B, if and only if Hand all the A(m)are\ndiagonal in B.\nTheorem 3 : If a time evolution is population preserv-\ning with respect to B, then the Γ ijfrom theorem 1 obey\nthe following inequality for n∈ {1,2,3,...}and for all\nbasis states |p0/an}bracketri}ht ∈ B,|p1/an}bracketri}ht ∈ B, ...,|pn/an}bracketri}ht ∈ B\nΓp0,pn≤nn/summationdisplay\nk=1Γpk−1,pk. (7)\nThis inequality becomes an equality, if and only if for\neachm, the expression λ(m)\npk−1−λ(m)\npkis independent of k,\nwhere theλ(m)\niare the eigenvalues of the A(m).\nThis theorem is applicable to the density matrix ρof\nan arbitrary state of nqubits. Let the states |↑/an}bracketri}htand|↓/an}bracketri}ht\ndenote a basis of the Hilbert space of each single qubit\nand assume that the time evolution is population pre-\nserving with respect to the basis Bwhich consists of all\nthe tensor products of the single-qubit states |↑/an}bracketri}htand|↓/an}bracketri}ht.\nThis situation corresponds, e.g., to the scenario with a\nmagnetic hold field discussed in the introduction. Ac-\ncording to theorem 1, each density matrix element ρijin\nthe preferred basis Bobeys Eq. (5). Specifically, therewill be phase damping, described by Γ ij. For any choice\nof the basis states |i/an}bracketri}htand|j/an}bracketri}htone can obviously construct\nasequenceofbasisstates |p0/an}bracketri}ht,|p1/an}bracketri}ht, ...,|pn/an}bracketri}htinawaythat\n|p0/an}bracketri}ht=|i/an}bracketri}ht,|pn/an}bracketri}ht=|j/an}bracketri}ht, and that (for all k∈ {1,2,...,n})\nthe states |pk−1/an}bracketri}htand|pk/an}bracketri}htdiffer by only one local spin\nflip between states |↑/an}bracketri}htand|↓/an}bracketri}ht. Application of Eq. (7)\nyields an upper bound for Γ ijand each Γ pk−1,pkon the\nrighthand side obviously describes a local phase damping\nrate.\nThisis thecentralresultofthe presentpaper. Interest-\ningly, the upper bound in the inequality (7) is a factor of\nnhigherthan the decoherenceratethat onewould obtain\nfor uncorrelated phase damping of individual qubits.\nThe experimental relevance of this upper bound arises\nfromthefactthatmeasurementsofthesingle-qubitphase\ndamping rates are fairly easy to perform because no en-\ntangledstateneedstobe preparedanddetected. Accord-\ning to the inequality (7), such measurements already set\na worst-case upper bound for the phase damping rate of\nany entangled state.\nIV. EXAMPLES\nTo illustrate this concept, consider an example of a\nGreenberger-Horne-Zeilinger(GHZ)-type entangled state\n[22] ofnqubits\n|ψGHZ/an}bracketri}ht=1√\n2/parenleftbig\n|↑/an}bracketri}ht⊗n+|↓/an}bracketri}ht⊗n/parenrightbig\n. (8)\nHere,|↑/an}bracketri}ht⊗n=|↑/an}bracketri}ht⊗···⊗|↑/an}bracketri}ht abbreviatesa tensor product\nofntimes the same single-qubit state and the inequality\n(7) can be applied to the basis states\n|pk/an}bracketri}ht=|↑/an}bracketri}ht⊗k⊗|↓/an}bracketri}ht⊗(n−k)(9)\nwithk∈ {0,1,...,n}. This yields an upper bound for\nthe phase damping rate of the GHZ state because Γ p0,pn\ndescribes the decay of /an}bracketle{tp0|ρ|pn/an}bracketri}ht=/an}bracketle{t↓ ··· ↓ |ρ| ↑ ··· ↑/an}bracketri}ht .\nThe other coefficients Γ pk−1,pkappearing in the inequal-\nity (7) describe decay of /an}bracketle{tpk−1|ρ|pk/an}bracketri}ht. The states |pk−1/an}bracketri}ht\nand|pk/an}bracketri}htdifferonlybyaspinflipofthe k-thqubit. Hence,\nthis coefficient can be determined experimentally from a\nmeasurement of the single-qubit phase damping rate of\nthek-th qubit.\nLet ΓGHZ= Γp0,pndenote the phase damping rate\ncoefficient of the GHZ state and let Γ k= Γpk−1,pkde-\nnote the phase damping rate coefficient of the k-th qubit,\nwhere the specific orientation of the other qubits was\ndropped from the notation for brevity. Then Eq. (7)\nyields\nΓGHZ≤n(Γ1+Γ2+···+Γn). (10)\nTo illustrate this inequality further, consider some ex-\namples forn= 2 qubits. Here, one typically denotes the3\nBell states as\n|ψ±/an}bracketri}ht=1√\n2(|↑↓/an}bracketri}ht±|↓↑/an}bracketri}ht ), (11)\n|φ±/an}bracketri}ht=1√\n2(|↑↑/an}bracketri}ht±|↓↓/an}bracketri}ht ). (12)\nFor 2 qubits, the GHZ-type state of Eq. (8) is obviously\nthe Bell state |φ+/an}bracketri}ht.\nExample 1 : Consider local phase damping generated\nby the Lindblad operators\nA(1)=|↑/an}bracketri}ht/an}bracketle{t↑|1⊗ /BD2, (13)\nA(2)= /BD1⊗|↑/an}bracketri}ht/an}bracketle{t↑| 2 (14)\nwhere /BDdenotes the identity matrix. This yields Γ 1=\nγ(1), Γ2=γ(2), and\nΓψ±= Γφ±= Γ1+Γ2. (15)\nLocal phase damping acts identically on all Bell states.\nThe phase damping rate of each Bell state is simply the\nsum of the local phase damping rates.\nExample 2 : Consider phase damping generated by the\nLindblad operator\nA(1)=|↑↑/an}bracketri}ht/an}bracketle{t↑↑|−|↓↓/an}bracketri}ht/an}bracketle{t↓↓| . (16)\nThe states |↑↑/an}bracketri}htand|↓↓/an}bracketri}htexperience phase damping rel-\native to the two-dimensional rest of the Hilbert space.\nHere, Γ 1= Γ2=γ(1)and\nΓψ±= 0,Γφ±= 2(Γ1+Γ2). (17)\nThe states |ψ±/an}bracketri}htspan a decoherence-free subspace,\nwhereas the states |φ±/an}bracketri}htexperience phase damping at a\nrate that reaches the upper bound (10) for n= 2.\nThis example is somewhat related to the experiment\nwith two trapped ions in Ref. [15]. In that experiment,\nthe phase damping is dominated by fluctuating ambient\nmagnetic fields with frequencies primarily at 60 Hz and\nits harmonics. These fields cause a fluctuating Zeeman\nenergy for the ions. The ions are separated by only a\nfew micrometers so that the ambient magnetic fields are\nroughly uniform across the trapping region. As a result,\nthe states |↑↓/an}bracketri}htand|↓↑/an}bracketri}htexperience no net Zeeman effect\nandthestates |ψ±/an}bracketri}htexperiencenophasedamping,atleast\nto lowest order. The states |↑↑/an}bracketri}htand|↓↓/an}bracketri}ht, however, expe-\nrience plus or minus twice the single-qubit Zeeman shift,\nleading to a phase damping that is twice as fast as for a\nsingle qubit.\nNote that technical fluctuations of a macroscopicmag-\nnetic field need not necessarily allow for a description in\nterms of Eq. (1). However, if a similar experiment were\nperformed with quantum dots in a solid, then the mag-\nnetic dipole-dipole interactions with the large number of\nsurrounding nuclear spins with thermal occupation may\ncreate an environment that can be described by Eq. (1).\nExample 3 : For comparison, consider phase damping\ngenerated by the Lindblad operators\nA(1)=|↑↑/an}bracketri}ht/an}bracketle{t↑↑|, A(2)=|↓↓/an}bracketri}ht/an}bracketle{t↓↓| (18)withγ(1)=γ(2). Here,A(1)creates phase damping for\nthe state |↑↑/an}bracketri}htrelativeto the three-dimensionalrestofthe\nHilbert space and A(2)creates an analogouseffect for the\nstate|↓↓/an}bracketri}ht. This yields Γ 1= Γ2=γ(1)and\nΓψ±= 0,Γφ±= Γ1+Γ2. (19)\nAgain, the states |ψ±/an}bracketri}htspan a decoherence-free subspace.\nBut now, the phase damping rate of the states |φ±/an}bracketri}htno\nlonger reaches the upper bound (10).\nSuch a phase damping might be obtained hypotheti-\ncally, if there are two different fields, one coupling only\nto state|↑↑/an}bracketri}htand another coupling only to state |↓↓/an}bracketri}ht. If\nthese fields are uncorrelated, they will each generate an\nindividual Lindblad operator, as in Eq. (18). This situa-\ntion differs from the phase damping caused by a common\nfield, which is expressed by one Lindblad operator as in\nEq. (16).\nAcknowledgments\nI would like to thank Simon Baur, Geza Giedke,\nMarkus Heyl, and Stefan Kehrein for stimulating dis-\ncussions. This work was supported by the German Ex-\ncellence Initiative through the Nanosystems Initiative\nMunich and by the Deutsche Forschungsgemeinschaft\nthrough SFB 631.\nAppendix A: Proofs\nProof of theorem 1 :A(m)\nij=λ(m)\niδijand Eq. (1)\nyieldDρij=/summationtext\nklmγ(m)\n2δikρklδlj(2λ(m)\ni(λ(m)\nj)∗−|λ(m)\ni|2−\n|λ(m)\nj|2), whereδijdenotes the Kronecker symbol. Com-\nbination with |λ(m)\ni−λ(m)\nj|2=|λ(m)\ni|2+|λ(m)\nj|2−\n2Re(λ(m)\ni(λ(m)\nj)∗) yields Eq. (2). Combination with\n(d/dt−D)ρij= (Eiρij−ρijEj)/i¯hcompletes the proof.\nProof of theorem 2 : IfHand all the A(m)are diagonal\ninB, then theorem 1 yields Γ ii=ωii= 0 and (d/dt)ρii=\n0 for alli.\nTo prove the other direction, assume that the time\nevolution is population preserving with respect to B. I\nwill first show that all the A(m)are diagonal in B. To\nthis end, it suffices to consider ( d/dt)ρjj= 0 for the\nspecial case where ρ=|j/an}bracketri}ht/an}bracketle{tj|att= 0 where |j/an}bracketri}htis any\npure basis state that belongs to B. Eq. (1) yields 0 =\n(d/dt)ρjj=−/summationtext\nmγ(m)/summationtext\nk/ne}ationslash=j|A(m)\nkj|2. This holds for all\nj. This yields A(m)\nkj= 0 for allj,k,mwithj/ne}ationslash=k. Hence,\nall theA(m)are diagonal in B.\nIt remains to be shown that His diagonal in B. To\nthis end, select an arbitrary pair of indices i,jwith\ni/ne}ationslash=j. It needs to be shown that Hij= 0. Here, it\nsuffices to consider only initial density matrices of the\nformρ=|ψ/an}bracketri}ht/an}bracketle{tψ|with|ψ/an}bracketri}ht= (|i/an}bracketri}ht+eiϕ|j/an}bracketri}ht)/√\n2 with a4\nreal parameter ϕ. As seen above, all the A(m)are diag-\nonal inBso that theorem 1 yields Dρii= 0. The pop-\nulation preserving character of the time evolution then\nyields 0 = i¯h(d/dt)ρii=/summationtext\nk∈{i,j}(Hikρki−ρikHki) =\nHijρji−c.c.=iIm(Hijeiϕ). This holds for all real ϕ,\nespecially for ϕ= 0 andϕ=π/2. HenceHij= 0.\nProof of theorem 3 : Assume that the time evolution\nis population preserving. According to theorem 2, all\ntheA(m)are diagonal in B. Hence, theorem 1 is appli-\ncable. The Cauchy-Schwarz inequality |/summationtextn\nk=1xky∗\nk|2≤(/summationtextn\nk=1|xk|2)(/summationtextn\nk=1|yk|2) withxk=λ(m)\npk−1−λ(m)\npkand\nyk= 1 reads |λ(m)\np0−λ(m)\npn|2=|/summationtextn\nk=1(λ(m)\npk−1−λ(m)\npk)|2≤\nn/summationtextn\nk=1|λ(m)\npk−1−λ(m)\npk|2. Multiply this by γ(m)/2. Then\nsum overmto obtain the inequality (7). It is well known\nthat the Cauchy-Schwarzinequality becomes an equality,\nif and only if the vectors ( x1,...,xn) and (y1,...,yn) are\nlinearly dependent. Here, this is equivalent to x1=x2=\n···=xn.\n[1] M. A. Nielsen and I. L. Chuang, Quantum Computation\nand Quantum Information (Cambridge University Press,\nCambridge, UK, 2000).\n[2] P. W. Shor, In Proceedings of the 35th Annual Sympo-\nsium on Foundations of Computer Science , edited by\nS. Goldwasser (IEEE Computer Society, Los Alamitos,\nCA, 1994) p. 116.\n[3] R. Feynman, Int. J. Theor. Phys. 21, 467 (1982).\n[4] L. K. Grover, Phys. Rev. Lett. 79, 325 (1997).\n[5] W. H. Zurek, Phys. Today 44(10), 36 (1991).\n[6] S. Haroche, Phys. Today 51(7), 36 (1998).\n[7] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).\n[8] W. H. Zurek, Nat. Phys. 5, 181 (2009).\n[9] G. M. Palma, K.-A. Suominen, and A. K. Ekert, Proc.\nR. Soc. London A 452, 567 (1996).\n[10] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306\n(1997).\n[11] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953\n(1997).\n[12] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev.\nLett.81, 2594 (1998).\n[13] R. Blume-Kohout, H. K. Ng, D. Poulin, and L. Viola,\nPhys. Rev. Lett. 100, 030501 (2008).\n[14] P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G.\nWhite, Science 290, 498 (2000).\n[15] D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett,\nW. M. Itano, C. Monroe, and D. J. Wineland, Science\n291, 1013 (2001).\n[16] E. M. Fortunato, L. Viola, J. Hodges, G. Teklemariam,\nand D. G. Cory, New J. Phys. 4, 5 (2002).[17] M. Carravetta, O. G. Johannessen, and M. H. Levitt,\nPhys. Rev. Lett. 92, 153003 (2004).\n[18] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).\n[19] H. J. Carmichael, Statistical Methods in Quantum Op-\ntics 1: Master Equations and Fokker-Planck Equations\n(Springer, Berlin, 1999).\n[20] H.-P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (OxfordUniversityPress, Oxford, UK,\n2002).\n[21] Note that pure phase rotation can lead to a decay of\nthe expectation value of an observable. As an example,\nconsider a large number of spin-1 /2 particles, initially\naligned to maximize the expectation value of the xcom-\nponentofthetotalspin /angbracketleftSx/angbracketright.Assumethatthetimeevolu-\ntion is population preserving with respect to the zcom-\nponents of each individual spin-1 /2 particle. It is well-\nknown from spin-echo experiments that in this scenario,\nunitary phase rotation generated by a Hamiltonian may\ncause a decay of /angbracketleftSx/angbracketright. Obviously, pure phase damping\ncould also cause such a decay of /angbracketleftSx/angbracketright. Hence, phase rota-\ntion and phase damping can sometimes generate similar\neffets, at least for a specific observable. The difference be-\ntween phase rotation and phase damping becomes clear\nwhen reconstructing the full density matrix, because for\npure phase rotation all the |ρij|are time independent,\nwhereas in the presence of phase damping the |ρij|de-\ncay.\n[22] D. M. Greenberger, M. A. Horne, A. Shimony, and\nA. Zeilinger, Am. J. Phys. 58, 1131 (1990)."    },    {        "title": "1712.01032v1.Resonance_oscillation_of_a_damped_driven_simple_pendulum.pdf",        "content": "arXiv:1712.01032v1  [physics.class-ph]  4 Dec 2017Resonance oscillation of a damped driven simple pendulum\nD. Kharkongor1,2, and Mangal C. Mahato1,∗\n1Department of Physics, North-Eastern Hill University, Shi llong-793022, India and\n2Department of Physics, St. Anthony’s College, Shillong-79 3003, India\nAbstract\nThe resonance characteristics of a driven damped harmonic o scillator are well known. Unlike\nharmonic oscillators which are guided by parabolic potenti als, a simple pendulum oscillates under\nsinusoidal potentials. The problem of an undamped pendulum has been investigated to a great\nextent. However, the resonance characteristics of a driven damped pendulum have not been re-\nported so far due to the difficulty in solving the problem analy tically. In the present work we\nreport the resonance characteristics of a driven damped pen dulum calculated numerically. The\nresults are compared with the resonance characteristics of a damped driven harmonic oscillator.\nThe work can be of pedagogic interest too as it reveals the ric hness of driven damped motion of a\nsimple pendulumin comparison to and how strikingly it differs from the motion of a driven damped\nharmonic oscillator. We confine our work only to the nonchaot ic regime of pendulum motion.\nPACS numbers: 45.20.-d; 46.40.Ff; 07.05Tp; 05.45.-a\n∗Electronic address: mangal@nehu.ac.in\n1I. INTRODUCTION\nThe motion of a simple pendulum has the same equation as the motion of a particle in a\nsinusoidal potential. And the motion of the simple pendulum with very s mall amplitude x\n(so that sin xcan be safely approximated as x) is equivalent to that of a harmonic oscillator.\nHowever, as the amplitude becomes larger, the motion of the simple p endulum differs from\nthat of a harmonic oscillator. The natural frequency of oscillation o f a harmonic oscillator\n(ω0=/radicalBig\nk\nm) is independent of its amplitude, with mas the mass of the oscillator and kis\nthe stiffness constant of the spring. However, since a simple pendu lum with large amplitude\nis different from a harmonic oscillator its frequency of oscillation is not independent of the\namplitude of its free oscillation. The period or the frequency, ω, of oscillation of a freely\noscillating pendulum of finite amplitude, x0, is given in terms of an elliptic integral of the\nfirst kind and has been given in many text books, as for example[1–3]. The frequencies in\nterms of simple series expansions have also been given[2, 3] and ther e are many attempts to\nimprove upon the series expansion in order to get to as close as the e xact (elliptic integral)\nvalue using only few terms, to cite a few[4–9]. For comparison of the f requency of oscillation\nof harmonic oscillator, ω0, and various expresssions for the frequency, ω1, of oscillations of\na simple pendulum we use the expressions used by Kittel, et. al.[2]\nω1\nω0≈1−x2\n0\n16(1.1)\nand the expression arrived at by Bel´ endez, et. al.[8]\nω1\nω0≈1\n4/parenleftbigg\n1+/radicalbigg\ncosx0\n2/parenrightbigg2\n(1.2)\nin addition to the exact result[8],\nω1\nω0=π\n2K(k), (1.3)\nwhere\nK(k) =/integraldisplayπ\n2\n0dφ/radicalbig\n1−ksin2φ(1.4)\nand\nk= sin2x0\n2(1.5)\nand plotted in Fig. 1, using standard tables[10] for the elliptic integra ls (1.4). It is to\nbe noted that whereas the frequency of free oscillation of a simple h armonic oscillator is\n2independent of amplitude, the frequency of free oscillation of a simp le pendulum decreases\nwith its amplitude.\nIt is also well known that if a harmonic oscillator is driven by an externa l periodic drive\nof frequency ω, then the oscillator responds with larger and larger amplitude x0asω→ω0\nand the amplitude becomes infinitely large at ω=ω0. The condition ω=ω0is termed\nas the resonance condition. Note, however, that a simple pendulum cannot have a similar\nresonance oscillation at a fixed frequency ω=ω0asω(x0) is not independent of x0.\nWhen the harmonic oscillator is (viscous) damped so that it satisfies t he equation of\nmotion[3, 11]\nd2x\ndt2+γdx\ndt+ω2\n0x= 0, (1.6)\nthe oscillator, for small damping coefficient γ <2ω0, oscillates with displacement\nx(t) =Ae−γt\n2cos(ω1t+φ), (1.7)\nwith the frequency of oscillation ω1=/radicalBig\nω2\n0−γ2\n2and diminishing amplitude Ae−γt\n2, whereA\nandφarearbitraryconstants. Noticeagainthateventhoughfrequen cyofoscillationdepends\non the damping factor γ, it remains fixed for all time, independent of the (diminishing)\namplitude. The motion of (under)damped pendulum can be described by\nd2x\ndt2+γdx\ndt+ω2\n0sinx= 0. (1.8)\nAn approximate solution has been derived by Johannessen[12]. The a pproximate solution\ncompares quite well with the numerical solution. The solution is oscillat ory in nature with\ndiminishing amplitude as time progresses, Fig. 1 of Ref.[12]. The time pe riods of oscil-\nlations are plotted in Fig. 3 of the same refrence as a function of time . One can clearly\nnotice that initially the frequency of oscillations are small but keep inc reasing with time\nand the frequency approaches the value appropriate for the dam ped harmonic oscillator,\nω1=/radicalBig\nω2\n0−γ2\n2. This is as it should be because as the amplitude of oscillation approach es\nzero the motion of simple pendulum becomes closer to that of a simple h armonic oscillator.\nThe above description reiterates the well-known result that, in pre sence of damping, the\nharmonic oscillator as well as the simple pendulum asymptotically appro ach the stationary\nstatex(t→∞) = 0. However, if the damped harmonic oscillator is, in addition, subje cted\nto a periodic forcing F(t) =F0cosωt, then the equation of motion\nd2x\ndt2+γdx\ndt+ω2\n0x=F0\nmcosωt (1.9)\n3has a solution\nx(t) =x0cos(ωt+φ), (1.10)\nwith\nx0=F0\nm1\n((ω2−ω2\n0)2+(ωγ)2)1\n2(1.11)\nand\nφ= tan−1(γω\nω2−ω2\n0), (1.12)\nisthephasedifferencebetween F(t)andtheresponse x(t). Owingtothepresenceofdamping\na mean power loss of\nP=F2\n0\n2mγω\n(ω2−ω2\n0)2+(ωγ)2(1.13)\noccurs.Pshows a peak exactly at the frequency ω=ω0, even though the amplitude peaks\nat a lower frequency ω=/radicalBig\nω2\n0−γ2\n2.Pbecomes maximum at ω=ω0because the phase\ndifference isπ\n2at this frequency. One can take this frequency ω=ω0as the resonance\nfrequency of the forced damped harmonic oscillator. Obviously, th e resonance frequency is\nindependent of the damping coefficient or the amplitude of oscillation. The corresponding\nresonance condition for a damped simple pendulum driven by a periodic force does not have\na known analytical expression. We, therefore, numerically obtain t he resonance frequency\nof the forced damped simple pendulum as discussed in the following.\nII. RESONANCE FREQUENCY\nWe investigate the behavior of a forced damped simple pendulum in the present work.\nIn particular, we explore the resonance behavior of such a pendulu m by calculating the\ncorresponding mean power loss or equivalently the hysteresis loss. We, in fact, investigate\nthe equivalent problem of motion of an underdamped particle in a sinus oidal potential under\nthe influence of an external periodic force F(t) =F0cosωt.\nTheequationofmotioninaperiodicpotential V(x) =−sinxisgivenby(indimensionless\nform)\nd2x\ndt2+γdx\ndt−cosx=F0cosωt. (2.1)\nNote that this is exactly the equation of a damped driven simple pendu lum if one iden-\ntifies the angular displacement θ=x−π\n2. By solving this equation we obtain the mean\nhysteresis loop, its area, and the amplitudes of the corresponding trajectories x(t) in order\n4to characterize the resonance behavior of the system. We identif y, as explained earlier, the\nresonance frequency as one for which the hysteresis loop area is t he largest and calculate\nthe corresponding mean amplitude x0of oscillation x(t). The mean hysteresis loop area is\njust the integral/integraltext\nFdxover a large number Nof periods of F(t) and calculate the average\nfor one cycle of F. This quantity is again averaged over Minitial conditions x(t= 0) but\nwithv(t= 0) =dx\ndt(t= 0) = 0, giving the mean hysteresis loop area A, for various values of\nfrequency ωfor a given γ. We set F0= 0.2 throughout in this work except in the inset of\nFig. 2. We keep the dimensionless mean amplitude F0small so that the pendulum always\nremain in the nonchaotic regime. We, thus, identify the frequency ω0which gives the largest\nhysteresis loop area. The corresponding mean amplitude of oscillatio nx0is also calculated.\nFig. 2 gives the plot of the resonance frequency ω0versus the mean amplitude x0. Each\npoint on the plot corresponds to a fixed value of γ. For comparison, we also plot the res-\nonance frequency ω=ω0of oscillation of damped driven harmonic oscillator in the same\ngraph. Also plotted in the same graph is the exact frequency of fre e (γ= 0) oscillation of\nthe simple pendulum as a guide to the eye. A similar graph is plotted for d rive amplitude\nF0= 0.1 as an inset to Fig. 2.\nIn Fig. 2 at least three qualitatively different regions can be clearly ide ntified. One, in\nthe intermediate range 0 .165≤γ≤0.38 ofγ, the obtained graph shows the usual tendency\nof increasing resonance frequency as the amplitude of oscillation de creases, as though to\napproach the harmonic oscillator limit, as γincreases. And the tendency is more so for the\nsmaller drive amplitude F0= 0.1 as shown in the inset of the figure. Of course, the range\nofγin this case does not lie in 0 .165≤γ≤0.38 but 0.099≤γ≤0.28. Two, for large\nγ >0.38, the obtained ω0decreases rapidly with decreasing amplitude x0asγincreases. ω0\nthus moves away from the harmonic oscillator limit of resonance freq uencyω0. In both these\nregionsof γtheresonance frequencies remainlower thanthefreeoscillation fr equencies. And\nthirdly, in the small γregime, the resonance frequency abruptly drops to a small value j ust\nbelowγ≈0.165, thus we get a disjoint branch of the resonance amplitude freq uency curve.\nThereafter ω0decreases much slowly with increasing x0. The slow decrease of ω0allows\nthe curve to ultimately cross the amplitude-frequency curve of fr ee oscillation of the simple\npendulum. Thus, at smaller γvalues the resonance frequency ω0becomes larger than the\nfree oscillation frequency. In the following, we discuss the resonan ce oscillations in the three\nregions of γseparately.\n5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0  0.5  1  1.5  2  2.5  3ω1\nx0Exact Solution\nBelendez's Approximation\nKittel's Approximation\nFIG. 1: The figure shows the frequency of oscillation ω1as a function of initial position x0for\nan undamped simple pendulum. As is shown in the figure, notice that the approximations as\nobtained by Kittel and Bel´ endez depart from the exact resul t at large amplitudes. Also, shown in\nthe figure (by the dash-dotted horizontal line) is the corres ponding frequency of oscillation ω0of\nan driven-damped harmonic oscillator as a function of initi al position x0.\nFigure 2 is the main result of our paper. It shows the contrasting re sonance behavior of\na linear and a nonlinear system. Usual physical intuition often does n ot work in nonlinear\nsystems. It is, therefore, hard to explain offhand why a simple pend ulum behaves so differ-\nently from a harmonic oscillator as far as the resonance character istics are concerned. We\nonly try to provide a rough explanation using our numerical results.\nThe hysteresis loop area/integraltext\nFdxdepends on both the amplitude < x0>of the response\ntrajectory x(t) as well as the phase difference φbetween F(t) andx(t). For intermediate as\nwell as large damping γ, the phase difference φdecreases with increasing period τofF(t)\nfor allγvalues. Also, the amplitude < x0>is smaller for larger γvalues for any period\nτof the drive. These are understandable. However, as we can see f rom Figs. 3 and 4, the\n6 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1\n 0  0.5  1  1.5  2−ω0\n−x0γ = 0.010.1640.1650.38\n1.14\n1.70.150.24\n1.0\n 0.75 0.8 0.85 0.9 0.95 1\n 0 0.5  1 1.5  2γ = 0.010.0980.099\n1.261.00.28\nFIG. 2: The figure shows the plot of the resonance frequency ω0versus the mean amplitude x0\nof the damped-driven pendulum with each point correspondin g to a particular value of damping\nγ(a few of them labelled) for a forcing amplitude F0= 0.2. The inset shows the same but with\na forcing amplitude F0= 0.1. Also shown, in both the main plot as well as in the inset, are the\nexact result of the frequency of oscillation ω1of an undamped simple pendulum (by broken line)\nand (by the dash-dotted horizontal line) the corresponding resonance frequency of oscillation ω0\nof an driven-damped harmonic oscillator.\namplitude < x0>shows a nonmonotonic behavior, it peaks at some intermediate τvalue.\nThis is a peculiar feature in the present case of periodically driven und erdamped sinusoidal\n(nonlinear) potential system.\nA. Intermediate range of γvalues\nThis range refers to 0 .165≤γ≤0.38, ofγforF0= 0.2 and 0.099≤γ <0.28 for\nF0= 0.1. Both present similar features though we consider Fig. 3 for the f ormer case only.\n7 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9\n 6.5  7  7.5  8 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6<x0>\nφ/π\nτγ = 0.24\nγ = 0.32\nγ = 0.24\nγ = 0.32 0.2 0.3 0.4 0.5\n 6.5  7 7.5  8φ/π\nτγ = 0.24\nγ = 0.32\nFIG. 3: The figure shows the variation of the amplitude of the e nsemble averaged hysteresis loop\nand the phase of the loop as a function of the driving period τ. Note that the two graphs indicated\nby the horizontal arrows (y-axis label is towards the right h and side) signify the phase of the loop\nfor the two γvalues as indicated. The phase variation as a function of dri ving period is replotted\nin the inset. The right vertical arrow and the right vertical line on< x0>indicates the resonant\nperiod,τ0= 6.66 forγ= 0.24 whereas the left vertical arrow and the left vertical line on< x0>\nis forγ= 0.32 withτ0= 6.542.\nFigure 3 shows both the amplitude < x0>and the phase lag φtogether, for the two typical\nvalues of γequal to 0.24 and 0.32, for easy comparison. However, the inset of the figure has\nonly the plot of phase lag as a function of the drive period τ. The peculiar features to be\nnoticed are that < x0>shows a peak in the lower τrange for both the γvalues. However,\nthe peak corresponding to γ= 0.32 occurs at a slightly lower value of τthanγ= 0.24.\nMoreover, in the same small range of τthe phase lag φis larger in the case of γ= 0.24 than\nγ= 0.32. On the other hand, in the larger range of τthe phase lag φis larger for the larger\nγ= 0.32. Thus in the larger range of τthe phase lag shows the usual behaviour. The small\n8 0.235 0.24 0.245 0.25 0.255 0.26 0.265 0.27 0.275\n 6.6 6.8  7 7.2 7.4 7.6 7.8  8 8.2 8.4 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46<x0>\nφ/π\nτγ = 0.80\nγ = 0.90\nγ = 0.80\nγ = 0.90\nFIG. 4: The figure shows the variation of the amplitude of the e nsemble averaged hysteresis loop\nand the phase of the loop as a function of the driving period τ. Note that the two graphs indicated\nby the horizontal arrows (y-axis label is towards the right h and side) signify the phase of the loop\nfor the two γvalues as indicated. The right vertical arrow and the right v ertical line on < x0>\nindicates the resonant period, τ0= 7.02 forγ= 0.90 whereas the left vertical arrow and the left\nvertical line on < x0>is forγ= 0.80 withτ0= 6.865.\nvalues of φin this range shows that when the system is driven at a slow rate the s ystem\nx(t) follows the drive F(t) closely. However, the same does not hold in the range of smaller\nτ(or larger frequency ω) where inertia seems to play an important role.\nIn the larger frequency ωrange it is difficult for the system x(t) to follow the drive F(t)\nand consequently the phase lag φbetween them is large, φclose toπ\n2. Moreover, x(t) seems\nto overshoot F(t). That is, while F(t) turns back from its maximum x(t) continues on its\nway for a longer while, before it retraces its path back due to its iner tia. And inertia is\nmore effective at lower damping. However, at still larger frequencie s the system again fails\nto respond to the field and the amplitude begins to decrease. With th is plausible qualitative\n9explanation of the peaking behaviour of < x0>as a function of τit becomes easier to see\nwhy the resonance frequency ω0increases with decreasing amplitude x0.\nAs mentioned earlier, ω0corresponds to the drive frequency ωat which the mean power\nloss is maximum and x0is the corresponding amplitude of x(t). And,F(t) being sinusoidal\nandx(t) also being roughly sinusoidal, given an amplitude < x0>ofx(t) the power loss\nor the hysteresis F(x) loop area becomes maximum when φ=π\n2. Ifφis nearlyπ\n2it is\nthe amplitude < x0>that determines the maximum of the hysteresis loss. Therefore in\nthis case maximum of power loss occurs at a frequency close to wher e< x0>becomes\nmaximum. From Fig. 3, one can, therefore, see that ω0is smaller for γ= 0.24 thanγ= 0.32\nwith correspondingly larger x0forγ= 0.24 than γ= 0.32. Thus, considering these two\ntypical values of γ, in this intermediate range of γ, the curve ω0(x0) should have the same\nqualitative nature as given in Fig. 2, that is, ω0decreases with increasing x0. However, the\nvariation is not so sharp as in the large γcase (γ >0.38 forF0= 0.2).\nB. Large γregime\nIn this large damping ( γ >0.38 forF0= 0.2 andγ >0.28 forF0= 0.1) regime naturally\namplitudes < x0>of oscillation are relatively small compared to those in the smaller γ\nregimes. It is also intuitively obvious that the response amplitude sho uld decrease with\nincreasing damping, for example, < x0>(γ= 0.8) is greater than < x0>(γ= 0.9). That\nthe resonance frequency should also decrease with damping can be qualitatively explained\nagain from the obtained variation of < x0>andφwithτas in Fig. 4.\nWe choose two large values of γequal to 0.8 and 0.9 for illustration. Here, the phase lags\nφare small ( φ <π\n2) andφis consistently smaller for γ= 0.8 thanγ= 0.9 at any value of\nτ. The effect of inertia gradually diminishes as γincreases. The consequence of this can be\nseen from the diminishing sharpness of < x0>peaks with increasing γ. Also to be noticed\nfrom Fig. 4 is that in case of γ= 0.8 the< x0>peak occurs at larger frequency (smaller τ)\nthanγ= 0.9. The variation of < x0>is much smaller than the variation of φin the plotted\nrelevant region of τ. The resonance peak occurs close to where the hysteresis loop ar ea is\nthe largest, that is where φis large (≈π\n2), thus from the figure, close to small values of τ\n(but larger than in case of intermediate range of γconsidered earlier). Both < x0>andφ\nconspire together to maximize the hysteresis loop area to determin eω0and corresponding\n10x0at resonance. Fig. 4, thus helps in getting a plausible qualitative idea t hat at larger\ndamping not only the resonance response amplitudes is smaller but th e system becomes\nslower to respond too. The rapid rise of ω0withx0with decreasing γis an outcome that\ndoes not seem unusual as γdecreases ω0tends towards the natural frequency of oscillation\nω1atγ= 0, Fig. 2.\nC. Small γregime\nFor small γvalues, for example 0 .06≤γ <0.165 forF0= 0.2, the explanation of the\nnature of the resonance curve ω0(x0) has an entirely different origin. The resonance curve\nis disjoint from the earlier two regimes (in which the curves were cont iguous). In this range\nofγ, the hysteresis loop area maximizes as a function of frequency in a r egion of frequency\nwhere there exists not one kind of particle trajectory but two for a givenγand amplitude\nF0ofF(t) [13–15]. Of course, if the amplitude F0is large (say, >0.25), the trajectories\nbecome chaotic[16]. We consider only amplitudes F0that give nonchaotic trajectories.\nThese two states of trajectory have the status of distinct dyna mical states having well\ndefined basins of attraction in the ( x(0),v(0)) space. One of the two states of trajectories\nhas a comparatively small amplitude (SA) and has a small phase lag ( φ <π\n3) with respect\nto the external drive F(t) whereas the other state has a large amplitude (LA) and a large\nphase lag. Figures 5 and 6 show the SA and LA states with their respe ctive mean hysteresis\nloops shown in the bottom panels of both the figures. To obtain the h ysteresis loop, as\nexplained earlier, the trajectory of the pendulum is averaged over the entire duration of the\ntrajectory as x(F(t)). The area bounded by the hysteresis loops corresponds to the energy\ndissipated to the surrounding medium. The basins of attraction of t hese two dynamical\nstates with parameter values γ= 0.12,τ= 8.0 andF0= 0.2 are shown earlier in [13]. We\nreproduce an updated plot for explanation purposes in Fig. 7. In Fig . 7 we also show the\nstroboscopic plots in the ( x(0)−v(0)) plane when the initial velocity of the particles at\nt= 0 isv(t= 0) = 0 and in the absence and presence of minute thermal fluctuat ions. These\nstroboscopic plots appear in Fig. 7 as colored regions. In the absen ce of fluctuations, these\nstroboscopic regions reduce to a single point, corresponding to an attractor, as shown by the\ncross-mark. The upper stroboscopic plot corresponds to the LA attractor while the lower\nstroboscopic plot corresponds to the SA attractor. The basins o f attraction shown in Fig. 7\n11 0.8 1.2 1.6 2 2.4\n 5000  5005  5010  5015  5020  5025  5030x\ntx(t)\nF(t)\n 0.8 1.2 1.6 2 2.4\n-0.2 -0.15 -0.1 -0.05  0  0.05  0.1  0.15  0.2−x\nF(t)\nFIG. 5: The top panel represents an SA state with a small phase lag with respect to the forcing\nF(t) with amplitude F0= 0.2. Notice that the amplitude F0in the top panel is magnified so\nthat comparison with x(t) can be visualised easily. The bottom panel represents the c orresponding\nhysteresis loop.\nhave the usual physical significance. For example, if the initial posit ion of the particle were\nto lie in the range of, say, 0 .2π≤x(0)≤1.1π, then on evolving the system after initial\ntransients have died out, the particle will home into the SA attracto r and will remain in its\ninitial well only. But on the other hand, suppose if the initial position o f the particle were\nto lie in the range of, say, −0.3π≤x(0)≤0, then the particle will oscillate with larger\namplitude and get fixed to the LA attractor. The (transient) time e volutions towards the\nfixed centres of attractions have been shown schematically in Fig. 7 and marked by arrows.\nWe note that the illustration given is when the initial velocity of the par ticle at any position\nwithin one period of the potential well is zero. This zero initial velocity is represented in\nFig. 7 as a horizontal line.\n12-1 0 1 2 3 4\n 5000  5005  5010  5015  5020x\ntx(t)\nF(t)\n-1 0 1 2 3 4\n-0.2 -0.15 -0.1 -0.05  0  0.05  0.1  0.15  0.2−x\nF(t)\nFIG. 6: The top panel represents an LA state with a larger phas e lag with respect to the forcing\nF(t) with amplitude F0= 0.2. Notice that the amplitude F0in the top panel is magnified so\nthat comparison with x(t) can be visualised easily. The bottom panel represents the c orresponding\nhysteresis loop.\nFor a given γthe domain of the basins of attraction depends on the value of ω, Fig. 8.\nFor small ω(for example, for γ= 0.13, ω <0.74) only SA states appear whereas for large ω\n(for the same γ, ω >0.85) the domain of the basins of attraction of SA states shrink to ze ro\nand only LA states appear and for the intermediate ωboth states coexist. We choose initial\nvelocityv(0) = 0 always and hence the occurrence of LA or SA states depend s on the initial\nposition ( x(0)) within a period of the sinusoidal potential. In the region of coex istence of\nthe two states we choose two hundred initial positions lying in the ran ge−π\n2< x(0)≤3π\n2\nat equal intervals in order to calculate the mean values that takes in to account the presence\nof the two dynamical states in right proportions. At this point it sho uld be noted that\nthe hysteresis loop area corresponding to the LA states are large r compared to the area\n13-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2\n-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6v(t=0)\nx(t=0)/πSA(1)\nLA(1)\nSA(0)\nLA(0)\nLA(-1)SA(0)SA(1)\nLA(1)\nLA(0)\nLA(-1)0 0 0 0τ\nτ\nττ\n2τ2τ3τ\nFIG. 7: The basins of attraction of the LA and SA attractor in t he absence (cross-mark) and\npresence of minute fluctuations is shown. The bracketed numb ers on LA and SA indicate the well\nnumber of the periodic potential where the trajectory settl es down in the LA attractor or the SA\nattractor. For this figure, γ= 0.12,τ= 8.0 andF0= 0.2 andV(x) =−sinx. The horizontal line\nin the middle is when the particles initially have velocity v(t= 0) = 0. Corresponding to this zero\ninitial velocity, the LA attractors are the upper colored re gions whereas the SA attractors are the\nlower colored regions. The green regions correspond to T= 0.0001 and the red regions correspond\ntoT= 0.001.\ncorresponding to the SA states and hence the need to consider all possible values of x(0) in\norder to get the mean values of the hysteresis loop area and the me an amplitude < x0>of\nthe trajectories.\nForγ >0.165, in the intermediate range of γforF0= 0.2, there is only one kind of\ntrajectory for all values of ωand initial positions x(0), the mean amplitude < x0>varies\ncontinuously with the varation of ω. It turns out that γ≈0.164 is the critical value of γ\n14 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9\n 0.06  0.08  0.1  0.12  0.14  0.16ω\nγREGION\nOF\nCOEXISTENCE\nSALA\nFIG. 8: Phase diagram in the ω-γspace denoting the regions of pure LA, pure SA, and coexisten ce\nregion. Here, F0= 0.2.\nabove which there is no distinction between the LA and SA states. Ho wever, for γ <0.164,\nas mentioned earlier, for large value of ωwe obtain only LA states but as ωis decreased\nsome LA states corresponding to some initial positions x(0) give way to SA states. The\nparticular value of ω, depending on the value of γ, at which SA states begin appearing also\ngives the largest hysteresis loop area and can be identified with ω0and the corresponding\n< x0>as thex0. The locus of these ( x0,ω0) is shown in Fig. 9 by the right-hand side\ndash-open circle thick boundary line. This boundary line is the resona nce line shown in Fig.\n2 forγ <0.165. At γ≈0.164,ω0abruptly drops to a smaller value due to the sudden\nappearance of the SA states of trajectories (in place of some ear lier LA trajectories) with\nmean hysteresis loop area of SA states being much smaller than LA st ates. The slopes of\n(x0,ω0) lines of Fig. 9 abruptly change indicating the beginning of the coexist ence region of\nthe two states.\n15 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1\n 0.5  1 1.5  2 2.5  3 3.5  4ω\n<x0>Coexistence RegionLA\nSAγ = 0.13\nγ = 0.14\nγ = 0.15\nγ = 0.16\nγ = 0.164\nγ = 0.17\nCoexistence Boundary\nFIG. 9: The figure shows the variation of the amplitudes of the LA state and the SA state as\na function of the driving frequency ω. The boundary line (dash-open circle line) separate the\ncoexistence region from the regions where either SA state is present or LA state is present only.\nThe amplitudes between the boundary line are calculated in t he coexistence region for γ= 0.13,\nγ= 0.14 andγ= 0.15.\nD. A superficial analogy\nFig. 9 is quite instructive. Apart from the thick boundary line on the la rge< x0>\nside of the abscissa, we have an another thick boundary line on the s mall< x0>side. For\nsmaller< x0>values we obtain only SA states of the pendulum. These two boundar y\nlines, one separating the LA states from the coexistence region an d the other separating\nthe coexistence region from the purely SA region, meet at a point wh ere the slope of the\n(< x0>,ω) line is zero for γ≈0.164. This point is somewhat analogous to the critical point\nin the (P−ρ) diagram of the liquid-gas system. The two thick boundary lines thus enclose\nthe region of coexistence of the two states and separate the LA s tates from the SA states of\n16trajectories. We show in Fig. 9, for γ= 0.13,γ= 0.14 andγ= 0.15, the mean amplitude\nof the trajectories when the particles are in the pure LA state, pu re SA state and when\nthe states coexists. For calculating the amplitude in the coexistenc e region, we ensemble\naverage over all initial conditions taken and obtain the ensemble ave raged hysteresis loop\nwhereby the ensemble averaged amplitude can be calculated.\nThough analogy of the SA and LA states of trajectories of the pen dulum with the liquid-\ngas phase is quite superficial it is suggestively tempting to draw furt her analogy between the\nsystem of two dynamical states, LA and SA, of trajectories with t he phases of the liquid-gas\nsystem. The frequency ωappears analogous to the pressure P, the friction coefficient γ\nto the temperature Tand the amplitude < x0>of the trajectory to the density ρof the\nliquid. And finally the hysteresis loop area A(ω,γ) seems analogous to the Gibbs free energy\ng(P,T) of the liquid.\nω←→P\nγ←→T\n< x0>←→ρ\nA(ω,γ)←→g(P,T)(2.2)\nHowever, these analogies are not exact but only superficial and ca nnot be stretched far.\nIII. CONCLUDING REMARKS\nIn this work we have investigated the dynamics of a driven damped sim ple pendulum\nor equivalently the motion of a particle in a sinusoidal potential in a med ium that offers\ndissipation. Note that we have investigated the motion of the simple p endulum at the\ntemperature T= 0, that is, without considering the effect of thermal fluctuations . Also, we\nhave kept the drive amplitude F0small so that the motion is nonchaotic.\nIn order to investigate the motion of a damped driven simple pendulum one needs to go\nmuch beyond the usual LCR circuit problem. In fact, no analytical s olution has so far been\nfound for this problem. The numerical solutions obtained, however , offer interesting insight.\nIn the underdamped regime, in certain range of γand amplitude and frequency of the drive\nF(t), two distinct solutions exist for the same periodic driving force F(t). This leads to an\nentirely new relationship between the amplitude of motion with the res onance frequency of\nthe simple pendulum having no relationship with the corresponding driv en damped simple\n17harmonicoscillator. That is, theresonance characteristics ofthe dampedharmonicoscillator\ncannotbeextrapolatedtoobtaintheresonancecharacteristics ofadampedsimplependulum.\nAcknowledgement\nWe thank the Computer Centre, North-Eastern Hill University, Sh illong, for providing\nthe high performance computing facility, SULEKOR.\n[1] A. Sommerfeld, Lectures on Theoretical Physics: Mechan ics, Levant Books (Indian Reprint),\nKolkata (2003).\n[2] C.Kittel, W. D. Knight, andM.A. Ruderman, Berkeley Phys icsCourse-Volume I,Mechanics\nMcGraw-Hill Book Company, Inc., New York (1962).\n[3] J. B. Marion, Classical Dynamics of Particles and System s, Second Edition, Academic Press\nInc., Orlando, 1970.\n[4] R. B. Kidd, and S. L. Fogg, The Physics Teacher 40, 81(2002 ).\n[5] R. R. Parwani, Eur. J. Phys. 25, 37(2004).\n[6] F. M. S. Lima, and P. Arun, Am. J. Phys. 74, 892(2006).\n[7] K. Johannessen, Eur. J. Phys. 32, 407(2011).\n[8] A. Bel´ endez, E. Arribas, A. M´ arquez, M.Orta˜ no, and S. Gallego, Eur. J. Phys. 32, 1303(2011).\n[9] A. H. Salas, https://www.researchgate.net/Publicati on/290946080 (January 2016).\n[10] Handbook of Mathematical Functions, Edited by M. Abram owitz, and I. A. Stegun, Dover\nPublications, Inc, New York, 1965.\n[11] D. Kleppner, and R. Kolenkow, An Introduction to Mechan ics, Tata McGraw Hill, Special\nIndian Edition, New Delhi, 2007.\n[12] K. Johannesen, Eur. J. Phys. 35, 035014(2014).\n[13] S. Saikia, A. M. Jayannavar, and M. C. Mahato, Phys. Rev. E 83, 061121 (2011).\n[14] W. L. Reenbohn and M. C. Mahato, Phys. Rev. E 88, 032143 (2 013).\n[15] D. Kharkongor, W. L. Reenbohn, and M. C. Mahato, Phys. Re v. E 94, 022148 (2016).\n[16] W. L. Reenbohn, and M. C. Mahato, Phys. Rev. E 91, 052151 ( 2015).\n18"    },    {        "title": "1908.07148v1.Damping_Mechanisms_of_the_Solar_Filament_Longitudinal_Oscillations_in_Weak_Magnetic_Field.pdf",        "content": "Draft version November 10, 2021\nTypeset using L ATEXpreprint style in AASTeX63\nDamping Mechanisms of the Solar Filament Longitudinal Oscillations in Weak\nMagnetic Field\nL. Y. Zhang,1, 2C. Fang,1, 2and P. F. Chen1, 2\n1School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China\n2Key Laboratory of Modern Astronomy & Astrophysics, Nanjing University, China\n(Received June 1, 2019; Revised August 18, 2019; Accepted November 10, 2021)\nABSTRACT\nLongitudinal oscillations of solar \flament have been investigated via numerical sim-\nulations continuously, but mainly in one dimension (1D), where the magnetic \feld\nline is treated as a rigid \rux tube. Whereas those one-dimensional simulations can\nroughly reproduce the observed oscillation periods, implying that gravity is the main\nrestoring force for \flament longitudinal oscillations, the decay time in one-dimensional\nsimulations is generally longer than in observations. In this paper, we perform a two-\ndimensional (2D) non-adiabatic magnetohydrodynamic simulation of \flament longitu-\ndinal oscillations, and compare it with the 2D adiabatic case and 1D adiabatic and\nnon-adiabatic cases. It is found that, whereas both non-adiabatic processes (radiation\nand heat conduction) can signi\fcantly reduce the decay time, wave leakage is another\nimportant mechanism to dissipate the kinetic energy of the oscillating \flament when\nthe magnetic \feld is weak so that gravity is comparable to Lorentz force. In this case,\nour simulations indicate that the pendulum model might lead to an error of \u0018100% in\ndetermining the curvature radius of the dipped magnetic \feld using the longitudinal\noscillation period when the gravity to Lorentz force ratio is close to unity.\nKeywords: magnetohydrodynamics (MHD), methods: numerical | Sun: \flaments,\nprominences | Sun: oscillations\n1.INTRODUCTION\nSolar \flaments, or called prominences when appearing above the solar limb, are cold dense plasma\nconcentrations embedded in the hot tenuous corona. In the early era when the spatial resolution\nof observations was not high, \flaments seem to be static, hence they were thought to be held in\nequilibrium with the Lorentz force balancing the gravity. Therefore, it was proposed that the local\nmagnetic \feld of \flaments should have a dipped con\fguration, either of a normal-polarity type\n(Kippenhahn & Schl uter 1957) or inverse-polarity type (Kuperus & Raadu 1974). After the dynamic\nfeatures were discovered, e.g., the counterstreamings (Zirker et al. 1998), it was suggested that a\nCorresponding author: P. F. Chen\nchenpf@nju.edu.cnarXiv:1908.07148v1  [astro-ph.SR]  20 Aug 20192 Zhang et al.\ndipped magnetic con\fguration is not necessary, and a prominence could be be a completely dynamic\nstructure, with existing plasma draining down and new plasma replenishing it (Karpen et al. 2001).\nSuch a possibility was validated in some observations (Wang 1999; Zou et al. 2016).\nEven in the case with magnetic dips, some \flaments are never static. They oscillate in response\nto any perturbations, which are ubiquitous in the solar atmosphere. From the physical point of\nview, \flament oscillations can be divided into transverse and longitudinal modes (Shen et al. 2014;\nZhang et al. 2017; Arregui et al. 2018), where the oscillation direction is perpendicular and parallel\nto the local magnetic \feld, respectively. Similar to other oscillating phenomena, \flament oscillations,\ncharacterized by oscillation period and decay time, can also be utilized to estimate some physical\nparameters, mainly the magnetic \feld. This is called prominence seismology (Arregui et al. 2018).\nHence, it is important to understand what determines the oscillation period and what determines\nthe decay time.\nFilament longitudinal oscillations were \frst reported by Jing et al. (2003), and in fact, the coun-\nterstreamings existing in many solar \flaments might be due to longitudinal oscillations of \flament\nthreads (Lin et al. 2003), although the counterstreamings in some \flaments are alternating unidi-\nrectional \rows (Zou et al. 2017) or are the combination of \flament thread longitudinal oscillations\nand unidirectional \rows (Chen et al. 2014). Several mechanisms were put forward to account for the\nrestoring force of the \flament longitudinal oscillations, such as the magnetic \feld-aligned component\nof gravity, gas pressure, and magnetic tension force (Jing et al. 2003; Vr\u0014 snak et al. 2007). With\none-dimensional (1D) hydrodynamic simulations, Luna & Karpen (2012) and Zhang et al. (2012)\nveri\fed that the \feld-aligned component of gravity can explain the observed periods, which are\naround 1 hour. In particular, the magnetic con\fguration in Zhang et al. (2012) was extracted from\nhigh-resolution observations, the agreement of the oscillation period between the simulations and\nthe observations is strongly indicative of that the \feld-aligned component of gravity is the restoring\nforce for the \flament longitudinal oscillations, i.e., the \flament longitudinal oscillations can be ac-\ncounted for with the pendulum model. Hence, the longitudinal oscillation period of solar \flaments\nwas considered to be able to diagnose the curvature of the dipped magnetic \feld, an important part\nof prominence seismology.\nIt is noted that the above-mentioned pendulum model is somewhat simpli\fed, where the magnetic\n\feld is assumed to be a rigid \rux tube. In real observations, the magnetic \feld might be deformed\nby the moving \flaments (Li & Zhang 2012). The deformed magnetic \feld results in change of the\n\feld-aligned gravity, which then alters the oscillation period. The deformation would make the\nprominence seismology not so straightforward. Luna et al. (2016) performed two-dimensional (2D)\nmagnetohydrodynamic (MHD) simulations of \flament longitudinal oscillations, where they found\nthat the magnetic \feld is slightly deformed by the oscillating \flament. It was suggested that the\ndeformation was so small that the oscillation period is similar to that estimated from the simpli\fed\npendulum model. However, their result might be due to that their magnetic \feld is not weak\nenough. In order to quantify whether the \flament gravity can signi\fcantly deform the shape of\nthe magnetic \feld, Zhou et al. (2018) de\fned a dimensionless parameter, plasma \u000e=\u001agL\nB2=2\u00160=\n11:5n\n1011cm\u00003L\n100 Mm(B\n10 G)\u00002, wherenis the number density of the prominence, Lis the length of the\nprominence thread, and Bis the magnetic \feld. If \u000eis much smaller than unity, the deformation of\nthe magnetic \feld would be small; If \u000eis comparable to or larger than unity, the deformation of the\nmagnetic \feld would be signi\fcant. Based on the parameters used in Luna et al. (2016), we foundDamping of filament longitudinal oscillations 3\nthat their\u000eis\u00180.2. Hence, it is expected that the oscillating \flament would not modify the magnetic\n\feld signi\fcantly, hence the pendulum model should be valid. In this paper, we plan to extend the \u000e\nparameter to be around unity, a regime where magnetic \feld would be deformed against the \flament\ngravity, and to investigate whether the pendulum model still works \fne.\nMore importantly, although Zhang et al. (2012) successfully reproduced the observed oscillation\nperiod of a prominence, the decay time of the oscillation in their simulation is 1.5 times larger than\nin the observations. They attributed the longer decay time in their simulations to the absence of\nother energy loss mechanisms, such as wave leakage, which can be modeled only in 2D or 3D MHD\nsimulations. Therefore, in this paper, we also intend to investigate whether the decay time of \flament\noscillations in 2D would be signi\fcantly reduced in contrast to the corresponding 1D case.\nThis paper is organized as follows: The numerical method is described in x2, the numerical results\nare presented inx3, which are followed by discussions in x4. A summary is given in x5.\n2.NUMERICAL METHOD\nWe solve the following 2D ideal MHD equations in the x-zplane to investigate the dynamics of\n\flament oscillations in this work:\n@\u001a\n@t+r\u0001(\u001av) = 0; (1)\n@(\u001av)\n@t+r\u0001\u0012\n\u001avv\u0000BB\n\u00160\u0013\n+r\u0012\np+B2\n2\u00160\u0013\n=\u0000\u001ag; (2)\n@etot\n@t+r\u0001\u0012\netotv+\u0012\np+B2\n2\u00160\u0013\nv\u0000BB\n\u00160\u0001v\u0013\n=\u0000\u001av\u0001g+r\u0001q\u0000nenH\u0003(T) +H(z);(3)\n@B\n@t+r\u0001(vB\u0000Bv) = 0; (4)\nwhere\u001a= 1:4mpnHis the mass density, v= (vx,vz) is the velocity, B= (Bx,Bz) is the magnetic\n\feld,p= 2:3nHkBTis the thermal pressure ( mpis the proton mass and kBis the Boltzmann constant),\nandetot=\u001av2=2 +p=(\r\u00001) +B2=(2\u00160) is the total energy density, where \r= 5=3 is the ratio of the\nspeci\fc heats. The gravity is set to be uniform, i.e., g=g\f^ez, whereg\fis 274 m s\u00002. In the energy\nequation, i.e., Equation (3), we consider thermal conduction, the optically thin radiative cooling, and\nthe background heating. The \feld-aligned heat conduction is described as follows,\nq=\u0014k(b\u0001rT)b; (5)\nwhere qis the heat \rux vector, b=B=jBjis the unit vector along the magnetic \feld, \u0014k=\n10\u00006T5=2erg cm\u00001s\u00001K\u00001is the Spitzer heat conductivity. \u0003( T) is the radiative loss coe\u000ecient for the\noptically thin emission, which is obtained by interpolating a cooling table based on the SPEX package\n(see Schure et al. 2009, for details). The radiative loss coe\u000ecient is set to 0 when T < 8\u0002103K.\nH(z) =H0e\u0000z=H mis the steady heating term, where the amplitude H0= 1:5\u000210\u00004erg cm\u00003s\u00001\nandHm= 40 Mm is the scale height. This term is introduced in order to maintain the background\ncorona. The MHD equations are numerically solved with the MPI Adaptive Mesh Re\fnement Ver-\nsatile Advection Code (MPI-AMRVAC; Keppens et al. 2012; Porth et al. 2014; Xia et al. 2018). In\nparticular, the heat conduction term in Equation (3) is solved with an implicit scheme in order to4 Zhang et al.\navoid too small time steps. The computational domain is a rectangular region in the range of the\nCartesian coordinates x2[\u0000100;100] Mm and z2[0;100] Mm. The resolution of the \fnest mesh\nlayer is 156 km.\nThe simulated normal polarity \flament is placed in a dipped magnetic \feld. Following Luna\net al. (2016), we adopt a quadrupolar magnetic con\fguration, which is described by Bx=\nB0(cosk1x e\u0000k1(z\u0000z0)\u0000cosk2x e\u0000k2(z\u0000z0)) andBz=B0(\u0000sink1x e\u0000k1(z\u0000z0)+ sink2x e\u0000k2(z\u0000z0)).\nHere, we take k1=k2=3 =\u0019=200 Mm\u00001,z0= 4 Mm, and B0= 10 G. This con\fguration has a\nbald patch structure near the magnetic neutral line x= 0. The bottom boundary is treated to be\na line-tied one, with all quantities being \fxed, the top boundary is a free one, and the re\recting\nconditions are set on the two side boundaries. Similar to Zhou et al. (2018), the initial conditions\nare realized via the following two steps:\n(1) To set up a quiet Sun atmosphere: The temperature distribution from the photosphere to the\nbottom of the corona is prescribed as follows:\nT(z) =8\n<\n:Tph+ (Tco\u0000Tph)(1 + tanh( z\u0000htr\u0000c1)=wtr)=2z\u0014htr;\n(7Fc(z\u0000htr)=(2\u0014) +Ttr7=2)2=7z >htr;(6)\nwhere the height of our transition region, htr, is set to a value of 1 :5 Mm, and its thickness, wtr, is\ntaken to be 0 :2 Mm. The temperatures of the photosphere, transition region, and the corona are\nTph= 9\u0002103K,Ttr= 1:6\u0002105K, andTco= 1:5\u0002106K. With a bottom number density of 1 :2\u0002\n1014cm\u00003, the density distribution is calculated based on the hydrostatic equilibrium, where\ngravity is balanced by the gas pressure gradient. Such analytical distributions, together with\nthe quadrupolar magnetic \feld described above, evolve gradually, and reach a real equilibrium\nstate after\u0018110 minutes in the MHD simulations, when the vertical temperature distribution\nis shown in Figure 1.\n(2) We place a bulk of dense plasma around the magnetic dips to represent a \flament thread,\nwhose density pro\fle takes the following form:\n\u001a=\u001aco+\u000e\u001ae\u0000(x\u0000x0)4\nw4x\u0000(z\u0000z0)4\nw4z; (7)\nwhere (x0;z0) = (0;19:8) Mm is the position of the mass center, wx= 4 Mm,wz= 3 Mm;\n\u001acois the plasma density in the ambient corona obtained from Step (1) and \u000e\u001a= 99\u001aco. The\ncorresponding temperature, T, is also modi\fed so that \u001aTremains unchanged. We then perform\nMHD simulations again.\nIt is seen that the whole system evolves gradually. While plasmas is sucked into the \flament thread\ndue to thermal instability, the \flament oscillates with an initial amplitude of 16 km s\u00001. After\u0018290\nminutes, the amplitude of the \flament transverse oscillation is still around 0.2 km s\u00001. In order\nto speed up the decay, we set the velocity to be 0 everywhere in the simulation domain when the\n\flament is at its equilibrium position. As a test, we continue the simulation and \fnd the residual\nmaximum velocity is only 0.06 km s\u00001. Now, the system approaches a new equilibrium state, where\nthe \flament becomes \u00185.5 Mm long in the x-direction and\u001810 Mm thick in the z-direction. The\ncenter of mass drops to zc= 17:7 Mm. The distributions of the magnetic \feld and plasma density areDamping of filament longitudinal oscillations 5\n020406080100\nz(Mm)102\n101\n100T(MK)\nFigure 1. The vertical distribution of the temperature before the \flament is introduced.\nFigure 2. The distributions of the magnetic \feld (black lines) and the plasma density (color scale) used as\nthe initial conditions for this paper. The blue line marks the single magnetic \feld line threading the centroid\nof the \flament.\ndisplayed in Figure 2 as solid lines and color scales, respectively. Such a state is taken to be the real\ninitial conditions for the simulation work to be presented in this paper. Note that there is a second6 Zhang et al.\nplasma condensation near the original point as shown in Figure 2, which is formed due to thermal\ninstability, and is not relevant to the study in this paper.\nAccording to Zhang et al. (2013), the oscillation properties do not depend on the type of perturba-\ntions, no matter it is due to impulsive momentum or localized heating. In this work we choose the\nformer one. Similar to Luna et al. (2016), the \flament is perturbed with a magnetic \feld-aligned\nvelocity, which is expressed as\nv0=v0be\u0000(x\u0000x0)4\n~wx\u0000(z\u0000z0)4\n~wz; (8)\nwherev0= 20 km s\u00001andbis the same as in Equation (5), while ~ wx= 15 Mm and ~ wz= 10 Mm.\nThe ensuing evolution is calculated with the MPI-ARMVAC code. It is noted here that we also\nsimulate adiabatic cases for comparison, where the radiation and heat conduction in Equation (3)\nare removed.\n3.RESULTS\n3.1. 2D non-adiabatic case\nFigure 3. Evolution of the magnetic \feld (solid lines) and the plasma density (color scale). For comparison,\nthe initial magnetic \feld is overplotted as the dashed lines.\nIn this subsection, we present the numerical results in the 2D non-adiabatic case. The evolution of\nthe \flament oscillation is depicted in Figure 3, where the color represents the density and the solid\nlines correspond to the evolving magnetic \feld. To illustrate how the magnetic \feld is deformed, the\ninitial magnetic \feld is overplotted as the dotted lines. It is seen that, after the velocity perturbation\nis imposed on the \flament, the \flament begins to oscillate, but the amplitude decays rapidly. It is\nalso noticed that the upper and lower parts of the \flament do not oscillate in phase since they haveDamping of filament longitudinal oscillations 7\ndi\u000berent oscillation periods, which are determined by di\u000berent curvature radii of the local magnetic\n\feld (see also Luna et al. 2016). As a result, the \flament changes its shape continuously.\n020406080100120140160\nTime(min)-10 -5  0  5 10 15s(Mm)\n0.250.500.751.00\nT(MK)\n020406080100120140160\nTime(min)20\n10\n01020v||(km/s)\nsimulation\nbest-fit\nFigure 4. Left: Time-distance diagram of of the temperature distribution along the magnetic \feld line\nthreading the \flament centroid. Right: Evolution of the \feld-aligned velocity of the \flament centroid.\nIn order to analyze the oscillation behavior more quantitatively, we extract the magnetic \feld line\nthat passes through the initial centroid of the \flament, which is indicated by the blue line in Figure\n2. This \feld line is rooted at the position with x=\u000676:8 Mm and z= 0. The time-distance\ndiagram of the temperature distribution along this line is plotted in the left panel of Figure 4, where\nthe origin of the distance is set at x= 0. The \flament corresponds to the blue area, whose center\nis indicated by the yellow dashed line, and whose boundaries are marked by the two dot-dashed\nlines. The \flament velocity along the \feld line ( vk) is determined by the velocity of the \flament\ncentroid, and its evolution is displayed in the right panel of Figure 4 as the blue circles. It has the\ntypical damped sinusoidal pro\fle. Therefore, we \ft vkwith the following damped sine function via\nthe least-square method,\nvk=v0e\u0000t=\u001csin\u0000\n2\u0019t=P +'\u0001\n; (9)\nwherev0is the initial amplitude, Pis the oscillation period, \u001cis the decay time, and 'is the phase\nangle. It turns out that v0= 20 km s\u00001,P= 49 minutes, \u001c= 38 minutes, and '=\u0019=2. The \ftted\ncurve is overplotted in the right panel of Figure 4 as the black line. It is seen that the \ftting is\nreasonable for the \frst 1.5 periods, and the deviation becomes remarkable after that. It is evident\nthat the oscillation period should be shorter and shorter, rather being a constant, in the late stage\nof the evolution.\n3.2. 1D non-adiabatic case\nIn order to \fnd out what new e\u000bects are brought by the two dimensions, we perform a 1D non-\nadiabatic hydrodynamic simulation for comparison. In order to make the comparison meaningful,\nthe magnetic con\fguration in the 1D case is taken from the initial magnetic \feld line across the\n\flament centroid, i.e., the blue line in Figure 2. Similar to the initial condition-producing procedure\nfor the 2D simulation, a segment of \flament is inset into the magnetic dip, with the length and\ndensity identical to the 2D case. After the same velocity perturbation is imposed on the \flament,\nthe \flament thread starts to oscillate. The time-distance diagram of the temperature distribution is\ndisplayed in the left panel of Figure 5. The evolution of the velocity at the \flament center is shown\nin the right panel as the the blue circles. Similar to the 2D case, the velocity evolution is also \ftted8 Zhang et al.\n020406080100120140160\nTime(min)-10 -5  0  5 10 15s(Mm)\n0.250.500.751.00\nT(MK)\n020406080100120140160\nTime(min)20\n10\n01020v||(km/s)\nsimulation\nbest-fit\nFigure 5. Left: Time-distance diagram of the temperature distribution in the 1D non-adiabatic case. Right:\nEvolution of the velocity of the \flament centroid in the 1D non-adiabatic case.\nwith a damped sine function the same as Equation (9) with the least-square method. It is revealed\nthat the oscillation period is 30 minutes and the decay time is 76 minutes. In this case, the variation\nof the period is more obvious since the deviation from the \ftted curve with a \fxed period becomes\nlarger and larger.\n3.3. 1D and 2D adiabatic cases\nFor comparison, we also numerically simulated the adiabatic cases in both 1D and 2D. Since the\nevolutions are similar, the details are not described here. However, the \ftting results are presented as\nfollows: In the 1D adiabatic case, the oscillation period is P=37 minutes, and it is almost decayless; In\nthe 2D adiabatic case, the oscillation period is P=44 minutes, and the decay time is \u001c=211 minutes.\n4.DISCUSSIONS\nBased on the analysis of 196 \flament longitudinal oscillation events observed in 2014, Luna et al.\n(2018) found that the ratio of the damping time to the period \u001c=P can be as small as 0.6. Several\nfactors may contribute to the damping. The primary one is the non-adiabatic processes including\nradiation and heat conduction, as investigated via simulations by Zhang et al. (2012). However,\ntheir results indicate that radiation and heat conduction are not enough to account for the observed\ndamping, and other factors should play a role as well. One possible candidate is the mass change.\nAccording to Luna & Karpen (2012), the mass accretion due to continual thermal condensation would\nspeed up the damping. This is understandable since increasing mass leads to deceasing velocity\nin order to conserve the total momentum. Interestingly, according to Zhang et al. (2013), mass\ndrainage would also lead to stronger damping. They showed that when the amplitude of the \flament\nlongitudinal oscillation is too large, part of the \flament material drains down across the apex of the\nmagnetic dip. The mass drainage takes away part of the mechanical energy of the \flament, leading\nto stronger damping as well. The second additional candidate is the thread-thread interaction. As\ndemonstrated by Zhou et al. (2018), when there are two dips (hence two \flament threads) along one\nmagnetic \feld line, the two threads exchange kinetic energy, leading to weaker decay for one thread\nand stronger decay for the other. The third additional candidate is the deformation of the magnetic\n\feld lines, which would generate kink waves propagating outward. In this case, the oscillation energy\nis taken away via wave leakage. The signi\fcant feature of the \flament longitudinal oscillations in\nour 2D non-adiabatic case is the rapid decay, where the decay time \u001cis only 0.7 times the oscillation\nperiodP, i.e.,\u001c= 0:7P, in contrast to \u001c= 2:5Pin the 1D non-adiabatic case (Zhang et al. 2012).Damping of filament longitudinal oscillations 9\nSince there are no signi\fcant mass change and thread-thread interaction here, we discuss how the\nnon-adiabatic processes and wave leakage a\u000bect the damping.\n4.1. Understanding the damping due to non-adiabatic processes\n40\n20\n02040\ns(Mm)0.030.040.050.060.07p(ergcm3)\nTime= 0.0 (min)\n40\n20\n02040\ns(Mm)0.030.040.050.060.07p(ergcm3)\nTime= 5.7 (min)\n40\n20\n02040\ns(Mm)0.030.040.050.060.07p(ergcm3)\nTime=22.9 (min)\n40\n20\n02040\ns(Mm)0.030.040.050.060.07p(ergcm3)\nTime=34.3 (min)\nFigure 6. Snapshots of the gas pressure distribution along the magnetic \feld line crossing the \flament\ncentroid, where the thatched areas corresponds to the \flament, and the magenta arrows indicate the velocity\nof the \flament.\nFrom the energy point of view, it is straightforward to understand how the \flament longitudinal\noscillations decay faster because of radiation and heat conduction (see Zhang et al. 2013, for details).\nIt is simply because radiation and heat conduction take away the energy of the oscillating \flament.\nIn this subsection, we try to understand the damping process from the force point of view.\nFor longitudinal oscillations, Lorentz force does not play a role directly, and gravity is always a\nrestoring force. Therefore, from the dynamics point of view, the only possible force for the decay is\nthe gas pressure di\u000berence between the left and right boundaries of the \flament. Figure 6 displays\nfour snapshots of the gas pressure distribution along the blue line marked in Figure 2. It is seen that,\ndi\u000berent from the \feld-aligned gravity, the pressure gradient force is mostly opposite to the \flament\nvelocity, rather than to the \flament displacement. To see this more clearly, Figure 7 displays the\nevolution of the \flament displacement in panel (a), the evolution of the \flament velocity in panel10 Zhang et al.\n(b), and the evolution of the pressure gradient force per square centimeter ( Fp) in panel (c). It is\nrevealed that the pressure gradient force is almost antiphase with the \flament velocity, with a small\nphase di\u000berence. Considering the slight phase shift, we decompose the pressure gradient force Fpinto\ntwo parts. The \frst part, Fp1, is obtained by shifting the Fppro\fle so that the new pro\fle is exactly\nantiphase with vk. This is done by taking the maximum running correlation coe\u000ecient between Fp1\nandvk. Therefore, Fp1can be considered as the viscous force. Fp2is simply the residual. Panel (d)\nof Figure 7 displays the evolutions of Fp1(black solid line) and Fp2(red dashed line). It is seen that\nFp1is dominant, i.e., for the non-adiabatic case, the pressure gradient force acts mainly as a damping\nforce, and only a minor part of it, Fp2, contributes slightly to the restoring force since it is antiphase\nwith the \flament displacement ( s).\n0 50 100 150\nTime(min) -3  0  3  6  9 s(Mm)(a)\n0 50 100 150\nTime(min)-10  0 10 20v||(km/s)(b)\n0 50 100 150\nTime(min)-0.0100.01Fp(erg/cm)(c)\n0 50 100 150\nTime(min)-0.0100.01(d)\nFp1\nFp2\nFigure 7. Evolution of several quantities in the 2D non-adiabatic case. Panel (a) is for the \flament\ndisplacement along the magnetic \feld line, panel (b) is for the \flament velocity along the magnetic \feld\nline, panel (c) is for Fp, the pressure gradient force across the left and right boundaries of the \flament per\nsquare centimeter, and panel (d) is for Fp1andFp2, the two decomposed parts of Fp.\nFor comparison, Figure 8 displays the evolutions of the \flament displacement (panel a), \feld-aligned\nvelocity (panel b), gas pressure gradient force Fp(panel c), and the decomposed two forces (panel d)\nin the 2D adiabatic case. It is seen that the pressure gradient force Fpis roughly proportional to the\n\flament displacement swith a negative coe\u000ecient, rather than proportional to the \flament velocity\nas in the non-adiabatic case. That is to say, the pressure gradient force Fpacts mainly as a restoring\nforce in the 2D adiabatic case, though it is weaker than the gravity. Similar to that in the previous\nparagraph, we decompose Fpinto two components, Fp1(black solid line) and Fp2(red dashed line),\nwhereFp1is obtained by shifting Tpto be exactly antiphase with s, andFp2is the residual, i.e.,Damping of filament longitudinal oscillations 11\n0 50 100 150\nTime(min) -5  0  5s(Mm)(a)\n0 50 100 150\nTime(min)-20-10  0 10 20v||(km/s)(b)\n0 50 100 150\nTime(min)-0.0200.02Fp(erg/cm)(c)\n0 50 100 150\nTime(min)-0.0200.02(d)\nFp1\nFp2\nFigure 8. The same as Figure 7, but for the 2D adiabatic case.\nFp2=Fp\u0000Fp1. It is seen that in the non-adiabatic case, the dominant component of the pressure\ngradient,Fp1, contributes to the restoring force (see Luna et al. 2016, as well), and only the minor\ncomponent, Fp2, contributes to the damping. This explains why the damping time is much longer in\nthe 2D adiabatic case.\nFrom the above analysis, it is revealed that in the adiabatic case, the gas pressure gradient force\nmainly acts as a restoring force, i.e., when the \flament is to the right away from the equilibrium\nposition, the coronal plasma on the right part is compressed with a higher gas pressure. However,\nwhen radiation and heat conduction are included, it is always the upstream side of the \flament\nthat has higher gas pressure, making the pressure gradient force almost a viscous force. In order\nto understand the reason for the di\u000berence, we examine the evolution of the thermal parameters in\nthe two coronal segments on the two sides of the \flament along the magnetic \rux tube. It is found\nthat, as the \flament moves to the right, the right-hand coronal segment is compressed (and the left-\nhand coronal segment is rare\fed), leading to higher gas pressure and density on the right (and lower\npressure and density on the left). As a result of the higher density, the radiative cooling is enhanced\non the right side (and weakened on the left). As time goes on, the temperature decreases on the\nright side (and increases on the left side), leading to lower gas pressure on the right side soon after\nthe \flament reaches its rightmost position (and higher gas pressure on the left side). Consequently,\nthe pressure gradient force mainly acts as a viscous force. Note that radiation and heat conduction\nare important since the timescales of radiation and heat conduction in the corona, tens of minutes,\nare comparable to the period of the \flament longitudinal oscillations. It is expected that they are\nmuch less important for \flament transverse oscillations since the period of the latter is several times\nshorter.12 Zhang et al.\n4.2. Damping due to wave leakage\nComparing the 2D non-adiabatic and 1D non-adiabatic cases, it is revealed that the decay time in\nthe 2D case is only half that in the 1D case. It is seen that the 2D \flament longitudinal oscillations\ndecay faster than the 1D oscillations. Such a di\u000berence is also seen between the 2D adiabatic and\n1D adiabatic cases, where the latter is decayless and the former has a decay time of 76 minutes. We\nconjecture that the additional energy loss mechanism of the 2D e\u000bect is wave leakage. The reason\nis that the plasma \u000ein our paper, i.e., the ratio of gravity to Lorentz force, is close to unity, so the\nlongitudinal oscillations of the \flament would deform the magnetic \feld, and the deforming magnetic\n\feld would generate transverse waves, which then propagate away from the \flament.\n10\n 5\n 0 5 10\nx(Mm)17181920z(Mm)\nFigure 9. The trajectory of the \flament centroid ( colored circles ) and the magnetic \feld lines across the\n\flament centroid, where the red line corresponds to the initial state, the dash-dotted line corresponds to\nthe moment when the \flament reaches its rightmost position, and the blue line corresponds to t= 171 :75\nminutes. The dashed line is a \ftted circular arc with a radius of 70 Mm. Note that the horizontal and\nvertical axes are not to scale.\nIn order to con\frm it, we plot the evolution of the magnetic \feld line threading the \flament centroid\nat the initial ( red line ) and the \fnal ( blue line ) times in Figure 9. In this \fgure, the trajectory of\nthe \flament centroid is overplotted as small colored circles, where the color coding from red to blue\nindicates the time elapse. We can see that, as the \flament moves, the magnetic \feld line is deformed\nby at least\u00180.5 Mm.\nAs expected, such deformation would excite \flament transverse oscillations. Figure 10 displays the\ntemporal evolution of the vertical velocity of the \flament centroid in the 2D non-adiabatic case. It\nis seen that short-period oscillations are superposed on a quickly damped oscillations with a larger\nperiod. The short-period oscillation has a period of 6.38 minutes, which is in the typical range of\n\flament transverse oscillations (Tripathi et al. 2009; Hillier et al. 2013), whereas the quickly-dampedDamping of filament longitudinal oscillations 13\n020406080100120140160\nTime(min)0.2\n0.00.2v(km/s)\nFigure 10. Evolution of the vertical velocity of the \flament, showing a decayed transverse oscillation.\noscillation has a period of 25 minutes, which is about half of the \flament longitudinal oscillation. The\nhalf period is expected when a \flament is oscillating along a dipped magnetic \feld. Note that the\namplitude of transverse oscillation induced by the longitudinal oscillation is only 0.2 km s\u00001, which\nis in the lower range detected by the Hinode satellite (Hillier et al. 2013), and much smaller than the\nvelocity amplitude of \flament transverse oscillations indcued by coronal waves, e.g., 8.8 km s\u00001in\nLiu et al. (2012) and 65{89 km s\u00001in Shen et al. (2014).\n0 5 10 15 20\nTime(min) 20 40 60 80z(Mm)\n1.0\n0.5\n0.00.51.0\nv(km/s)\nFigure 11. Time-distance diagram of the vertical velocity ( v?) along the z-axis at x= 0.14 Zhang et al.\nAs the magnetic \feld line across the \flament oscillates vertically, it is anticipated to see fast-mode\nwaves being excited. To demonstrate the fast-mode wave generation, we plot the time evolution of\nthevzdistribution along the z-axis in Figure 11, where blue color means upward velocity and red\ncolor means downward velocity. It is seen that quasi-periodic waves propagate both in the upward\nand the downward directions. The period of the wave train is calculated to be \u00186.38 minutes based\non wavelet analysis, which is almost identical to that of the \flament transverse oscillation. The\npropagation speed of the fast-mode waves as indicated by the slope of the oblique ridges is measured\nto be 810\u000660 km s\u00001, the typical fast-mode wave speed in the background corona.\nIn order to quantify how much energy is taken away by the fast-mode MHD waves, we select a \fxed\nrectangular region around the oscillating \flament as indicated by the pink box in Figure 3. Note that\nthe box is large enough so that the \flament is always inside the box. Within the box, we calculate\nthe kinetic energy convected into the box ( Eek=\u0000H\nLekv\u0001ndl, whereekis the kinetic energy density,\nnis the normal vector, and lis the length along the box boundary), the work done by Lorentz force\n(EB=RR\n\u0006v\u0001(j\u0002B)d\u001b, where the integral is done over the whole area of the box), and the work\ndone by gas pressure ( Ep=\u0000RR\n\u0006v\u0001rpd\u001b, where the integral is done over the whole area of the box).\nWe also calculate the work done by gravity inside the box ( Eg=RR\n\u0006v\u0001Fgd\u001b, where the integral is\ndone over the whole area of the box). The evolution of the four parts of the accumulating energy\nis displayed in Figure 12, where the pink area corresponds to the in\rux of the kinetic energy, the\nyellow area corresponds to the work done by Lorentz force, the blue area represents the work done\nby gas pressure, the green area represents the work done by gravity, and the red line corresponds to\nthe sum of all the terms. It is seen that near the end of the simulation when the oscillation is almost\ncompletely attenuated, the work done by Lorentz force and gas pressure is dominantly negative,\nmeaning that most of the initial kinetic energy of the \flament longitudinal oscillation is lost by the\nwork done by Lorentz force and gas pressure. Since fast-mode magnetoacoustic waves are due to the\nsynchronous action of the Lorentz force and the gas pressure, it further implies that the major part\nof the initial kinetic energy of the \flament is taken away by wave leakage.\n020406080100120140160\nTime(min)  -2  -1   0Timeintegral(1016erg/cm)\nTotal\nInfowofkineticenergy\nWorkbyLorentzforce\nWorkbypressuregradientforce\nWorkbygravity\nFigure 12. Time-integral of the energy loss from the \fxed region indicated by the pink box in Figure 3.\nWave leakage was proposed to be able to lose the energy of an oscillating coronal loop to the\nsurroundings (Cally 1986). Further simulations (Brady & Arber 2005; Selwa et al. 2006) and mode\nanalysis (D\u0013 \u0010az et al. 2006; Verwichte et al. 2006) indicated that in a 2D slab model, wave energy\ncannot be trapped, and wave leakage can e\u000eciently attenuate the coronal loop oscillations. TerradasDamping of filament longitudinal oscillations 15\net al. (2006), however, pointed out that with a \rux tube in a 3D magnetic con\fguration, wave leak-\nage is not e\u000bective anymore, and resonant absorption becomes the dominant damping mechanism.\nThe di\u000berence between the 2D slab model and the 3D \rux tube model can be understood in an\nintuitional way: In the 2D slab model, whenever the coronal loop oscillates vertically, all the back-\nground magnetic \feld lines are pushed or pulled to oscillate. Hence, it is nor surprising that wave\nleakage is an e\u000ecient mechanism for the energy loss. On the other hand, in the 3D model where the\n\rux tube is embedded in a magnetic \feld roughly aligned with the \rux tube, when the \rux tube\noscillates, the ambient \feld lines may be pushed aside because of the third degree of freedom, and\nare not necessary to oscillate following the oscillating \rux tube. That is to say, the oscillating slab\nin the 2D model is similar to a piston, which is not the case in 3D if the background magnetic \feld\nis roughly parallel to the \rux tube. It is pointed out here that, however, if the background magnetic\n\feld is quasi-perpendicular to the \rux tube, the oscillating \rux tube would serve as a piston even\nin the 3D case. In this situation, all the ambient magnetic \feld lines would be pushed or pulled to\noscillate, leading to signi\fcant wave leakage. The magnetic con\fguration of solar \flaments is exactly\nlike this, i.e., a sheared (and maybe twisted) core \feld overlain by unsheared envelope magnetic \feld\n(Chen 2011; Parenti 2014). As a result, when the \flament, which is held by the core \feld, oscillates\nvertically, it would generate kink motions of the envelope \feld, leading to wave leakage.\n4.3. Validity of the pendulum model\nFor \flament longitudinal oscillations, it is often believed that the restoring force is the \feld-aligned\ncomponent of gravity. Therefore, pendulum model was used to relate the oscillation period to the\ncurvature radius of the dipped magnetic \feld, as con\frmed by the 1D hydrodynamic simulations\n(Luna & Karpen 2012; Zhang et al. 2012). In order to verify the pendulum model in 2D where\nmagnetic \feld may react to the \feld-aligned motion of the \flament, Luna et al. (2016) performed 2D\nMHD simulations, where the magnetic \feld around the \flament is as weak as 10 G. It was found that\nthe pendulum model still holds and there is no strong coupling between longitudinal and transverse\noscillations of the \flament. It is noted that, according to Zhou et al. (2018), whether the magnetic\n\feld may signi\fcantly react or not depends on the dimensionless parameter \u000e, i.e., the gravity to\nLorentz force ratio, rather than the plasma \f, i.e., the gas to magnetic pressure ratio. We checked\nthe plasma \u000ein Luna et al. (2016), and found that it is about 0.2, i.e., its gravity is several times\nsmaller than the Lorentz force. Therefore, it is natural to see that the magnetic \feld deformation\nwas small and the \flament longitudinal oscillation did not excite signi\fcant transverse oscillation in\ntheir simulations.\nIn this paper, we extended the plasma \u000eregime to near unity. We found that fast-mode wave\ntrains are excited and propagate upward and downward away from the oscillating \flament, taking\naway a signi\fcant portion of the oscillation energy. With radiation and heat conduction considered,\nthe decay time of the longitudinal oscillation is 76 minutes in the 1D simulation, and becomes 38\nminutes in the 2D simulations. The deformation of the magnetic \feld is not trivial, which is up to 0.5\nMm as evidenced in Figure 9. Because of the deformation of the magnetic \feld, even the oscillation\nperiod is di\u000berent between the 2D and 1D cases. As seen from Figure 9, as the \flament moves to\nthe right, the corresponding portion of the magnetic \feld is pressed to be \ratter as indicated by\nthe dash-dotted line in Figure 9, which results in a smaller component of the gravity along the \feld\nline. As a result, the corresponding period in the 2D case becomes longer. This explains why the\noscillation period in the 1D non-adiabatic case is 30 minutes, whereas the period becomes 49 minutes16 Zhang et al.\nin the 2D non-adiabatic case. The relative error is more than 50%. Since the curvature radius of\nthe magnetic dip is proportional to the square of the oscillation period according to the pendulum\nmodel, the relative error of the curvature radius would be 100%. Because the gravity is comparable\nto the Lorentz force, the \feld line is strongly curved near the \flament and becomes almost straight\nfurther away as evidenced by the red line in Figure 9. Such an extremely non-uniform curvature\ndistribution accounts for the decreasing period as the oscillation attenuates seen in the right panels\nof Figures 4 and 5.\nIt is interesting to notice that, although the magnetic \feld line across the \flament centroid is not\ncircular, the actual trajectory of the \flament is almost circular. As revealed by the colored circles in\nFigure 9, the trajectory \fts into a circular arc ( dashed line ) very well.\n5.SUMMARY\nIn this paper, we investigated the \flament longitudinal oscillations in the weak magnetic \feld\nregime. The main results are summarized as follows:\n(1) Our simulations veri\fed the suggestion proposed by Zhou et al. (2018), i.e., whether the dense\nplasma of a \flament may modify the magnetic \feld is not determined by the plasma \f, it is determined\nby the gravity to Lorentz force ratio \u000e=\u001agL\nB2=2\u00160. In our case where \fis small but \u000eis close to unity,\nthe magnetic \feld is substantially changed by the oscillating \flament. That is, low plasma \fdoes\nnot guarantee that the magnetic \feld lines are not changed by the \flament gravity. In the high \u000e\ncase as in this paper, tha application of the pendulum model would lead to an error of \u0018100% in\nestimating the curvature radius of the dipped magnetic \feld.\n(2) In the framework of 2D simulations, the inclusion of heat conduction and radiation signi\fcantly\nreduce the decay time from 211 minutes in the adiabatic case to 34 minutes in the non-adiabatic case,\nimplying that the non-adiabatic processes are the primary agent that dissipates the kinetic energy of\nthe \flament.\n(3) With heat conduction and radiation being included, the decay time is remarkably reduced from\n113 minutes in the 1D case to 34 minutes in the 2D case. It is found that the \flament longitudinal\noscillations excite transverse fast-mode waves, and the wave leakage is one important agent that can\nlose the kinetic energy of the \flament.\nACKNOWLEDGMENTS\nThis research was supported by the Chinese foundations (NSFC 11533005, 11961131002, and\nU1731241) and Jiangsu 333 Project (No. BRA2017359). The numerical calculations in this pa-\nper were performed on the cluster system in the High Performance Computing Center (HPCC) of\nNanjing University. PFC thanks ISSI and ISSI-Beijing for supporting team meetings on solar \fla-\nments.\nREFERENCES\nArregui, I., Oliver, R., & Ballester, J. L. 2018,\nLiving Reviews in Solar Physics, 15, 3,\ndoi: 10.1007/s41116-018-0012-6Brady, C. S., & Arber, T. D. 2005, A&A, 438,\n733, doi: 10.1051/0004-6361:20042527\nCally, P. S. 1986, SoPh, 103, 277,\ndoi: 10.1007/BF00147830Damping of filament longitudinal oscillations 17\nChen, P. 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Ballester1, and M.Goossens2\n1DepartamentdeF´ ısica,Universitatdeles IllesBalears,E -07122,PalmadeMallorca,Spain\n[roberto.soler;ramon.oliver;joseluis.ballester]@uib .es\n2CentreforPlasmaAstrophysicsand Leuven MathematicalMod elingandComputational\nScienceCenter, K. U.Leuven, Celestijnenlaan200B,3001He verlee, Belgium\nmarcel.goossens@wis.kuleuven.ac.be\nABSTRACT\nTransverse oscillations of small amplitude are commonly se en in high-\nresolution observations of filament threads, i.e. the fine-s tructures of solar fila-\nments/prominences, and are typically damped in a few periods. Kink wave modes\nsupported by the thread body o ffer a consistent explanation of these observed oscilla-\ntions. Among the proposed mechanisms to explain the kink mod e damping, resonant\nabsorptionintheAlfv´ encontinuumseemstobethemoste fficientasitproducesdamp-\ningtimesofabout3periods. However,foranonzero- βplasmaandtypicalprominence\nconditions, the kink mode is also resonantly coupled to slow (or cusp) continuum\nmodes, which could further reduce the damping time. In this L etter, we explore for\nthefirsttimebothanalyticallyandnumericallythee ffectoftheslowcontinuumonthe\ndamping of transverse thread oscillations. The thread mode l is composed of a homo-\ngeneous and straightcylindricalplasma,an inhomogeneous transitionallayer, and the\nhomogeneous coronal plasma. We find that the damping of the ki nk mode due to the\nslowresonance ismuchless e fficient thanthat dueto theAlfv´ enresonance.\nSubject headings: Sun: oscillations — Sun: magnetic fields — Sun: corona — Sun:\nprominences\n1. Introduction\nThe existence of the fine-structure of solar filaments /prominences is clearly shown in high-\nresolutionobservationsbybothground-basedandsatellit e-on-boardtelescopes. Thesefine-structures,\nhere called threads, have a typical width, W, and length, L, in the ranges 0.2arcsec<W<– 2 –\n0.6arcsec and 5arcsec <L<20arcsec (Linet al. 2005). Filament and prominence threads are\nbelieved to be thin magnetic flux tubes partially filled with p rominence-like plasma and whose\nfootpointsareanchoredinthephotosphere. Observersusua llyreportthepresenceoftransverseos-\ncillationsandwavesinthesefine-structures(e.g.Linet al .2003,2005,2007;Okamotoet al.2007),\nwhichhavebeeninterpretedintermsofmagnetohydrodynami c(MHD)waves(e.g.,Terradas et al.\n2008). The reader is referred to Oliver&Ballester (2002), B allester (2006), and Banerjee et al.\n(2007) for reviews of these events. In addition, a typical pr operty of small-amplitudeprominence\noscillationsobservedinDopplerseriesisthattheyarequi cklydamped,withdampingtimesofthe\norderofseveralperiods(Molowny-Horaset al.1999;Terrad as et al.2002). Sincesmall-amplitude\noscillations are of local nature and related to periodic mot ions of threads, there must exist some\nmechanismwhichisableto dampthread oscillationsinan e fficient way.\nThedampingofprominenceoscillationshasrecentlybeenin vestigatedinanumberofpapers\n(this topic has been reviewed by Oliver 2008). Soleret al. (2 008) modeled a filament thread as a\nhomogeneous cylinder embedded in an unbounded corona and st udied the effect of nonadiabatic\nmechanisms (thermal conduction, radiative losses, and hea ting) on the wave damping. As in pre-\nviousworks(Carbonell et al.2004; Soleret al.2007), theyc oncludedthatthermale ffectsareonly\nefficient in damping the slow mode, the fast kink mode being almos t undamped. Subsequently,\nArreguiet al. (2008) used a more realistic thread model and i ncluded an inhomogeneous transi-\ntionregionbetweenthecylindricalthreadandtheexternal medium. Heneglectedplasmapressure\nand adopted the so-called β=0 approximation, where βis the ratio of the plasma pressure to\nthe magnetic pressure. These authors found that the kink mod e is resonantly coupled to Alfv´ en\ncontinuummodesdueto thepresence ofthetransverseinhomo geneouslayer, and obtainedaratio\nof the damping time to the period τD/P≈3 for typical prominence conditions. In the case of\ncoronalloops(see,forexample,recentreviewsbyGoossens et al.2006;Goossens2008),resonant\nabsorptionhasbeen extensivelyinvestigatedas adampingm echanismforthekinkmode.\nAlthoughtheplasma βinsolarprominencesisprobablysmall,itisdefinitelynonz ero. Hence,\nif gas pressure is taken into account in the model of Arreguie t al. (2008), i.e. the β/nequal0 case, it\nturns out that the kink mode phase speed, namely ck, is also within the slow (or cusp) continuum\nthat extends between the internal, csi, and external, cse, sound speeds, i.e. csi<ck<cse. In this\ncase, the kink mode is not only resonantly coupled to Alfv´ en continuum modes but also to slow\ncontinuum modes. The frequency of the kink mode is both withi n the Alfv´ en and slow continua\nbecause of the high density and low temperature of the promin ence material in comparison with\nthe coronal values. Hence, the slow continuum damping arise s as an additional mechanism to\ndampthekinkmodeinfilamentthreads,anditse fficiencyneedstobeassessed. Althoughtheslow\nresonance has been previously investigated in the context o f absorption of driven MHD waves\nin the solar atmosphere (e.g., ˇCadeˇ z et al. 1997; Erd´ elyiet al. 2001) and sunspots (e.g., Keppens\n1996),thepresentworkstudiesforthefirsttimethee ffectoftheslowresonanceonthedampingof– 3 –\nthekinkmodeinfilamentthreads. Forcoronalloopsthekinks peedisoutsidetheslowcontinuum.\nA coronal loop is presumably hotter and denser than its surro unding corona, so the ordering of\nsound,Alfv´ en, andkinkspeedsis cse<csi<vAi<ck<vAe. Therefore, thereisno slowresonance\nforthekinkmodeincoronal loopsand thepresent studydoesn ot applytocoronal loops.\nThisLetterisorganizedas follows: §2containsadescriptionofthemodelconfigurationand\nthemathematicalmethod. Theresultsare presented and disc ussedin§3. Finally,our conclusions\nare givenin§4.\n2. Model andmethod\nThemodelfortheequilibriumconfigurationconsideredhere isequivalenttothatofArreguiet al.\n(2008). Wemodelafilamentthreadasastraightcylinderwith prominence-likeconditionsembed-\nded in an unbounded corona, with a transverse inhomogeneous layer between both media (see\nFig.1). Weusecylindricalcoordinates,namely r,ϕ,andzfortheradial,azimuthal,andlongitudi-\nnal coordinates. Weadopt thedensityprofile byRuderman & Ro berts (2002) whichonly depends\non theradial direction,\nρ0(r)=ρi,ifr≤a−l/2,\nρtr(r),ifa−l/2<r<a+l/2,\nρe,ifr≥a+l/2,(1)\nwith\nρtr(r)=ρi\n2/braceleftBigg/parenleftBigg\n1+ρe\nρi/parenrightBigg\n−/parenleftBigg\n1−ρe\nρi/parenrightBigg\nsin/bracketleftbiggπ\nl(r−a)/bracketrightbigg/bracerightBigg\n. (2)\nIn these expressions, ρiis the internal density, ρeis the external (coronal) density, ais the tube\nmean radius, and lis the transitional layer width. The limits l/a=0 andl/a=2 correspond to a\nhomogeneous tube and a fully inhomogeneous tube, respectiv ely. We use the following densities:\nρi=5×10−11kg m−3andρe=2.5×10−13kg m−3. Therefore, the density contrast between\nthe internal and external plasma is ρi/ρe=200. The plasma temperature is related to the density\nthrough the usual ideal gas equation. We consider Ti=8000 K and Te=106K for the internal\nand external temperatures, respectively. The magnetic fiel d is taken homogeneous and orientated\nalongthe z-direction, B0=B0ˆz, withB0=5G everywhere. Withtheseconditions, β≈0.04.\n2.1. Analyticaldispersion relation\nWe consider the linear, ideal MHD equations for the β/nequal0 case. Sinceϕandzare ignor-\nablecoordinates,theperturbedquantitiesareputproport ionaltoexp (iωt+imϕ−ikzz),whereωis– 4 –\nFig. 1.—Sketch ofthemodelconfiguration.\nthe frequency, and mandkzare the azimuthal and longitudinal wavenumbers, respectiv ely. In the\nabsenceofatransitionallayer,i.e. l/a=0,thedispersionrelationofmagnetoacousticwavesisob-\ntainedbyimposingthecontinuityoftheradialdisplacemen t,ξr,andthetotalpressureperturbation,\npT, atr=a. Thisdispersionrelation(Edwin&Roberts 1983)is Dm(ω,kz)=0, with\nDm(ω,kz)=ρemi/parenleftBig\nω2−k2zv2\nAi/parenrightBigJ′\nm(mia)\nJm(mia)−ρime/parenleftBig\nω2−k2zv2\nAe/parenrightBigK′\nm(mea)\nKm(mea), (3)\nwherevAi,e=B0/√µρi,eis the internal/external Alfv´ en speed, and the quantities miandmeare\ngivenby\nm2\ni=/parenleftBig\nω2−k2\nzc2\nsi/parenrightBig/parenleftBig\nω2−k2\nzv2\nAi/parenrightBig\n/parenleftBig\nc2\nsi+v2\nAi/parenrightBig/parenleftBig\nc2\nTik2\nz−ω2/parenrightBig,m2\ne=/parenleftBig\nω2−k2\nzc2\nse/parenrightBig/parenleftBig\nω2−k2\nzv2\nAe/parenrightBig\n/parenleftBig\nc2\nse+v2\nAe/parenrightBig/parenleftBig\nω2−c2\nTek2\nz/parenrightBig, (4)\nwherecTi,e=csi,evAi,e//radicalBig\nc2\nsi,e+v2\nAi,eistheinternal/externalcusp(ortube)speed.\nWhen an inhomogeneous transitional layer is present in the m odel, we cannot obtain an an-\nalytical expression for the dispersion relation unless som e assumptions are made. An elegant\nmethod for obtaining an analytical dispersion relation is t o combine the jump conditions and the\napproximation of the thin boundary (TB). The jump condition s were derived by Sakurai et al.\n(1991a) and Goossenset al. (1995) for the driven problem and by Tirry& Goossens (1996) for\nthe eigenvalue problem. This method was used, for example, b y Sakurai etal. (1991b) for deter-\nminingthe absorption of sound waves in sunspots,Goossense t al. (1992) for determiningsurface\neigenmodes in incompressible and compressible plasmas, an d Van Doorsselaereet al. (2004) for– 5 –\nkinkeigenmodesinpressurelesscoronalloops. Thismethod isvalidiftheinhomogeneouslength-\nscalewithintheresonantlayerissu fficientlysmallincomparisonwiththetuberadius. Apartfrom\ntheAlfv´ enresonance,Sakurai et al.(1991a)alsoprovided jumpconditionsfortheslowresonance,\nwhich can be applied to the present situation. Hence, the TB f ormalism allows us to assess the\nparticularcontributionoftheslowresonancetothekinkmo dedamping.\nIn our model the magnetic field is straight and constant so tha t the variations of the local\nAlfv´ en frequency and the local cusp (or slow) frequency are only due to the variation of density.\nIn thatcase thejumpconditionsat theAlfv´ enresonancepoi nt,namely r=rA, can bewrittenas\n/bracketleftbigξr/bracketrightbig=−iπm2/r2\nA\nω2\nA|∂rρ0|ApT,/bracketleftbigpT/bracketrightbig=0,atr=rA, (5)\nwhereωAand|∂rρ0|Aare the Alfv´ en frequency and the modulus of the radial deriv ative of the\ndensityprofile,respectively,bothquantitiesevaluateda tr=rA. Therespectivejumpconditionsat\ntheslowresonancepoint,namely rs, are\n/bracketleftbigξr/bracketrightbig=−iπk2\nz\nω2c|∂rρ0|s/parenleftBiggc2\ns\nc2s+v2\nA/parenrightBigg2\npT,/bracketleftbigpT/bracketrightbig=0,atr=rs, (6)\nwhereωcand|∂rρ0|sarethecuspfrequency andthemodulusoftheradial derivati veofthedensity\nprofile, respectively, both quantities evaluated at r=rs, while the factor/parenleftbigg\nc2\ns\nc2s+v2\nA/parenrightbigg\n=/parenleftBigβ\nβ+2/γ/parenrightBig\n≈0.034\nis a constant everywhere in the equilibrium, with γ=5/3 the adiabatic index. The jump condi-\ntions (6) are independent of the azimuthal wavenumber, m, but they depend on the longitudinal\nwavenumber, kz. So, forkz=0, jumpconditions(6) become/bracketleftbigξr/bracketrightbig=/bracketleftbigpT/bracketrightbig=0, meaningthatthere is\nno slowresonancein suchacase.\nNext, we use jump conditions (5) and (6) to obtain a correctio n to the dispersion relation (3)\nduetobothresonances in theTBapproximation,\nDm(ω,kz)=−iπm2/r2\nA\nω2\nAρiρe\n|∂rρ0|A−iπk2\nz\nω2c/parenleftBiggc2\ns\nc2s+v2\nA/parenrightBigg2ρiρe\n|∂rρ0|s. (7)\nThe first term on the right-hand side of equation (7) correspo nds to the effect of the Alfv´ en reso-\nnance, whilethesecond termis presentduetotheslowresona nce.\n2.1.1. Expressionsin thethintubeapproximation\nFurther analytical progress can be made by combining the thi n boundary (TB) and the thin\ntube (TT) approximations. This was done extensively by Goos sensetal. (1992). We perform a– 6 –\nasymptoticexpansion of the Bessel functions present in the dispersion relation (7) by considering\nthe long-wavelength limit, i.e., kza≪1, and only keep the lowest order, nonzero term of the\nexpansion. Thus,thedispersionrelation (7)becomes,\nρem/a/parenleftBig\nω2−k2zv2\nAi/parenrightBig+ρim/a/parenleftBig\nω2−k2zv2\nAe/parenrightBig+iπρiρem2/r2\nA\nω2\nA1\n|∂rρ0|A+k2\nz\nω2\nc/parenleftBiggc2\ns\nc2s+v2\nA/parenrightBigg21\n|∂rρ0|s=0.(8)\nNow, we write the frequency as ω=ωR+iωI. It follows from the resonant conditions that\nωA=ωc=ωR. The position of the Alfv´ en, rA, and slow, rs, resonance points can be computed\nby equating the real part of the kink mode frequency to the loc al Alfv´ en and slow frequencies,\nrespectively. By thisprocedure, weobtain\nrA=a+l\nπarcsin/bracketleftBiggρi−ρe\nρi+ρe−2v2\nAik2\nz\nω2\nRρi\n(ρi+ρe)/bracketrightBigg\n, (9)\nfortheAlfv´ enresonance point,and\nrs=a+l\nπarcsin/bracketleftBiggρi−ρe\nρi+ρe−2c2\nsik2\nz\nω2\nRρi\n(ρi+ρe)/bracketrightBigg\n, (10)\nfor the slow resonance point. In order to derive equation (10 ), we have used the approximation\nω2\nc≈c2\nsk2\nz, which is valid for c2\ns≪v2\nA. Note that we need the value of ωRto determine rAandrs.\nExpressionsfor|∂rρ0|Aand|∂rρ0|sare obtainedfromthedensityprofile(eq. [1]),\n|∂rρ0|A=/parenleftbiggρi−ρe\nl/parenrightbiggπ\n2cosαA,|∂rρ0|s=/parenleftbiggρi−ρe\nl/parenrightbiggπ\n2cosαs. (11)\nwithαA=π(rA−a)/landαs=π(rs−a)/l. We insert these expressions in equation (8) and\nneglect terms with ω2\nI, i.e., low-damping situation. Then we obtain an expression for the ratio\nωR/ωIafter somealgebraic manipulations. Since the oscillatory period,P, and the damping time,\nτD, are related tothefrequencyas follows\nP=2π\nωR, τD=1\nωI, (12)\nitis thenstraight-forwardto givean expressionfor τD/P,\nτD\nP=F1\n(l/a)/parenleftBiggρi+ρe\nρi−ρe/parenrightBiggm\ncosαA+(kza)2\nm/parenleftBiggc2\ns\nc2\ns+v2\nA/parenrightBigg21\ncosαs−1\n, (13)\nwhereFis a numerical factor that depends on the density profile ( F=2/πin our case). As\nin previous expressions, the term with kzcorresponds to the contribution of the slow resonance.\nIf this term is dropped and m=1 and cosαA=1, equation (13) is equivalent to equation (3)– 7 –\nof Arregui et al. (2008), which only takes the Alfv´ en resona nce into account and was first ob-\ntained by Goossenset al. (1992) (see also equivalent expres sions in Ruderman &Roberts 2002;\nGoossenset al.2002).\nNext, we assume rA≈rs≈afor simplicity, so cos αA≈cosαs≈1. The ratio of the two\ntermsin equation(13)is\n(τD)A\n(τD)s≈(kza)2\nm2/parenleftBiggc2\ns\nc2\ns+v2\nA/parenrightBigg2\n, (14)\nwhere(τD)Aand(τD)sare the damping times exclusively due to the Alfv´ en and slow resonances,\nrespectively. To make a simple calculation we note that the o bserved wavelengths of promi-\nnence oscillations correspond to 10−3<kza<10−1. Form=1 andkza=10−2we obtain\n(τD)A/(τD)s≈10−7, meaning that the Alfv´ en resonance is much more e fficient than the slow res-\nonance for damping the kink mode. Note that even in the extrem e case of a very large β, i.e.,\nc2\ns→∞and so/parenleftbigg\nc2\ns\nc2s+v2\nA/parenrightbigg\n→1,(τD)A/(τD)s≪1 for typical values of kza. By means of this simple\ncalculation,wecananticipatethattheslowresonancewill beirrelevantforthekinkmodedamping.\n2.2. Numerical computations\nIn addition to the analytical approximations, we also numer ically solve the full eigenvalue\nproblem by means of the PDE2D code (Sewell 2005). We consider the resistive version of the\nlinear MHD equations. The magnetic Reynolds number, Rm, in the corona is considered to be\naround1012,butusingthisvaluerequires takingan enormousnumberofg ridpointsin thenumer-\nical computations. We therefore use a smaller valueof Rm, and consequently a smallernumber of\ngrid points, but we make sure that the resonant plateau is rea ched and the damping time becomes\nindependent of the magnetic di ffusivity. We use a nonuniform grid with a large density of grid\npoints within the inhomogeneous layer in order to correctly describe the small spatial scales that\ndevelopduetobothresonances. InthenextSection,thesenu mericalresultsarecomparedwiththe\nanalyticalTBapproximations.\n3. Results\nWe focus our investigation on the damping of the kink mode ( m=1). First, we fix the\nlongitudinal wavenumber to kza=10−2and consider l/a=0.2. We numerically compute (see\nFig.2)theeigenfunctions: thevelocityperturbation( vr,vϕ,vz),themagneticfieldperturbation( B1r,\nB1ϕ,B1z), the density perturbation ( ρ1), and the gas pressure perturbation ( p1). The behavior of vϕ\nandB1ris similar to that of B1ϕandvr, respectively, and hence they are not displayed in Figure 2.– 8 –\nFig. 2.— Modulus in arbitrary units of kink mode perturbatio ns:a)vr,b)B1ϕ,c)ρ1,d)vz,e)B1z,\nandf)p1, as functions of r/aforl/a=0.2 andkza=10−2. The small panel in each graphic\ncorrespondsto an enhancementoftheeigenfunctionwithint hetransitionallayer.\nBecause of the resonances the perturbations show large peak s in the transitional layer. The small\npanels of Figure 2 display the behavior of the eigenfunction s within the inhomogeneous layer,\nwhichallowsustoascertainthepositionofthepeaksinmore detail. Thepeaksoftheperturbations\nvr,B1ϕ,andρ1arerelatedtotheAlfv´ enresonance, whiletheperturbatio nsvz,B1z,andp1aremore\naffectedbytheslowresonanceandtheirpeaksappearatadi fferentposition. Moreover,weseethat\npeaks related totheAlfv´ en resonanceare widerthanthoser elated totheslowresonance, meaning\nthat the slow resonance produces smaller spatial scales wit hin the resonant layer. Considering the\nnumericalvalueof ωR, wegetrA/a≈1andrs/a≈1.08fromequations(9)and(10), respectively.\nBoth valuesarein agoodagreement withthepositionofpeaks in Figure2.\nNext,weplotinFigure3theratioofthedampingtimetothepe riod,τD/P,asafunctionof kza\ncorrespondingtothekinkmodefor l/a=0.2. InthisFigure,wecomparethenumericalresultswith\nthose obtained from the TB approximation. At first sight, we s ee that the slow resonance (dashed\nlineinFig.3)ismuchlesse fficientthantheAlfv´ enresonance(symbols)indampingtheki nkmode.– 9 –\nFig. 3.— Ratio of the damping time to the period, τD/P, as a function of the dimensionless\nwavenumber, kza,correspondingtothekinkmodefor l/a=0.2. Thesolidlineisthecompletenu-\nmericalsolutionobtainedwiththePDE2Dcode. Thesymbolsa ndthedashedlinearetheresultsof\ntheTB approximationcorrespondingto theAlfv´ en and slowr esonances (theeffect ofthefirst and\nsecond terms on the right-hand side of equation [7], respect ively). The shaded zone corresponds\ntotherangeoftypicallyobservedwavelengthsin prominenc eoscillations.\nThe individual contribution of each resonance in the TB appr oximation has been determined by\nsolvingequation(7)andonlytakingthetermontheright-ha ndsiderelatedtothedesiredresonance\ninto account. For the wavenumbers relevant to prominence os cillations, 10−3<kza<10−1, the\nvalueofτD/Prelatedtotheslowresonanceisbetween4and8ordersofmagn itudelargerthanthe\nratioobtainedbytheAlfv´ enresonance. Theresultoftheth intubelimit(eq.[14])agreeswiththis.\nOntheotherhand,thecompletenumericalsolution(solidli ne)isclosetotheresultfortheAlfv´ en\nresonance. In agreement with Arreguiet al. (2008), we obtai nτD/P≈3 in the relevant range\nofkza. As was stated by Arreguiet al. (2008, Fig. 2 d), the discrepancy between the numerical\nresult and the TB approximation increases with l/a, the difference being around 20% for l/a=1.\nHowever, for small, realistic values of l/athis discrepancy is less important and the TB approach\nis a very good approximationto the numerical result. For sho rt wavelengths ( kza/greaterorsimilar100) the value\nofτD/Pincreases and the e fficiency of the Alfv´ en resonance as a damping mechanism decre ases.\nForkza≈102both the slow and the Alfv´ en resonances produce similar (an d inefficient) damping\ntimes.– 10 –\n4. Conclusion\nIn this Letter, we have studied the kink mode damping in filame nt threads due to resonant\ncoupling to slowcontinuum modes. As far as we know, thisis th efirst timethat thisphenomenon\nis studied in the context of solar prominences. By consideri ng the thin boundary approximation,\nwe have found that, contrary to the Alfv´ en resonance, the sl ow resonance is very ine fficient in\ndamping the kink mode for typical prominence-like conditio ns. This conclusion also holds for\nfluting modes ( m≥2). The very small damping due to the slow resonance is compar able to that\ndue to thermal effects studied in previous papers (Soleret al. 2008). Therefo re, we conclude that\nthe effect of the slow resonance is not relevant to the damping of tra nsverse thread oscillations,\nwhichis probablygovernedby theAlfv´ enresonance.\nRS, RO, and JLBacknowledgethefinancial supportreceived fr om theSpanish MICINN and\nFEDER funds, and the Conselleria d’Economia, Hisenda i Inno vaci´ o of the CAIB under Grants\nNo. AYA2006-07637 and PCTIB-2005GC3-03, respectively. RS , RO, and JLB also want to ac-\nknowledge the International Space Science Institute teams “Coronal waves and Oscillations” and\n“Spectroscopy and Imaging of quiescent and eruptive solar p rominences from space” for useful\ndiscussions. RS thanks the Conselleria d’Economia, Hisend a i Innovaci´ o for a fellowship. MG\nacknowledgessupportfrom thegrantGOA /2009-009. MG isgratefulto UIBforfinancial support\nduringhisstay atUIB and toJLBforthehospitalityat theSol arPhysicsGroup.\nREFERENCES\nArregui,I., Terradas, J. Oliver,R., &Ballester, J.L. 2008 ,ApJ, 682,L141\nBallester, J. 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L. 2007,A&A,471,1023\nSoler, R., Oliver,R., &Ballester, J. L. 2008,ApJ,684,725\nTerradas, J., Molowny-Horas, R., Wiehr, E., Balthasar, H., Oliver, R., & Ballester, J. L. 2002,\nA&A,393,637\nTerradas, J., Arregui,I., Oliver,R., & Ballester, J.L. 200 8,ApJ,678, L153– 12 –\nTirry,W.J., &Goossens,M.ApJ, 471,501\nVan Doorsselaere, T.,Andries,J., Poedts,S., & Goossens,M . 2004,ApJ,606,1223\nThispreprintwaspreparedwiththeAAS L ATEXmacrosv5.2."    },    {        "title": "2111.08982v1.Spectral_asymptotics_for_the_vectorial_damped_wave_equation.pdf",        "content": "Spectral asymptotics for the vectorial damped wave equation\nGuillaume Klein\nAbstract\nThe eigenfrequencies associated to a scalar damped wave equation are known to belong to a\nband parallel to the real axis. In [Sjö00] J. Sjöstrand showed that up to a set of density 0, the\neigenfrequencies are confined in a thinner band determined by the Birkhoff limits of the damping\nterm. In this article we show that this result is still true for a vectorial damped wave equation.\nIn this setting the Lyapunov exponents of the cocycle given by the damping term play the role of\nthe Birkhoff limits of the scalar setting.\n1 Introduction\n1.1 Setting\nLet(M;g)be a compact Riemannian manifold of dimension dwithout boundary, we will more-\nover assume that (M;g)is connected and C1. Let \u0001be the Laplace-Beltrami operator on Mfor\nthe metricgand letabe aC1-smooth function from MtoHn(C), the space of hermitian matrices\nof dimension n. We are interested in the following system of equations\n\u001a\n(@2\nt\u0000\u0001 + 2a(x)@t)u= 0 inD0(R\u0002M)n\nujt=0=u02H1(M)nand@tujt=0=u12L2(M)n.(1)\nLetH=H1(M)n\bL2(M)nand define on Hthe unbounded operator\nAa=\u00120 Idn\n\u0001\u00002a\u0013\nof domainD(Aa) =H2(M)n\bH1(M)n.\nBy application of Hille-Yosida theorem to Aathe system (1) has a unique solution in the space\nC0(R;H1(M)n)\\C1(R;L2(M)n). Ifuis a solution of (1) then we can define the energy of uat\ntimetby the formula\nE(u;t) =1\n2Z\nMj@tu(t;x)j2+jru(t;x)j2\ngdx (2)\nwherejvj2\ng=gx(v;v). An integration by parts shows that\nd\ndtE(u;t) =\u0000Z\nMh2a(x)@tu(t;x);@tu(t;x)iCndx, (3)\nso ifais not null then the energy is not constant. In particular if ais positive semi-definite the\nenergy is non-increasing and acan be seen as a dampener. In the problem of stabilisation of the\nwave equation one is interested with the long time behavior of the energy of solutions to (1).\nThe spectrum of Aais obviously related to this long time behavior although it is not sufficient to\ncompletely describe it (see for example [Leb93]). In this article we will thus be interested in the\nasymptotic distribution of the eigenvalues of Aa. Besides the possible applications to stabilisation\nthe problem of determining an asymptotic distribution of eigenvalues for a non-self-adjoint oper-\nator is an interesting problem by itself but it can also be seen as one of the many generalization\nof Weyl’s law. This problem has already been studied in the scalar case ( n= 1) for example in\n[Sjö00]. The aim of this article is to expend some results of [Sjö00] to the case where n\u00151.\n1arXiv:2111.08982v1  [math.AP]  17 Nov 20211.2 Results\nIt is a classical fact that the spectrum of Aais symmetric with respect to the real axis and that it\ncontains only discrete eigenvalues with finite multiplicities, this is proved using Fredholm theory.\nIfuis a stationary solution of (1) we can write u(t;x) =eit\u001cv(x)and we are lead to the equation\n(\u0000\u0001\u0000\u001c2+ 2ia\u001c)v= 0. (4)\nWe say that \u001c2Cis an eigenvalue for (4) if there is a non-zero solution vof (4). Note that \u001cis an\neigenvalue of (4) if and only if i\u001cis an eigenvalue of Aa.\nDefinition. We note sp(a(x))the set of all the eigenvalues of a(x)and define\na\u0000= inf\nx2Mmin sp(a(x))anda+= sup\nx2Mmax sp(a(x)).\nBy reasoning on the energy with (3) it is easy to see that Im(\u001c)can only be in the interval\n[2a\u0000; 2a+]. We are now interested in the asymptotic repartition of Im(\u001c)whenj\u001cjgoes to infinity.\nSince the spectrum of Aais invariant under complex conjugation we will only be interested in the\nlimit Re(\u001c)!+1.\nWe define the function p:T\u0003M!Rbyp(x;\u0018) =gx(\u0018;\u0018), notice that pis the principal symbol\nof\u0000\u0001. We also define \u001eas the Hamiltonian flow generated by p, it can also be interpreted as the\ngeodesic flow on T\u0003Mtravelled at twice the speed. In what follows (x0;\u00180)will be a point of T\u0003M\nand we will write (xt;\u0018t)for\u001et(x0;\u00180).\nDefinition. Lettbe a real number, we define the function Gt:p\u00001(1=2)!Mn(C)as the solution of\nthe differential equation\u001a\nG0(x0;\u00180) = Idn\n@tGt(x0;\u00180) =\u0000a(xt)Gt(x0;\u00180).(5)\nThis definition naturally extends to any point (x0;\u00180)2T\u0003M.\nNotice that Gis a cocycle map, this means that for every s;t2Rand every (x;\u0018)2T\u0003Mwe\nhave the equality Gt+s(x;\u0018) =Gt(\u001es(x;\u0018))Gs(x;\u0018). Whenn= 1the function ais real valued and\nwe simply have\nGt(x0;\u00180) = exp\u0012\n\u0000Zt\n0a(xs)ds\u0013\n. (6)\nThis formula however ceases to be correct when nis greater than 1. Throughout the entire article\nifEis a vector space and k\u0001k\u0003is a norm on Ewe will also write k\u0001k\u0003for the associated operator\nnorm onL(E).\nDefinition. For every positive time twe define the following quantities\nC\u0000\nt=\u00001\ntsup\n(x0;\u00180)2p\u00001(1=2)ln (kGt(x0;\u00180)k2)\nandC+\nt=\u00001\ntinf\n(x0;\u00180)2p\u00001(1=2)ln\u0000\nkGt(x0;\u00180)\u00001k\u00001\n2\u0001\n.\nWe will also note C\u0006\n1= limt!+1C\u0006\nt.\nIt is easy to show that this limit always exists using a sub-additivity argument. Note that the\nexistence and the value of the limit does not depend on the choice of the norm since they are all\nequivalent, however it is sometimes easier to work with the operator norm associated with the\neuclidian norm on Cn. Again, ifn= 1we get the simpler expression\nC\u0000\nt= inf\n(x;\u0018)2p\u00001(1=2)1\ntZt\n0a(xs)dsandC+\nt= sup\n(x;\u0018)2p\u00001(1=2)1\ntZt\n0a(xs)ds.\nFor an analogue to this expressions when n\u00151see [Kle18] .\n2Theorem 1.1. For every\" >0there is only a finite number of eigenvalues of (4)outside the strip\nR+i]C\u0000\n1\u0000\";C+\n1+\"[.\nIn the scalar case this was first proved by Lebeau [Leb93] using microlocal defect measures.\nIn fact, because of the setting of his article, he only proved it for the upper bound and a\u00140but\nhis proof easily extends to Theorem 1.1 with n= 1. Theorem 1.1 was also proved by Sjöstrand in\n[Sjö00] for n= 1using different techniques. The argument of Sjöstrand relies on a conjugation by\npseudo-differential operators to replace the damping term aby its average on geodesics of length\nT. Because of the non commutativity of matrices it seems that this argument cannot be modified\nin a straight forward manner to prove the vectorial case ( n\u00151). In [Kle17], using the same\ntechnique as Lebeau, the upper bound of Theorem 1.1 was proved in the general case n\u00151for a\nfunctionavalued in H+\nn(C), the space of Hermitian positive semi-definite matrices. Once again\nthe argument used there can easily be adapted to prove Theorem 1.1 in its full generality (see\n[Kle18]).\nThis result is the best possible in the sense that there can be infinitely many eigenvalues of\n(4) outside of the strip R+i[C\u0000\n1;C+\n1]. However we are going to show that in a way, “most” of\nthe eigenvalues of (4) are in fact in a narrower strip. In order to give a meaning to the previous\nstatement we need the following result.\nTheorem 1.2. The number of eigenvalues \u001cwithRe(\u001c)2[0;\u0015]is equivalent to\nn\u0012\u0015\n2\u0019\u0013dZZ\np\u00001([0;1])1dxd\u0018\nwhen\u0015goes to +1. Moreover the remainder is a O(\u0015d\u00001).\nForn= 1this result was first proved in [MaMa82] by Markus and Matsaev and then indepen-\ndently in [Sjö00] by Sjöstrand. Once again the proof can easily be adapted to our case n\u00151.\nWe now want to make further estimations of the asymptotics of the imaginary part of the eigen-\nvalues of (4). Recall that the matrix Gis a cocycle and that the geodesic flow on p\u00001(1=2)preserves\nLiouville’s measure. Thus we can use Kingman’s subadditive ergodic theorem to show that the lim-\nits\n\u0015n(x;\u0018) = lim\nt!11\ntlogkGt(x;\u0018)k2and\u00151(x;\u0018) = lim\nt!11\ntlog\u0010\r\rGt(x;\u0018)\u00001\r\r\u00001\n2\u0011\nexist for almost every (x;\u0018)2p\u00001(1=2). Moreover the functions (x;\u0018)7!\u00151(x;\u0018)and(x;\u0018)7!\n\u0015n(x;\u0018)are both measurable and bounded. Note that the existence and the value of the limit does\nnot depend on the choice of the norm because they are all equivalent. Note also that \u00151and\u0015n\nare respectively the smallest and largest Lyapunov exponents defined by the multiplicative ergodic\ntheorem of Oseledets. The statement of Oseledets theorem can be found in Annex B. We now\ndefine\n\u0003\u0000= ess inf\u00151(x;\u0018)and\u0003+= ess sup\u0015n(x;\u0018).\nIn the scalar case n= 1so obviously \u00151=\u0015nand we have\n\u0003+=\u0000 ess inf\n(x;\u0018)2p\u00001(1=2)lim\nt!11\ntZt\n0a(xs)dsand \u0003\u0000=\u0000ess sup\n(x;\u0018)2p\u00001(1=2)lim\nt!11\ntZt\n0a(xs)ds.\nNotice the minus sign in comparison to the definition of C\u0006\n1. In general we have\nC\u0000\n1\u0014\u0000\u0003+\u0014\u0000\u0003\u0000\u0014C+\n1\nand every inequality is usually strict. We are now ready to state the main result of this article.\nTheorem 1.3. For every\">0the number of eigenvalues \u001csatisfying Re(\u001c)2[\u0015;\u0015+ 1]andIm(\u001c)=2\n]\u0000\u0003+\u0000\";\u0000\u0003\u0000+\"[iso(\u0015d\u00001)when\u0015tends to infinity.\n3In view of Theorem 1.2 and Theorem 1.3 we see that, up to a null density subset, all of the\neigenvalues of (4) have their imaginary part in the interval ]\u0000\u0003+\u0000\";\u0000\u0003\u0000+\"[. Theorem 1.3 was\nproved by Sjöstrand [Sjö00] when n= 1and the asymptotics was then refined by Anantharaman\nfor a negatively curved manifold in [Ana10]. The main goal of this article is to prove Theorem 1.3\nin the general case n\u00151. As for Theorem 1.1 the arguments used by Sjöstrand in [Sjö00] seems\nnot to work anymore when n\u00151. This mainly comes from the fact that matrices do not commute\nand thus formula (6) is no longer true when n\u00151.\n1.3 Two open questions\nIf we taken= 1then there is only one Lyapunov exponent for Gwhich is simply the opposite of\nthe Birkhoff average of a:\n\u00151(x0;\u00180) = lim\nt!1\u00001\ntZt\n0a(xs)ds.\nIf we make the assumption that the geodesic flow is ergodic on Mwe get\n\u0003\u0000= \u0003+=\u00001\nvol(M)Z\nMa(x)dx=\u00151(x;\u0018)a.e.\nand Theorem 1.3 tells us that most of the eigenvalues of Aaare concentrated around the vertical\nline of imaginary part\u00001\nvol(M)R\nMa(x)dx. Now if we drop the assumption n= 1 and keep the\nergodic assumption we do not necessarily have \u0003+= \u0003\u0000but the Lyapunov exponents \u0015idefined\nby Theorem B.1 will be constant almost everywhere. We write \u0015ifor the almost sure value of the\nfunction (x;\u0018)7!\u0015i(x;\u0018)and we thus have \u00151= \u0003\u0000and\u0015n= \u0003+. Theorem 1.3 tells us that most\nof the eigenvalues of Aawill be concentrated around the strip fz2C:Im(z)2[\u0000\u0015n;\u0000\u00151]gbut it\nseems natural to ask if the following stronger property holds.\nQuestion 1. Is it true that for every \">0the number of eigenvalues \u001csatisfying Re(\u001c)2[\u0015;\u0015+ 1]\nand\nIm(\u001c)=2n[\ni=1]\u0015i\u0000\";\u0015i+\"[\nis ao(\u0015d\u00001)when\u0015tends to infinity ?\nIt seems that at the present time, the techniques developed in this article do not allow us to\nanswer this question. It was however proved in [Kle18] that the answer is yes when the manifold\nMis just a circle. The proof for this result relies on microlocal deffect measures and the fact that\non the circle the Lyapunov exponents are constant everywhere and not almost everywhere.\nQuestion 2. If the answer to the first question is yes, is it true that the eigenvalues are equally\ndistributed between the Lyapunov exponents ?\nLet us rephrase this more precisely. Let \u00151\u0014:::\u0014\u0015nbe the constant almost everywhere\nLyapunov exponents of G. We want to know if for \">0small enough and for every i2f1;:::;ng\nthe number of eigenvalues of (4) in the box [0;\u0015] +i]\u0015i\u0000\";\u0015i+\"[is equivalent to\nki\u0012\u0015\n2\u0019\u0013dZZ\np\u00001([0;1])1dxd\u0018\nwhen\u0015goes to +1and where kiis the multiplicity of the Lyapunov exponent \u0015i.\nUnfortunately the techniques used to answer Question 1 for M=R=2\u0019Zare not powerful\nenough to answer Question 2 even in that simple setting.\n41.4 Plan of the article\nSection 2 is dedicated to the proof of Theorem 1.3 which starts by a semi-classical reduction. The\ngeneral idea of the proof is to express the eigenvalues of (4) as zeros of some Fredholm determinant\ndepending holomorphically in zand then to use Jensen’s formula to bound the number of these\nzeros.\nIn order to construct the aforementioned Fredholm determinant and to get the appropriate\nbound we need to construct some approximate resolvent for \u0000\u0001\u0000\u001c2+ 2ia(x)\u001c, this is the object\nof Proposition 1. The proof of Proposition 1 is postponed to Section 3 and represents the core of\nthis article. Section 3 starts by a sketch of the proof of Proposition 1 for easier understanding.\nThe article then ends with two annexes. The first one presents the semi-classical anti-Wick\nquantization and its basic properties. The second annex presents the multiplicative ergodic theo-\nrem of Oseledets.\nAcknowledgments\nThis article is for the vast majority a reproduction of the last chapter of my PhD Thesis (see\n[Kle18]). I would like to thank again my advisor, Nalini Anantharaman. There is no doubt that\nthis article would not have seen the light of day without her help and support.\n2 Proof of Theorem 1.3\nThe first step of the proof is to perform a semi-classical reduction borrowed from [Sjö00]. Recall\nfrom the Introduction that i\u001cis an eigenvalue of Aaif and only if there exists some non zero\nv:M!Cnsuch that\n(\u0000\u0001\u0000\u001c2+ 2ia\u001c)v= 0.\nWe are interested in the asymptotic behaviour of the eigenvalues of Aaand since its spectrum is\ninvariant by complex conjugation we can restrict ourself to the case Re(\u001c)!+1. Let us call h\nour semiclassical parameter tending to zero and let i\u001cbe an eigenvalue of Aa, depending on h,\nsuch thath\u001c= 1 +o(1)whenhgoes to zero. If we write \u001c=\u0014=hthe previous equation becomes\n(\u0000h2\u0001\u0000\u00142+ 2ia\u0014h)v= 0.\nNow if we write z=\u00142, and\u0014=pzwithRe(z)>0the equation becomes\n(\u0000h2\u0001 + 2ihapz\u0000z)v= 0.\nWe might finally rewrite it as\n(P\u0000z)v= 0 (7)\nwithP=P(z) =P+ihQ(z), whereP=\u0000h2\u0001is the semiclassical Laplacian and Q(z) = 2apz.\nNote thatPis self adjoint, Qdepends holomorphically on zin a neighbourhood of 1and it is\nself adjoint whenever zis a positive real number. Throughout the rest of the article we will use\ndifferential operators depending on the semi-classical parameter h, an exposition of the theory of\nh-pseudo-differential operators is given for example in [Zwo12].\nRemark. Notice that z=\u00142and that (1 +x)2= 1 + 2x+o(x)so we have\nh\u00001Im(z) =h\u000012Im(\u0014) +o(1) = 2 Im(\u001c) +o(1).\nThis explains the appearance of some multiplications by two in the rest of the article.\n5According to this semi-classical reduction, finding an upper bound on the number of eigenvalues\nofPin an open set 1 +he\nyields an upper bound on the number of eigenvalues of (4) in an open\nseth\u00001+e\n=2 +o(1).\nDefinition. Let\">0be fixed, we then put\ne\n =fz2C:Re(z)2]\u00002; 2[;Im(z)2]2a\u0000\u00003;\u0000\u0003+\u0000\"=2[g\ne!=fz2C:Re(z)2]\u00001; 1[;Im(z)2]2a\u0000\u00002;\u0000\u0003+\u0000\"[g\nez0=i(2a\u0000\u00001).\n0\u00001\u00002 1 2 3 \u00003\n\u0000\u0003+\nC\u0000\n1\n2a\u0000\nez0e\ne!\"=2\n\"=2\nFigure 1: Drawing of e\n,e!andez0.\nWe are going to prove that the number of eigenvalues of Pin!h= 1 + 2he!is ao(h1\u0000d). Since\nthere are no eigenvalues of (4) with imaginary part smaller than 2a\u0000this will prove that for every\n\">0the number of eigenvalues \u001csatisfying Re(\u001c)2[h\u00001\u00001;h\u00001+ 1] andIm(\u001c)\u0014\u0000\u0003+\u0000\"is\nao(h1\u0000d)whenhtends to zero. The proof is exactly the same for eigenvalues satisfying Im(\u001c)\u0015\n\u0000\u0003\u0000+\"and this will thus prove Theorem 1.3.\nThe key ingredient here is the next proposition but, in order to improve clarity, its proof is\npostponed until the next section.\nProposition 1. For every complex number zin\nh= 1 + 2he\nthere exists an operator R(z)2L(L2)\ndepending holomorphically on z2\nhsuch thatR(z)(P\u0000z) = Id +R1(z) +R2(z)whereR1;R22\nL(L2),kR1(z)kL2<1=2andkR2(z)ktr=o(h1\u0000d). Moreover for z0= 1 + 2hez02!hthe operator\nR(z0)(P(z0)\u0000z0)is invertible in L2and\r\r\r(R(z0)(P(z0)\u0000z0))\u00001\r\r\r\nL2is uniformly bounded in h.\n6Using this proposition we want to bound the number of eigenvalues of Pin\nh. First of all\nnotice that ifP\u0000zhas a non zero kernel then so does R(z)(P\u0000z), thus we only need to bound\nthe dimension of the kernel of R(z)(P\u0000z). Now sincekR1(z)kL2\u00141=2the operator Id +R1(z)is\ninvertible and there exists an invertible operator Q(z)such thatQ(z)R(z)(P\u0000z) = Id+K(z)where\nK(z) =Q(z)R2(z). SinceQis invertible we have dim ker(R(z)(P\u0000z)) = dim ker(Id + K(z)).\nThe operator R2is of trace class and thus Kis also of trace class with\nkKktr\u0014kQkL2kR2ktr\u00142kR2ktr=o(h1\u0000d).\nIt follows that zis an eigenvalue of Ponly ifD(z)def= det(1 +K(z))is equal to 0. Moreover the\nmultiplicity of an eigenvalue zofPis less than the multiplicity of the zero of D. Using a general\nestimate on Fredholm determinants we get\njD(z)j\u0014exp(kK(z)ktr)\nOn the other hand for z0we have\njD(z0)j\u00001= det((1 + K(z0))\u00001)and(1 +K(z0))\u00001= 1\u0000K(z0)(1 +K(z0))\u00001.\nUsing this and the estimate on Fredholm determinant we get\njD(z0)j\u00001=jdet(1\u0000K(z0)(1 +K(z0))\u00001)j\u0014exp\u0000\nkK(z0)(1 +K(z0))\u00001ktr\u0001\n\u0014exp\u0000\nk(1 +K(z0))\u00001kkK(z0)ktr\u0001\nwhich, according to Proposition 1, is smaller than exp(CkK(z0)ktr)for some constant C > 0\nindependent of h. So far we have proved that\nlogjD(z0)j\u0015\u0000CkK(z0)ktrand8z2\nh;logjD(z)j\u0014kK(z)ktr. (8)\nAs we will see in section 3 the operators R(z)andR2(z)both depend holomorphically in zon\nh.\nSinceP\u0000zis holomorphic we get that Id +R1(z)andQ(z)are also holomorphic and finally that\nK(z)is holomorphic in zon\nh. Consequently the function z7!D(z)is holomorphic on \nh, we\nwant to apply Jensen’s inequality in order to bound the number of zeros of Don a subset of \nh\ncontaining z0.\nSincee\nis a simply connected open set there exists a Riemannian mapping e\t :e\n!D(0; 1)\nbetweene\nand the open unit disk which also satisfy e\t(ez0) = 0 . If we put \th:z7!e\t(z=2\u00001\nh)then\n\th: \nh!D(0;1)is a Riemannian mapping which maps z0to0. For every 0< t < 1let us call\nn(t)the number of zeros (with multiplicity) of D\u000e\t\u00001\nhinD(0;t); Jensen’s formula states that\n1\n2\u0019Z2\u0019\n0logjD\u000e\t\u00001\nh(tei\u0012)jd\u0012\u0000logjD\u000e\th(0)j=Zt\n0n(s)\nsds.\nForrclose enough to 0we have \th(!h)\u001aD(0; 1\u0000r), we then use (8) to obtain\n1\n2\u0019Z2\u0019\n0logjD\u000e\t\u00001\nh((1\u0000r=2)ei\u0012)jd\u0012\u0014sup\nz2\nhkK(z)ktr=o(h1\u0000d)\nand logjD\u000e \u00001\nh(0)j\u0015\u0000CkK(z0)ktr.\nCombining this estimates with Jensen’s formula for D\u000e\t\u00001\nhgives us\nZ1\u0000r=2\n0n(s)\nsds\u0014(1 +C) sup\nz2\nhkK(z)ktr=o(h1\u0000d)\nand since the map s7!n(s)is increasing we have\nn(1\u0000r)\u00142\nrZ1\u0000r=2\n0n(s)\nsds\u00142\nr(1 +C) sup\nz2\nhkK(z)ktr\nThe number of zeros of Din!his equal to the number of zeros of D\u000e\t\u00001\nhin\th(!h)which is\nobviously less than n(1\u0000r).\n7Figure 2: If ris sufficiently close to 0thene\t(e!)\u001aD(0;1\u0000r).\nNotice that rdoes not depend on hbecause \nh,!h,z0and\thare just rescaled versions of e\n,\ne!,ez0ande\t. We therefore obtain the desired bound : the number of zeros of Din!his ao(h1\u0000d)\nand the proof of Theorem 1.3 is complete.\n3 Proof of Proposition 1\nIn order to ease the notations we will use the Landau notation (or “big O” notation) directly for\noperators throughout all this section. It must always be interpreted as a Landau notation for the\nL2norm of the operator when the semi-classical parameter hgoes to zero. For example if we write\nP\u0000z=P\u00001 +O(h)we mean that\nkP\u0000z\u0000P+ 1kL2=O(h).\n3.1 Idea of the proof\nNotice thatP\u0000z=P\u00001 +O(h)and that the principal symbol of P\u00001isj\u0018j2\ng\u00001. Using\nfunctional calculus (see (15)) it is easy to find some pseudo-differential operator A3such that the\nprincipal symbol of A3(P\u0000z)is1wherejj\u0018j2\ng\u00001j\u0015Chfor some fixed but large enough C. On\nthe set wherejj\u0018j2\ng\u00001j<Ch we use a different approach. We start with the formula\n\u0000i\nhZT\n0eit(P\u0000z)=hdt(P\u0000z) = 1\u0000eiT(P\u0000z)=h\nand we would like to have keiT(P\u0000z)=hkL2<1for some time Tso1\u0000eiT(P\u0000z)=hwould be invert-\nible. As we will show in Lemma 3.2 the operator eiT(P\u0000z)=hbehaves like\ne\u0000iTz=heiTh\u0001OpAW\nh(G2T(x;\u0018=2))\n8for functions that are concentrated around the energy layer p\u00001(1). Here OpAW\nhdenotes the anti-\nWick quantization1and\u001etis the Hamiltonian flow associated with p. We thus see that we cannot\nhope to havekeiT(P\u0000z)=hkL2<1for everyz2\nh. More precisely we cannot hope to have\nkeiT(P\u0000z)=hkL2<1when Im(z=h)\u00152C\u0000\n1. However, for almost every (x;\u0018)inp\u00001(1)we have\nlimt!11\ntlogkG2t(x;\u0018=2)k\u00142\u0003+and so forTlarge enough we have\n1\nTlogkG2T(x;\u0018=2)k\u00142\u0003++\"=2 (9)\nfor every (x;\u0018)of a compact set K\u001ap\u00001(1)with large measure. If \u001fis aC1function with\ncompact support in \u0018such that\u001f(x;\u0018) = 0 for every (x;\u0018)that does not satisfy (9) then\nkeiT(P\u0000z)=hOpAW\nh(\u001f)kL2<1.\nIn order to localize this to the energy layer jj\u0018jg\u00001j< Ch we will use operators of the form\nf\u0000P\u00001\nh\u0001\nwheref2S(R)is properly chosen. For instance we will show that the operator\nf\u0012P\u00001\nh\u0013\nOpAW\nh(1\u0000\u001f)f\u0012P\u00001\nh\u0013\nhas a small trace norm compared to h1\u0000d. Then, by choosing carefully f,\u001fandTdepending on h\nwe can define an operator R(z)which has the properties presented in Proposition 1.\n3.2 Construction of the operator R(z)\nAccording to Oseledec’s theorem for almost every (x;\u0018)2T\u0003Mthe Lyapunov exponents of\n(Gt(x;\u0018))t\u00150are well defined. Recall that\n\u0003\u0000= ess inf\n(x;\u0018)2T\u0003M\u00151(x;\u0018)and\u0003+= ess sup\n(x;\u0018)2T\u0003M\u0015n(x;\u0018),\nwhere\u00151(x;\u0018)\u0014:::\u0014\u0015n(x;\u0018)are (if they exist) the Lyapunov exponents of (Gt(x;\u0018))t\u00150. So for\nalmost every (x;\u0018)inf(x;\u0018)2T\u0003M:j\u0018jg= 1]gwe have\nlim\nt!+11\ntlog(kG2t(x;\u0018=2)k) = 2\u0015n(x;\u0018)\u00142\u0003+.\nAccording to Egorov’s theorem for every \";\u0011 > 0there exist a compact set K\u001ap\u00001(1)and a time\nT0such that for every T\u0015T0and every (x;\u0018)inKwe have\n1\nTlog(kG2T(x;\u0018=2)k)<2\u0003++\"=2\nand the Liouville’s measure of p\u00001(1)nKis smaller than \u0011. Now remark that the geodesic flow is\ncontinuous and so, for a fixed T, the matrix GTdepends continuously on (x;\u0018). This means that if\na point (x;\u0018)is close enough to Kthen we still have\n1\nTlog(kG2T(x;\u0018=2)k)<2\u0003++\"=2.\nConsequently there exists some C1smooth function e\u001f(T)\nK:T\u0003M!R+such thate\u001f(T)\nKequals 1\nonK,supp(e\u001f(T)\nK)\u001af(x;\u0018)2T\u0003M:j\u0018jg2[1=2; 3=2]gand\n(x;\u0018)2supp(e\u001f(T)\nK) =)1\nTlog(kG2T(x;\u0018=2)k)<2\u0003++\"=2.\nWe then define the function \u001f(T)\nK=e\u001f(T)\nK\u000ej\u000e\u001eTwherej(x;\u0018) = (x;\u0000\u0018)and\u001eis the hamiltonian\nflow generated by p. Notice that the functions jand\u001eTboth preserve the Liouville measure on\np\u00001(1).\n1See Annex A for a definition.\n9We now choose some non negative function f2S(R)such thatf(0) = 1 andsupp( ^f)is compact.\nLetCbe some positive constant which will be fixed later and let us define the three operators A1,\nA2andA3as\nA1(z) =\u0000f\u0012P\u00001\nCh\u0013\nOpAW\nh(\u001f(T)\nK)f\u0012P\u00001\nCh\u0013i\nhZT\n0eit(P\u0000z)=hdt,\nA2=\u0000f\u0012P\u00001\nCh\u0013\nOpAW\nh(1\u0000\u001f(T)\nK)f\u0012P\u00001\nCh\u0013i\nhZT\n0eit(P\u0000z0)=hdt,\nandA3=\u0012\n1\u0000f\u0012P\u00001\nCh\u0013\u0013\nw(P\u00001)\u0012\n1\u0000f\u0012P\u00001\nCh\u0013\u0013\n.\nHere OpAW\nh(b)is theh-anti-Wick quantization of a symbol bandwis defined by\nw:x7!1\nx\u0010\n1\u0000\u001f\u0010x\nCh\u0011\u0011\nwhere\u001fis some smooth cut-off function with \u001f(x) = 1 around zero and some small support\nthat will be chosen later. For a definition of the anti-Wick quantization see Annex A. We finally\ndefineR(z)as the sum of A1,A2andA3and we are now ready to state a more precise version of\nProposition 1.\nProposition 2. We have the following equalities.\nA1(z)(P\u0000z) =f\u0012P\u00001\nCh\u0013\nOpAW\nh(\u001f(T)\nK)f\u0012P\u00001\nCh\u0013\n+O\u00121\nC\u0013\n+O\u0010\ne\u0000T\"=2\u0011\n+OT(h1=2)(10)\nA2(P\u0000z0) =f\u0012P\u00001\nCh\u0013\nOpAW\nh(1\u0000\u001f(T)\nK)f\u0012P\u00001\nCh\u0013\n+O\u00121\nC\u0013\n+O\u0010\ne\u0000T\"=2\u0011\n+OT(h1=2)(11)\n\r\r\r\r\rA3(P\u0000z)\u0000\u0012\n1\u0000f\u0012P\u00001\nCh\u0013\u00132\r\r\r\r\r\nL2\u0014O\u00121\nC\u0013\n+\u000e (12)\nWhere\u000e > 0depends only on \u001fandfand can be made arbitrarily small. We moreover have the\nfollowing bounds\nkA2(P\u0000z)kTr\u0014 Z\np\u00001(1)1\u0000\u001f(T)\nKdL0!\n\u0001OC(hd\u00001) (13)\n\r\r\r\rf\u0012P\u00001\nCh\u0013\nOpAW\nh(1\u0000\u001f(T)\nK)f\u0012P\u00001\nCh\u0013\r\r\r\r\nTr\u0014 Z\np\u00001(1)1\u0000\u001f(T)\nKdL0!\n\u0001OC(hd\u00001) (14)\nwhereL0is the Liouville measure on p\u00001(1).\n3.3 Proof of Proposition 2\nWe start by proving a lemma on commutators with f\u0000P\u00001\nCh\u0001\n.\nLemma 3.1. LetUbe a pseudo-differential operator on Mwith symbol ucompactly supported in the\n\u0018variable. We have the following bound.\n\r\r\r\r\u0014\nf\u0012P\u00001\nCh\u0013\n;U\u0015\r\r\r\r\nL2=O\u00121\nC\u0013\n.\n10Proof. We first recall the formula\nf\u0012P\u00001\nCh\u0013\n=1p\n2\u0019Z\nR^f(t)eitP\u00001\nChdt. (15)\nAccording to this formula\n\u0014\nf\u0012P\u00001\nCh\u0013\n;U\u0015\n=Z\nR^f(t)h\neitP\u00001\nCh;Ui\ndt=Z\nR^f(t)e\u0000it\nChh\neitP\nCh;Ui\ndt.\nSinceeitP\nChis an isometry we have\n\r\r\rh\neitP\nCh;Ui\r\r\r\nL2=\r\r\reitP\nChUe\u0000itP\nCh\u0000U\r\r\r\nL2\nand by Egorov’s theorem we know that eitP\nChUe\u0000itP\nCh= Oph(u\u000e\u001et=C) +O(h)where\u001eis the\nHamiltonian flow on T\u0003Massociated with p. Notice that if we write (y;\u0011) =\u001et=C(x;\u0018)then\nj\u0018j2\nx=j\u0011j2\nysouandu\u000e\u001et=Care both compactly supported in \u0018. Consequently we have\n\r\ru\u0000u\u000e\u001et=C\r\r\n1=O\u00121\nC\u0013\nuniformly in t2supp( ^f)and the same estimates goes with all the derivatives of u\u0000u\u000e\u001et=C.\nAccording to Calderon-Vaillancourt’s theorem we then have\n\r\r\rh\neitP\nCh;Ui\r\r\r\nL2=O\u00121\nC\u0013\nuniformly in t2supp( ^f)and we finally get\n\r\r\r\r\u0014\nf\u0012P\u00001\nCh\u0013\n;U\u0015\r\r\r\r\nL2=O\u00121\nC\u0013\n.\nWe also need a lemma to approximate eit(P\u0000z)=h.\nLemma 3.2. Lett2[0;T]be a fixed positive real number and ube a symbol compactly supported, we\nhave\r\r\rOpAW\nh(u)h\neitP=h\u0000OpAW\nh(G2t\u000ej\u000e\u001et(x;\u0018=2))e\u0000ith\u0001i\r\r\r\nL2(M)=OT(h1=2)\nwherej(x;\u0018) = (x;\u0000\u0018)for every (x;\u0018)2T\u0003Mand\u001etis the Hamiltonian flow generated by p.\nIn other words the damped propagator can be, at the lowest order, factorized by the undamped\npropagator and a damping part which principal symbol is given by the function G. We here give\nthe same proof as in [Kle18] but an arguably more digest proof of a similar result can be found for\nexample in Lemma 6 of [Non11].\nProof. We are only going to prove this result with M=Rdwith a metric g, the extension to any\ncompact Riemannian manifold is straightforward. The first step of the proof is to precisely describe\nthe action of eitP=hon coherent state, this is a classical result and we will follow the presentation\nand notations of [Rob06]. Let g:Rd!Cbe the function defined by\ng:x7!1\n\u0019d=4exp\u0000\n\u0000kxk2\n2=2\u0001\n.\nand we note '0= \u0003hgwhere \u0003his the dilatation operator defined by \u0003hf(x) =h\u0000d=4f(h\u00001=2x).\nIn other words we have\n'0(x) =1\n(\u0019h)d=4exp\u0012\u0000kxk2\n2h\u0013\n.\n11If\u001a= (x0;\u00180)is a point of T\u0003Rd=R2dwe define'\u001aby\n'\u001a=T(\u001a)'0\nwhereT(\u001a) = exp\u0000i\nh(\u00180\u0001x+ihx0\u0001@x)\u0001\nis the Weyl operator, in other words\n'\u001a(x) =eix\u0001\u00180=h'0(x\u0000x0).\nThe function '\u001ais called the coherent state associated with (x0;\u00180). Finally, if v2Cnwe define\n'\u001a;v='\u001a\u0001v. According to [Rob06], for every integer N, every\u001ain a compact set Kand every\nt2[0;T]we have\r\r\reitP=h'\u001a;v\u0000 (N)\n\u001a;v(t)\r\r\r\nL2=OK;T(hN+1\n2)\nwhere\n (N)\n\u001a;v(t) =ei\u000et=hT(\u001at)\u0003heG2t(j(e\u001a))M[Ft]0\n@X\n0\u0014j\u0014Nhj=2bj(t)g1\nA.\nWe will only give a partial description of the terms of  (N)\n\u001a;v(t)here, for a complete definition see\n[Rob06]. The point \u001at2T\u0003Rdis simply given by the inverse Hamiltonian flow : \u001at= (x\u0000t;\u0018\u0000t) =\n\u001e\u0000t(x0;\u00180) =\u001e\u0000t\u001aande\u001a= (x0;\u00180=2). The quantity \u000etis real and only depends on t. The function\neGt:T\u0003Rd!Mn(C)is defined as the solution of the following differential equation :\n(\neG0(x0;\u00180) = Idn\n@teGt(x0;\u00180) =\u0000a(xt)pzeGt(x0;\u00180)(16)\nwe thus have @teG2t(j(e\u001a)) =\u00002a(x\u0000t)pzeG2t(j(e\u001a)). The function j:T\u0003Rd!T\u0003Rdis defined by\nj(x;\u0018) = (x;\u0000\u0018). The functions bj(t)are polynomial functions in the xvariable with coefficients\ninCn, the first term b0(t)is constant and equal to v. The termM[Ft]describe the sprawl of the\nGaussiangunder the action of the propagator, Ftis a flow of linear symplectic transformation and\nMis a realization of the metaplectic representation. We can apply the same result when a= 0and\nwe find\n\r\r\r\r\r\re\u0000iht\u0001'\u001a;v\u0000ei\u000et=hT(\u001at)\u0003hM[Ft]0\n@X\n0\u0014j\u0014Nhj=2cj(t)g1\nA\r\r\r\r\r\r\nL2=OK;T(hN+1\n2),\nthe only difference with eitP=h'\u001a;vis in the polynomials cjand in the absence of eGt, note that we\nalso havec0(t) =v. Since the function v7! (N)\n\u001a;v(t)is linear and since b0(t) =c0(t) =vwe can\ndefine by induction some matrices q1(t;\u001a);:::;qN(t;\u001a)depending polynomially on xsuch that\n\u0010\n1 +h1=2q1(t;\u001a) +:::+hN=2qN(t;\u001a)\u00110\n@X\n0\u0014j\u0014Nhj=2cj(t)g1\nA=\n0\n@X\n0\u0014j\u0014Nhj=2bj(t)g1\nA+O(hN+1\n2) (17)\nfor everyv2Cn,t2[0;T]and\u001a2K. Indeed the matrix qi(t;\u001a)must satisfy the relation\nci(t) +iX\nj=1qj(t;\u001a)ci\u0000j(t) =bi(t)()qi(t;\u001a)v=bi(t)\u0000ci(t)\u0000i\u00001X\nj=1qj(t;\u001a)ci\u0000j(t)\n12for everyv2Cnand since all the polynomials bj(t)andcj(t)depend linearly on vthis defines\nuniquely the matrix qi(t;\u001a)if theqj(t;\u001a); j= 1;:::;i\u00001are fixed. Now let f1;f2:R2d!Mn(C)\nbe two symbols of order \u0014m2R; then we have\nM[Ft]OpW\n1(f1) = OpW\n1\u0000\nf1\u000eF\u00001\nt\u0001\nM[Ft]\nwhere OpW\n1is the Weyl quantization for h= 1. A proof of this result can be found in [Rob06]. We\nalso have\n\u0003h(f1\u0001f2) =hd=4\u0003h(f1)\u0001\u0003h(f2),\nandT(zt)(f1\u0001f2) =T(zt)(f1)\u0001T(zt)(f2).\nUsing the previous relations and (17) we see that there exist some matrices Q1(t;\u001a);:::;QN(t;\u001a)\ndepending polynomially on xsuch that\n\r\r\r\r\r\r (N)\n\u001a;v\u0000 \neG2t(j(e\u001a)) +NX\ni=1hi=2Qi(t;\u001a)!\nei\u000et=hT(\u001at)\u0003hM[Ft]0\n@X\n0\u0014j\u0014Nhj=2cj(t)g1\nA\r\r\r\r\r\r\nL2=O(hN+1\n2).\nNotice that the matrix eGt(j(\u001a))does not depend on x, this is due to the fact that b0(t) =c0(t).\nConsequently we have\n\r\r\r\r\reitP=h'\u001a;v\u0000 \neG2t(j(e\u001a)) +NX\ni=1hi=2Qi(t;\u001a)!\ne\u0000iht\u0001'\u001a;v\r\r\r\r\r\nL2=O(hN+1\n2) (18)\nand theO(hN+1\n2)is uniform in \u001a2Kandt2[0;T]. If we take some symbol uwith supp(u)bK\nthen the estimates become uniform in \u001a:\n\r\r\r\r\rOpAW\nh(u)eitP=h'\u001a;v\u0000OpAW\nh(u) \neG2t(j(e\u001a)) +NX\ni=1hi=2Qi(t;\u001a)!\ne\u0000iht\u0001'\u001a;v\r\r\r\r\r\nL2=O(hN+1\n2)(19)\nuniformly in t2[0;T]and\u001a2R2d. The next step is to remark that e\u0000iht\u0001'\u001a;vis a sum of\nderivatives of anisotropic coherent states associated with \u001at. So, by using \u001a=\u001et\u001at, there exist\nsome symbols g(i)\ntsuch that\n\r\r\rOpAW\nh(u)h\neG2t(j(e\u001a))e\u0000iht\u0001\u0000\u0010\nOpAW\nh(eG2t\u000ej\u000e\u001e2t)+\nNX\ni=1hj=2OpAW\nh(g(i)\nt)!\ne\u0000iht\u0001#\n'\u001a;v\r\r\r\r\r\nL2=O(hN+1\n2)\nuniformly in \u001a2R2dandt2[0;T]. The same is true for the matrices Qiand thus there exist some\nsymbolsG(i)\ntsuch that\n\r\r\r\r\rOpAW\nh(u)\"\neitP=h\u0000 \nOpAW\nh(eG2t\u000ej\u000e\u001e2t) +NX\ni=1hj=2OpAW\nh(G(i)\n2t)!\ne\u0000iht\u0001#\n'\u001a;v\r\r\r\r\r\nL2=O(hN+1\n2).\n(20)\nWe now use the fact that for every function f= (f1;:::;fn)2L2(Rd)nwe have\nfi=1\n(2\u0019h)dZ\nR2dhfi;'z;eiiL2(Rd)fidz (21)\nwhereei= (0;:::; 0;1;0:::;0)is thei-th vector of the canonical basis of Cn. Combining (20) and\n(21) we get that\n\r\r\r\r\rOpAW\nh(u)\"\neitP=h\u0000 \nOpAW\nh(eG2t\u000ej\u000e\u001e2t) +NX\ni=1hj=2OpAW\nh(G(i)\n2t)!\ne\u0000iht\u0001#\r\r\r\r\r\nL2=O(hN+1\n2\u0000d).\n13Notice thatkOpAW\nh(G(i)\n2t)kL2is bounded uniformly in hand so, if we take N+ 1>2dand only\nkeep the principal terms we get\n\r\r\rOpAW\nh(u)\u0010\neitP=h\u0000OpAW\nh(eG2t\u000ej\u000e\u001e2t(x;\u0018=2))e\u0000iht\u0001\u0011\r\r\r\nL2=OT(h1=2).\nThe last final step is to remark that eGtdepends smoothly on z= 1+O(h): we haveeGt=Gtwhen\nz= 1and soeGt=Gt+OT(h). Plugging this in the previous equality, we get what we wanted :\n\r\r\rOpAW\nh(u)\u0010\neitP=h\u0000OpAW\nh(Gt\u000ej\u000e\u001et)e\u0000iht\u0001\u0011\r\r\r\nL2=OT(h1=2).\nThis finishes the proof of the lemma.\nWe can now prove (10); we start by writing\n\u0000i\nhZT\n0eit(P\u0000z)=hdt(P\u0000z) = Id\u0000eiT(P\u0000z)=h\nand so\nA1(z)(P\u0000z) =f\u0012P\u00001\nCh\u0013\nOpAW\nh(\u001f(T)\nK)f\u0012P\u00001\nCh\u0013\u0010\nId\u0000eiT(P\u0000z)=h\u0011\n.\nUsing the fact that OpAW\nh(\u001f(T)\nK) = Oph(\u001f(T)\nK) +OT(h)twice and applying Lemma 3.1 we get\nf\u0012P\u00001\nCh\u0013\nOpAW\nh(\u001f(T)\nK)f\u0012P\u00001\nCh\u0013\ne\u0000iT(P\u0000z)=h=\nf\u0012P\u00001\nCh\u00132\nOpAW\nh(\u001f(T)\nK)e\u0000iT(P\u0000z)=h+OT(h) +O\u00121\nC\u0013\n.\nThe operator norm of f\u0000P\u00001\nCh\u00012is bounded bykf2k1and it only remains to estimate the operator\nnorm of OpAW\nh(\u001fK)eiT(P\u0000z)=h. According to Lemma 3.2 we have\nOpAW\nh(\u001f(T)\nK)eiT(P\u0000z)=h=e\u0000itz=hOpAW\nh(\u001f(T)\nK)OpAW\nh(G2T\u000ej\u000e\u001eT(x;\u0018=2))e\u0000ihT\u0001\n+OT(h1=2)\n=e\u0000itz=hOpAW\nh(\u001f(T)\nKG2T\u000ej\u000e\u001eT(x;\u0018=2))e\u0000ihT\u0001+OT(h1=2)\nSincee\u0000ihT\u0001is an isometry we have\n\r\r\rOpAW\nh(\u001f(T)\nK)eiT(P\u0000z)=h\r\r\r\nL2\u0014k\u001f(T)\nKG2T\u000ej\u000e\u001eT(x;\u0018=2)k1eTIm(z=h)+OT(h1=2),\nrecall that\u001f(T)\nK=e\u001f(T)\nK\u000ej\u000e\u001eTso\nk\u001f(T)\nKG2T\u000ej\u000e\u001eT(x;\u0018=2)k1=ke\u001f(T)\nKG2T(x;\u0018=2)k1.\nIf we then use the definition of e\u001f(T)\nK,GTand\nhwe see that\nke\u001f(T)\nKG2T(x;\u0018=2)k1e\u0000TIm(z=h)\u0014e\u0000T\"=2,\nwhich finishes the proof of (10). The same technique is used to prove (11) except that we don’t\neven have to use Lemma 3.2 because Im(z0=h)is small enough :\n\r\r\reiT(P\u0000z0)\r\r\r\nL2\u0014e\u0000T.\n14We continue by proving (12). Recall that P\u0000z=P\u00001 +O(h)and thatkwk1=O\u00001\nCh\u0001\nso\nA3(z)(P\u0000z) =A3(z)(P\u00001) +O\u00121\nC\u0013\n=\u0012\n1\u0000f\u0012P\u00001\nCh\u0013\u0013\nw(P\u00001)(P\u00001)\u0012\n1\u0000f\u0012P\u00001\nCh\u0013\u0013\n+O\u00121\nC\u0013\n.\nAccording to the definition of wwhe havew(P\u00001)(P\u00001) = 1\u0000\u001f\u0000P\u00001\nCh\u0001\nand so it only remains\nto estimate the operator norm of\n\u0012\n1\u0000f\u0012P\u00001\nCh\u0013\u0013\n\u001f\u0012P\u00001\nCh\u0013\u0012\n1\u0000f\u0012P\u00001\nCh\u0013\u0013\n.\nThe norm of this operator is bounded by k(1\u0000f)\u001f(1\u0000f)k1, recall that f(0) = 1 and thatfis\ncontinuous so by choosing \u001fwith sufficiently small support around 0we get\nk(1\u0000f)\u001f(1\u0000f)k1<\u000e\nfor any fixed positive \u000eand (12) is proved.\nIt only remains now to prove (13) and (14), we start with (14) and we proceed as in [Sjö00].\nSince the Anti-Wick quantification is positive and since 1\u0000\u001f(T)\nK\u00150we know that\n\r\r\r\rf\u0012P\u00001\nCh\u0013\nOpAW\nh(1\u0000\u001f(T)\nK)f\u0012P\u00001\nCh\u0013\r\r\r\r\nTr\n= Tr\u0014\nf\u0012P\u00001\nCh\u0013\nOpAW\nh(1\u0000\u001f(T)\nK)f\u0012P\u00001\nCh\u0013\u0015\n= Tr\"\nf\u0012P\u00001\nCh\u00132\nOpAW\nh(1\u0000\u001f(T)\nK)#\n=1p\n2\u0019Tr\u0014Z\nRcf2(t)eitP\u00001\nChOpAW\nh(1\u0000\u001f(T)\nK)dt\u0015\n=Cp\n2\u0019Tr\u0014Z\nRcf2(Ct)eitP\u00001\nhOpAW\nh(1\u0000\u001f(T)\nK)dt\u0015\n.\nForClarge enough we have\nsuppcf2(C\u0001)\u001a]\u00001\n2Tmin;1\n2Tmin[ (22)\nwhereTminis the smallest possible length of a closed trajectory in p\u00001(1)for the Hamiltonian flow\ngenerated by p. Whenever (22) is satisfied we know that\nlim\nh!0Tr\u0014Z\nRcf2(Ct)eitP\u00001\nhOpAW\nh(1\u0000\u001f(T)\nK)dt\u0015\n=Cdh1\u0000dcf2(0)Z\np\u00001(1)1\u0000\u001f(T)\nKL0(d\u001a) (23)\nwhereL0is the Liouville measure on p\u00001(1)andCdonly depends on d, the dimension of M. One\ncan find a proof of this classical fact in [DiSj99] for instance. We now use the formula kABkTr\u0014\nkAkkBkTrand the fact that\r\rf\u0000P\u00001\nCh\u0001\r\r\nL2\u0014kfk1to get (14). We use the same technique for (13)\nand so it only remains to show that\n\u0000i\nhZT\n0eit(P\u0000z0)=hdt(P\u0000z)\nis uniformly bounded in h. We recall thatP\u0000z0is invertible and so we can write\n\u0000i\nhZT\n0eit(P\u0000z0)=hdt(P\u0000z) =\u0010\nId\u0000eiT(P\u0000z0)=h\u0011\n(P\u0000z0)\u00001(P\u0000z).\nBy using the energy formula (3) we see that\r\reiT(P\u0000z0)\r\r\nL2\u0014e\u0000T\u00141and that\r\r(P\u0000z0)\u00001\r\r\nL2is\nuniformly bounded in h. Consequently the operator (P\u0000z0)\u00001(P\u0000z)is uniformly bounded in L2\nnorm when hgoes to 0. This finishes the proof of Proposition 2.\n153.4 End of the proof of Proposition 1\nWe now use Proposition 2 to prove Proposition 1. According to Proposition 2 we have\nR(z)(P\u0000z) = Id\u0000f\u0012P\u00001\nh\u0013\nOpAW\nh(1\u0000\u001f(T)\nK)f\u0012P\u00001\nh\u0013\n+A2(P\u0000z)\n+O\u00121\nC\u0013\n+O(\u000e) +O(e\u0000\"T=2) +OT(h1=2).\nWe define\nR2(z) =\u0000f\u0012P\u00001\nh\u0013\nOpAW\nh(1\u0000\u001f(T)\nK)f\u0012P\u00001\nh\u0013\n+A2(P\u0000z)\nandR1(z) =R(z)(P\u0000z)\u0000R2(z).\nWe start by fixing a constant Clarge enough so the remainder O\u00001\nC\u0001\nis smaller than 1=100. We\nthen chose \u001fin the definition of A3so that the remainder O(\u000e)is smaller than 1=100. Fix some\narbitrary\u0011>0, forTlarge enough the remainder O(e\u0000\"T=2)is smaller than 1=100and there exists\nsome\u001f(T)\nKsuch that\nlim sup\nh!0hn\u00001kR2(z)kTr<\u0011.\nWe finally take hsmall enough so that the remainder OT(h1=2)is also smaller than 1=100. By\ndoing so we have constructed an operator R(z)such thatR(z)(P\u0000z) = Id +R1(z) +R2(z)with\nkR1(z)kL2<1=10andkR2(z)kTr\u0014\u0011h1\u0000dforhsmall enough. When hgoes to 0we can repeat\nthe same process and make \u0011arbitrarily small and get kR2(z)kTr=o(h1\u0000d). Moreover according\nto Proposition 2 we have\nkR2(z0)kL2=O\u00121\nC\u0013\n+O(e\u0000T\"=2) +OT(h1=2)\nand as before we can choose CandTin order to also have kR2(z0)kL2<1=10forhsmall enough.\nThis implies that R(z0)(P\u0000z0)is invertible and that\r\r\r(R(z0)(P\u0000z0))\u00001\r\r\ris uniformly bounded\ninh. The proof of Proposition 1 is thus finished.\n16A Semi-classical anti-Wick quantization\nIn this appendix we present, without any proof, a construction of the h-Anti-Wick quantization\nand some of its basic properties. The proofs for h= 1 can be found in [Gos11]. We start by\nconstructing it on Rdfor scalar valued symbols.\nDefinition. Let(x;\u0018)be a point of T\u0003Rd=R2d, we define the function e(x;\u0018):Rd!Rdby\ne(x;\u0018):y7!1\n(h\u0019)d=4e\u0000kx\u0000yk2\n2=2heiy\u0001\u0018=h.\nA standard calculation shows that ke(x;\u0018)kL2(Rd)= 1. We now define \u0005(x;\u0018):L2(Rd)!L2(Rd)\nas the orthogonal projector on the vector subspace generated by e(x;\u0018).\nDefinition. Leta2S0\n1;0(R2d), we define its h-anti-Wick quantization OpAW\nh(a) :L2(Rd)!L2(Rd)\nby\nOpAW\nh(a) =1\n(2\u0019h)dZ\nR2da(x;\u0018)\u0005(x;\u0018)dxd\u0018 .\nThe anti-Wick quantization has a few convenient properties.\nProposition 3. Ifais real valued and non negative then OpAW\nh(a)is self-adjoint and positive:\n8u2L2(Rd);\nOpAW\nh(a)u;u\u000b\nL2(Rd)\u00150.\nMoreover we have the following estimates\n\r\rOpAW\nh(a)\r\r\nL2(Rd)\u0014kak1\nand\nOpAW\nh(1) = IdL2(Rd).\nTheh-anti-Wick quantization is linked to the h-Weyl quantization in the following way.\nProposition 4. OpAW\nh(a)is ah-pseudo differential operator of order \u00140and we have\nOpAW\nh(a) = OpW\nh(a\u0003\")\nwhere\": (y;\u0011)7!(h\u0019)\u0000n=2e\u0000k(y;\u0011)k2\n2=h.\nConsequently we know that\r\rOpAW\nh(a)\u0000OpW\nh(a)\r\r\nL2(Rd)=O(h)whenhgoes to 0. The same\nconstruction can be used for symbols avalued in Mn(C), the operator OpAW\nh(a)then acts on\nL2(Rd)n. The previous results still hold in this case but Proposition 3 needs to be slightly modified:\nProposition 5. Ifais valued in H+\nn(C)the space of Hermitian positive semi-definite matrices then\nOpAW\nh(a)is self-adjoint and positive:\n8u2L2(Rd)n;\nOpAW\nh(a)u;u\u000b\nL2(Rd)n\u00150.\nMoreover we have the following estimates\n\r\rOpAW\nh(a)\r\r\nL2(Rd)n\u0014sup\n(x;\u0018)2Rdka(x;\u0018)k2\nand\nOpAW\nh(Idn) = IdL2(Rd)n.\nWe can then define a h-anti-Wick quantization on a manifold Musing a partition of unity.\n17B Multiplicative ergodic theorem of Oseledets\nIn this appendix we present, without any proofs, the multiplicative ergodic theorem of Oseledets\nand some related results. The proofs can be found in [Led84] and [BaPe13]. Let (X;\u0016)be a\nprobability space and let ('t)t2Rbe a one parameter group of measure preserving functions from\nXtoX. LetG:R\u0002X!Mn(C)be a cocycle, this means that Gsatisfies the following conditions\n:\n-8x2X,G(0;x) = Idn,\n-8x2X,8s;t2R,G(s+t;x) =G(s;'t(x))G(t;x).\nTheorem B.1 (Oseledets) .Assume that logkG(t;\u0001)kandlogkG(t;\u0001)\u00001kare both in L1(X;\u0016)for\neveryt2[0; 1]. For\u0016-almost every x2Xthere exists real numbers \u00151(x)< ::: < \u0015 k(x)(x)and a\ndecomposition Cn=Vx\n1\b:::\bVx\nk(x)such that for every v2Vx\ninf0g\nlim\nt!\u000611\ntlogkG(t;x)vk=\u0015i(x).\nMoreover, the \u0015iare invariant by 't:\u0015i(x) =\u0015i('(x))and\nG(t;x)Vx\ni=V't(x)\ni .\nThe numbers \u0015iare called the Lyapunov exponents of GanddimVx\niis called the multiplicity of\nthe Lyapunov exponent \u0015i(x). If the dynamical system (X;\u0016; ('t)t2R)is ergodic then the Lyapunov\nexponents and their multiplicity are constant on a full measure set of X. Note that the choice of\nthe norm over the space Mn(C)does not matter since they are all equivalent. Let x2Xbe a point\nfor which the Lyapunov exponents are well defined, and let \u00161\u0014:::\u0014\u0016n(x)be the Lyapunov\nexponents counted with multiplicity, then\ni\u00001X\nj=0\u0016n\u0000j(x) = lim\nt!\u000611\ntlogk\u0003iGt(t;x)k\nwhere \u0003iGacts on \u0003iCnby\n\u0003iG(t;x)(u1^:::^ui) =G(t;x)u1^:::^G(t;x)ui.\nIn particular, the greatest Lyapunov exponent is given by the norm of G:\n\u0016n(x) = lim\nt!\u000611\ntlogkG(t;x)k:\nIf the matrix G(t;x)is invertible for every tthen we also have\n\u00161(x) = lim\nt!\u000611\ntlog\u0000\r\rG(t;x)\u00001\r\r\u0001\n.\n18Bibliography\n[Ana10] Anantharaman, N. Geom. Funct. Anal. (2010) 20: 593. https://doi.org/10.1007/s00039-\n010-0071-x\n[BaPe13] L. Barreira, Y. Pesin, Introduction to smooth ergodic theory , Graduate studies in mathe-\nmatics volume 148 (2013), AMS.\n[DiSj99] Dimassi, M. and Sjöstrand, J., Spectral asymptotics in the semi-classical limit, London\nMath. Soc. Lecture Notes 268 , Cambridge Univ. Press, 1999.\n[Gos11] Maurice A. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical\nPhysics , Pseudo-Differential Operators Theory and Applications Vol. 7, Birkhäuser.\n[Kle17] Guillaume Klein. Best exponential decay rate of energy for the vectorial damped wave equa-\ntion. SIAM J. Control Optim., 56(5), 3432–3453. (22 pages)\n[Kle18] Guillaume Klein. Stabilisation et asymptotique spectrale de l’équation des ondes amor-\nties vectorielle . PhD thesis, defended on 12 december 2018 at Strasbourg university.\nhttps://tel.archives-ouvertes.fr/tel-01943093v2.\n[Leb93] G. Lebeau. Equation des ondes amorties . Algebraic and Geometric Methods in Mathemati-\ncal Physics. Volume 19 of the series Mathematical Physics Studies pp 73-109.\n[Led84] F. Ledrappier, Quelques propriétés des exposants caractéristiques , Volume XII of the series\nÉcole d’Été de Probabilités de Saint-Flour (1984), 305-396.\n[MaMa82] A. S. Markus and V. I. Matsaev Comparison theorems for spectra of linear operators and\nspectral asymptotics Tr. Mosk. Mat. Obs., 45 (1982), 133–181\n[Non11] S. Nonnenmacher, Spectral theory of damped quantum chaotic systems ,\nhttps://www.imo.universite-paris-saclay.fr/ \u0018nonnenma/publis/Damped-wave-lectures3.pdf.\n[Rob06] D. Robert, Propagation of coherent states in quantum mechanics and applications,\nhttp://www.math.sciences.univ-nantes.fr/ \u0018robert/proc_cimpa.pdf.\n[Sjö00] J. Sjöstrand, Asymptotic Distribution of Eigenfrequencies for Damped Wave Equations , Publ.\nRIMS, Kyoto Univ. 36 (2000), 573-611.\n[Zwo12] M. Zworski, Semiclassical analysis , Graduate Studies in Mathematics volume 138 (2012),\nAMS.\n19"    },    {        "title": "2109.02044v1.Regularity_of_the_semigroups_associated_with_some_damped_coupled_elastic_systems_II__a_nondegenerate_fractional_damping_case.pdf",        "content": "arXiv:2109.02044v1  [math.AP]  5 Sep 2021REGULARITY OF THE SEMIGROUPS ASSOCIATED WITH SOME DAMPED CO UPLED\nELASTIC SYSTEMS II: A NONDEGENERATE FRACTIONAL DAMPING CAS E\nKA¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\nABSTRACT . In this paper, we examine regularity issues for two damped a bstract elastic systems;\nthe damping and coupling involve fractional powers µ,θ, with0≤µ,θ≤1, of the principal\noperators. The matrix defining the coupling and damping is no ndegenerate. This new work is\na sequel to the degenerate case that we discussed recently in [1]. First, we prove that for\n1/2≤µ,θ≤1, the underlying semigroup is analytic. Next, we show that fo rmin(µ,θ)∈(0,1/2),\nthe semigroup is of certain Gevrey classes. Finally, some ex amples of application are provided.\nCONTENTS\n1. Introduction 1\n2. Analyticity: case1\n2≤µ,θ≤1 5\n3. Gevrey semigroup: case min(µ,θ)∈(0,1\n2) 8\n4. An optimal result 11\n5. Examples of application 13\nReferences 14\n1. I NTRODUCTION\nLetHbe a Hilbert space with inner product /a\\}bracke⊔le{⊔.,./a\\}bracke⊔ri}h⊔and norm |.|. LetA1andA2be two\nself-adjoint on the Hilbert space H, strictly positive, with dense domains D(A1)andD(A2)\nrespectively. Let B1andB2be other self-adjoint, positive operators on the Hilbert sp aceH,\nwith dense domains D(B1)andD(B2)respectively, satisfying for some positive constants α0,\nα1,α2,β1,β2andµ,θ∈(0,1],\nB1≤α0B2,\nβ1Aθ\n2≤B2≤β2Aθ\n2,\nβ1Aθ\n2≤B2≤β2Aθ\n2,\nDate : September 7, 2021.\n2010 Mathematics Subject Classification. 47D06, 35B40.\nKey words and phrases. Semigroup regularity, fractional damping, structural dam ping, Kelvin-V oigt damping,\ncoupled elastic systems.\n12 KA ¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\nin the sense that,\n|B1\n2\n1u|2≤α0|B1\n2\n2u|2, u∈D(B1\n2\n2)⊆D(B1\n2\n1), (1.1)\nα1|Aµ\n2\n1u|2≤ |B1\n2\n1u|2≤α2|Aµ\n2\n1u|2, u∈D(B1\n2\n1) =D(Aµ\n2\n1), (1.2)\nβ1|Aθ\n2\n2u|2≤ |B1\n2\n2u|2≤β2|Aθ\n2\n2u|2, u∈D(B1\n2\n2) =D(Aθ\n2\n2). (1.3)\nNote that from (1.1)-(1.3), one has D(Aθ\n2\n2)⊆D(Aµ\n2\n1).\nSetVj=D(A1\n2\nj),j= 1,2. We assume that for each j= 1,2,Vj֒→H ֒→V′\nj, each injection\nbeing dense and compact, where V′\njdenotes the topological dual of Vj.\nLetαandγbe positive constants, and let βbe a nonzero real constant.\nConsider the evolution system\n(1.4)ytt+A1y+αB1yt+βB1zt= 0 in(0,∞)\nztt+A2z+βB1yt+γB2zt= 0 in(0,∞)\ny(0) =y0∈V1, yt(0) =y1∈H, z(0) =z0∈V2, zt(0)∈H.\nIntroduce the Hilbert space H=V1×H×V2×H, over the field Cof complex numbers, equipped\nwith the norm\n||Z||2=|A1\n2\n1u|2+|v|2+|A1\n2\n2w|2+|z|2,∀Z= (u,v,w,z)∈ H.\nDefine the operator\nAµ,θ=\n0I0 0\n−A1−αB10−βB1\n0 0 0 I\n0−βB1−A2−γB2\n\nwith domain\nD(A) =/braceleftig\n(u,v,w,z)∈V1×V1×V2×V2;A1u+αB1v+βB1z∈H, andA2w+βB1v+γB2z∈H/bracerightig\n.\nThen, by denoting v=utandz=wt, system (1.4) can be rewritten as an abstract linear evoluti on\nequation on the Hilbert space H,\n(1.5)/braceleftbiggdU\ndt(t) =Aµ,θU, t≥0\nU(0) = (u0,u1,z0,z1)\nIn the sequel we suppose that the constants α,β,α0andγsatisfy the following inequality\n(1.6) αγ > β2α0\nWe have,\nProposition 1. The operator Aµ,θis dissipative.\nProof. One easily checks that for every Z= (u,v,w,z)∈D(Aµ,θ),\nℜ(Aµ,θZ,Z) =−α|B1\n2\n1v|2−2βℜ(B1\n2\n1v,B1\n2\n1z)−γ|B1\n2\n2z|2\n≤ −α|B1\n2\n1v|2+2|β||B1\n2\n1v||B1\n2\n1z|−γ|B1\n2\n2z|2\n≤ −α|B1\n2\n1v|2+|β|\nc|B1\n2\n1v|2+α0|β|c|B1\n2\n2z|2−γ|B1\n2\n2z|2,REGULARITY OF SEMIGROUP ... COUPLED ELASTIC SYSTEMS 3\nfor some positive constant c. Then\n(1.7) ℜ(Aµ,θZ,Z)≤ −(α−|β|\nc)|B1\n2\n1v|2−(γ−α0|β|c)|B1\n2\n2z|.\nSinceαγ > β2α0, it is possible to choose csuch that k1:=α−|β|\nc>0andk2:=γ−α0|β|c >0\n(equivalently|β|\nα< c <γ\nα0|β|). So that the operator Aµ,θis dissipative. ⊔ ⊓\nFurther, the operator Aµ,θis densely defined, so Aµ,θis closable on H. Therefore, the\nLumer-Phillips Theorem shows that the operator Aµ,θgenerates a strongly continuous semigroup\nof contractions (S(t))t≥0on the Hilbert space H, which leads to the well-posedness of the system\n(1.4).\nMoreover, as in [1], the operator Aµ,θsatisfies\n(1.8) iR⊂ρ(Aµ,θ)\nwhereρ(Aµ,θ)denotes the resolvent set of Aµ,θ.\nAs a consequence, the semigroup etAis strongly stable [2].\nOur main goal is to study some regularity properties for the s olutions of the system (1.5).\nBefore going on, let us recall some definitions relevant to th e regularity of C0-semigroups.\nDefinition 1.1. LetT(t) :=etAbe aC0-semigroup on a Hilbert space H.\n(1)The semigroup T(t)is said to be analytic if\n•for some ϕ∈(0,π\n2),T(t)can be extended to Σϕ, where\nΣϕ={0}∪{τ∈C:|arg(τ)|< ϕ},\nso that for any x∈ H,τ/ma√s⊔o→T(τ)xis continuous on Σϕ, and for each τ1,τ2∈Σϕ,\nT(τ1+τ2) =T(τ1)T(τ2).\n•The map τ/ma√s⊔o→T(τ)is analytic over Σϕ\\{0}, in the sense of the uniform operator\ntopology of L(H).\n(2)The semigroup etAis said to be differentiable if for any x∈ H,t/ma√s⊔o→etAxis differentiable\non(0,∞).\n(3)The semigroup etAis said to be of Gevrey class δ(withδ >1) ifetAis infinitely\ndifferentiable and for any compact subset K ⊂(0,∞)and any λ >0, there exists a\nconstant C=C(λ,K)such that\n/bardblAnetA/bardblL(H)≤Cλn(n!)δ,∀t∈ K, n≥0.\nIn this paper we will use the following standard results to id entify analytic or Gevrey class\nsemigroups, based on the estimation for the resolvent of the generator of the semigroup.\nLemma 1.1. [9]LetA:D(A)⊂ H → H generate a C0-semigroup of contractions etAonH.\nSuppose that\n(1.9) iR⊂ρ(A)\nwhereρ(A)denotes the resolvent of A.\nThe semigroup etAis analytic if and only if\n(1.10) limsup\n|λ|→∞|λ|/bardbl(iλI−A)−1/bardblL(H)<∞.4 KA ¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\nLemma 1.2. [10] LetA:D(A)⊂ H → H generate a bounded C0-semigroup etAonH. Suppose\nthatAsatisfies the following estimate, for some 0< α <1,\n(1.11) limsup\n|λ|→∞|λ|α/bardbl(iλI−A)−1/bardblL(H)<∞.\nThenetAis of Gevrey class δfort >0, for every δ >1\nα.\nIn their work [3], G. Chen and Russell considered the followi ng system\nytt+Ay+Byt= 0 in(0,∞)\nwhereAis a self-adjoint operator on the Hilbert space H, strictly positive, with domain dense\ninHandBis a positive self-adjoint operator with dense domain in H. SetE=D(A1\n2)×H.\nThey show that if for every ρ >0, there exists ε(ρ)>0such that\n(2ρ−ε(ρ))A1\n2≤B≤(2ρ+ε(ρ))A1\n2,\nthen the underlying semigroup is analytic on E.\nThen, they conjecture the analyticity of the semigroup prov ided that\n∃0< ρ1< ρ2<∞:ρ1A1\n2≤B≤ρ2A1\n2,\nor else\nρ2\n1A≤B2≤ρ2\n2A.\nLater on, S.P. Chen and Triggiani responded to those conject ures by proving that if\n∃µ∈(0,1],∃0< ρ1< ρ2<∞:ρ1Aµ≤B≤ρ2Aµ,\nthen the semigroup is\n(1) analytic for1\n2≤µ≤1, but not analytic for 0< µ <1\n2, [4]\n(2) of Gevrey class δfor allδ >1\n2µfor0< µ <1\n2, [5]\nand this on a wide range of energy spaces, not only E. See also the works by Huang [6], and\nHuang and Liu [7].\nIn particular, [5] generalizes the work of Taylor [10] where the author discusses Gevrey semigroups,\nand illustrates his work with several examples including th e caseB= 2ρAµfor some positive\nconstant ρ.\nOur purpose in this note is to examine the following question s: Assume a coercive dissipative\nmechanism: αγ > β2α0. Do we have results similar to Chen-Triggiani [4, 5], namely , is the\nsemigroup (S(t))t≥0analytic for1\n2≤µ,θ≤1? And if s:= min(µ,θ)lies in(0,1\n2), is the\nsemigroup of Gevrey class δfor allδ >1\n2s?\nBefore answering those questions, we would like to mention t he work [8] where the authors\ndiscuss an abstract evolutionary system of the form\n˙Z=AZ\nwithAgiven by\nA=/parenleftbigg\n−A0B\nC−A1/parenrightbigg\n,\nwhereC=−B∗. They establish sufficient conditions on the operators A0,A1,BandCfor the\noperator Ato generate an exponentially stable, analytic, differenti able or Gevrey class semigroup.REGULARITY OF SEMIGROUP ... COUPLED ELASTIC SYSTEMS 5\nTheir results apply to many dynamical systems. However, the condition C=−B∗excludes the\nabstract system considered in this note, where we have B=C, if we use the notations in [8].\n2. A NALYTICITY :CASE1\n2≤µ,θ≤1\nTheorem 2.1. If1\n2≤µ,θ≤1the associated semigroup (S(t))t≥0is analytic.\nProof. SinceiR⊂ρ(Aµ,θ),then, using Lemma 1.1, it suffices to show (1.10). Suppose the result\nis false, then there exist a sequence λnof real numbers with |λn|going to ∞asn→ ∞ , and\na sequence Zn= (un,vn,wn,zn)inD(Aµ,θ), with||Zn||= 1 and\n(2.1) lim\n|λn|→∞|λn|−1||(iλnI−Aµ,θ)Zn||= 0.\nFirst, taking the real part of the inner product of |λn|−1(iλnI−Aµ,θ)ZnwithZn, then using\n(2.1), (1.7) and the condition (1.6), we obtain\n(2.2) |λn|−1\n2|B1\n2\n1vn|=o(1),|λn|−1\n2|B1\n2\n2zn|=o(1) and|λn|−1\n2|B1\n2\n1zn|=o(1).\nNow equation (2.1) can be rewritten explicitly as follows\n|λn|−1/parenleftbigg\niλnA1\n2\n1un−A1\n2\n1vn/parenrightbigg\n=o(1), (2.3)\n|λn|−1(iλnvn+A1un+αB1vn+βB1zn) =o(1), (2.4)\n|λn|−1/parenleftbigg\niλnA1\n2\n2wn−A1\n2\n2zn/parenrightbigg\n=o(1), (2.5)\n|λn|−1(iλnzn+βB1un+A2wn+γB2zn) =o(1). (2.6)\nTaking the inner product of (2.3) and (2.4) with A1\n2\n1unandvnrespectively, and (2.5) and (2.6)\nwithA1\n2\n2wnandznrespectively, one obtains\n|λn|−1/parenleftbigg\niλn|A1\n2\n1un|2−/a\\}bracke⊔le{⊔vn,A1un/a\\}bracke⊔ri}h⊔/parenrightbigg\n=o(1), (2.7)\n|λn|−1/parenleftbigg\niλn|vn|2+/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔+α|B1\n2\n1vn|2+β/a\\}bracke⊔le{⊔B1zn,vn/a\\}bracke⊔ri}h⊔/parenrightbigg\n=o(1), (2.8)\n|λn|−1/parenleftbigg\niλn|A1\n2\n2wn|2−/a\\}bracke⊔le{⊔zn,A2wn/a\\}bracke⊔ri}h⊔/parenrightbigg\n=o(1), (2.9)\n|λn|−1/parenleftbigg\niλn|zn|2+/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔+β/a\\}bracke⊔le{⊔B1vn,zn/a\\}bracke⊔ri}h⊔+γ|B1\n2\n2zn|2/parenrightbigg\n=o(1). (2.10)\nTaking into account estimations (2.2), the equations (2.8) and (2.10) can be rewritten as follows\n|λn|−1/parenleftbig\niλn|vn|2+/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔/parenrightbig\n=o(1), (2.11)\n|λn|−1/parenleftbig\niλn|zn|2+/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔/parenrightbig\n=o(1). (2.12)\nCombining conjugates of (2.11) and (2.12) with (2.7) and (2. 9), one get after dividing by λn|λn|−1\n|A1\n2\n1un|2+|A1\n2\n2wn|2−|vn|2−|zn|2=o(1),6 KA ¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\nwhich, thanks to ||Zn||= 1, leads to\n(2.13) |vn|2+|zn|2−1\n2=o(1).\nWe will prove that |vn|=o(1)and|zn|=o(1)which contradicts (2.13). For this, dividing (2.11)\nand (2.12) by |λn|−1λnto get\n(2.14) i|vn|2+1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔=o(1),\nand\n(2.15) i|zn|2+1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔=o(1).\nThus, it suffices to prove that1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔=o(1) and1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔=o(1).\nTo start, we have the following estimate by applying the Cauc hy-Schwarz inequality,\n(2.16)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|A1−µ\n2\n1un|\n|λn|1\n2|Aµ\n2\n1vn|\n|λn|1\n2.\nNote that A1−µ\n1unis bounded, since A1−µ\n1un=A1\n2−µ\n1/parenleftbigg\nA1\n2\n1un/parenrightbigg\nandA1\n2−µ\n1 is a bounded operator\nandA1\n2\n1unis also bonded. Then taking the inner product of (2.4) with A1−µ\n1un, and dividing\nthe obtained result by |λn|−1λn, we get\n(2.17)−i/angbracketleftig\nA1−µ\n1un,vn/angbracketrightig\n+1\nλn|A1−µ\n2\n1un|2+α/angbracketleftbigg\nA1−µ\n1un,1\nλnB1vn/angbracketrightbigg\n+β/angbracketleftbigg\nA1−µ\n1un,1\nλnB1zn/angbracketrightbigg\n=o(1).\nSinceA1−µunandvnare bounded, the first term in the left hand side of (2.17) is bo unded.\nOn the other hand, from (2.3) and the boundedness of |A1\n2\n1un|, we deduce that1\nλnA1\n2\n1vnis\nbounded.\nThe estimate of/angbracketleftig\nA1−µ\n1un,1\nλnB1vn/angbracketrightig\nin (2.17) will be proved as follows:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\nA1−µ\n1un,1\nλnB1vn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg\n1\nλ1\n2nB1\n2\n1/parenleftig\nA1−µ\n1un/parenrightig\n,1\nλ1\n2nB1\n2\n1vn/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleB1\n2\n1/parenleftig\nA1−µ\n1un/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n|λn|1\n2|B1\n2\n1vn|\n|λn|1\n2\n≤1\n4/parenleftigg\n|A1−µ\n2\n1un|\n|λn|1\n2/parenrightigg2\n+C\n|B1\n2\n1vn|\n|λn|1\n2\n2\n, (2.18)\nwhere we have used (1.2), and hereafter, Cdenotes a generic positive constant that is independent\nofn.REGULARITY OF SEMIGROUP ... COUPLED ELASTIC SYSTEMS 7\nSimilarly, we give an estimate of/angbracketleftig\nA1−µ\n1un,1\nλnB1zn/angbracketrightig\nas follows:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\nA1−µ\n1un,1\nλnB1zn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg\n1\nλ1\n2nB1\n2\n1/parenleftig\nA1−µ\n1un/parenrightig\n,1\nλ1\n2nB1\n2\n1zn/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|B1\n2\n1/parenleftig\nA1−µ\n1un/parenrightig\n|\n|λn|1\n2|B1\n2\n1zn|\n|λn|1\n2\n≤1\n4/parenleftigg\n|A1−µ\n2\n1un|\n|λn|1\n2/parenrightigg2\n+C\n|B1\n2\n1zn|\n|λn|1\n2\n2\n.\nFurthermore, recall that|B1\n2\n1vn|\n|λn|1\n2=o(1) and|B1\n2\n1zn|\n|λn|1\n2=o(1) (by (2.2)). All that leads to the\nboundedness of|A1−µ\n2\n1un|\n|λn|1\n2.\nReturning to (2.16), we get, (keeping in mind again|Aµ\n2\n1vn|\nλ1\n2n=o(1), from (2.2) and (1.2)),\n(2.19)1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔=o(1).\nSimilarly, we have\n(2.20)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|A1−θ\n2\n2wn|\n|λn|1\n2|Aθ\n2\n2zn|\n|λn|1\n2.\nTaking the inner product of (2.6) with A1−θ\n2wn, which is also bounded, since θ≥1\n2, we get\nafter dividing by |λn|−1λn\n(2.21)−i/angbracketleftig\nA1−θ\n2wn,zn/angbracketrightig\n+1\nλn|A1−θ\n2\n2wn|2+α/angbracketleftbigg\nA1−θ\n2wn,1\nλnB1vn/angbracketrightbigg\n+γ/angbracketleftbigg\nA1−θ\n2wn,1\nλnB2zn/angbracketrightbigg\n=o(1).\nAs in (2.17), the first term in (2.21) is bounded. Moreover, as in (2.18), using (1.1) and (1.3),\nwe have the following estimate of/angbracketleftig\nA1−θ\n2wn,1\nλnB1vn/angbracketrightig\n:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\nA1−θ\n2wn,1\nλnB1vn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftigg\n1\nλ1\n2nB1\n2\n1/parenleftig\nA1−θ\n2wn/parenrightig\n,1\nλ1\n2nB1\n2\n1vn/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleB1\n2\n1/parenleftig\nA1−θ\n2un/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n|λn|1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingleB1\n2\n1vn/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n|λn|1\n2\n≤C|A1−θ\n2\n2wn|\n|λn|1\n2|B1\n2\n1vn|\n|λn|1\n2\n≤1\n4\n|A1−θ\n2\n2wn|\n|λn|1\n2\n2\n+C\n|B1\n2\n1vn|\n|λn|1\n2\n2\n.\nSimilarly, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\nA1−θ\n2wn,1\nλnB2zn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n4\n|A1−θ\n2\n2wn|\n|λn|1\n2\n2\n+C\n|B1\n2\n2zn|\n|λn|1\n2\n2\n.8 KA ¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\nThen, using that|B1\n2\n1vn|\n|λn|1\n2=o(1)and|B1\n2\n2zn|\n|λn|1\n2=o(1), (2.21) implies that|A1−θ\n2\n2wn|\n|λn|1\n2=O(1). Keeping\nin mind|Aθ\n2\n2zn|\nλ1\n2n=o(1) (from (2.2) and (1.3)), it follows\n1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔=o(1).\nReporting (2.19) and (2.20) in (2.14) and (2.15) respective ly to derive\n|vn|2+|zn|2=o(1)\nwhich contradicts (2.13). ⊔ ⊓\n3. G EVREY SEMIGROUP :CASEmin(µ,θ)∈(0,1\n2)\nTheorem 3.1. For every µ,θ∈(0,1]such that s:= min(µ,θ)∈(0,1\n2), the associated semigroup\nis of Gevrey class δfor every δ≥1\n2s. More precisely, there exists a positive constant Csuch\nthat we have the resolvent estimate:\n(3.1) |λ|2s/bardbl(iλI−Aµ,θ)−1/bardblL(H)≤C,∀λ∈R.\nProof. Suppose the result is false, then there exist a sequence λnof real numbers where |λn|\ngoes to ∞asn→ ∞ , and a sequence Zn= (un,vn,wn,zn)inD(Aµ,θ), with||Zn||= 1 such\nthat\n(3.2) lim\n|λn|→∞|λn|−2s|(iλI−Aµ,θ)Zn|= 0.\nTaking the real part of the inner product of |λn|−2s(iλnI−Aµ,θ)ZnwithZn, we derive from\nthe dissipativity estimate:\n(3.3) |λn|−s|B1\n2\n1vn|=o(1),|λn|−s|B1\n2\n2zn|=o(1) and|λn|−s|B1\n2\n1zn|=o(1).\nEquation (3.2) may be rewritten as:\n|λn|−2s/parenleftbigg\niλnA1\n2\n1un−A1\n2\n1vn/parenrightbigg\n=o(1), (3.4)\n|λn|−2s(iλnvn+A1un+αB1vn+βB1zn) :=hn=o(1), (3.5)\n|λn|−2s/parenleftbigg\niλnA1\n2\n2wn−A1\n2\n2zn/parenrightbigg\n=o(1), (3.6)\n|λn|−2s(iλnzn+βB1vn+A2wn+γB2zn) :=kn=o(1). (3.7)\nAs in the proof of the last theorem, one can deduce:\ni|vn|2+1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔=o(1), (3.8)\ni|zn|2+1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔=o(1), (3.9)\nand\n|vn|2+|zn|2−1\n2=o(1). (3.10)REGULARITY OF SEMIGROUP ... COUPLED ELASTIC SYSTEMS 9\nAgain as in the first case, we will prove that |vn|=o(1)and|zn|=o(1)which would contradict\n(3.10). To this end, we shall prove that1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔=o(1) and1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔=o(1).\nThe application of the Cauchy-Schwarz inequality yields\n(3.11) |1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔| ≤|A1−µ\n2\n1un|\n|λn|1−s|Aµ\n2\n1vn|\n|λn|s.\nThe major difficulty is to prove that|A1−µ\n2\n1un|\n|λn|1−sis bounded. Taking the inner product of (3.5) with\n|λn|2sλ−1\nn1\nλ1−2s\nnA1−µ\n1unto get\n1\nλ1−2sn/angbracketleftig\nA1−µ\n1un,ivn/angbracketrightig\n+1\nλ2−2sn|A1−µ\n2\n1un|2\n+/angbracketleftbigg1\nλ1−snA1−µ\n1un,1\nλ1−sn(αB1vn+βB1zn)/angbracketrightbigg\n=1\nλ1−2sn/angbracketleftig\nA1−µ\n1un,|λn|2sλ−1\nnhn/angbracketrightig\n.\nConsequently, it follows\n1\nλ2−2sn|A1−µ\n2\n1un|2=−/angbracketleftbigg1\nλ1−snA1−µ\n1un,1\nλ1−sn(αB1vn+βB1zn)/angbracketrightbigg\n+/angbracketleftbigg1\nλ1−2snA1−µ\n1un,(|λn|2sλ−1\nnhn−ivn)/angbracketrightbigg\n. (3.12)\nFirst, we have the following estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−snA1−µ\n1un,1\nλ1−sn(αB1vn+βB1zn)/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleB1\n2\n1/parenleftig\nA1−µ\n1un/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n|λn|1−sα|B1\n2\n1vn|+|β||B1\n2\n1zn|\n|λn|1−s\n≤C|A1−µ\n2\n1un|\n|λn|1−s/parenleftbigg\n|B1\n2\n1vn|+|B1\n2\n1zn|/parenrightbigg\n|λn|s,\nwhere we have used (1.2), and the elementary inequality: 1−s > s , sinces∈(0,1\n2).\nUsing Young inequality and the estimates\n|λn|−s|B1\n2\n1vn|=o(1) and||λn|−s|B1\n2\n1zn|=o(1)\nwe deduce\n(3.13)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−snA1−µ\n1un,1\nλ1−sn(αB1vn+βB1zn)/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n3/parenleftigg\n|A1−µ\n2\n1un|\n|λn|1−s/parenrightigg2\n+o(1).\nWe are going to estimate the last term in (3.12). First, note t hat ifµ≥1\n2, then1\nλ1−2s\nnA1−µ\n1un\nis bounded (because A1/2\n1unis bounded and s <1\n2), so that/angbracketleftig\n1\nλ1−2s\nnA1−µ\n1un,(|λn|2sλ−1\nnhn−ivn)/angbracketrightig\nis bounded. Next, let us assume that 0< µ <1\n2. We have the following two estimates\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−2snA1−µ\n1un,(|λn|2sλ−1\nnhn−ivn)/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|A1−µ\n1un|\n|λn|1−2s(|hn|+|vn|),10 KA ¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\nthe second one is obtained by applying an interpolation ineq uality:\n|A1−µ\n1un|\n|λn|1−2s≤C\n|A1−µ\n2\n1un|\n|λn|(1−µ)(1−2s)\n1−2µ\n1−2µ\n1−µ\n|A1\n2\n1un|µ\n1−µ.\nThe combination of those two estimates leads to the followin g estimate, (keeping in mind that\n(1−µ)(1−2s)\n1−2µ≥1−s):\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−2snA1−µ\n1un,(|λn|2sλ−1\nn|λn|2sλ−1\nnhn−ivn)/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/parenleftigg\n|A1−µ\n2\n1un|\n|λn|1−s/parenrightigg1−2µ\n1−µ\n|A1\n2\n1un|µ\n1−µ(|hn|+|vn|).\nAgain, using Young inequality and the boundedness of |A1\n2\n1un|,|hn|and|vn|, we find\n(3.14)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−2snA1−µ\n1un,(|λn|2sλ−1\nnhn−ivn)/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n3/parenleftigg\n|A1−µ\n2\n1un|\n|λn|1−s/parenrightigg2\n+C.\nNow, using (3.13) and (3.14) in (3.12), we derive that/parenleftbigg\n|A1−µ\n2\n1un|\n|λn|1−s/parenrightbigg2\nis bounded. Finally, recall\nthat|Aµ\n2\n1vn|\n|λn|s=o(1) (by (3.3) and (1.2)), then we get, using (3.11), the estimate\n(3.15)1\nλn/a\\}bracke⊔le{⊔A1un,vn/a\\}bracke⊔ri}h⊔=o(1).\nSimilarly, we will prove that1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔=o(1).We start by the following estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|A1−θ\n2\n2wn|\n|λ1−sn||Aθ\n2\n2zn|\n|λsn|.\nAs for (3.15), it suffices to show that|A1−θ\n2\n2wn|\n|λ1−s\nn|is bounded (since|Aθ\n2\n2zn|\n|λn|s=o(1) by (3.3) and\n(1.3) ).\nTaking the inner product of (3.5) with |λn|2sλ−1\nn1\nλ1−2s\nnA1−θ\n2wnwe deduce that\n1\nλ2−2sn|A1−θ\n2\n2wn|2=−/angbracketleftbigg1\nλ1−snA1−θ\n2wn,1\nλ1−snαB1vn/angbracketrightbigg\n−/angbracketleftbigg1\nλ1−snA1−θ\n2wn,1\nλ1−snβB2zn/angbracketrightbigg\n+/angbracketleftbigg1\nλ1−2snA1−θ\n2un,(|λn|2sλ−1\nnkn−izn)/angbracketrightbigg\n. (3.16)\nFirst, we have the following two estimates\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−snA1−θ\n2wn,1\nλ1−snαB1vn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−snB1\n2\n1/parenleftig\nA1−θ\n2wn/parenrightig\n,1\nλ1−snαB1\n2\n1vn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C|A1−θ\n2\n2wn|\n|λn|1−s.|B1\n2\n1vn|\n|λn|1−s(3.17)REGULARITY OF SEMIGROUP ... COUPLED ELASTIC SYSTEMS 11\nand/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−snA1−θ\n2wn,1\nλ1−snβB2zn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−snB1\n2\n2/parenleftig\nA1−θ\n2wn/parenrightig\n,1\nλ1−snβB1\n2\n2zn/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C|A1−θ\n2\n2wn|\n|λn|1−s.|B1\n2\n2zn|\n|λn|s, (3.18)\nwhere we have used that 1−s > s , sinces∈(0,1\n2).\nTo estimate the last term in (3.16) we proceed as in (3.12): if θ≥1\n2, then1\nλ1−2s\nnA1−θ\n2wnis\nbounded, so that/angbracketleftig\n1\nλ1−2s\nnA1−θ\n2wn,(|λn|2sλ−1\nnkn−izn)/angbracketrightig\nis bounded. If 0< θ <1\n2: we have the\nfollowing two estimates: the first is\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−2snA1−θ\n2wn,(|λn|2sλ−1\nnkn−izn)/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|A1−θ\n2wn|\n|λn|1−2s(|kn|+|zn|),\nthe second, is obtained by applying an interpolation inequa lity:\n|A1−θ\n2wn|\n|λn|1−2s≤C\n|A1−θ\n2\n2wn|\n|λn|(1−θ)(1−2s)\n1−2θ\n1−2θ\n1−θ\n|A1\n2\n2wn|θ\n1−θ,\nto get, (keeping in mind that(1−θ)(1−2s)\n1−2θ≥1−s), the following estimate\n(3.19)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg1\nλ1−2snA1−θ\n2wn,(|λn|2sλ−1\nnkn−izn)/angbracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C\n|A1−θ\n2\n2wn|\n|λn|1−s\n1−2θ\n1−θ\n|A1\n2\n2wn|θ\n1−θ(|kn|+|zn|).\nAgain, using appropriate Young inequality at (3.17,3.18) a nd (3.19) we deduces from (3.16) the\nboundedness of/parenleftigg\n|A1−θ\n2\n2wn|\n|λn|1−s/parenrightigg2\nis bounded. Finally, recall that|Aθ\n2\n2zn|\n|λn|s=o(1), then we get by (3),\nthe estimate\n(3.20)1\nλn/a\\}bracke⊔le{⊔A2wn,zn/a\\}bracke⊔ri}h⊔=o(1).\n⊔ ⊓\n4. A N OPTIMAL RESULT\nIn this section we suppose that A1=A2:=A,B1:=Aµ,B2:=Aθ, and instead of (1.1), we\ntakeµ≤θ, which implies (1.1).\nTheorem 4.1. The resolvent estimate (3.1) is optimal, in the sense that, f or every µ,θ∈(0,1]\nsuch that µ∈(0,1\n2)andµ≤θ, for every r∈(2µ,1], we have:\n(4.1) limsup\n|λ|→∞|λ|r/bardbl(iλI−Aµ,θ)−1/bardblL(H)=∞.\nProof. We are going to show that there exist a sequence of positive re al numbers (λn)n≥1, and\nfor each n, an element Zn∈ D(A)such that for every r∈(2µ,1], one has:\nlim\nn→∞λn=∞,||Zn||= 1,lim\nn→∞λ−r\nn||(iλn−Aµ,θ)Zn||= 0. (4.2)12 KA ¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\nIndeed, if we have sequences λnandZnsatisfying (4.2), then we set\nVn=λ−r\nn(iλn−Aµ,θ)Zn, U n=Vn\n||Vn||. (4.3)\nTherefore, ||Un||= 1 and\nlim\nn→∞λr\nn||(iλn−Aµ,θ)−1Un||= lim\nn→∞1\n||Vn||=∞, (4.4)\nwhich would establish the claimed result. Thus, it remains t o prove the existence of such\nsequences.\nFor each n≥1, letenbe the eigenfunction of the operator A, andωnbe its corresponding\neigenvalue as in the proof of [1, Theorem 1.2]. As in that proo f, we seek Znin the form\nZn= (anen,iλnanen,cnen,iλncnen), withλnand the complex numbers anandcnchosen such\nthatZnfulfills the desired conditions.\nSet\nλn=√ωn. (4.5)\nWith that choice, we readily check that:\n(4.6)(iλn−Aµ,θ)Zn=\n0/bracketleftbig\n(ωn−λ2\nn)an+iλnωµ\nn(αan+βcn)/bracketrightbig\nen\n0\n(−λ2\nn+ωn)cn+iλn(βωµ\nnan+γwθ\nncn))en\n\n=\n0\niωµ+1\n2n[αan+βcn]en\n0/parenleftbigg\niγωθ−1\n2ncn+iβωµ−1\n2nan/parenrightbigg\nωnen\n,by (4.5).\nFor each n≥1, we set\nan=−γβ−1ωθ−µ\nncn. (4.7)\nIt then follows from (4.6):\n(4.8) (iλn−Aµ,θ)Zn=\n0/parenleftig\n−αγβ−1wθ−µ\nn+β/parenrightig\niωµ+1\n2ncnen\n0\n0\n\nThere exists k >0such that\nlim\nn→∞λ−2r\nn||(iλn−A)Zn||2\n= lim\nn→∞k2|cn|2λ−2r\nnω2θ+1\nn|en|2=k2lim\nn→∞ω2µ−r\nnω2θ−2µ+1\nn|cn|2(4.9)\n= 0,forr >2µ,andREGULARITY OF SEMIGROUP ... COUPLED ELASTIC SYSTEMS 13\nprovided that the sequence (ω2θ−2µ+1\nn|cn|2)converges to some nonzero real number, and ||Zn||= 1.\nOne checks that\n||Zn||2=ωn|an|2+λ2\nn|an|2+ωn|cn|2+λ2\nn|cn|2\n= 2ωn|an|2+2ωn|cn|2\n= 2γ2β−2w2θ−2µ\nnωn|cn|2+2ωn|cn|2(4.10)\n= 2/parenleftig\nγ2β−2+w−2(θ−µ)\nn/parenrightig\nω2θ−2µ+1\nn|cn|2,\nso that we might just choose\ncn=ωµ−θ−1\n2n/radicalbigg\n2/parenleftig\nγ2β−2+w2µ−2θ\nn/parenrightig(4.11)\nto get||Zn||= 1 as desired. ⊔ ⊓\n5. E XAMPLES OF APPLICATION\nLetΩbe a bounded domain in RNwith smooth boundary Γ. Typical examples of application\ninclude, but are not limited to\n(1)Interacting membranes\nytt−a∆y+α(−∆)µyt+β(−∆)µzt= 0 inΩ×(0,∞)\nztt−b∆z+β(−∆)µyt+γ(−∆)θzt= 0 inΩ×(0,∞)\ny= 0, z= 0 onΓ×(0,∞),\nwhereα,β,γ,µ andθare as in Section 1.\nHere,H=L2(Ω),A1=A2:=A=−∆,B1= (−∆)µ,B2= (−∆)θwithD(A) =\nH2(Ω)∩H1\n0(Ω). ThenAis a densely defined, positive unbounded operator on the Hilb ert\nspaceH. Moreover, V:=D(A1\n2) =H1\n0(Ω) and the injections V ֒→H ֒→V′=H−1(Ω)\nare dense and compact.\nThen the corresponding semigroup is\n•analytic, according to Theorem 2.1, for 1/2≤µ,θ≤1.\n•in Gevrey class δ >1\n2µ, according to Theorem 3.1, for µ∈(0,1/2),θ∈(0,1]and\nµ≤θ.\n(2)Interacting membrane and plate\nytt−∆y+α(−∆)µyt+β(−∆)µzt= 0 inΩ×(0,∞)\nztt+∆2z+β(−∆)µyt+γ∆2θzt= 0 inΩ×(0,∞)\ny= 0, z= 0,∂z\n∂ν= 0 onΓ×(0,∞),\nwhereα,β,γ,µ andθas in Section 1.\nHere,H=L2(Ω),A1=−∆,A2= ∆2,B1= (−∆)µ,B2= ∆2θwithD(A1) =\nH2(Ω)∩H1\n0(Ω),D(A2) =H4(Ω)∩H2\n0(Ω). ThenAi,i= 1,2,are densely defined, positive\nunbounded operators on the Hilbert space H. Moreover, V1:=D(A1\n2\n1) =H1\n0(Ω),V2=H2\n0(Ω)\nand the injections Vi֒→H ֒→V′\ni,V′\n1=H−1(Ω),V2=H−2(Ω),are dense and compact.\nThen the corresponding semigroup is\n•analytic, according to Theorem 2.1, for 1/2≤µ,θ≤1.14 KA ¨IS AMMARI, FARHAT SHEL, AND LOUIS TEBOU\n•in Gevrey class δ >1\n2min(µ,θ), according to Theorem 3.1, for µ,θ∈(0,1]and\nmin(µ,θ)∈(0,1/2).\n(3)Interacting plates\nytt+∆2y+α∆2µyt+β∆2µzt= 0 inΩ×(0,∞)\nztt+∆2z+β∆2µyt+γ∆2θzt= 0 inΩ×(0,∞)\ny= 0,∂y\n∂ν= 0, z= 0,∆z= 0 onΓ×(0,∞),\nwhereα,β,γ,µ andθare as in Section 1.\nHere,H=L2(Ω),A1= ∆2withD(A1) =H4(Ω)∩H2\n0(Ω),andB1=Aµ\n1,whileA2= ∆2\nwithD(A2) ={u∈H4(Ω);u= ∆u= 0 onΓ}, andB2=Aθ\n2. The operators A1,A2,B1and\nB2satisfy all the desired requirements.\nThe the corresponding semigroup is\n•analytic, according to Theorem 2.1, for 1/2≤µ≤θ≤1.\n•in Gevrey class δ >1\n2µ, according to Theorem 3.1, for µ∈(0,1/2),θ∈(0,1]andµ≤θ.\nREFERENCES\n[1] K. Ammari, F. Shel and L. Tebou, Regularity and stability of the semigroup associated with some interacting\nelastic systems I: A degenerate damping case, J. Evol. Equ. ( 2021). https://doi.org/10.1007/s00028-021-00738-7.\n[2] W. Arendt and C. J. K. Batty, Tauberian theorems and stabi lity of one-parameter semigroups, Trans. Amer.\nMath. Soc., 306 (1988), 837-852.\n[3] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart.\nAppl. Math. 39 (1982), 433-454.\n[4] S. P. Chen and R. Triggiani, Proof of extensions of two con jectures on structural damping for elastic systems.\nPacific J. Math. 136 (1989), 15-55.\n[5] S. P. Chen and R. Triggiani, Gevrey class semigroups aris ing from elastic systems with gentle dissipation:\nthe case 0< α <1/2. Proc. Am. Math. Soc. 110 (1990), 401-415.\n[6] F. Huang, On the mathematical model for linear elastic sy stems with analytic damping, SIAM J. Control\nOptim., 26 (1988), 714-724.\n[7] F. Huang and K. Liu, Holomorphic property and exponentia l stability of the semigroup associated with linear\nelastic systems with damping, Ann. Diff. Eqs., 4(1988), 411 -424.\n[8] Z. Liu and J. Yong, Qualitative properties of certain C0semigroups arising in elastic systems with various\ndampings. Adv. Differential Equations 3 (1998), 643-686.\n[9] Z. Liu and S. Zheng, Semigroups associated with dissipat ive systems, Chapman and Hall/CRC, 1999.\n[10] S. Taylor, Gevrey regularity of solutions of evolution equations and boundary controllability, Gevrey semigroup s\n(Chapter 5), Ph.D Thesis, School of Mathematics, Universit y of Minnesota, 1989.\nUR A NALYSIS AND CONTROL OF PDE S, UR 13ES64, D EPARTMENT OF MATHEMATICS , F ACULTY OF ,\nSCIENCES OF MONASTIR , U NIVERSITY OF MONASTIR , 5019 M ONASTIR , T UNISIA\nEmail address :kais.ammari@fsm.rnu.tn\nUR A NALYSIS AND CONTROL OF PDE S, UR 13ES64, D EPARTMENT OF MATHEMATICS , F ACULTY OF ,\nSCIENCES OF MONASTIR , U NIVERSITY OF MONASTIR , 5019 M ONASTIR , T UNISIA\nEmail address :farhat.shel@fsm.rnu.tn\nDEPARTMENT OF MATHEMATICS AND STATISTICS , FLORIDA INTERNATIONAL UNIVERSITY , M ODESTO MAIDIQUE ,\nCAMPUS , M IAMI , F LORIDA 33199, USA\nEmail address :teboul@fiu.edu"    },    {        "title": "1204.5612v1.CPT__Lorentz_invariance__mass_differences__and_charge_non_conservation.pdf",        "content": "arXiv:1204.5612v1  [hep-ph]  25 Apr 2012CPT, Lorentz invariance, mass differences, and charge\nnon-conservation\nA.D. Dolgova,b,c,d, V.A. Novikova,b\nSeptember 11, 2018\naNovosibirsk State University, Novosibirsk, 630090, Russia\nbInstitute of Theoretical and Experimental Physics, Moscow, 113 259, Russia\ncDipartimento di Fisica, Universit` a degli Studi di Ferrara, I-4410 0 Ferrara, Italy\ndIstituto Nazionale di Fisica Nucleare, Sezione di Ferrara, I-44100 Ferrara, Italy\nAbstract\nA non-local field theory which breaks discrete symmetries, i ncluding C, P, CP,\nand CPT, but preserves Lorentz symmetry, is presented. We de monstrate that at\none-loop level the masses for particle and antiparticle rem ain equal due to Lorentz\nsymmetry only. An inequality of masses implies breaking of t he Lorentz invariance\nand non-conservation of the usually conserved charges.\n11 Introduction\nThe interplay of Lorentz symmetry and CPT symmetry was consider ed in the literature\nfor decades. The issue attracted an additional interest recently due to a CPT-violating sce-\nnario in neutrino physics with different mass spectrum of neutrinos a nd antineutrinos [1].\nTheoretical frameworks of CPT breaking in quantum field theories, in fact in string theo-\nries, and detailed phenomenology of oscillating neutrinos with differen t masses of νand ¯ν\nwas further studied in papers [2].\nOn the other hand, it was argued in ref. [3] that violation of CPT auto matically leads\nto violation of the Lorentz symmetry [3]. This might allow for some more freedom in\nphenomenology of neutrino oscillations.\nVery recently this conclusion was revisited in our paper [4]. We demons trated that field\ntheories with different masses for particle and antiparticle are extr emely pathological ones\nand can’t be treated as healthy quantum field theories. Instead we constructed a class\nof slightly non-local Lorentz invariant field theories with the explicit b reakdown of CPT\nsymmetry and with the same masses for particle and antiparticle.\nAn example of such theory is a non-local QED with the Lagrangian L=L0+Ln.l.,\nwhereL0is the usual QED Lagrangian:\nL0=−1\n4F2\nµν(x)+¯ψ(x)[iˆ∂−eˆA(x)−m]ψ(x), (1)\nandLn.lis a small non-local addition:\nLn.l.(x) =g/integraldisplay\ndy¯ψ(x)γµψ(x)Aµ(y)K(x−y), (2)\nHereFµν(x) =∂µAν(x)−∂νAµ(x) is the electromagnetic field strength tensor, Aµ(x) is\nthe four-potential, and ψ(x) is the Dirac field for electrons.\nNon-local form-factor K(x−y) is chosen in such a way that it explicitly breaks T-\ninvariance, e.g.\nK(x−y) =θ(x0−y0)θ[(x−y)2]e−(x−y)2/l2, (3)\nwherelis a scale of the non-locality and the Heaviside functions θ(x0−y0)θ[(x−y)2] are\nequal to the unity for the future light-cone and are identically zero for the past light-cone.\nNon-local interaction, eq. (2), breaks T-invariance, preserves C- and P-invariance and,\nas a result, breaks CPT-invariance. This construction demonstra tes that CPT-symmetry\ncan be broken in Lorentz-invariant non-local field theory! The mas ses of an electron, m,\nand of a positron, ˜ m, remain identical to each other in this theory despite breaking of\nCPT-symmetry. The evident reason is that the interaction Ln.l.(x) is C-invariant and its\nexact C-symmetry preserves the identity of masses and anti-mas ses.\nIn this note we would like to study further the relation between mass difference for a\nparticle and an antiparticle and CPT-symmetry. We start from the s tandard local free\nfield theory of electrons with the usual dispersion relation between energy and momentum:\np2\nµ=p2\n0−p2=m2= ˜m2(4)\nand introduce a non-local interaction that breaks the whole set of discrete symmetries,\ni.e. C, P, CP, T, and CPT. So there is no discrete symmetry which pres erves equality\n1ofmto ˜min this case. Hence in principle the interaction can shift mfrom ˜m. But an\nexplicit one-loop calculation demonstrates that this is not true. So w e conclude that it is\nLorentz-symmetry that keeps the identity\nm= ˜m . (5)\nThis conclusion invalidates the experimental evidence for CPT-symm etry based on the\nequality of masses of particles and antiparticles. CPT may be strong ly broken in a Lorentz\ninvariant way and in such a case the masses must be equal. Another w ay around, if we\nassume that the masses are different, then Lorenz invariance mus t be broken. Lorentz and\nCPT violating theories would lead not only to mass difference of particle s and antiparticles\nbut to much more striking phenomena such as violation of gauge invar iance, current non-\nconservation, and even to a breaking of the usual equilibrium statis tics (for the latter see\nref. [5]).\n2 C, CP and CPT violating QFT\nTo formulate a model we start with the standard QED Lagrangian:\nL0=−1\n4Fµν(x)Fµν(x)+¯ψ(x)[iˆ∂−eˆA(x)−m]ψ(x), (6)\nand add the interaction of a photon, Aµ, with an axial current\nL1=g1¯ψ(x)γµγ5ψ(x)Aµ(x) (7)\nand with the electric dipole moment of an electron\nL2=g2¯ψ(x)σµνγ5ψ(x)Fµν(x). (8)\nThe first interaction, L1, breaks C and P-symmetry and conserves CP-symmetry. The\nsecond interaction breaks P- and CP-symmetry. Still the sum of La grangians\nL=L0+L1+L2 (9)\npreserves CPT-symmetry. To break the CPT we modify the interac tionL1to a non-local\none˜L1:\nL1→˜L1(x) =/integraldisplay\ndyg1¯ψ(x)γµγ5ψ(x)K(x−y)Aµ(y). (10)\nWith this modification the model\nL=L0+¯L1+L2 (11)\nbreaks all discrete symmetries.\n23 One-loop calculation\nIn general to calculate high order perturbative contributions of a non-local interaction into\nS-matrix one has to modify the Dyson formulae for S-matrix with T-ordered exponential\nS=T/braceleftbigg\nexp/parenleftbigg\ni/integraldisplay\nd4xLint/parenrightbigg/bracerightbigg\n(12)\nand the whole Feynman diagram techniques.\nBut in the first order in the non-local interaction one can work with t he usual Feynman\nrules in the coordinate space. The only difference is that one of the v ertices becomes\nnon-local.\n4 Mass and wave function renormalization for parti-\ncle and antiparticle\nWe start with the standard free field theory for an electron, i.e.\nL=¯ψ[iˆ∂−m]ψ (13)\nthat fixes the usual dispersion law\np2=p2\n0−p2=m2. (14)\nThe self-energy operator, Σ( p), contributes both to the mass renormalization and to\nthe wave function renormalization. In general one-loop effective L agrangin can be written\nin the form:\nL(1)\neff=¯ψ[i(Aγµ+Bγµγ5)∂µ−(m1+im2γ5)]ψ . (15)\nIt is useful to rewrite the same one-loop effective Lagrangian in ter ms of the field for\nantiparticle ψc:\nψc= (−i)[¯ψγ0γ2]T, (16)\nL(1)\neff=¯ψc[i(Aγ5−Bγµγ5)∂µ−(m1+im2γ5)]ψc. (17)\nWe see that the mass term is the same for ψand forψc, but the wave function renor-\nmalization is different: the coefficient in front of the pseudovector c hanges its sign. This\nchange is unobservable since one can remove Bγµγ5andim2γ5terms by redefining of\nvariables. Indeed\n¯ψ(A+Bγ5)γµψ≡¯ψ′√\nA2+B2γµψ′, (18)\nwhere\nψ= (coshα+iγ5sinhα)ψ′, (19)\ntanh2α=B/A , (20)\nand\n¯ψ(m1+iγ5m2)ψ≡/radicalBig\nm2\n1+m2\n2¯ψ′ψ′, (21)\n3eg2 g1eγ\nFigure 1: The diagram contributing to the mass difference of electro n and\npositron. The blob represents a non-local form-factor.\nwhere\nψ= exp(iγ5β)ψ′, (22)\ntan2β=m2/m1. (23)\nThissimpleobservationissufficient toconcludethattechnicallythere isnopossibilityto\nwriteone-loopcorrections thatproducedifferent contributions f orparticleandantiparticle.\nStill it is instructive to check directly that the difference is zero.\n5 Explicit one-loop calculation\nWe are looking for a one-loop contribution into self-energy operato r Σ(p) that breaks C,\nCP, and CPT symmetries and that changes the chirality of the fermio n line. It is clear\nthat this contribution potentially can be different (opposite in sign) f or particle ψand\nantiparticle ψc.\nTo construct such contribution we need bothanomalous interactio ns˜L1andL2. Indeed\ninteraction L2changes chirality and breaks CP symmetry, while non-local interact ion˜L1\nbreaks C and CPT and leaves the chirality unchanged. In combination they break all\ndiscrete symmetries and change chirality. There are two diagrams t hat are proportional\ntog1g2(see Fig. 1).\nWe will calculate these diagrams in two steps. The first step is a pure a lgebraic one.\nSelf-energy Σ( p) is 4×4 matrix that was constructed from a product of three other 4 ×4\nmatrices, i.e. two vertices and one fermion propagator. Notice tha t any 4×4 matrix can\nbe decomposed as a sum over complete set of 16 Dirac matrices. In t his decomposition of\nΣ(p) we need terms that are odd in Cand changes chirality. Fortunately there is only one\nDirac matrix with these properties. That is σµν. So\nΣ(p) =σµνIµν(p), (24)\nwhereIµνrepresents Feynman (divergent) integral. We could obtain eq. (24 ) after some\nlong explicit algebraic transformation, but the net result is determin ed by the symmetry\nonly.\nThe second step is the calculation of Feynman integrals. Again fortu nately we do not\nneed actual calculations. Indeed due to the Lorentz symmetry of the theory this Iµνshould\n4be a tensor that depends only on the momentum of fermion line p. The general form for\nIµνis\nIµν=Agµν+Bpµpν. (25)\nAs a result we get\nΣ(p) =σµνIµν≡0 (26)\nand we conclude that the one-loop contribution into possible mass diff erence is identically\nzero.1\n6 CPT and charge non-conservation\nThere is widely spread habit to parametrize CPTviolation by attributing different masses\nto particle and antiparticle. This tradition is traced to an old time of th e first observation\nofK−¯K-mesons oscillation.\nForK-mesons with a given momenta qthe theory of oscillation is equivalent to a non-\nhermitian Quantum Mechanics (QM) with two degrees of freedom. Dia gonal elements of\n2×2 Hamiltonian matrix represent masses for particle and antiparticle. Their unequality\nbreaks CPT-symmetry. Experimental bounds on mass difference a re considered as bounds\non the CPT-symmetry violation parameters. Such strategy has no explicit loop-holes and\nis still used for parametrization of CPT-symmetry violation in DandBmeson oscillations.\nQuantum Field Theory (QFT) deals not with onemode for a given momen ta but rather\nwith an infinite sum over all momenta. The set of plane waves with all po ssible momenta\nfor particle and antiparticle is a complete set of orthogonal modes a nd an arbitrary field\noperator can be decomposed over this set.\nNaive generalization of CPT-conserving QFT to CPT-violating QFT was to attribute\ndifferent masses for particle and antiparticle [1, 2]). Say for a comple x scalar field they use\nthe infinite sum [1, 2]\nφ(x) =/summationdisplay\nq/braceleftBigg\na(q)1√\n2Ee−i(Et−qx)+b+(q)1√\n2˜Eei(˜Et−qx)/bracerightBigg\n, (27)\nwhere (a(q),a+(q)), (b(q),b+(q)) are annihilation and creation operators, and ( m,E) and\n(˜m,˜E) are masses and energies of particle and antiparticle respectively.\nGreenberg [3] found that this construction runs into trouble. Th e dynamic of fields\ndetermined according to eq. (27) cannot be a Lorentz-invariant o ne.\nWe’d like to notice that for charged particles (say for electrons and positrons) similar\ngeneralization of the field theory breaks not only the Lorentz symm etry but the electric\ncharge conservation as well. The reason is very simple. For the stan dard QED the operator\nof electric charge ˆQ(t) can be written in the form\nˆQ(t) =/summationdisplay\nq/braceleftBig\na+(q)a(q)−b+(q)b(q)/bracerightBig\n. (28)\n1Recently our former collaborators published a paper where they de monstrated that for a particle with\na non-standard dispersion law the quantity which they define as mas s can be different for particle and\nantiparticle [4].\n5Operator ˆQ(t)isadiagonalone, i.e. therearenomixed termswithdifferent moment a. The\nmodes with different momenta areorthogonal to each other and dis appear after integration\nover space. This is a technical explanation why one can construct a time-independent\noperator.\nIf one shifts the mass of electron from the mass of positron the sit uation drastically\nchanges. For electron the modes with different momenta are still or thogonal to each other.\nThe same is true for the modes of positron, they are also orthogon al among themselves.\nBut there is no reason for wave function of electron with mass mbe orthogonal to wave\nfunctions for positron with mass ˜ m. As a result one obtaines\nQ(t) =/summationdisplay\nq/braceleftBig\na+(q)a(q)−b+(q)b(q)/bracerightBig\n+C/summationdisplay\nq(E−˜E)√\n4E˜E/bracketleftBig\nb(q)a(−q)e−i(E+˜E)t+h.c./bracketrightBig\n,(29)\nwhere constant Cdepends on the sorts of particles and on the definition of the charg e.\nWe can conclude from this equation that non-conservation of char ge exhibits itself only\nin annihilation processes but not in the scattering processes. So th ere is no immediate\nproblem with the Coulomb law. Nevertheless non-conservation of th is type is also abso-\nlutely excluded by the experiment. In a case of charge-nonconser vation annihilation of\nparticle and antiparticle with a creation of the infinite number of soft massless photons\ncreates a terrible infrared problem. Infrared catastrophe can n ot be avoided by usual sum-\nmation over infrared photons. On the other hand, as is argued in re f. [7], the electron\ndecay might be exponentially suppressed due to vanishing of the cor responding formfactor\ncreated by virtual longitudinal photons.\nSimilar arguments lead to the conclusion that conservation of energ y cannot survive as\nwell in a theory with different masses of particles and antiparticles.\n7 Conclusion\nWe have shown that in the framework of a Lorentz invariant field the ory it is impossible\nto have different masses of particles and antiparticles, even if CPT ( together with C and\nP) invariance is broken. On the other hand, unequal masses of par ticles and antiparticles\nimply breaking of the Lorentz invariance. Moreover, in such theorie s charge and energy\nconservation seem to be broken as well.\n8 Acknowledgements\nThisworkwassupportedbytheGrantofGovernmentofRussianFe deration(11.G34.31.0047),\nby NSh-3172.2012.2, and by the Grant RFBR 11-02-00441.\nReferences\n[1] H. Murayama, T. Yanagida. Phys. Lett. B 520(2001) 263.\n6[2] G. Barenboim, L. Borissov, J.D. Lykken, A.V. Smirnov, JHEP 0210(2002) 001;\nG. Barenboim, L. Borissov, J.D. Lykken, Phys. Lett. B 534(2002) 106;\nG. Barenboim, J.F. Beacom, L. Borissov, B. Kayser Phys. Lett. B 537(2002) 227;\nG. Barenboim, L. Borissov, J.D. Lykken, hep-ph/0212116.\n[3] O.W. Greenberg, Phys.Rev.Lett. 89(2002) 231602.\n[4] M. Chaichian, A.D. Dolgov, V.A. Novikov, A. Tureanu, Phys. Lett. B699(2011) 177.\n[5] A.D. Dolgov Phys. Atom. Nucl. 73(2010) 588, e-Print: arXiv:0903.4318 [hep-ph].\n[6] M. Chaichian, K. Fujikawa, A. Tureanu, hep-th/1203.0267.\n[7] M.B. Voloshin, L.B. Okun, Pis’ma ZhETF, 28(1978) 156.\n7"    },    {        "title": "1805.01815v2.Effective_damping_enhancement_in_noncollinear_spin_structures.pdf",        "content": "Effective damping enhancement in noncollinear spin structures\nLevente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nDamping mechanisms in magnetic systems determine the lifetime, diffusion and transport prop-\nerties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological\nLandau–Lifshitz–Gilbert equation, here the effective damping parameter in noncollinear magnetic\nsystems is determined describing the linewidth in resonance experiments or the decay parameter\nin time-resolved measurements. It is shown how the effective damping can be calculated from the\nelliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded\nthat the effective damping is larger than the Gilbert damping, and it may significantly differ be-\ntween excitation modes. Numerical results for the effective damping are presented for the localized\nmagnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system.\nSpinwaves(SW)ormagnonsaselementaryexcitations\nof magnetically ordered materials have attracted signifi-\ncant research attention lately. The field of magnonics[1]\nconcerns the creation, propagation and dissipation of\nSWs in nanostructured magnetic materials, where the\ndispersion relations can be adjusted by the system ge-\nometry. A possible alternative for engineering the prop-\nerties of magnons is offered by noncollinear (NC) spin\nstructures[2] instead of collinear ferro- (FM) or antifer-\nromagnets (AFM). SWs are envisaged to act as informa-\ntion carriers, where one can take advantage of their low\nwavelengths compared to electromagnetic waves possess-\ning similar frequencies[3]. Increasing the lifetime and the\nstability of magnons, primarily determined by the relax-\nation processes, is of crucial importance in such applica-\ntions.\nThe Landau–Lifshitz–Gilbert (LLG) equation[4, 5] is\ncommonly applied for the quasiclassical description of\nSWs, where relaxation is encapsulated in the dimen-\nsionless Gilbert damping (GD) parameter α. The life-\ntime of excitations can be identified with the resonance\nlinewidth in frequency-domain measurements such as fer-\nromagnetic resonance (FMR)[6], Brillouin light scatter-\ning (BLS)[7] or broadband microwave response[8], and\nwith the decay speed of the oscillations in time-resolved\n(TR) experiments including magneto-optical Kerr effect\nmicroscopy (TR-MOKE)[9] and scanning transmission x-\nray microscopy (TR-STXM)[10]. Since the linewidth is\nknowntobeproportionaltothefrequencyofthemagnon,\nmeasuring the ratio of these quantities is a widely ap-\nplied method for determining the GD in FMs[3, 6]. An\nadvantage of AFMs in magnonics applications[11, 12] is\ntheir significantly enhanced SW frequencies due to the\nexchange interactions, typically in the THz regime, com-\npared to FMs with GHz frequency excitations. However,\nit is known that the linewidth in AFM resonance is typ-\nically very wide because it scales with a larger effective\ndamping parameter αeffthan the GD α[13].\nThe tuning of the GD can be achieved in magnonic\ncrystals by combining materials with different values of\nα. It was demonstrated in Refs. [14–16] that this leadsto a strongly frequency- and band-dependent αeff, based\non the relative weights of the magnon wave functions in\nthe different materials.\nMagnetic vortices are two-dimensional NC spin config-\nurations in easy-plane FMs with an out-of-plane magne-\ntized core, constrained by nanostructuring them in dot-\norpillar-shapedmagneticsamples. Theexcitationmodes\nofvortices, particularlytheirtranslationalandgyrotropic\nmodes, havebeeninvestigatedusingcollective-coordinate\nmodels[17] based on the Thiele equation[18], linearized\nSW dynamics[19, 20], numerical simulations[21] and ex-\nperimental techniques[22–24]. It was demonstrated theo-\nretically in Ref. [21] that the rotational motion of a rigid\nvortex excited by spin-polarized current displays a larger\nαeffthan the GD; a similar result was obtained based on\ncalculating the energy dissipation[25]. However, due to\nthe unbounded size of vortices, the frequencies as well\nas the relaxation rates sensitively depend on the sample\npreparation, particularly because they are governed by\nthe magnetostatic dipolar interaction.\nIn magnetic skyrmions[26], the magnetic moment di-\nrections wrap the whole unit sphere. In contrast to vor-\ntices, isolated skyrmions need not be confined for stabi-\nlization, and are generally less susceptible to demagneti-\nzation effects[3, 27]. The SW excitations of the skyrmion\nlattice phase have been investigated theoretically[28–30]\nand subsequently measured in bulk systems[3, 8, 31]. It\nwas calculated recently[32] that the magnon resonances\nmeasured via electron scattering in the skyrmion lattice\nphase should broaden due to the NC structure. Calcula-\ntions predicted the presence of different localized modes\nconcentrated on the skyrmion for isolated skyrmions\non a collinear background magnetization[33–35] and for\nskyrmions in confined geometries[20, 36, 37]. From the\nexperimental side, the motion of magnetic bubbles in a\nnanodisk was investigated in Ref. [38], and it was pro-\nposed recently that the gyration frequencies measured in\nIr/Fe/Co/Pt multilayer films is characteristic of a dilute\narray of isolated skyrmions rather than a well-ordered\nskyrmion lattice[6]. However, the lifetime of magnons in\nskyrmionic systems based on the LLG equation is appar-arXiv:1805.01815v2  [cond-mat.mtrl-sci]  15 Oct 20182\nently less explored.\nIt is known that NC spin structures may influence the\nGD via emergent electromagnetic fields[29, 39, 40] or via\nthe modified electronic structure[41, 42]. Besides deter-\nmining the SW relaxation process, the GD also plays\na crucial role in the motion of domain walls[43–45] and\nskyrmions[46–48] driven by electric or thermal gradients,\nboth in the Thiele equation where the skyrmions are\nassumed to be rigid and when internal deformations of\nthe structure are considered. Finally, damping and de-\nformations are also closely connected to the switching\nmechanisms of superparamagnetic particles[49, 50] and\nvortices[51], as well as the lifetime of skyrmions[52–54].\nTheαeffin FMs depends on the sample geometry due\nto the shape anisotropy[13, 55, 56]. It was demonstrated\nin Ref. [56] that αeffis determined by a factor describing\nthe ellipticity of the magnon polarization caused by the\nshape anisotropy. Elliptic precession and GD were also\ninvestigated by considering the excitations of magnetic\nadatomsonanonmagneticsubstrate[57]. Thecalculation\nof the eigenmodes in NC systems, e.g. in Refs. [6, 20, 35],\nalso enables the evaluation of the ellipticity of magnons,\nbut this property apparently has not been connected to\nthe damping so far.\nAlthough different theoretical methods for calculating\nαeffhave been applied to various systems, a general de-\nscriptionapplicabletoallNCstructuresseemstobelack-\ning. Here it is demonstrated within a phenomenological\ndescription of the linearized LLG equation how magnons\nin NC spin structures relax with a higher effective damp-\ning parameter αeffthan the GD. A connection between\nαeffand the ellipticity of magnon polarization forced by\nthe NC spin arrangement is established. The method\nis illustrated by calculating the excitation frequencies\nof isolated skyrmions, considering experimentally deter-\nmined material parameters for the Pd/Fe/Ir(111) model\nsystem[58]. It is demonstrated that the different local-\nized modes display different effective damping parame-\nters, with the breathing mode possessing the highest one.\nThe LLG equation reads\n∂tS=−γ/primeS×Beff−αγ/primeS×/parenleftBig\nS×Beff/parenrightBig\n,(1)\nwithS=S(r)the unit-length vector field describing\nthe spin directions in the system, αthe GD and γ/prime=\n1\n1+α2ge\n2mthe modified gyromagnetic ratio (with gbeing\ntheg-factor of the electrons, ethe elementary charge and\nmthe electron mass). Equation (1) describes the time\nevolution of the spins governed by the effective magnetic\nfieldBeff=−1\nMδH\nδS, withHthe Hamiltonian or free\nenergy of the system in the continuum description and\nMthe saturation magnetization.\nThe spins will follow a damped precession relaxing\nto a local minimum S0ofH, given by the condition\nS0×Beff=0. Note that generally the Hamiltonian rep-\nresents a rugged landscape with several local energy min-\nima, corresponding to e.g. FM, spin spiral and skyrmionlattice phases, or single objects such as vortices or iso-\nlated skyrmions. The excitations can be determined by\nswitching to a local coordinate system[20, 34, 47] with\nthe spins along the zdirection in the local minimum,\n˜S0= (0,0,1), and expanding the Hamiltonian in the\nvariablesβ±=˜Sx±i˜Sy, introduced analogously to spin\nraising and lowering or bosonic creation and annihila-\ntion operators in the quantum mechanical description of\nmagnons[59–61]. The lowest-order approximation is the\nlinearized form of the LLG Eq. (1),\n∂tβ+=γ/prime\nM(i−α)/bracketleftbig\n(D0+Dnr)β++Daβ−/bracketrightbig\n,(2)\n∂tβ−=γ/prime\nM(−i−α)/bracketleftbig\nD†\naβ++ (D0−Dnr)β−/bracketrightbig\n.(3)\nFor details of the derivation see the Supplemental\nMaterial[62]. The term Dnrin Eqs. (2)-(3) is respon-\nsible for the nonreciprocity of the SW spectrum[2]. It\naccounts for the energy difference between magnons\npropagating in opposite directions in in-plane oriented\nultrathin FM films[63, 64] with Dzyaloshinsky–Moriya\ninteraction[65, 66] and the splitting between clockwise\nand counterclockwise modes of a single skyrmion[20].\nHere we will focus on the effects of the anomalous\nterm[34]Da, which couples Eqs. (2)-(3) together. Equa-\ntions (2)-(3) may be rewritten as eigenvalue equations by\nassuming the time dependence\nβ±(r,t) =e−iωktβ±\nk(r). (4)\nForα= 0, the spins will precess around their equilib-\nriumdirection ˜S0. Iftheequationsareuncoupled, the ˜Sx\nand ˜Syvariables describe circular polarization, similarly\nto the Larmor precession of a single spin in an exter-\nnal magnetic field. However, the spins are forced on an\nelliptic path due to the presence of the anomalous terms.\nThe effective damping parameter of mode kis defined\nas\nαk,eff=/vextendsingle/vextendsingle/vextendsingle/vextendsingleImωk\nReωk/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5)\nwhich is the inverse of the figure of merit introduced in\nRef. [15]. Equation (5) expresses the fact that Im ωk,\nthe linewidth in resonance experiments or decay coeffi-\ncient in time-resolved measurements, is proportional to\nthe excitation frequency Re ωk.\nInterestingly, there is a simple analytic expression con-\nnectingαk,effto the elliptic polarization of the modes at\nα= 0. Forα/lessmuch1, the effective damping may be ex-\npressed as\nαk,eff\nα≈/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr=/integraltext\na2\nk(r) +b2\nk(r)dr/integraltext\n2ak(r)bk(r)dr,\n(6)3\n0.0 0.2 0.4 0.6 0.8 1.00246810\nFIG. 1. Effective damping parameter αk,effas a function of\ninverseaspectratio bk/akofthepolarizationellipse, assuming\nconstantakandbkfunctions in Eq. (6). Insets illustrate the\nprecession for different values of bk/ak.\nwhere the (0)superscript denotes that the eigenvectors\nβ±\nk(r)defined in Eq. (4) were calculated for α= 0, while\nak(r)andbk(r)denote the semimajor and semiminor\naxes of the ellipse the spin variables ˜Sx(r)and ˜Sy(r)\nare precessing on in mode k. Details of the derivation\nare given in the Supplemental Material[62]. Note that\nan analogous expression for the uniform precession mode\nin FMs was derived in Ref. [56]. The main conclusion\nfrom Eq. (6) is that αk,effwill depend on the considered\nSW mode and it is always at least as high as the GD\nα. Although Eq. (6) was obtained in the limit of low\nα, numerical calculations indicate that the αk,eff/αratio\ntends to increase for increasing values of α; see the Sup-\nplementalMaterial[62]foranexample. Theenhancement\nof the damping from Eq. (6) is shown in Fig. 1, with the\nspace-dependent ak(r)andbk(r)replaced by constants\nfor simplicity. It can be seen that for more distorted po-\nlarization ellipses the spins get closer to the equilibrium\ndirectionafterthesamenumberofprecessions, indicating\na faster relaxation.\nSince the appearance of the anomalous terms Dain\nEqs. (2)-(3) forces the spins to precess on an elliptic\npath, it expresses that the system is not axially sym-\nmetric around the local spin directions in the equilib-\nrium state denoted by S0. Such a symmetry breaking\nnaturally occurs in any NC spin structure, implying a\nmode-dependent enhancement of the effective damping\nparameter in NC systems even within the phenomeno-\nlogical description of the LLG equation. Note that the\nNC structure also influences the electronic properties of\nthe system, which can lead to a modification of the GD\nitself, see e.g. Ref. [42].\nIn order to illustrate the enhanced and mode-\ndependent αk,eff, we calculate the magnons in isolated\nchiralskyrmionsinatwo-dimensionalultrathinfilm. Thedensity of the Hamiltonian Hreads[67]\nh=/summationdisplay\nα=x,y,z/bracketleftBig\nA(∇Sα)2/bracketrightBig\n+K(Sz)2−MBSz\n+D(Sz∂xSx−Sx∂xSz+Sz∂ySy−Sy∂ySz),(7)\nwithAthe exchange stiffness, Dthe Dzyaloshinsky–\nMoriya interaction, Kthe anisotropy coefficient, and B\nthe external field.\nIn the following we will assume D>0andB≥0\nwithout the loss of generality, see the Supplemental\nMaterial[62] for discussion. Using cylindrical coordi-\nnates (r,ϕ)in real space and spherical coordinates S=\n(sin Θ cos Φ ,sin Θ sin Φ,cos Θ)in spin space, the equi-\nlibrium profile of the isolated skyrmion will correspond\nto the cylindrically symmetric configuration Θ0(r,ϕ) =\nΘ0(r)andΦ0(r,ϕ) =ϕ, the former satisfying\nA/parenleftbigg\n∂2\nrΘ0+1\nr∂rΘ0−1\nr2sin Θ 0cos Θ 0/parenrightbigg\n+D1\nrsin2Θ0\n+Ksin Θ 0cos Θ 0−1\n2MBsin Θ 0= 0 (8)\nwith the boundary conditions Θ0(0) =π,Θ0(∞) = 0.\nThe operators in Eqs. (2)-(3) take the form (cf.\nRefs. [34, 35, 47] and the Supplemental Material[62])\nD0=−2A/braceleftBigg\n∇2+1\n2/bracketleftbigg\n(∂rΘ0)2−1\nr2/parenleftbig\n3 cos2Θ0−1/parenrightbig\n(∂ϕΦ0)2/bracketrightbigg/bracerightBigg\n−D/parenleftbigg\n∂rΘ0+1\nr3 sin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−K/parenleftbig\n3 cos2Θ0−1/parenrightbig\n+MBcos Θ 0, (9)\nDnr=/parenleftbigg\n4A1\nr2cos Θ 0∂ϕΦ0−2D1\nrsin Θ 0/parenrightbigg\n(−i∂ϕ), (10)\nDa=A/bracketleftbigg\n(∂rΘ0)2−1\nr2sin2Θ0(∂ϕΦ0)2/bracketrightbigg\n+D/parenleftbigg\n∂rΘ0−1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n+Ksin2Θ0.(11)\nEquation (11) demonstrates that the anomalous terms\nDaresponsible for the enhancement of the effective\ndamping can be attributed primarily to the NC arrange-\nment (∂rΘ0and∂ϕΦ0≡1) and secondarily to the\nspins becoming canted with respect to the global out-\nof-plane symmetry axis ( Θ0∈ {0,π}) of the system.\nTheDnrtermintroducesanonreciprocitybetweenmodes\nwith positive and negative values of the azimuthal quan-\ntum number (−i∂ϕ)→m, preferring clockwise rotat-\ning modes ( m < 0) over counterclockwise rotating ones\n(m > 0) following the sign convention of Refs. [20, 34].\nBecauseD0andDnrdepend onmbutDadoes not, it is\nexpected that the distortion of the SW polarization el-\nlipse and consequently the effective damping will be more\nenhanced for smaller values of |m|.\nThe different modes as a function of external field\nare shown in Fig. 2(a), for the material parameters de-\nscribing the Pd/Fe/Ir(111) system. The FMR mode at4\n0.7 0.8 0.9 1.0 1.1 1.20255075100125150175\n(a)\n0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.53.0\n(b)\nFIG. 2. Localized magnons in the isolated skyrmion, with the\ninteraction parameters corresponding to the Pd/Fe/Ir(111)\nsystem[58]:A = 2.0pJ/m,D =−3.9mJ/m2,K =\n−2.5MJ/m3,M= 1.1MA/m. (a) Magnon frequencies f=\nω/2πforα= 0. Illustrations display the shapes of the excita-\ntion modes visualized on the triangular lattice of Fe magnetic\nmoments, with red and blue colors corresponding to positive\nand negative out-of-plane spin components, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6).\nωFMR =γ\nM(MB−2K), describing a collective in-phase\nprecession of the magnetization of the whole sample, sep-\narates the continuum and discrete parts of the spectrum,\nwith the localized excitations of the isolated skyrmion\nlocated below the FMR frequency[34, 35]. We found a\nsingle localized mode for each m∈{0,1,−2,−3,−4,−5}\nvalue, so in the following we will denote the excita-\ntion modes with the azimuthal quantum number. The\nm=−1mode corresponds to the translation of the\nskyrmion on the field-polarized background, which is a\nzero-frequency Goldstone mode of the system and not\nshown in the figure. The m=−2mode tends to zero\naroundB= 0.65T, indicating that isolated skyrmions\nbecome susceptible to elliptic deformations and subse-\nquently cannot be stabilized at lower field values[68].\nThe values of αm,effcalculated from Eq. (6) for the\ndifferent modes are summarized in Fig. 2(b). It is impor-\ntant to note that for a skyrmion stabilized at a selected\n0 20 40 60 80 100-0.04-0.020.000.020.040.06\n-0.03 0.00 0.03-0.030.000.03FIG. 3. Precession of a single spin in the skyrmion in the\nPd/Fe/Ir(111) system in the m= 0andm=−3modes at\nB= 0.75T, from numerical simulations performed at α=\n0.1. Inset shows the elliptic precession paths. From fitting\nthe oscillations with Eq. (4), we obtained |Reωm=0|/2π=\n39.22GHz,|Imωm=0|= 0.0608ps−1,αm=0,eff= 0.25and\n|Reωm=−3|/2π= 40.31GHz,|Imωm=−3|= 0.0276ps−1,\nαm=−3,eff= 0.11.\nfield value, the modes display widely different αm,effval-\nues, with the breathing mode m= 0being typically\ndamped twice as strongly as the FMR mode. The ef-\nfective damping tends to increase for lower field values,\nand decrease for increasing values of |m|, the latter prop-\nerty expected from the m-dependence of Eqs. (9)-(11)\nas discussed above. It is worth noting that the αm,eff\nparameters are not directly related to the skyrmion size.\nWealsoperformedthecalculationsfortheparametersde-\nscribing Ir|Co|Pt multilayers[69], and for the significantly\nlargerskyrmionsinthatsystemweobtainedconsiderably\nsmaller excitation frequencies, but quantitatively similar\neffective damping parameters; details are given in the\nSupplemental Material[62].\nThe different effective damping parameters could pos-\nsibly be determined experimentally by comparing the\nlinewidths of the different excitation modes at a selected\nfield value, or investigating the magnon decay over time.\nAn example for the latter case is shown in Fig. 3, dis-\nplaying the precession of a single spin in the skyrmion,\nobtained from the numerical solution of the LLG Eq. (1)\nwithα= 0.1. AtB= 0.75T, the frequencies of the\nm= 0breathing and m=−3triangular modes are close\nto each other (cf. Fig. 2), but the former decays much\nfaster. Because in the breathing mode the spin is follow-\ning a significantly more distorted elliptic path (inset of\nFig. 3) than in the triangular mode, the different effective\ndamping is also indicated by Eq. (6).\nIn summary, it was demonstrated within the phe-\nnomenological description of the LLG equation that the\neffective damping parameter αeffdepends on the consid-\nered magnon mode. The αeffassumes larger values if5\nthe polarization ellipse is strongly distorted as expressed\nby Eq. (6). Since NC magnetic structures provide an\nanisotropic environment for the spins, leading to a dis-\ntortion of the precession path, they provide a natural\nchoice for realizing different αeffvalues within a single\nsystem. The results of the theory were demonstrated for\nisolated skyrmions with material parameters describing\nthe Pd/Fe/Ir(111) system. 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Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nIn the Supplemental Material the derivation of the linearized equations of motion and the effective\ndamping parameter are discussed. Details of the numerical determination of the magnon modes in\nthe continuum model and in atomistic spin dynamics simulations are also given.\nS.I. LINEARIZED\nLANDAU–LIFSHITZ–GILBERT EQUATION\nHere we will derive the linearized form of the Landau–\nLifshitz–Gilbert equation given in Eqs. (2)-(3) of the\nmaintextanddiscussthepropertiesofthesolutions. The\ncalculation is similar to the undamped case, discussed in\ndetail in e.g. Refs. [1–3]. Given a spin configuration sat-\nisfying the equilibrium condition\nS0×Beff=0, (S.1)\nthe local coordinate system with ˜S0= (0,0,1)may be\nintroduced, andtheHamiltonianbeexpandedinthevari-\nables ˜Sxand˜Sy. Thelineartermmustdisappearbecause\nthe expansion is carried out around an equilibrium state.\nThe lowest-order nontrivial term is quadratic in the vari-\nables and will be designated as the spin wave Hamilto-\nnian,\nHSW=/integraldisplay\nhSWdr, (S.2)\nhSW=1\n2/bracketleftbig˜Sx˜Sy/bracketrightbig/bracketleftbiggA1A2\nA†\n2A3/bracketrightbigg/bracketleftbigg˜Sx\n˜Sy/bracketrightbigg\n=1\n2/parenleftBig\n˜S⊥/parenrightBigT\nHSW˜S⊥. (S.3)\nThe operator HSWis self-adjoint for arbitrary equi-\nlibrium states. Here we will only consider cases where\nthe equilibrium state is a local energy minimum, mean-\ning thatHSW≥0; the magnon spectrum will only be\nwell-defined in this case. Since hSWis obtained as an\nexpansion of a real-valued energy density around the\nequilibrium state, and the spin variables are also real-\nvalued, fromtheconjugateofEq.(S.3)onegets A1=A∗\n1,\nA2=A∗\n2, andA3=A∗\n3.\nThe form of the Landau–Lifshitz–Gilbert Eq. (1) in\nthe main text may be rewritten in the local coordinates\nby simply replacing Sby˜S0everywhere, including the\ndefinitionoftheeffectivefield Beff. TheharmonicHamil-\ntonianHSWin Eq. (S.2) leads to the linearized equation\nof motion\n∂t˜S⊥=γ/prime\nM(−iσy−α)HSW˜S⊥,(S.4)\n∗rozsa.levente@physnet.uni-hamburg.dewithσy=/bracketleftbigg\n0−i\ni0/bracketrightbigg\nthe Pauli matrix.\nBy replacing ˜S⊥(r,t)→˜S⊥\nk(r)e−iωktas usual, for\nα= 0the eigenvalue equation\nωk˜S⊥\nk=γ\nMσyHSW˜S⊥\nk (S.5)\nis obtained. If HSWhas a strictly positive spectrum,\nthenH−1\n2\nSWexists, and σyHSWhas the same eigenvalues\nasH1\n2\nSWσyH1\n2\nSW. Since the latter is a self-adjoint ma-\ntrix with respect to the standard scalar product on the\nHilbert space, it has a real spectrum, consequently all ωk\neigenvalues are real. Note that the zero modes of HSW,\nwhich commonly occur in the form of Goldstone modes\ndue to the ground state breaking a continuous symme-\ntry of the Hamiltonian, have to be treated separately.\nFinally, we mention that if the spin wave expansion is\nperformed around an equilibrium state which is not a\nlocal energy minimum, the ωkeigenvalues may become\nimaginary, meaning that the linearized Landau–Lifshitz–\nGilbert equation will describe a divergence from the un-\nstable equilibrium state instead of a precession around\nit.\nEquations (2)-(3) in the main text may be obtained\nby introducing the variables β±=˜Sx±i˜Syas described\nthere. The connection between HSWand the operators\nD0,Dnr, andDais given by\nD0=1\n2(A1+A3), (S.6)\nDnr=1\n2i/parenleftBig\nA†\n2−A2/parenrightBig\n, (S.7)\nDa=1\n2/bracketleftBig\nA1−A3+i/parenleftBig\nA†\n2+A2/parenrightBig/bracketrightBig\n.(S.8)\nAn important symmetry property of Eqs. (2)-(3) in\nthe main text is that if (β+,β−) =/parenleftbig\nβ+\nke−iωkt,β−\nke−iωkt/parenrightbig\nis an eigenmode of the equations, then (β+,β−) =/parenleftBig/parenleftbig\nβ−\nk/parenrightbig∗eiω∗\nkt,/parenleftbig\nβ+\nk/parenrightbig∗eiω∗\nkt/parenrightBig\nis another solution. Following\nRefs. [1, 3], this can be attributed to the particle-hole\nsymmetry of the Hamiltonian, which also holds in the\npresence of the damping term. From these two solutions\nmentioned above, the real-valued time evolution of the\nvariables ˜Sx,˜Symay be expressed as\n˜Sx\nk=eImωktcos (ϕ+,k−Reωkt)/vextendsingle/vextendsingleβ+\nk+β−\nk/vextendsingle/vextendsingle,(S.9)\n˜Sy\nk=eImωktsin (ϕ−,k−Reωkt)/vextendsingle/vextendsingleβ+\nk−β−\nk/vextendsingle/vextendsingle,(S.10)arXiv:1805.01815v2  [cond-mat.mtrl-sci]  15 Oct 20182\nwithϕ±,k= arg/parenleftbig\nβ+\nk±β−\nk/parenrightbig\n. As mentioned above, the\nImωkterms are zero in the absence of damping close to\na local energy minimum, and Im ωk<0is implied by\nthe fact that the Landau–Lifshitz–Gilbert equation de-\nscribes energy dissipation, which in the linearized case\ncorresponds to relaxation towards the local energy min-\nimum. In the absence of damping, the spins will precess\non an ellipse defined by the equation\n/parenleftBig\n˜Sx\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)\n+2˜Sx\nk˜Sy\nksin (ϕ+,k−ϕ−,k)/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsinglecos2(ϕ+,k−ϕ−,k)\n+/parenleftBig\n˜Sy\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)= 1,(S.11)\nwhere the superscript (0)indicatesα= 0. The semima-\njor and semiminor axes of the ellipse akandbkmay be\nexpressed from Eq. (S.11) as\nakbk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (S.12)\na2\nk+b2\nk= 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n.(S.13)\nNote thatβ+\nkandβ−\nk, consequently the parameters of\nthe precessional ellipse akandbk, are functions of the\nspatial position r.\nS.II. CALCULATION OF THE EFFECTIVE\nDAMPING PARAMETER FROM\nPERTURBATION THEORY\nHere we derive the expression for the effective damping\nparameter αeffgiven in Eq. (6) of the main text. By\nintroducingβk=/parenleftbig\nβ+\nk,−β−\nk/parenrightbig\n,\nD=/bracketleftbiggD0+Dnr−Da\n−D†\naD0−Dnr/bracketrightbigg\n,(S.14)\nand using the Pauli matrix σz=/bracketleftbigg\n1 0\n0−1/bracketrightbigg\n, Eqs. (2)-(3)\nin the main text may be rewritten as\n−ωkσzβk=γ/prime\nM(D+iασzD)βk(S.15)\nin the frequency domain. Following standard perturba-\ntion theory, we expand the eigenvalues ωkand the eigen-\nvectorsβkin the parameter α/lessmuch1. For the zeroth-order\nterms one gets\n−ω(0)\nkσzβ(0)\nk=γ\nMDβ(0)\nk, (S.16)\n0.0 0.1 0.2 0.3 0.4 0.50.00.51.01.52.02.5FIG. S1. Effective damping coefficients αm,effof the isolated\nskyrmion in the Pd/Fe/Ir(111) system at B= 1T, calcu-\nlated from the numerical solution of the linearized Landau–\nLifshitz–Gilbert equation (S.15), as a function of the Gilbert\ndamping parameter α.\nwith realω(0)\nkeigenvalues as discussed in Sec. S.I. The\nfirst-order terms read\n−ω(0)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n−ω(1)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n=γ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleD/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n+iαγ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσzD/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n,\n(S.17)\nafter taking the scalar product with β(0)\nk. The first terms\non both sides cancel by letting Dact to the left, then\nusing Eq. (S.16) and the fact that the ω(0)\nkare real. By\napplying Eq. (S.16) to the remaining term on the right-\nhand side one obtains\nω(1)\nk=−iαω(0)\nk/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr,(S.18)\nby writing in the definition of the scalar product. By\nusing the definition αk,eff=|Imωk/Reωk|≈/vextendsingle/vextendsingle/vextendsingleω(1)\nk/ω(0)\nk/vextendsingle/vextendsingle/vextendsingle\nand substituting Eqs. (S.12)-(S.13) into Eq. (S.18), one\narrives at Eq. (6) in the main text as long as/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndoes not change sign under the integral.\nIt is worthwhile to investigate for which values of α\ndoes first-order perturbation theory give a good estimate\nforαk,effcalculated from the exact solution of the lin-\nearized equations of motion, Eq. (S.15). In the materials\nwhere the excitations of isolated skyrmions or skyrmion\nlattices were investigated, significantly different values of\nαhave been found. For example, intrinsic Gilbert damp-\ning parameters of α= 0.02-0.04were determined experi-\nmentallyforbulkchiralmagnetsMnSiandCu 2OSeO 3[4],\nα= 0.28was deduced for FeGe[5], and a total damp-\ning ofαtot= 0.105was obtained for Ir/Fe/Co/Pt mag-\nnetic multilayers[6], where the latter value also includes3\nvarious effects beyond the Landau–Lifshitz–Gilbert de-\nscription. Figure S1 displays the dependence of αm,eff\nonαfor the eigenmodes of the isolated skyrmion in the\nPd/Fe/Ir(111) system, shown in Fig. 2 of the main text.\nMost of the modes show a linear correspondence between\nthe two quantities with different slopes in the displayed\nparameter range, in agreement with Eq. (6) in the main\ntext. For the breathing mode m= 0the convex shape\nof the curve indicates that the effective damping param-\neter becomes relatively even larger than the perturbative\nexpression Eq. (6) as αis increased.\nS.III. EIGENMODES OF THE ISOLATED\nSKYRMION\nHere we discuss the derivation of the skyrmion profile\nEq. (8) and the operators in Eqs. (9)-(11) of the main\ntext. The energy density Eq. (7) in polar coordinates\nreads\nh=A/bracketleftbigg\n(∂rΘ)2+ sin2Θ (∂rΦ)2+1\nr2(∂ϕΘ)2\n+1\nr2sin2Θ (∂ϕΦ)2/bracketrightbigg\n+D/bracketleftbigg\ncos (ϕ−Φ)∂rΘ\n−1\nrsin (ϕ−Φ)∂ϕΘ + sin Θ cos Θ sin ( ϕ−Φ)∂rΦ\n+1\nrsin Θ cos Θ cos ( ϕ−Φ)∂ϕΦ/bracketrightbigg\n+Kcos2Θ−MBcos Θ.\n(S.19)\nThe Landau–Lifshitz–Gilbert Eq. (1) may be rewritten\nas\nsin Θ∂tΘ =γ/primeBΦ+αγ/primesin ΘBΘ,(S.20)\nsin Θ∂tΦ =−γ/primeBΘ+αγ/prime1\nsin ΘBΦ,(S.21)\nwith\nBχ=−1\nMδH\nδχ\n=−1\nM/bracketleftbigg\n−1\nr∂r/parenleftbigg\nr∂h\n∂(∂rχ)/parenrightbigg\n−∂ϕ∂h\n∂(∂ϕχ)+∂h\n∂χ/bracketrightbigg\n,\n(S.22)\nwhereχstands for ΘorΦ. Note that in this form it is\ncommon to redefine BΦto include the 1/sin Θfactor in\nEq. (S.21)[7]. The first variations of Hfrom Eq. (S.19)may be expressed as\nδH\nδΘ=−2A/braceleftbigg\n∇2Θ−sin Θ cos Θ/bracketleftbigg\n(∂rΦ)2+1\nr2(∂ϕΦ)2/bracketrightbigg/bracerightbigg\n−2Ksin Θ cos Θ +MBsin Θ\n−2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΦ + cos (ϕ−Φ)1\nr∂ϕΦ/bracketrightbigg\n,\n(S.23)\nδH\nδΦ=−2A/braceleftbigg\nsin2Θ∇2Φ + sin 2Θ/bracketleftbigg\n∂rΘ∂rΦ +1\nr2∂ϕΘ∂ϕΦ/bracketrightbigg/bracerightbigg\n+ 2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΘ + cos (ϕ−Φ)1\nr∂ϕΘ/bracketrightbigg\n,\n(S.24)\nTheequilibriumconditionEq.(8)inthemaintextmay\nbe obtained by setting ∂tΘ =∂tΦ = 0in Eqs. (S.20)-\n(S.21) and assuming cylindrical symmetry, Θ0(r,ϕ) =\nΘ0(r)and Φ0(r,ϕ) =ϕ. In the main text D>0\nandB≥0were assumed. Choosing D<0switches\nthe helicity of the structure to Φ0=ϕ+π, in which\ncaseDshould be replaced by |D|in Eq. (8). For the\nbackground magnetization pointing in the opposite di-\nrectionB≤0, one obtains the time-reversed solutions\nwith Θ0→π−Θ0,Φ0→Φ0+π,B→−B. Time rever-\nsal also reverses clockwise and counterclockwise rotating\neigenmodes; however, the above transformations do not\ninfluence the magnitudes of the excitation frequencies.\nFinally, we note that the frequencies remain unchanged\neven if the form of the Dzyaloshinsky–Moriya interaction\nin Eq. (S.19), describing Néel-type skyrmions common in\nultrathin films and multilayers, is replaced by an expres-\nsion that prefers Bloch-type skyrmions occurring in bulk\nhelimagnets – see Ref. [3] for details.\nFordeterminingthelinearizedequationsofmotion,one\ncan proceed by switching to the local coordinate system\nas discussed in Sec. S.I and Refs. [1, 3]. Alternatively,\nthey can also directly be derived from Eqs. (S.20)-(S.21)\nby introducing Θ = Θ 0+˜Sx,Φ = Φ 0+1\nsin Θ 0˜Syand\nexpanding around the skyrmion profile from Eq. (8) up\nto first order in ˜Sx,˜Sy– see also Ref. [2]. The operators\nin Eq. (S.3) read\nA1=−2A/parenleftbigg\n∇2−1\nr2cos 2Θ 0(∂ϕΦ0)2/parenrightbigg\n−2D1\nrsin 2Θ 0∂ϕΦ0−2Kcos 2Θ 0+MBcos Θ 0,\n(S.25)\nA2=4A1\nr2cos Θ 0∂ϕΦ0∂ϕ−2D1\nrsin Θ 0∂ϕ,(S.26)\nA3=−2A/braceleftbigg\n∇2+/bracketleftbigg\n(∂rΘ0)2−1\nr2cos2Θ0(∂ϕΦ0)2/bracketrightbigg/bracerightbigg\n−2D/parenleftbigg\n∂rΘ0+1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−2Kcos2Θ0+MBcos Θ 0, (S.27)4\nwhich leads directly to Eqs. (9)-(11) in the main text via\nEqs. (S.6)-(S.8).\nThe excitation frequencies of the ferromagnetic state\nmay be determined by setting Θ0≡0in Eqs. (9)-(11) in\nthe main text. In this case, the eigenvalues and eigenvec-\ntors can be calculated analytically[1],\nωk,m=γ/prime\nM(1−iα)/bracketleftbig\n2Ak2−2K+MB/bracketrightbig\n,(S.28)\n/parenleftBig\nβ+\nk,m(r),β−\nk,m(r)/parenrightBig\n= (0,Jm−1(kr)),(S.29)\nwithJm−1theBesselfunction ofthefirstkind, appearing\ndue to the solutions being regular at the origin. Equa-\ntion (S.28) demonstrates that the lowest-frequency exci-\ntation of the background is the ferromagnetic resonance\nfrequencyωFMR =γ\nM(MB−2K)atα= 0. Since the\nanomalous term Dadisappears in the out-of-plane mag-\nnetized ferromagnetic state, all spin waves will be circu-\nlarly polarized, see Eq. (S.29), and the effective damping\nparameterwillalwayscoincidewiththeGilbertdamping.\nRegarding the excitations of the isolated skyrmion, for\nα= 0the linearized equations of motion in Eq. (S.15)\nare real-valued; consequently, β±\nk,m(r)can be chosen to\nbe real-valued. In this case Eqs. (S.9)-(S.10) take the\nform\n˜Sx\nk,m= cos (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r) +β−\nk,m(r)/parenrightBig\n,(S.30)\n˜Sy\nk,m= sin (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r)−β−\nk,m(r)/parenrightBig\n.(S.31)\nThis means that modes with ωk,m>0form> 0will\nrotate counterclockwise, that is, the contours with con-\nstant ˜Sx\nk,mand ˜Sy\nk,mwill move towards higher values of\nϕastis increased, while the modes with ωk,m>0for\nm < 0will rotate clockwise. Modes with m= 0corre-\nspond to breathing excitations. This sign convention for\nmwas used when designating the localized modes of the\nisolated skyrmion in the main text, and the kindex was\ndropped since only a single mode could be observed be-\nlow the ferromagnetic resonance frequency for each value\nofm.\nS.IV. NUMERICAL SOLUTION OF THE\nEIGENVALUE EQUATIONS\nThe linearized Landau–Lifshitz–Gilbert equation for\nthe isolated skyrmion, Eqs. (2)-(3) with the operators\nEqs.(9)-(11)inthemaintext, weresolvednumericallyby\na finite-difference method. First the equilibrium profile\nwas determined from Eq. (8) using the shooting method\nfor an initial approximation, then obtaining the solution\non a finer grid via finite differences. For the calculationswe used dimensionless parameters (cf. Ref. [8]),\nAdl= 1, (S.32)\nDdl= 1, (S.33)\nKdl=KA\nD2, (S.34)\n(MB)dl=MBA\nD2, (S.35)\nrdl=|D|\nAr, (S.36)\nωdl=MA\nγD2ω. (S.37)\nThe equations were solved in a finite interval for\nrdl∈[0,R], with the boundary conditions Θ0(0) =\nπ,Θ0(R) = 0. For the results presented in Fig. 2 in the\nmain text the value of R= 30was used. It was confirmed\nbymodifying Rthattheskyrmionshapeandthefrequen-\ncies of the localized modes were not significantly affected\nby the boundary conditions. However, the frequencies of\nthe modes above the ferromagnetic resonance frequency\nωFMR =γ\nM(MB−2K)did change as a function of\nR, since these modes are extended over the ferromag-\nnetic background – see Eqs. (S.28)-(S.29). Furthermore,\nin the infinitely extended system the equations of mo-\ntion include a Goldstone mode with/parenleftbig\nβ+\nm=−1,β−\nm=−1/parenrightbig\n=/parenleftbig\n−1\nrsin Θ 0−∂rΘ0,1\nrsin Θ 0−∂rΘ0/parenrightbig\n, corresponding to\nthe translation of the skyrmion on the collinear\nbackground[1]. This mode obtains a finite frequency in\nthe numerical calculations due to the finite value of R\nand describes a slow clockwise gyration of the skyrmion.\nHowever, this frequency is not shown in Fig. 3 of the\nmain text because it is only created by boundary effects.\nIn order to investigate the dependence of the effective\ndamping on the dimensionless parameters, we also per-\nformed the calculations for the parameters describing the\nIr|Co|Pt multilayer system[9]. The results are summa-\nrized in Fig. S2. The Ir|Co|Pt system has a larger di-\nmensionless anisotropy value ( −KIr|Co|Pt\ndl = 0.40) than\nthe Pd/Fe/Ir(111) system ( −KPd/Fe/Ir(111)\ndl = 0.33). Al-\nthough the same localized modes are found in both cases,\nthe frequencies belonging to the m= 0,1,−3,−4,−5\nmodes in Fig. S2 are relatively smaller than in Fig. 2\ncompared to the ferromagnetic resonance frequency at\nthe elliptic instability field where ωm=−2= 0. This\nagrees with the two limiting cases discussed in the lit-\nerature: it was shown in Ref. [1] that for Kdl= 0the\nm= 1,−4,−5modes are still above the ferromagnetic\nresonance frequency at the elliptic instability field, while\nin Ref. [2] it was investigated that all modes become soft\nwithfrequenciesgoingtozeroat (MB)dl= 0inthepoint\n−Kdl=π2\n16≈0.62,belowwhichaspinspiralgroundstate\nis formed in the system. Figure S2(b) demonstrates that\nthe effective damping parameters αm,effare higher at the\nellipticinstabilityfieldinIr|Co|PtthaninPd/Fe/Ir(111),\nshowing an opposite trend compared to the frequencies.\nRegarding the physical units, the stronger exchange\nstiffness combined with the weaker Dzyaloshinsky–5\n0.03 0.04 0.05 0.06 0.07 0.080246810\n(a)\n0.03 0.04 0.05 0.06 0.07 0.081.01.52.02.53.03.5\n(b)\nFIG. S2. Localized magnons in the isolated skyrmion, with\nthe interaction parameters corresponding to the Ir|Co|Pt\nmultilayer system from Ref. [9]: A= 10.0pJ/m,D=\n1.9mJ/m2,K=−0.143MJ/m3,M = 0.96MA/m. The\nanisotropy reflects an effective value including the dipolar in-\nteractions as a demagnetizing term, −K =−K 0−1\n2µ0M2\nwithK0=−0.717MJ/m3. (a) Magnon frequencies f=ω/2π\nforα= 0. Illustrations display the shapes of the excitation\nmodes visualized as the contour plot of the out-of-plane spin\ncomponentsona 1×1nm2grid,withredandbluecolorscorre-\nsponding to positive and negative Szvalues, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6)\nin the main text.Moriya interaction and anisotropy in the multilayer sys-\ntem leads to larger skyrmions stabilized at lower field val-\nues and displaying lower excitation frequencies. We note\nthat demagnetization effects were only considered here\nas a shape anisotropy term included in K; it is expected\nthat this should be a relatively good approximation for\nthe Pd/Fe/Ir(111) system with only a monolayer of mag-\nnetic material, but it was suggested recently[6] that the\ndipolar interaction can significantly influence the excita-\ntion frequencies of isolated skyrmions in magnetic multi-\nlayers.\nS.V. SPIN DYNAMICS SIMULATIONS\nFor the spin dynamics simulations displayed in Fig. 3\nin the main text we used an atomistic model Hamiltonian\non a single-layer triangular lattice,\nH=−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightJSiSj−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightDij(Si×Sj)−/summationdisplay\niK(Sz\ni)2\n−/summationdisplay\niµBSz\ni, (S.38)\nwith the parameters J= 5.72meV for the Heisenberg\nexchange,D=|Dij|= 1.52meV for the Dzyaloshinsky–\nMoriya interaction, K= 0.4meV for the anisotropy,\nµ= 3µBfor the magnetic moment, and a= 0.271nm\nfor the lattice constant. For the transformation be-\ntween the lattice and continuum parameters in the\nPd/Fe/Ir(111) system see, e.g., Ref. [10]. The simula-\ntionswereperformedbynumericallysolvingtheLandau–\nLifshitz–Gilbert equation on an 128×128lattice with\nperiodic boundary conditions, which was considerably\nlarger than the equilibrium skyrmion size to minimize\nboundary effects. The initial configuration was deter-\nmined by calculating the eigenvectors in the continuum\nmodel and discretizing it on the lattice, as shown in the\ninsets of Fig. 2 in the main text. It was found that such\na configuration was very close to the corresponding exci-\ntation mode of the lattice Hamiltonian Eq. (S.38), simi-\nlarly to the agreement between the continuum and lattice\nequilibrium skyrmion profiles[10].\n[1] C. Schütte and M. Garst, Phys. Rev. B 90, 094423\n(2014).\n[2] V. P. Kravchuk, D. D. Sheka, U. K. Rössler, J. van den\nBrink, andYu.Gaididei, Phys.Rev.B 97, 064403(2018).\n[3] S.-Z. Lin, Phys. Rev. B 96, 014407 (2017).\n[4] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I.\nStasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler,\nNat. Mater. 14, 478 (2015).\n[5] M. Beg, M. Albert, M.-A. Bisotti, D. Cortés-Ortuño, W.\nWang, R. Carey, M. Vousden, O. Hovorka, C. Ciccarelli,\nC. S. Spencer, C. H. Marrows, and H. Fangohr, Phys.\nRev. B95, 014433 (2017).\n[6] B. Satywali, F. Ma, S. He, M. Raju, V. P. Kravchuk,\nM. Garst, A. Soumyanarayanan, and C. Panagopoulos,arXiv:1802.03979 (2018).\n[7] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys.\nRev. E90, 023203 (2014).\n[8] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku-\nbetzka, A. N. Bogdanov, and R. Wiesendanger, New J.\nPhys.18, 065003 (2016).\n[9] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sam-\npaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K.\nGarcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M.\nGeorge, M. Weigand, J. Raabe, V. Cros, and A. Fert,\nNat. Nanotechnol. 11, 444 (2016).\n[10] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme-\ndenko, and R. Wiesendanger, Phys. Rev. B 97, 174436\n(2018)."    },    {        "title": "1411.7219v1.Geometry_of_world_sheets_in_Lorentz_Minkowski_space.pdf",        "content": "arXiv:1411.7219v1  [math.DG]  26 Nov 2014Geometry of world sheets in Lorentz-Minkowski space\nShyuichi IZUMIYA∗\nSeptember 25, 2018\nAbstract\nA world sheet in Lorentz-Minkowski space is a timelike subma nifold consisting of\na one-parameter family of spacelike submanifolds in Lorent z-Minkowski space. In this\npaper we investigate differential geometry of world sheets in Lorentz-Minkowski space as\nan application of the theory of big wave fronts.\n1 Introduction\nIn this paper we consider differential geometry of world sheets in Lo rentz-Minkowski space. A\nworld sheet is a timelike submanifold consisting of a one-parameter fa mily of spacelike sub-\nmanifolds in a Lorentz manifold. Since we do not have the notion of con stant time in the\nrelativity theory, we consider one-parameter families of spacelike s ubmanifolds depending on\nthe time-parameter (i.e., world sheets). In this case, the spacelike submanifold with the con-\nstant parameter is not necessarily the constant time in the ambient space. If we observe a\nsurface in our space, then it is moving around the sun. Moreover, t he solar system itself is\nmoving depending on the Galaxy movement. Therefore, even if it look s a fixed surface (for\nexample, a surface of a solid body) in Euclidean 3-space, it is a three d imensional world sheets\nin Lorentz-Minkowski 4-space. Moreover, there appeared highe r dimensional Lorentz mani-\nfolds in the theoretical physics (i.e., the super string theory, the b rane world scenario etc.). So\nwe consider world sheets with general codimension in general dimens ional Lorentz-Minkowski\nspace. In [11] lightlike flat geometry on a spacelike submanifold with ge neral codimension has\nbeen investigated. Their method is quite useful for the study of th e geometry of world sheets.\nOn the other hand, Lorentz-Minkowski space gives a geometric fr amework of the special\nrelativity theory. Although there are no gravity in Lorentz-Minkow ski space, it provides a\nsimple model of general Lorentz manifolds. In this paper we investig ate the lightlike geometry\nofworld sheets inLorentz-Minkowski space with general codimens ion fromtheview point ofthe\ncontact with lightlike hyperplanes. The natural connection betwee n geometry and singularities\nrelies on the basic fact that the contact of a submanifold with the mo dels of the ambient space\ncan be described by means of the analysis if the singularities of appro priate families of contact\nfunctions, or equivalently, of their associated Lagrangian/Legen drian maps. For the lightlike\ngeometry the models are lightlike hyperplanes or lightcones. The light like flat geometry is\nthe lightlike geometry which adopts lightlike hyperplanes as model hyp ersurfaces. Since we\n∗Department of Mathematics, Hokkaido University, Sapporo 060-0 810, Japan.\ne-mail:izumiya@math.sci.hokudai.ac.jp\n1consider world sheets (i.e., one parameter families of spacelike subma nifolds), the models are\nfamilies of lightlike hyperplanes and the theory of one parameter bifu rcations of Legendrian\nsingularities is essentially useful. Such a theory was initiated by Zakaly ukin [16, 17] as the\ntheory of big wave fronts. There have been some developments on this theory during past\ntwo decades[5, 6, 8, 9, 10, 18, 19]. Several applications of the the ory were discovered in those\narticles. For applying this theory, some equivalence relations among big wave fronts were used.\nHere, we consider another equivalence relation among big wave fron ts which is different from\nthe equivalence relations considered in those articles. This equivalen ce relation is corresponding\nto the equivalence relation introduced in [2, 3] for applying the singula rity theory to bifurcation\nproblems.\nIn§2 basic notations and properties of Lorentz-Minkowski space are explained. Differential\ngeometry of world sheets in Lorentz-Minkowski space is construc ted in§3. We introduce the\nnotion of (world and momentary) lightcone Gauss maps and induce th e corresponding curva-\ntures of world sheets respectively. In §4 we define the lightcone height functions family and the\nextended lightcone height functions family of a world sheet. We calcu late the singular points of\nthese families of functions andinduce the notionoflightcone pedal m aps andunfolded lightcone\npedal maps respectively. We investigate the geometric meanings of the singular points of the\nlightcone pedal maps from the view point of the contact with families o f lightlike hyperplanes\nin§5. We can show that the image of the unfolded lightcone pedal map is a big wave front of a\ncertain big Legendrian submanifold. Therefore, we apply the theor y of big wave fronts to our\nsituation and interpret the geometric meanings of the singularities o f the unfolded lightcone\npedal map in §6.\n2 Basic concepts\nWe introduce in this section some basic notions on Lorentz-Minkowsk i (n+1)-space. For basic\nconcepts and properties, see [14]. Let Rn+1={(x0,x1,...,x n)|xi∈R(i= 0,1,...,n)}be an\n(n+1)-dimensional cartesian space. For any x= (x0,x1,...,x n),y= (y0,y1,...,y n)∈Rn+1,\nthepseudo scalar product ofxandyis defined by /an}bracketle{tx,y/an}bracketri}ht=−x0y0+/summationtextn\ni=1xiyi.We call\n(Rn+1,/an}bracketle{t,/an}bracketri}ht)Lorentz-Minkowski (n+ 1)-space. We write Rn+1\n1instead of ( Rn+1,/an}bracketle{t,/an}bracketri}ht). We say\nthat a non-zero vector x∈Rn+1\n1isspacelike, lightlike or timelike if/an}bracketle{tx,x/an}bracketri}ht>0,/an}bracketle{tx,x/an}bracketri}ht= 0 or\n/an}bracketle{tx,x/an}bracketri}ht<0 respectively. The norm of the vector x∈Rn+1\n1is defined to be /bardblx/bardbl=/radicalbig\n|/an}bracketle{tx,x/an}bracketri}ht|.We\nhave the canonical projection π:Rn+1\n1−→Rndefined byπ(x0,x1,...,x n) = (x1,...,x n).Here\nwe identify {0}×RnwithRnand it is considered as Euclidean n-space whose scalar product is\ninduced from the pseudo scalar product /an}bracketle{t,/an}bracketri}ht.For a non-zero vector v∈Rn+1\n1and a real number\nc,we define a hyperplane with pseudo normal vby\nHP(v,c) ={x∈Rn+1\n1| /an}bracketle{tx,v/an}bracketri}ht=c}.\nWe callHP(v,c) aspacelike hyperplane , atimelike hyperplane or alightlike hyperplane ifvis\ntimelike, spacelike or lightlike respectively.\nWe now define Hyperbolicn-spaceby\nHn\n+(−1) ={x∈Rn+1\n1|/an}bracketle{tx,x/an}bracketri}ht=−1,x0>0}\nandde Sittern-spaceby\nSn\n1={x∈Rn+1\n1|/an}bracketle{tx,x/an}bracketri}ht= 1}.\n2We define\nLC∗={x= (x0,x1,...,x n)∈Rn+1\n1|x0/ne}ationslash= 0,/an}bracketle{tx,x/an}bracketri}ht= 0}\nand we call it the(open)lightcone at the origin. In the lightcone, we have the canonical unit\nspacelike sphere defined by\nSn−1\n+={x= (x0,x1,...,x n)| /an}bracketle{tx,x/an}bracketri}ht= 0, x0= 1}.\nWe callSn−1\n+thelightcone unit (n−1)-sphere.Ifx= (x0,x1,...,x n) is a lightlike vector, then\nx0/ne}ationslash= 0.Therefore we have\n/tildewidex=/parenleftbigg\n1,x1\nx0,...,xn\nx0/parenrightbigg\n∈Sn−1\n+.\nIt follows that we have a projection πL\nS:LC∗−→Sn−1\n+defined byπL\nS(x) =/tildewidex.\nFor anyx1,x2,...,xn∈Rn+1\n1,we define a vector x1∧x2∧···∧xnby\nx1∧x2∧···∧xn=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−e0e1···en\nx1\n0x1\n1···x1\nn\nx2\n0x2\n1···x2\nn......···...\nxn\n0xn\n1···xn\nn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwheree0,e1,...,enis the canonical basis of Rn+1\n1andxi= (xi\n0,xi\n1,...,xi\nn).We can easily check\nthat/an}bracketle{tx,x1∧x2∧···∧xn/an}bracketri}ht= det(x,x1,...,xn),so thatx1∧x2∧···∧xnis pseudo orthogonal\nto anyxi(i= 1,...,n).\n3 World sheets in Lorentz-Minkowski space\nIn this section we introduce the basic geometrical framework for t he study of world sheets in\nLorentz-Minkowski ( n+ 1)-space. Let Rn+1\n1be a time-oriented space (cf., [14]). We choose\ne0= (1,0,...,0) as the future timelike vector field. The world sheet is defined to be a timelike\nsubmanifold foliated by a codimension one spacelike submanifolds. Her e, we only consider the\nlocal situation, so that we considered a one-parameter family of sp acelike submanifolds. Let\nX:U×I−→Rn+1\n1bea timelike embedding ofcodimension k−1,whereU⊂Rs(s+k=n+1)\nis an open subset and Ian open interval. We write W=X(U×I) and identify WandU×I\nthrough the embedding X.The embedding Xis said to be timelikeif the tangent space TpW\nofWis a timelike subspace (i.e., Lorentz subspace of TpRn+1\n1) at any point p∈W. We write\nSt=X(U×{t}) for eacht∈I.We have a foliation S={St|t∈I}onW. We say that Stis\nspacelike if the tangent space TpStconsists only spacelike vectors (i.e., spacelike subspace) for\nany pointp∈ St.We say that ( W,S) (or,X) is aworld sheet ifWis time-orientable and each\nStis spacelike. We call Stamomentary space of (W,S). For anyp=X(u,t)∈W⊂Rn+1\n1,we\nhave\nTpW=/an}bracketle{tXt(u,t),Xu1(u,t),...,Xus(u,t)/an}bracketri}htR,\nwhere we write ( u,t) = (u1,...,u s,t)∈U×I,Xt=∂X/∂tandXuj=∂X/∂uj.We also have\nTpSt=/an}bracketle{tXu1(u,t),...,Xus(u,t)/an}bracketri}htR.\nSinceWis time-orientable, there exists a timelike vector field v(u,t) onW[14, Lemma 32].\nMoreover, we can choose that visfuture directed which means that /an}bracketle{tv(u,t),e0/an}bracketri}ht<0.\n3LetNp(W)bethepseudo-normalspaceof Watp=X(u,t)inRn+1\n1.SinceTpWisatimelike\nsubspace of TpRn+1\n1,Np(W) is a (k−1)-dimensional spacelike subspace of TpRn+1\n1(cf.,[14]). On\nthe pseudo-normal space Np(W),we have a ( k−2)-unit sphere\nN1(W)p={ξ∈Np(W)| /an}bracketle{tξ,ξ/an}bracketri}ht= 1}.\nTherefore, we have a unit spherical normal bundle over W:\nN1(W) =/uniondisplay\np∈WN1(W)p.\nOn the other hand, we write Np(St) as the pseudo-normal space of Statp=X(u,t) in\nRn+1\n1.ThenNp(St) isak-dimensional Lorentz subspace of TpRn+1\n1. Onthe pseudo-normal space\nNp(St),we have two kinds of pseudo spheres:\nNp(St;−1) ={v∈Np(St)| /an}bracketle{tv,v/an}bracketri}ht=−1}\nNp(St;1) ={v∈Np(St)| /an}bracketle{tv,v/an}bracketri}ht= 1}.\nWe remark that Np(St;−1) is the (k−1)-dimensional hyperbolic space and Np(St;1) is the (k−\n1)-dimensional deSitter space. Therefore, we have two unit sphe rical normal bundles N(St;−1)\nandN(St;1) overSt. SinceSt=X(U×{t}) is a codimension one spacelike submanifold in W,\nthere exists a unique timelike future directed unit normal vector fie ldnT(u,t) ofStsuch that\nnT(u,t) is tangent to Wat any point p=X(u,t).It means that nT(u,t)∈Np(St)∩TpWwith\n/an}bracketle{tnT(u,t),nT(u,t)/an}bracketri}ht=−1 and/an}bracketle{tnT(u,t),e0/an}bracketri}ht<0.We define a ( k−2)-dimensional spacelike unit\nsphere inNp(St) by\nN1(St)p[nT] ={ξ∈Np(St;1)| /an}bracketle{tξ,nT(u,t)/an}bracketri}ht= 0,p=X(u,t)}.\nThen we have a spacelike unit (k−2)-spherical bundle N1(St)[nT]overStwith respect to nT.\nSince we have T(p,ξ)N1(St)[nT] =TpSt×TξN1(St)p[nT],we have the canonical Riemannian\nmetric onN1(St)[nT] which we write ( Gij((u,t),ξ))1/lessorequalslanti,j/lessorequalslantn−1.SincenTis uniquely determined,\nwe writeN1[St] =N1(St)[nT].Moreover, we remark that N1(W)|St=N1[St] for anyt∈I.\nWe now define a map LG:N1(W)−→LC∗byLG(X(u,t),ξ) =nT(u,t)+ξ. We call LG\naworld lightcone Gauss map ofN1(W), whereW=X(U×I). Amomentary lightcone Gauss\nmapofN1[St] is defined to be the restriction of the world lightcone Gauss map of N1(W):\nLG(St) =LG|N1[St] :N1[St]−→LC∗.\nThis map leads us to the notions of curvatures. Let T(p,ξ)N1[St] be the tangent space of N1[St]\nat (p,ξ).With the canonical identification\n(LG(St)∗TRn+1\n1)(p,ξ)=T(nT(p)+ξ)Rn+1\n1≡TpRn+1\n1,\nwe have\nT(p,ξ)N1[St] =TpSt⊕TξSk−2⊂TpSt⊕Np(St) =TpRn+1\n1,\nwhereTξSk−2⊂TξNp(St)≡Np(St) andp=X(u,t).Let\nΠt:LG(St)∗TRn+1\n1=TN1[St]⊕Rs+2−→TN1[St]\n4be the canonical projection. Then we have a linear transformation\nSℓ(St)(p,ξ)=−Πt\nLG(St)(p,ξ)◦d(p,ξ)LG(St) :T(p,ξ)N1[St]−→T(p,ξ)N1[St],\nwhich is called the momentary lightcone shape operator ofN1[St] at (p,ξ).\nOn the other hand, for t0∈I,we choose a spacelike unit vector field nSalongW=\nX(U×I) at least locally such that nS(u,t0)∈N1(St0).Then we have /an}bracketle{tnS,nS/an}bracketri}ht= 1 and\n/an}bracketle{tXt,nS/an}bracketri}ht=/an}bracketle{tXui,nS/an}bracketri}ht=/an}bracketle{tnT,nS/an}bracketri}ht= 0 at(u,t0)∈U×I.Clearly, the vector nT(u,t0)+nS(u,t0)\nis lightlike. We define a mapping\nLG(St0;nS) :U−→LC∗\nbyLG(St0;nS)(u) =nT(u,t0) +nS(u,t0),which is called a momentary lightcone Gauss map\nofSt0=X(U×{t0})with respect to nS.Under the identification of St0andU×{t0}through\nX,we have the linear mapping provided by the derivative of the momenta ry lightcone Gauss\nmapLG(St0;nS) at each point p=X(u,t0),\ndpLG(St0;nS) :TpSt0−→TpRn+1\n1=TpSt0⊕Np(St0).\nConsider the orthogonal projection πt:TpSt0⊕Np(St0)→TpSt0.We define\nSp(St0;nS) =−πt◦dpLG(St0;nS) :TpSt0−→TpSt0.\nWe call the linear transformation Sp(St0;nS) annS-momentary shape operator ofSt0=X(U×\n{t0}) atp=X(u,t0).Let{κi(St0;nS)(p)}s\ni=1be the eigenvalues of Sp(St0;nS), which are called\nmomentary lightcone principal curvatures of St0with respect to nSatp=X(u,t0). Then a\nmomentary lightcone Lipschitz-Killing curvature of St0with respect to nSatp=X(u,t0) is\ndefined as follows:\nKℓ(St0;nS)(p) = detSp(St0;nS).\nWe say that a point p=X(u,t0) is annS-momentary lightcone umbilical point ofSt0if\nSp(St0;nS) =κ(St0;nS)(p)1TpSt0.\nWe say that W=X(U×I) istotallynS-lightcone umbilical if each point p=X(u,t)∈W\nis annS-momentary lightcone umbilical point of St.Moreover,W=X(U×I) is said to be\ntotally lightcone umbilical if it is totally nS-lightcone umbilical for any nS.We deduce now the\nlightcone Weingarten formula. Since St0=X(U× {t0}) is spacelike submanifold, we have a\nRiemannian metric (the first fundamental form ) onSt0defined byds2=/summationtexts\ni=1gijduiduj, where\ngij(u,t0) =/an}bracketle{tXui(u,t0),Xuj(u,t0)/an}bracketri}htfor anyu∈U.We also have a lightcone second fundamental\ninvariant of St0with respect to the normal vector field nSdefined by hij(St0;nS)(u,t0) =\n/an}bracketle{t−(nT+nS)ui(u,t0),Xuj(u,t0)/an}bracketri}htfor anyu∈U.By the similar arguments to those in the proof\nof [7, Proposition 3.2], we have the following proposition.\nProposition 3.1 We choose a pseudo-orthonormal frame {nT,nS\n1,...,nS\nk−1}ofN(St0)with\nnS\nk−1=nS.Then we have the following lightcone Weingarten formula :\n(a)LG(St0;nS)ui=/an}bracketle{tnT\nui,nS/an}bracketri}ht(nT+nS)+/summationtextk−2\nℓ=1/an}bracketle{t(nT+nS)ui,nS\nℓ/an}bracketri}htnS\nℓ\n−/summationtexts\nj=1hj\ni(St0;nS)Xuj,\n(b)πt◦LG(St0;nS)ui=−/summationtexts\nj=1hj\ni(St0;nS)Xuj.\nHere,/parenleftbig\nhj\ni(St0;nS)/parenrightbig\n=/parenleftbig\nhik(St0;nS)/parenrightbig/parenleftbig\ngkj/parenrightbig\nand/parenleftbig\ngkj/parenrightbig\n= (gkj)−1.\n5SinceLG(St0;nS)ui=dLG(St0;nS)(Xui), we have\nSp(St0;nS)(Xui(u,t0)) =−πt◦LG(St0;nS)ui(u,t0),\nsothattherepresentationmatrixof Sp(St0;nS)withrespecttothebasis {Xui(u,t0)}s\ni=1ofTpSt0\nis (hi\nj(St0;nS)(u,t0)).Therefore, we have an explicit expression of the momentary lightco ne\nLipschitz-Killing curvature of St0with respect to nSas follows:\nKℓ(St0;nS)(u,t0) =det/parenleftbig\nhij(St0;nS)(u,t0)/parenrightbig\ndet(gαβ(u,t0)).\nSince/an}bracketle{t−(nT+nS)(u,t0),Xuj(u,t0)/an}bracketri}ht= 0, we have\nhij(St0;nS)(u,t0) =/an}bracketle{tnT(u,t0)+nS(u,t0),Xuiuj(u,t0)/an}bracketri}ht.\nTherefore the lightcone second fundamental invariants of St0at a pointp0=X(u0,t0) depend\nonly on the values nT(u0)+nS(u0) andXuiuj(u0), respectively. Therefore, we write\nhij(St0;nS)(u0,t0) =hij(St0)(p0,ξ0),\nwherep0=X(u0,t0) andξ0=nS(u0,t0)∈N1(W)p0.Thus, the nS-momentary shape operator\nandthe momentary lightcone curvatures also depend only on nT(u0,t0)+nS(u0,t0),Xui(u0,t0)\nandXuiuj(u0,t0), which are independent of the derivations of the vector fields nTandnS.It\nfollows that we write Sp0(St0;ξ0) =Sp0(St0;nS), κi(St0,ξ0)(p0) =κi(St0;nS)(p0) (i= 1,...,s)\nandKℓ(St0,ξ0)(p0) =Kℓ(St0;nS)(p0) atp0=X(u0,t0) with respect to ξ0=nS(u0,t0).We\nalso say that a point p0=X(u0,t0) isξ0-momentary lightcone umbilical ifSp0(St0;ξ0) =\nκi(St0)(p0,ξ0)1Tp0St0. We say that a point p0=X(u0,t0) is aξ0-momentary lightcone parabolic\npointofSt0ifKℓ(St0;ξ0)(p0) = 0.\nLetκℓ(St)i(p,ξ)betheeigenvalues ofthelightconeshapeoperator Sℓ(St)(p,ξ), (i= 1,...,n−\n1). We write κℓ(St)i(p,ξ), (i= 1,...,s) for the eigenvalues whose eigenvectors belong to TpSt\nandκℓ(St)i(p,ξ), (i=s+1,...n) for the eigenvalues whose eigenvectors belong to the tangent\nspace of the fiber of N1[St].\nProposition 3.2 Forp0=X(u0,t0)andξ0∈N1[St0]p0,we have\nκℓ(St0)i(p0,ξ0) =κi(St0,ξ0)(p0),(i= 1,...s), κℓ(St0)i(p0,ξ0) =−1,(i=s+1,...n).\nProof.Since{nT,nS\n1,...,nS\nk−1}isapseudo-orthonormalframeof N(St)andξ0=nS\nk−1(u0,t0)∈\nSk−2=N1[St0]p,we have/an}bracketle{tnT(u0,t0),ξ0/an}bracketri}ht=/an}bracketle{tnS\ni(u0,t0),ξ0/an}bracketri}ht= 0 fori= 1,...,k−2.Therefore,\nwe have\nTξ0Sk−2=/an}bracketle{tnS\n1(u0,t0),...,nS\nk−2(u0,t0)/an}bracketri}ht.\nBythisorthonormalbasisof Tξ0Sk−2,thecanonicalRiemannianmetric Gij(p0,ξ0)isrepresented\nby\n(Gij(p0,ξ)) =/parenleftbigg\ngij(p0) 0\n0Ik−2/parenrightbigg\n,\nwheregij(p0) =/an}bracketle{tXui(u0,t0),Xuj(u0,t0)/an}bracketri}ht.\nOn the other hand, by Proposition 3.1, we have\n−s/summationdisplay\nj=1hj\ni(St0,nS)Xuj=LG(St0,nS)ui=dp0LG(St0;nS)/parenleftbigg∂\n∂ui/parenrightbigg\n,\n6so that we have\nSℓ(St0)(p0,ξ0)/parenleftbigg∂\n∂ui/parenrightbigg\n=s/summationdisplay\nj=1hj\ni(St0,nS)Xuj.\nTherefore, the representation matrix of Sℓ(St0)(p0,ξ0)with respect to the basis\n{Xu1(u0,t0),...,Xus(u0,t0),nS\n1(u0,t0),...,nS\nk−2(u0,t0)}\nofT(p0,ξ0)N1[St0] is of the form\n/parenleftbigg\nhj\ni(St0,nS)(u0,t0)∗\n0 −Ik−2/parenrightbigg\n.\nThus, the eigenvalues of this matrix are λi=κi(St0,ξ0)(p0), (i= 1,...,s) andλi=−1,\n(i=s+1,...,n−1). This completes the proof. ✷\nWe callκℓ(St)i(p,ξ) =κi(St,ξ)(p), (i= 1,...,s)momentary lightcone principal curvatures\nofStwith respect to ξatp=X(u,t)∈W.\nOn the other hand, we define a mapping /tildewidestLG(St) :N1(St)−→Sn−1\n+by\n/tildewidestLG(St)(p,ξ) =πL\nS(LG(St)(p,ξ)),\nwhich is called a normalized momentary lightcone Gauss map ofN1(St).Anormalized momen-\ntary lightcone Gauss map of Stwith respect to nSisamapping/tildewidestLG(St;nS) :U−→Sn−1\n+defined\nto be/tildewidestLG(St;nS)(u) =πL\nS(LG(St;nS)(u)).The normalized momentary lightcone Gauss map\nofStwith respect to nSalso induces a linear mapping dp/tildewidestLG(St;nS) :TpSt−→TpRn+1\n1under\nthe identification of U×{t}andSt,wherep=X(u,t).We have the following proposition.\nProposition 3.3 With the above notations, we have the following normalized l ightcone Wein-\ngarten formula with respect to nS:\nπt◦/tildewidestLG(St;nS)ui(u) =−s/summationdisplay\nj=11\nℓ0(u,t)hj\ni(St;nS)(u,t)Xuj(u,t),\nwhereLG(St;nS)(u) = (ℓ0(u,t),ℓ1(u,t),...,ℓ nu,t)).\nProof.By definition, we have ℓ0/tildewidestLG(St;nS) =LG(St;nS).It follows that\nℓ0/tildewidestLG(St;nS)ui=LG(St;nS)ui−ℓ0ui/tildewidestLG(St;nS).\nSince/tildewidestLG(St;nS)(u)∈Np(St),we haveπt◦/tildewidestLG(St;nS)ui=1\nℓ0πt◦LG(St;nS)ui.Bythe lightcone\nWeingarten formula with respect to nS(Proposition 3.1), we have the desired formula. ✷\nWe call the linear transformation /tildewideSp(St;nS) =−πt◦dp/tildewidestLG(St;nS) anormalized momentary\nlightcone shape operator of Stwith respect to nSatp. The eigenvalues {/tildewideκi(St;nS)(p)}s\ni=1of\n/tildewideSp(St;nS)arecalled normalized momentary lightcone principal curvatures . Bytheabovepropo-\nsition, we have /tildewideκi(St;nS)(p) = (1/ℓ0(u,t))κi(St;nS)(p).Anormalized momentary Lipschitz-\nKilling curvature ofStwith respect to nSis defined to be /tildewideKℓ(u,t) = det/tildewideSp(St;nS).Then\n7we have the following relation between the normalized momentary light cone Lipschitz-Killing\ncurvature and the momentary lightcone Lipschitz-Killing curvature :\n/tildewideKℓ(St;nS)(p) =/parenleftbigg1\nℓ0(u,t)/parenrightbiggs\nKℓ(St;nS)(p),\nwherep=X(u,t).By definition, p0=X(u0,t0) is thenS\n0-momentary umbilical point if and\nonly if/tildewideSp0(St;nS\n0) =/tildewideκi(St0;nS)(p0)1Tp0St0.We have the following proposition.\nProposition 3.4 For anyt0∈I,the following conditions (1)and(2)are equivalent :\n(1)There exists a spacelike unit vector field nSalongW=X(U×I)such that nS(u,t0)∈\nN1(St0)and the normalized momentary lightcone Gauss map /tildewidestLG(St0;nS)ofSt0=X(U×{t0})\nwith respect to nSis constant.\n(2)There exists v∈Sn−1\n+and a real number csuch that St0⊂HP(v,c).\nSuppose that the above conditions hold. Then\n(3)St0=X(U×{t0})is totally nS-momentary flat.\nProof.Suppose that the condition (1) holds. We consider a function F:U−→Rdefined by\nF(u) =/an}bracketle{tX(u,t0),v/an}bracketri}ht.By definition, we have\n∂F\n∂ui(u) =/an}bracketle{tXui(u,t0),v/an}bracketri}ht=/an}bracketle{tXui(u,t0),/tildewidestLG(St0;nS)(u)/an}bracketri}ht= 0,\nfor anyi= 1,...,s.Therefore, F(u) =/an}bracketle{tX(u,t0),v/an}bracketri}ht=cis constant. It follows that St0⊂\nHP(v,c) forv∈Sn−1\n+.\nSuppose that St0is a subset of a lightlike hyperplane H(v,c) forv∈SN−1\n+.SinceSt0⊂\nHP(v,c),we haveTpSt0⊂H(v,0) for any p∈ St0.If/an}bracketle{tnT(u,t),v/an}bracketri}ht= 0,thennT(u,t)∈\nHP(v,0).We remark that HP(v,0) does not contain timelike vectors. This is a contradiction.\nSo we have /an}bracketle{tnT(u,t),v/an}bracketri}ht /ne}ationslash= 0.We now define a vector field along W=X(U×I) by\nnS(u,t) =−1\n/an}bracketle{tnT(u,t),v/an}bracketri}htv−nT(u,t).\nWe can easily show that /an}bracketle{tnS(u,t),nS(u,t)/an}bracketri}ht= 1 and /an}bracketle{tnS(u,t),nT(u,t)/an}bracketri}ht= 0.SinceTpSt0⊂\nH(v,0), we have /an}bracketle{tXui(u,t0),nS(u,t0)/an}bracketri}ht= 0.HencenSis a spacelike unit vector field nSalong\nW=X(U×I) such that nS(u,t0)∈N1(St0) and/tildewidestLG(St0;nS)(u) =v.By Proposition 3.3, if\n/tildewidestLG(St0;nS) is constant, then ( hj\ni(St0;nS)(u,t0)) =O. It follows that St0is lightcone nS-flat.\n✷\n4 Lightcone height functions\nInordertostudythegeometricmeaningsofthenormalizedlightcon eLipschitz-Killingcurvature\n/tildewideKℓ(St;nS) ofSt=X(U× {t}), we introduce a family of functions on M=X(U).A family\noflightcone height functions H:U×(Sn−1\n+×I)−→RonW=X(U×I) is defined to be\nH((u,t),v) =/an}bracketle{tX(u,t),v/an}bracketri}ht.The Hessian matrix of the lightcone height function h(t0,v0)(u) =\nH((u,t0),v0) atu0is denoted by Hess( h(t0,v0))(u0).The following proposition characterizes the\nlightlike parabolic points and lightlike flat points in terms of the family of lig htcone height\nfunctions.\n8Proposition 4.1 LetH:U×(Sn−1\n+×I)−→Rbe the family of lightcone height functions on\na world sheet W=X(U×I).Then\n(1) (∂H/∂u i)(u0,t0,v0) = 0 (i= 1,...,s)if and only if there exists a spacelike section nSof\nN1(St0)such that v0=/tildewidestLG(St0;nS\n0)(u0).\nSuppose that p0=X(u0,t0),v0=/tildewidestLG(St0;nS\n0)(u0).Then\n(2)p0is annS\n0-parabolic point of St0if and only if detHess(h(t0,v0))u0) = 0,\n(3)p0is a flatnS\n0-umbilical point of St0if and only if rankHess(h(t0,v0))u0) = 0.\nProof. (1) Since ( ∂H/∂u i)((u0,t0)v0) =/an}bracketle{tXui(u0,t0),v0),(∂H/∂u i)((u0,t0),v0) = 0 (i=\n1,...,s) if and only if v0∈Np0(St0) andv0∈Sn−1\n+.By the same construction as in the proof\nof Proposition 3.4, we have a spacelike unit normal vector field nSalongW=X(U×I)\nwithnS(u,t0)∈N1(St0) such that v0=/tildewidestLG(St0;nS)(u0) =/tildewidestLG(St0;nS\n0)(u0).The converse also\nholds. For the proof of the assertions (2) and (3), as a conseque nce of Proposition 3.1, we have\nHess(h(t0,v0))(u0) =/parenleftig\n/an}bracketle{tXuiuj(u0,t0),/tildewidestLG(St0;nS)(u0)/an}bracketri}ht/parenrightig\n=/parenleftbigg1\nℓ0/an}bracketle{tXuiuj(u0,t0),nT(u0,t0)+nS(u0,t0)/an}bracketri}ht/parenrightbigg\n=/parenleftbigg1\nℓ0/an}bracketle{tXui(u0,t0),(nT+nS)uj(u0,t0)/an}bracketri}ht/parenrightbigg\n=/parenleftigg\n1\nℓ0/an}bracketle{tXui(u0,t0),−s/summationdisplay\nk=1hk\nj(St0;nS)(u0)Xuk(u0,t0)/an}bracketri}ht/parenrightigg\n=/parenleftbigg\n−1\nℓ0hij(St0;nS)(u0)/parenrightbigg\n.\nBy definition, Kℓ(St0;nS)(u0) = 0 if and only if det( hij(St0;nS)(u0)) = 0.The assertion (2)\nholds. Here, p0is a flatnS\n0-umbilical point if and only if ( hij(St0;nS)(u0)) =O.So we have the\nassertion (3). ✷\nWe also define afamily offunctions /tildewideH:U×(LC∗×I)−→Rby/tildewideH((u,t),v) =/an}bracketle{tX(u,t),/tildewidev/an}bracketri}ht−\nv0,wherev= (v0,v1,...,v n).We call/tildewideHafamily of extended lightcone height functions of\nW=X(U×I).Since∂/tildewideH/∂u i=∂H/∂u ifori= 1,...,sand Hess(/tildewideh(t,v)) = Hess(h(t,/tildewidev)),we\nhave the following proposition as a corollary of Proposition 4.1.\nProposition 4.2 Let/tildewideH:U×(LC∗×I)−→Rbe the extended lightcone height function of a\nworld sheet W=X(U×I).Then\n(1)/tildewideH((u0,t0),v0) = (∂/tildewideH/∂u i)((u0,t0),v0) = 0 (i= 1,...,s)if and only if there exists a\nspacelike section nSofN1(St0)such that\nv0=/an}bracketle{tX(u0,t0),/tildewidestLG(St0;nS\n0)(u0)/an}bracketri}ht/tildewidestLG(St0;nS\n0)(u0).\nSuppose that p0=X(u0,t0),v0=/an}bracketle{tX(u0,t0),/tildewidestLG(St0;nS\n0)(u0)/an}bracketri}ht/tildewidestLG(St0;nS\n0)(u0). Then\n(2)p0is annS\n0-parabolic point of St0if and only if detHess(/tildewideh(t0,v0))(u0) = 0,\n(3)p0is a flatnS\n0-umbilical point of St0if and only if rankHess(/tildewideh(t0,v0)(u0) = 0.\nProof.It follows from Proposition 4.1, (1) that ( ∂/tildewideH/∂u i)((u0,t0),v0) = 0 (i= 1,...,s) if and\nonly if there exists a spacelike section nSofN1(St0) such that v0=/tildewidestLG(St0;nS\n0)(u0).Moreover,\n9thecondition /tildewideH((u0,t0),v0) = 0isequivalenttheconditionthat v0=/an}bracketle{tX(u0,t0),/tildewidestLG(St0;nS\n0)(u0)/an}bracketri}ht,\nwherev0= (v0,v1,...,v n).This means that\nv0=/an}bracketle{tX(u0,t0),/tildewidestLG(St0;nS\n0)(u0)/an}bracketri}ht/tildewidestLG(St0;nS\n0)(u0).\nThe assertions (2) and (3) directly follows from the assertions (2) and (3) of Proposition 4.1.\n✷\nInspired by the above results, we define a mapping LP(St) :N1(St)−→LC∗by\nLP(St)((u,t),ξ) =/an}bracketle{tX(u,t),/tildewidestLG(St;ξ/an}bracketri}ht/tildewidestLG(St)((u,t),ξ).\nWe call it a momentary lightcone pedal map ofSt.Moreover, we define a map LP:N1(W)−→\nLC∗×Iby\nLP((u,t),ξ) = (LP(St)((u,t),ξ),t),\nwhich is called an unfolded lightcone pedal map ofW.\n5 Contact viewpoint\nIn this section we interpret the results of Propositions 4.1 and 4.2 fr om the view point of the\ncontact with lightlike hyperplanes.\nFirstly, we consider the relationship between the contact of a one p arameter family of sub-\nmanifolds with a submanifold and P-K-equivalence among functions (cf., [3]). Let Ui⊂Rr,\n(i= 1,2) be open sets and gi: (Ui×I,(ui,ti))−→(Rn,yi) immersion germs. We define\ngi: (Ui×I,(ui,ti))−→(Rn×I,(yi,ti)) bygi(u,t) = (gi(u),t).We write that ( Yi,(yi,ti)) =\n(gi(Ui×I),(yi,ti)).Letfi: (Rn,yi)−→(R,0)besubmersion germsandwritethat( V(fi),yi) =\n(f−1\ni(0),yi).We say that the contact of Y1with the trivial family of V(f1)at(y1,t1) is of the\nsame type asthe contact of Y2with the trivial family of V(f2)at(y2,t2) if there is a diffeomor-\nphism germΦ : ( Rn×I,(y1,t1))−→(Rn×I,(y2,t2)) of theformΦ( y,t) = (φ1(y,t),φ2(t)) such\nthatΦ(Y1) =Y2andΦ(V(f1)×I) =V(f2)×I. Inthiscasewewrite K(Y1,V(f1)×I;(y1,t1)) =\nK(Y2,V(f2)×I;(y2,t2)). We can show one of the parametric versions of Montaldi’s theore m\nof contact between submanifolds as follows:\nProposition 5.1 We use the same notations as in the above paragraph. Then K(Y1,V(f1)×\nI;(y1,t1)) =K(Y2,V(f2)×I;(y2,t2))if and only if f1◦g1andf2◦g2areP-K-equivalent\n(i.e., there exists a diffeomorphism germ Ψ : (U1×I,(u1,t1))−→(U2×I,(u2,t2))of the form\nΨ(u,t) = (ψ1(u,t),ψ2(t))and a function germ λ: (U1×I,(u1,t1))−→Rwithλ(u1,t1)/ne}ationslash= 0\nsuch that (f2◦g2)◦Φ(u,t) =λ(u,t)f1◦g1(u,t)).\nSince the proof of Proposition 5.1 is given by the arguments just alon g the line of the proof of\nthe original theorem in [13], we omit the proof here.\nWe now consider a function /tildewidehv:Rn+1\n1−→Rdefined by/tildewidehv(w) =/an}bracketle{tw,/tildewidev/an}bracketri}ht −v0,where\nv= (v0,v1,...,v n).For anyv0∈LC∗, we have a lightlike hyperplane h−1\nv0(0) =HP(/tildewidev0,v0).\nMoreover, we consider the lightlike vector v0=LP(St0)((u0,t0),ξ0),then we have\n/tildewidehv0◦X(u0,t0) =/tildewideH(u0,LP(St0)((u0,t0),ξ0))) = 0.\n10By Proposition 4.2, we also have relations that\n∂/tildewidehv0◦X\n∂ui(u0,t0) =∂/tildewideH\n∂ui((u0,t0),LP(St0)((u0,t0),ξ0))) = 0.\nfori= 1,...,s.This means that the lightlike hyperplane /tildewideh−1\nv0(0) =HP(/tildewidev0,v0) is tangent\ntoSt0=X(U× {t0}) atp0=X(u0,t0).The lightlike hypersurface HP(/tildewidev0,v0) is said to\nbe atangent lightlike hyperplane ofSt0=X(U× {t0}) atp0=X(u0,t0), which we write\nTLP(St0,v0,ξ0)),wherev0=LP(St0)(u0,t0).Then we have the following simple lemma.\nLemma 5.2 LetX:U×I−→Rn+1\n1be a world sheet. Consider two points (p1,ξ1),(p2,ξ2)∈\nN1(St0),wherepi=X(ui,t0),(i= 1,2).Then\nLP(St0)((u1,t0),ξ1)) =LP(St0)((u2,t0),ξ2))\nif and only if\nTLP(St0,LP(St0)((u1,t0),ξ1)) =TLP(St0,LP(St0)((u2,t0),ξ2)).\nBy definition, LP((u1,t1),ξ1) =LP((u2,t2),ξ2) if and only if\nt1=t2andLP(St1)((u1,t1),ξ1)) =LP(St1)((u2,t1),ξ2)).\nEventually, we have tools for the study of the contact between sp acelike hypersurfaces and\nlightlike hyperplanes. Since we have /tildewidehv(u,t) =/tildewidehv◦X(u,t),we have the following proposition\nas a corollary of Proposition 5.1.\nProposition 5.3 LetXi: (U×I,(ui,ti))−→(Rn+1\n1,pi) (i= 1,2)be world sheet germs and\nvi=LP(Sti,LP(Sti)((ui,ti),ξi))andWi=Xi(U×I).Then the following conditions are\nequivalent:\n(1)K(W1,TLP(St1,v1,ξ1)×I;(p1,t1)) =K(W2,TLP(St2,v2,ξ2)×I;(p2,t2)),\n(2)/tildewideh1,v1and/tildewideh2,v2areP-K-equivalent.\n6 Big wave fronts\nInthissectionwe applythetheoryofbig wave frontstothegeomet ry ofworldsheets inLorentz-\nMinkowski space. Let F: (Rk×(Rn×R),0)→(R,0) be a function germ. We say that Fis a\nnon-degenerate big Morse family of hypersurfaces if\n∆∗(F)|Rk×Rn×{0}: (Rk×Rn×{0},0)−→(R×Rk) is non-singular ,\nwhere\n∆∗(F)(q,x,t) =/parenleftbigg\nF(q,x,t),∂F\n∂q1(q,x,t),...,∂F\n∂qk(q,x,t)/parenrightbigg\n.\nWe simply say that Fis abig Morse family of hypersurfaces if ∆∗(F) is non-singular. By defini-\ntion, a non-degenerate big Morse family of hypersurfaces is a big Mo rse family of hypersurfaces.\nThen Σ ∗(F) = ∆(F)−1(0) is a smooth n-dimensional submanifold germ.\n11Proposition 6.1 The extended height functions family /tildewideH:U×(LC∗×I)−→Rat any point\n(u0,(v0,t0))∈Σ∗(/tildewideH)is a non-degenerate big Morse family of hypersurfaces.\nProof.We write X= (X0,...,X n) andv= (v0,...,v n)∈LC∗. Without loss of generality,\nwe assume that vn>0. Thenv0=/radicalbig\nv2\n1+···+v2\nn.\nFor ∆∗/tildewideH= (/tildewideH,/tildewideHu1,...,/tildewideHus), we prove that the map ∆∗/tildewideH|U×(LC∗×{t0})is submersive at\n(u0,v0,t0)∈∆∗/tildewideH−1(0). Its Jacobian matrix J∆∗/tildewideH|U×(LC∗×{t0})is\nJ∆∗/tildewideH|U×(LC∗×{t0})=\n/parenleftbigg/tildewideHuj/parenrightbigg\nj=1,...,s/parenleftbigg/tildewideHvj/parenrightbigg\nj=1,...,n−1 /parenleftbigg/tildewideHuiuj/parenrightbigg\ni,j=1,...,s/parenleftbigg/tildewideHuivj/parenrightbigg\ni=1,...,s,j=1,...,n−1\n.\nWe write that\nB=\n/parenleftbigg/tildewideHvj/parenrightbigg\nj=1,...,n−1 /parenleftbigg/tildewideHuivj/parenrightbigg\ni=1,...,s,j=1,...,n−1\n.\nIt is enough to show that the rank of the matrix B(u0,v0,t0) iss+ 1. By straightforward\ncalculations, we have\n/tildewideHvj(u,v,t) =−vj\nv0+Xj\nv0−n/summationdisplay\nk=1vkvj\nv3\n0Xk,\n/tildewideHuivj(u,v,t) =−(Xj)ui\nv0−n/summationdisplay\nk=1vkvj\nv3\n0(Xk)ui,\nfori= 1,...,sandj= 1,...,n. By the condition that /tildewideH(u0,v0,t0) =/tildewideHui(u0,v0,t0) = 0\nfori, we have relations/summationtextn\nk=1v0,k\nv0,0Xk=X0+v0,0and/summationtextn\nk=1v0,k\nv0,0(Xk)ui= (X0)uiwherev0=\n(v0,0,...,v 0,n). Therefore, the above formulae are\n/tildewideHvj(u0,v0,t0) =1\nv0,0/parenleftbigg\nXj−2vj−X0v0,j\nv0,0/parenrightbigg\n,\n/tildewideHuivj(u0,v0,t0) =1\nv0,0/parenleftbigg\n(Xj)ui−(X0)uiv0,j\nv0,0/parenrightbigg\n,\nfori= 1,...,sandj= 1,...,n.\nSince/an}bracketle{tv0,v0/an}bracketri}ht=/an}bracketle{tv0,Xui/an}bracketri}ht= 0 fori= 1,...,s,v0andXui(u0,t0) belong to HP(v0,0). On\ntheother hand, wehave /an}bracketle{tX(u0,t0)−2v0+2v0,0e0,v0/an}bracketri}ht=−2v2\n0,0/ne}ationslash= 0where e0= (1,0,...,0). So,\nvectorsX(u0,t0)−2v0+2v0,0e0,v0andXui(u0,t0) (fori= 1,...,s) are linearly independent.\nTherefore the rank of following matrix\nC=\nv0\nX−2v0+2v0,0e0\nXu1...\nXus\n=\nv0,0v0,1···v0,n\nX0X1−2v1···Xn−2vn\n(X0)u1(X1)u1···(Xn)u1............\n(X0)us(X1)us···(Xn)us\n\n12iss+ 2 at (u0,v0,t0). We subtract the first row by multiplied by X0/v0,0from the second\nrow, and we also subtract the first row multiplied by ( X0)ui/v0,0from the (2 + i)-th row for\ni= 1,...,s. Then we have\nC′=\nv0,0v0,1···v0,n\n0\n...\n0B(u0,v0,t0)\n\nand rankC′=s+2.Therefore rank B(u0,v0,t0) =s+1. This completes the proof. ✷\nWenowconsider the( n+1)-space Rn+1=Rn×Randcoordinatesofthisspacearewrittenas\n(x,t) = (x1,...,x n,t)∈Rn×R,which we distinguish space and time coordinates. We consider\nthe projective cotangent bundle π:PT∗(Rn×R)→Rn×R.Because of the trivialization\nPT∗(Rn×R)∼=(Rn×R)×P((Rn×R)∗),we have homogeneous coordinates\n((x1,...,x n,t),[ξ1:···:ξn:τ]).\nThen we have the canonical contact structure KonPT∗(Rn×R). For the definition and the\nbasic properties of the contact manifold ( PT∗(Rn×R),K),see [4, Appendix]. A submanifold\ni:L⊂PT∗(Rn×R) is said to be a big Legendrian submanifold if dimL=nanddip(TpL)⊂\nKi(p)for anyp∈L.We also call the map π◦i=π|L:L−→Rn×Rabig Legendrian map\nand the set W(L) =π(L) abig wave front ofi:L⊂PT∗(Rm).We say that a point p∈Lis a\nLegendrian singular point if rankd(π|L)p<n.In this case π(p) is the singular point of W(L).\nWe call\nWt(L) =π1(π−1\n2(t)∩W(L)) (t∈R)\namomentary front (or, asmall front ) for each t∈(R,0),whereπ1:Rn×R→Rnand\nπ2:Rn×R→Rare the canonical projections defined by π1(x,t) =xandπ2(x,t) =t\nrespectively. In this sense, we call Labig Legendrian submanifold. We say that a point p∈L\nisaspace-singular point ifrankd(π1◦π|L)p<nandatime-singular point ifrankd(π2◦π|L)p= 0,\nrespectively. By definition, if p∈Lis a Legendrian singular point, then it is a space-singular\npoint ofL.\nThediscriminant of the family Wt(L) is defined as the imageof singular pointsof π1|W(L).In\nthe general case, the discriminant consists of three components :the caustic CL=π1(Σ(W(L)),\nwhere Σ(W(L)) is the set of singular points of W(L) (i.e, the critical value set of the Legendrian\nmappingπ|L),the Maxwell stratified set ML,the projection of self intersection points of W(L);\nand also of the critical value set ∆ of π|W(L)\\Σ(W(L))(for more detail, see [8, 12, 18]). We remark\nthat ∆ is not necessary the envelope of the family of smooth moment ary frontsWt(L).There\nis a case that π−1\n2(t)∩W(L) is non-singular but π1|π−1\n2(t)∩W(L)has singularities, so that ∆ is\nthe set of critical values of the family of mapping π1|π−1\n2(t)∩W(L)for smooth π−1\n2(t)∩W(L).\nActually, ∆ is the critical value set of π|W(L)\\Σ(W(L)).\nFor any Legendrian submanifold germ i: (L,p0)⊂(PT∗(Rn×R),p0),it is known there\nexists a generating family (cf., [1]). Let F: (Rk×(Rn×R),0)→(R,0) be a big Morse family\nof hypersurfaces. Then Σ ∗(F) = ∆(F)−1(0) is a smooth n-dimensional submanifold germ. We\nhave a big Legendrian submanifold LF(Σ∗(F)) (cf., [1, 15, 17]), where\nLF(q,x,t) =/parenleftbigg\nx,t,/bracketleftbigg∂F\n∂x(q,x,t) :∂F\n∂t(q,x,t)/bracketrightbigg/parenrightbigg\n,\n13and/bracketleftbigg∂F\n∂x(q,x,t) :∂F\n∂t(q,x,t)/bracketrightbigg\n=/bracketleftbigg∂F\n∂x1(q,x,t) :···:∂F\n∂xn(q,x,t) :∂F\n∂t(q,x,t)/bracketrightbigg\n.\nItisknown thatanybig Legendriansubmanifold germcanbeconstru cted bytheabovemethod.\nWith this notation, the big Morse family of hypersurfaces is non-deg enerate if and only if\n(π2◦π)−1(t)∩LF(Σ∗(F)) is an−1-dimensional submanifold germ of PT∗(Rn×R) for any\nt∈(R,0).SinceLF(Σ∗(F)) is Legendrian, ( π2◦π)−1(t)∩LF(Σ∗(F))is an integral submanifold\nof the canonical contact structure K.\nWe now consider an equivalence relation among big Legendrian subman ifolds which pre-\nserves the discriminant of families of small fronts. We now consider t he following equiva-\nlence relation among big Legendrian submanifold germs: Let i: (L,p0)⊂(PT∗(Rn×R),p0)\nandi′: (L′,p′\n0)⊂(PT∗(Rn×R),p′\n0) be big Legendrian submanifold germs. We say that\niandi′arespace-parametrized Legendrian equivalent (or, briefly s-P-Legendrian equivalent )\nif there exist diffeomorphism germs Φ : ( Rn×R,π(p0))→(Rn×R,π(p′\n0)) of the form\nΦ(x,t) = (φ1(x),φ2(x,t)) such that/hatwideΦ(L) =L′as set germs, where /hatwideΦ : (PT∗(Rn×R),p0)→\n(PT∗(Rn×R),p′\n0) is the unique contact lift of Φ. We can also define the notion of stabilit y of\nLegendrian submanifold germs with respect to s-P-Legendrian equivalence which is analogous\nto thestability ofLagrangiansubmanifold germs with respect to Lag rangianequivalence (cf. [1,\nPart III]). We investigate s-P-Legendrianequivalence by using the notion of generating families\nofLegendriansubmanifoldgerms. Let f,g: (Rk×R,0)→(R,0)befunctiongerms. Wesaythat\nfandgareP-K-equivalentifthereexistsadiffeomorphismgermΦ : ( Rk×R,0)→(Rk×R,0)of\ntheformΦ( q,t) = (φ1(q,t),φ2(t))suchthat /an}bracketle{tf◦Φ/an}bracketri}htEk+1=/an}bracketle{tg/an}bracketri}htEk+1.LetF,G: (Rk×(Rn×R),0)→\n(R,0) be function germs. We say that FandGarespace-P-K-equivalent (or, briefly, s-P-K-\nequivalent ) if there exists a diffeomorphism germ Ψ : ( Rk×(Rn×R),0)→(Rk×(Rn×R),0) of\nthe form Ψ( q,x,t) = (φ(q,x,t),φ1(x),φ2(x,t))) such that /an}bracketle{tF◦Ψ/an}bracketri}htEk+n+1=/an}bracketle{tG/an}bracketri}htEk+n+1.The notion\nofP-K-versal deformation plays an important role for our purpose which has been introduced\nin (cf.,[2, 3]). We define the extended tangent space of f: (Rk×R,0)→(R,0) relative to P-K\nby\nTe(P-K)(f) =/angbracketleftbigg∂f\n∂q1,...,∂f\n∂qk,f/angbracketrightbigg\nEk+1+/angbracketleftbigg∂f\n∂t/angbracketrightbigg\nE1.\nThen we say that Fisinfinitesimally P-K-versaldeformation of f=F|Rk×{0}×Rif it satisfies\nEk+1=Te(P-K)(f)+/angbracketleftbigg∂F\n∂x1|Rk×{0}×R,...,∂F\n∂xn|Rk×{0}×R/angbracketrightbigg\nR.\nWe can show the following theorem analogous to those in [6, 18]. We on ly remark here that\nthe proof is analogous to the proof of [1, Theorem in §21.4].\nTheorem 6.2 LetF: (Rk×(Rn×R),0)→(R,0)andG: (Rk′×(Rn×R),0)→(R,0)be big\nMorse families of hypersurfaces. Then\n(1)LF(Σ∗(F))andLG(Σ∗(G))ares-P-Legendrian equivalent if and only if FandGare stably\ns-P-K-equivalent.\n(2)LF(Σ∗(F))iss-P-Legendre stable if and only if Fis an infinitesimally P-K-versal defor-\nmation off=F|Rk×{0}×R.\nSince the Legendrian submanifold germ i: (L,p)⊂(PT∗(Rn×R),p) is uniquely determined\non the regular part of the big wave front W(L),we have the following simple but significant\nproperty of Legendrian immersion germs [17].\n14Proposition 6.3 (Zakalyukin) Leti: (L,p)⊂(PT∗(Rn×R),p),i′: (L′,p′)⊂(PT∗(Rn×\nR),p′)be Legendrian submanifold germs such that regular sets of π◦i,π◦i′are dense respectively.\nThen(L,p) = (L′,p′)if and only if (W(L),π(p)) = (W(L′),π(p′)).\nThe assumption in Proposition 6.3 is a generic condition for i,i′.Especially, if iandi′\nares-P-Legendre stable, then these satisfy the assumption. Concernin g the discriminant of\nthe families of momentary fronts, we define the following equivalence relation among big wave\nfront germs. Let i: (L,p0)⊂(PT∗(Rn×R),p0) andi′: (L′,p′\n0)⊂(PT∗(Rn×R),p′\n0) be\nbig Legendrian submanifold germs. We say that W(L) andW(L′) arespace-parametrized\ndiffeomorphic (briefly,s-P-diffeomorphic ) if there exists a diffeomorphism germ Φ : ( Rn×\nR,π(p0))→(Rn×R,π(p′\n0)) defined by Φ( x,t) = (φ1(x),φ2(x,t))) such that Φ( W(L)) =W(L′).\nRemark that an s-P-diffeomorphism among big wave front germs preserves the diffeomo rphism\ntypes the discriminants. By Proposition 6.3, we have the following pro position.\nProposition 6.4 Leti: (L,p0)⊂(PT∗(Rn×R),p0)andi′: (L′,p′\n0)⊂(PT∗(Rn×R),p′\n0)be\nbig Legendrian submanifold germs such that regular sets of π◦i,π◦i′are dense respectively.\nTheniandi′ares-P-Legendrian equivalent if and only if (W(L),π(p0))and(W(L′),π(p′\n0))\nares-P-diffeomorphic.\nRemark 6.5 If we consider a diffeomorphism germ Φ : ( Rn×R,0)→(Rn×R,0) defined\nby Φ(x,t) = (φ1(x,t),φ2(t)),we can define time-Legendrian equivalence among big Legen-\ndrian submanifold germs. We can also define time- P-K-equivalence among big Morse fami-\nlies of hypersurfaces. By the arguments similar to the above parag raphs, we can show that\nthese equivalence relations describe the bifurcations of momentar y fronts of big Legendrian\nsubmanifolds. In [17] Zakalyukin classified generic big Legendrian sub manifold germs by time-\nLegendrian equivalence . The notion of time-Legendrian equivalence is a complementary notio n\nof space-Legendrian equivalence.\nWe have the following theorem on the relation among big Legendrian su bmanifolds and big\nwave fronts.\nTheorem 6.6 LetF: (Rk×Rn×R,0)−→(R,0)andG: (Rk′×Rn×R,0)−→(R,0)be\nbig Morse families of hypersurface such that LF(Σ∗(F))andLG(Σ∗(G))ares-P-Legendrian\nstable. Then the following conditions are equivalent :\n(1)LF(Σ∗(F))andLG(Σ∗(G))ares-P-Legendrian equivalent,\n(2)FandGare stablys-P-K-equivalent,\n(3)f(q,t) =F(q,0,t)andg(q′,t) =G(q′,0,t)are stablyP-K-equivalent,\n(4)W(LF(Σ∗(F)))andW(LG(Σ∗(G)))ares-P-diffeomorphic.\nProof.By the assertion (1) o f Theorem 6.2, the conditions (1) and (2) are equivalent. By\ndefinition, thecondition(2)implies thecondition(3). Italsofollowsfr omthedefinitionthatthe\ncondition (1) implies (4). We remark that all these assertions hold wit hout the assumptions\nof theS-P-Legendrian stability. Generically, the condition (4) implies the condit ion (1) by\nProposition 6.4. Of course, it holds under the assumption of S-P-Legendrian stability. By the\nassumption of s-P-Legendrian stability, the big Morse families of hypersurface FandGare\ninfinitesimally P-K-versal deformations of fandg, respectively (cf., Theorem 6.2, (2)). By\nthe uniqueness result of the infinitesimally P-K-versal deformations (cf., [3]), the condition (3)\nimplies the condition (2). This completes the proof. ✷\n15Remark 6.7 (1) Ifk=k′andq=q′in the above theorem, we can remove the word “stably”\nin the conditions (2),(3).\n(2)s-P-Legendrian stability for LF(Σ∗(F)) is generic for n≤5.\n(3) By the remark in the proof of the above theorem, the condition s (1) and (4) are equivalent\ngenerically for a general dimension nwithout the assumption on s-P-Legendrian stability.\nTherefore, the conditions (1),(2) and (4) are all equivalent to eac h other generically.\nWe now return to our situation. Since the extended lightcone height functions family /tildewideH:\nU×(LC∗×I)−→Ris a non-degenerate big Morse family of hypersurfaces, we have th e\ncorresponding big Legendrian submanifold L/tildewideH(Σ∗(/tildewideH))⊂PT∗(LC∗×I).By Proposition 4.2,\nwe have\nΣ∗(/tildewideH)) ={(u,LP(St)((u,t),ξ),t)|(u,t)∈U×I,ξ∈N1[St]p,p=X(u,t)}\n={(u,LP((u,t),ξ))|(u,t)∈U×I,ξ∈N1[St]p,p=X(u,t)}.\nIt follows that π(L/tildewideH(Σ∗(/tildewideH))) =LP(N1(W))⊂LC∗×I.Therefore, the image of the unfolded\nlightcone pedal is a big wave front.\nWe apply the above theorem to our situation.\nTheorem 6.8 LetXi: (U×I,(ui,ti))−→(Rn+1\n1,pi) (i= 1,2)be world sheet germs and\nvi=LP(Sti)((ui,ti),ξi))andWi=Xi(U×I).Suppose that the Legendrian submanifold germs\nL/tildewideHi(Σ∗(/tildewideHi))⊂PT∗(LC∗×I)ares-P-Legendrian stable. Then the following conditions are\nequivalent:\n(1)L/tildewideH1(Σ∗(/tildewideH1))andL/tildewideH2(Σ∗(/tildewideH2))ares-P-Legendrian equivalent,\n(2)/tildewideh1,v1and/tildewideh2,v2areP-K-equivalent,\n(3)/tildewideH1and/tildewideH2ares-P-K-equivalent,\n(4)LP1(N1(W1))andLP2(N1(W2))ares-P-diffeomorphic,\n(5)K(W1,TLP(St1,v1,ξ1)×I;(p1,t1)) =K(W2,TLP(St1,v2,ξ2)×I;(p2,t2)).\nProof.SinceLPi(N1(Wi)) are big wave fronts of L/tildewideHi(Σ∗(/tildewideHi)) (i= 1,2) respectively, we can\napply Theorem 6.6 and obtain that the conditions (1), (2), (3) and ( 4) are equivalent. By\nProposition 5.3, the conditions (2) and (5) are equivalent. This comp letes the proof. ✷\nReferences\n[1] V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps\nvol. I. Birkh¨ auser, 1986.\n[2] M. Golubitsky and D. Schaeffer, A theory for imperfect bifu rcation theory via singularity\ntheory, Commun. Pure Appl. Math.. vol. 32 (1979), 21–98\n[3] S. 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Zakalyukin, Reconstructions of fronts and caustics depending one param eter and ver-\nsality of mappings , J. Sov. Math. 27 (1984), 2713–2735.\n[18] V.M. Zakalyukin, Envelope of Families of Wave Fronts and Control Theory. Proc. Steklov\nInst. Math. 209(1995), 114–123.\n[19] V.M. Zakalyukin, Singularities of Caustics in generic translation-invarian t control problems.\nJournal of Mathematical Sciences, 126(2005), 1354–1360.\n17"    },    {        "title": "1506.01629v2.Fourier_series_in_weighted_Lorentz_spaces.pdf",        "content": "arXiv:1506.01629v2  [math.FA]  15 Dec 2015FOURIER SERIES IN WEIGHTED LORENTZ SPACES\nJAVAD RASTEGARI AND GORD SINNAMON\nAbstract. The Fourier coefficient map is considered as an operator from a\nweighted Lorentz space on the circle to a weighted Lorentz se quence space.\nFor a large range of Lorentz indices, necessary and sufficient conditions on the\nweights are given for the map to be bounded. In addition, new d irect ana-\nlogues are given for known weighted Lorentz space inequalit ies for the Fourier\ntransform. Applications are given that involve Fourier coe fficients of functions\nin LlogL and more general Lorentz-Zygmund spaces.\n1.Introduction\nThe study of weighted Fourier inequalities has so far focused on the Fourier\ntransform of functions on R. Very little, for general weights, has been accomplished\nfor the Fourier coefficient map, f/ma√sto→ˆf, where\nˆf(n) =/integraldisplay1\n0e−2πinxf(x)dx, f∈L1(T),\nfor each n∈Z. Although there are many similarities between this map and the\nFourier transform on R, the compactness of the domain of fand the discreteness\nof the domain of ˆfmake the theory substantially different.\nThe Fourier transform and Fourier coefficient map are of fundamen tal impor-\ntance in harmonic analysis. Following the successful characterizat ion of weighted\nLebesgue-space inequalities for the Hilbert transform and related operators, B.\nMuckenhoupt proposed the characterizationof weighted Lebesg ue-space Fourier in-\nequalitiesasanimportantgoalforthe field. Fromsubsequentwork byJ.Benedetto,\nH. Heinig, and R. Johnson in [2, 4] and particularly in [3], weighted Loren tz-space\nFourierinequalitiesemergedasapowerfultechniqueforprovingwe ightedLebesgue-\nspace Fourier inequalities, as well as being significant in their own right . Here we\nextend and adapt work from [15, 16, 19] to give weighted Lorentz- space Fourier\ninequalities for the Fourier coefficient map. Our results include, for a large range\nof indices pandq, necessary and sufficient conditions on weights uandwfor which\nthe inequality,\n(1.1) /bardblˆf/bardblΓq(u)≤C/bardblf/bardblΓp(w), f∈L1(T),\nholds for some constant Cindependent of f. It also includes direct analogues, for\nthe Fourier coefficient map, of the Lorentz-norm Fourier inequalitie s given in [3].\nThe Lorentz Γ-spaces in the inequality above are defined at the end of this in-\ntroduction, along with the more classical Λ-spaces and the Θ-spac es that figure\n2010Mathematics Subject Classification. Primary 42B35, Secondary 46E30, 42B05.\nKey words and phrases. Fourier series, Fourier coefficients, weights, Lorentz spac e.\nSupported by the Natural Sciences and Engineering Research Council of Canada.\n12 JAVAD RASTEGARI AND GORD SINNAMON\nprominently in our weight characterization. Section 2 is devoted to p roving a rep-\nresentation theorem for certain generalized quasi-concave func tions that will be\nneeded to prove sufficient conditions for (1.1). Section 3 contains a rather technical\nconstruction of the test functions that give necessary condition s for (1.1). In Sec-\ntion 4 these are combined to prove our main results: Propositions 4.2 and 4.4 give\na number of conditions that are sufficient to imply (1.1) for various ind ex ranges.\nTheorems 4.5–4.7 show that in a certain index range these conditions are also nec-\nessary. Applications to Fourier inequalities between Lorentz space s with specific\nweights appear in Section 5.\nOur Fourier inequalities are stated and proved for f∈L1(T) only. Here the\ncircleTis identified with the real interval [0 ,1]. In most cases they involve a norm\nor quasi-norm in which the L1functions are dense. In such a cases it is standard\nto extend the Fourier coefficient map so that the inequality remains v alid. This is\nleft to the reader.\nWe will make use of the Fourier coefficient map’s well-known group invar iance\nproperties: If f∈L1(T),g(x) =e2πin0xf(x) andh(x) =f(x−x0), then\n(1.2) ˆ g(n) =ˆf(n−n0) and ˆh(n) =e−2πinx0ˆf(n)\nfor anyn0∈Zand any x0∈T.\nThroughout the paper, L+denotes the collection of non-negative Lebesgue mea-\nsurable functions on (0 ,∞) and, for 0 < p <∞andw∈L+, the weighted Lebesgue\nspaceLp(w) denotes the normed (or quasi-normed) space of Lebesgue meas urable\nfunctions hon (0,∞) for which\n/bardblh/bardblp,w=/parenleftbigg/integraldisplay∞\n0|h(t)|pw(t)dt/parenrightbigg1/p\n<∞.\nThe Lebesgue spaces, for 1 ≤p≤ ∞, over a general measure space ( X,µ), are\ndefined in the usual way and denoted by Lp\nµ.\nIf{(ai,bi),i∈I}is a (necessarily finite or countable) collection of disjoint subin-\ntervals of (0 ,∞) we define the averaging operator Aby,\n(1.3) Af(x) =/braceleftBigg\n1\nbi−ai/integraltextbi\naif(t)dt, x∈(ai,bi),\nf(x), x / ∈ ∪i∈I(ai,bi).\nThe class of all such operators Ais denoted A. It is an easy exercise to show that\neachA∈ AmapsL+toL+and is formally self-adjoint, that is, for all f,g∈L+,\n/integraldisplay∞\n0Af(t)g(t)dt=/integraldisplay∞\n0f(t)Ag(t)dt.\nForu∈L+,uodenotes the level function ofuwith respect to Lebesgue measure\non (0,∞). If the function/integraltextx\n0u(t)dtlies under any line on (0 ,∞) then it has a\nwell-defined least concave majorant. This majorant is absolutely co ntinuous and so\nmay be represented as an integral. The function uois defined by the requirement\nthat/integraltextx\n0uo(t)dtbe the least concave majorant of/integraltextx\n0u(t)dt. We may take uoto be\ndecreasing,since it necessarilyagreeswith a decreasingfunction a lmosteverywhere.\nMoreover, if uis decreasing then uo=u.\nIf/integraltextx\n0u(t)dtdoes not lie under any line on (0 ,∞) the level function uomay be\ndefined as the limit of an increasing sequence of level functions. For details, and\nadditional properties of the level function, see [12, 14, 15, 16, 17 , 18].FOURIER SERIES IN WEIGHTED LORENTZ SPACES 3\n1.1.Lorentz Spaces. Let (X,µ) be aσ-finite measure space. For f∈L1\nµ+L∞\nµ,\nthe rearrangement, f∗, offwith respect to µ(see [6]) is the generalized inverse of\nthe distribution function\nµf(λ) =µ{x∈X:|f(x)|> λ},\ndefined by\nf∗(t) = inf{λ >0 :µf(λ)≤t}.\nThe rearrangement is a non-negative, decreasing, Lebesgue mea surable function on\n(0,∞), and so is\nf∗∗(t) =1\nt/integraldisplayt\n0f∗.\nDefine\n/bardblf/bardblΛp(w)=/bardblf∗/bardblp,w,/bardblf/bardblΘp(w)= sup\nh∗∗≤f∗∗/bardblh∗/bardblp,w,and/bardblf/bardblΓp(w)=/bardblf∗∗/bardblp,w.\nHerehis a function on (0 ,∞) andh∗is its rearrangement with respect to Lebesgue\nmeasure. Since f∗≤f∗∗it is easy to verify that, for every f,\n(1.4) /bardblf/bardblΛp(w)≤ /bardblf/bardblΘp(w)≤ /bardblf/bardblΓp(w).\nDefine Λ p(w) to be the set of µ-measurable functions ffor which /bardblf/bardblΛp(w)is finite,\nand define Θ p(w) and Γ p(w) correspondingly. Clearly, Γ p(w)⊆Θp(w)⊆Λp(w).\nWhen 1< p <∞,/bardbl·/bardblΘp(w)and/bardbl·/bardblΓp(w)are norms for any non-trivial weight\nw∈L+(see [19]) and /bardbl· /bardblΛp(w)is a norm whenever wis decreasing. However, if\nthere exists a constant csuch that\n(1.5) /bardblf/bardblΓp(w)≤c/bardblf/bardblΛp(w)\nfor allµ-measurable f, then/bardbl·/bardblΛp(w)is equivalent to both of the norms, /bardbl·/bardblΘp(w)\nand/bardbl·/bardblΓp(w). According to [1] such a cexists whenever w∈Bp, that is, whenever\nthere exists a constant bp(w) such that\n/integraldisplay∞\ntw(s)\nspds≤bp(w)\ntp/integraldisplayt\n0w(s)ds, t > 0.\nWhenp= 1 the situation is different. If w∈B1,∞, that is, if there exists a constant\nb1(w) such that\n1\ny/integraldisplayy\n0w(t)dt≤b1(w)\nx/integraldisplayx\n0w(t)dt,0< x < y < ∞,\nthen/bardbl·/bardblΛ1(w)is equivalent to the norm /bardbl·/bardblΘ1(w)=/bardbl·/bardblΛ1(wo). This follows from\nLemma 2.2 and Lemma 2.5 of [16].\nWhen the underlying measure µis Lebesgue measure, or any other infinite non-\natomic measure, /bardbl · /bardblΛp(w)is a norm if and only if wis decreasing. Also, the Bp\ncondition (when p >1) and the B1,∞condition (when p= 1) are necessary and\nsufficient for /bardbl·/bardblΛp(w)to be equivalent to a norm. See [13] and [8].\nIf the underlying measure µis finite, and f∈L1\nµ+L∞\nµ=L1\nµ, thenf∗is\nsupported on (0 ,µ(X)). Thus, /bardblf/bardblΛp(w)depends only on the restriction of wto\n(0,µ(X)). However, f∗∗is not supported on (0 ,µ(X)) so/bardblf/bardblΓp(w)does depend on\nvalues of woutside (0 ,µ(X)), but only through the value of/integraltext∞\nµ(X)w(t)dt\ntp.\nIf the underlying measure µis counting measure on Z, the space L1\nµ+L∞\nµmay\nbe identified with a space of sequences. It is possible to define the re arrangement4 JAVAD RASTEGARI AND GORD SINNAMON\nof a sequence directly to obtain another sequence but we will stick w ith the above\ndefinition, viewing L1\nµ+L∞\nµas a space of µ-measurable functions, with decreasing\nfunctions on (0 ,∞) as their rearrangements. This is only a notational difference;\nthe decreasing functions we obtain are constant on the intervals, [n,n+ 1) for\nn= 0,1,...so each may be identified with the corresponding rearranged seque nce\nif desired.\n2.Quasi-Concave Functions\nFunctions with two monotonicity conditions arise naturally in our stud y of\nFourier series in Lorentz spaces. Let α+β >0. By Ω α,βwe mean the collection\nof all functions f∈L+such that xαf(t) is increasing and x−βf(t) is decreasing.\nNotice that Ω α,βis a cone, being closed under addition and under multiplication by\npositive scalars. Functions in Ω 0,1are called quasi-concave because they are equiva-\nlent to concave functions, and functions in Ω α,βare called generalized quasi-concave\nfunctions.\nOurchiefinterestwillbeintheconeΩ 2,0, butwebeginbylookingatallthecones\nΩα,βtogether because they are related by simple transformations. Fo r instance, if\nλ >0 andg(t) =tγf(t1/λ), theng∈Ωα,βif and only if f∈Ωλ(α+γ),λ(β−γ).\nBesides being in Ω 2,0, the functions we encounter are constant on the interval\n(0,1). To deal with this additional restriction in general terms we intro duce the\ncones,\nPr\nξ={f∈L+:t−rf(t) is constant on (0 ,ξ)}\nand setP=P0\n1.\nDefinition 2.1. MapsA:L+\nν→L+\nµandB:L+\nµ→L+\nνare called formal adjoints\nprovided\n/integraldisplay\nYAf(y)g(y)dµ(y) =/integraldisplay\nXf(x)Bg(x)dν(x)\nfor allf∈L+\nνandg∈L+\nµ.\nThe following lemma is a modification of Lemma 4 in [19]. The lemma was\napplied outside its scope in Theorem 6 of [19]. In the version below we wid en the\nscope to include all operators Ahaving formal adjoints. This includes the averaging\noperators introduced in (1.3) and fills the gap in the proof of Theore m 6 of [19].\nLemma 2.2. Let0< p≤1≤q <∞. Suppose (Y,µ),(X,ν),(T,λ)areσ-finite\nmeasure spaces, k(x,t)≥0is aν×λ-measurable function, and A:L+\nν→L+\nµhas\na formal adjoint. Define KbyKh(x) =/integraltext\nTk(x,t)h(t)dλ(t)and letkt(x) =k(x,t).\nThen, for any u∈L+\nµandv∈L+\nν,\n(2.1) sup\nh≥0/bardblAKh/bardblq,uµ\n/bardblKh/bardblp,vν≤esssup\nt∈T/bardblAkt/bardblq,uµ\n/bardblkt/bardblp,vν.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 5\nProof.LetCbe the right-hand side of (2.1) and fix h∈L+\nλ. Since 0 < p≤1,\nMinkowski’s integral inequality shows that\n/integraldisplay\nT/bardblkt/bardblp,vνh(t)dλ(t) =/integraldisplay\nT/parenleftbigg/integraldisplay\nXk(x,t)pv(x)dν(x)/parenrightbigg1/p\nh(t)dλ(t)\n≤/parenleftbigg/integraldisplay\nX/parenleftbigg/integraldisplay\nTk(x,t)h(t)dλ(t)/parenrightbiggp\nv(x)dν(x)/parenrightbigg1/p\n=/bardblKh/bardblp,vν.\nLetB:L+\nµ→L+\nνbe a formal adjoint of A. For any g∈L+\nµwith/bardblg/bardblq′,uµ≤1,\nTonelli’s theorem implies,/integraldisplay\nYAKh(y)g(y)u(y)dµ(y) =/integraldisplay\nXKh(x)B(gu)(x)dν(x)\n=/integraldisplay\nT/parenleftbigg/integraldisplay\nXkt(x)B(gu)(x)dν(x)/parenrightbigg\nh(t)dλ(t).\nBut, by H¨ older’s inequality,/integraldisplay\nXkt(x)B(gu)(x)dν=/integraldisplay\nYAkt(y)g(y)u(y)dµ(y)≤ /bardblAkt/bardblq,uµ≤C/bardblkt/bardblp,vν,\nforλ-almost every t. Therefore,/integraldisplay\nY(AKh)(y)g(y)u(y)dµ(y)≤C/integraldisplay\nT/bardblkt/bardblp,vνh(t)dλ(t)≤C/bardblKh/bardblp,vν.\nTaking the supremum over all such gyields\n/bardblAKh/bardblq,uµ≤C/bardblKh/bardblp,vν.\nSinceh∈L+\nλwas arbitrary, the conclusion follows. /square\nIn order to apply this result to Pβ\nξ∩Ωα,β, we show that the range of a certain\npositive operator is a large subset of this cone. The positive operat or isKα,β\nξ,\ndefined by\nKα,β\nξh(x) =/integraldisplay∞\nξkα,β(x,t)h(t)dt,\nwherekα,β(x,t) = min( xβt−α,x−αtβ). It is easy to check that for fixed t,kα,β\nt(x)\nis in Ω α,βand that Kα,β\nξh(x)∈Pβ\nξ∩Ωα,βwhenever h∈L+. Thus, the image of\nL+underKα,β\nξis a subset of Pβ\nξ∩Ωα,β. What we mean by “large subset” is in\nLemma 2.5.\nWe start with a lemma stating the geometrically obvious fact that if a f unction\nis linear on some interval, so is its least concave majorant.\nLemma 2.3. Suppose ˜gis the least concave majorant of g∈L+. Ifξ >0,c≥0\nandg(x) =cxon(0,ξ)then˜g(x) =x˜g(ξ)/ξon(0,ξ).\nProof.Letλ= ˜g(ξ)/ξ. Since ˜g≥0 is concave, λx≤˜g(x) on (0,ξ] andλx≥˜g(x)\non [ξ,∞). Since ˜gis continuous,\nλ= ˜g(ξ)/ξ= lim\nx→ξ−˜g(x)/x≥lim\nx→ξ−g(x)/x=c.\nThusλx≥cx=g(x) on (0,ξ) andλx≥˜g(x)≥g(x) on [ξ,∞). Soλxis a concave\nmajorant of gand therefore λx≥˜g(x) on (0,∞). In particular, ˜ g(x) =λxon\n(0,ξ). /square6 JAVAD RASTEGARI AND GORD SINNAMON\nThenextlemmashowsthateveryfunctionin P1\nξ∩Ω0,1isequal,uptoequivalence,\nto the limit of an increasing sequence of functions in the range of K0,1\nξ.\nLemma 2.4. Letξ≥0and letgbe a quasi-concave function such that g(x)/xis\nconstant on (0,ξ). If˜gis the least concave majorant of g, theng≤˜g≤2gand there\nexists a sequence of functions ℓn∈L+such that K0,1\nξℓnincreases to ˜gpointwise.\nProof.Proposition 2.5.10 of [6] shows that g≤˜g≤2g.\nRecallthat aconcavefunction on(0 ,∞) isabsolutelycontinuousonclosedsubin-\ntervals of (0 ,∞). It has left and right derivatives everywhere, the right derivativ e is\nright continuous, both are decreasing, and the right derivative is le ss than or equal\nto the left derivative at each point.\nLetϕdenote the right derivative of ˜ gand leta= ˜g(ξ)−ξϕ(ξ) ifξ >0 and\na= ˜g(0+) ifξ= 0. If ξ= 0 it is clear that a≥0 and if ξ >0, Lemma 2.3\nshows that ˜ g(ξ)/ξis the left derivative of ˜ gatξso in this case, too, a≥0. Let\nϕ(∞) = lim t→∞ϕ(t). Forn > ξandt >0, set\n(2.2) ℓn(t) =ϕ(∞)χ(n,n+1)(t)+(a/t)nχ(ξ,ξ+1\nn)(t)+ϕ(t)−ϕ(tn+1\nn)\ntlog(n+1\nn).\nSinceϕis decreasing, ℓn∈L+for all positive integers n > ξ.\nWe apply K0,1\nξto each of the three terms separately. The first term becomes/integraltextn+1\nnϕ(∞)min(x,t)dt. For each x, this is a moving average of the increasing\nfunction ϕ(∞)min(x,t) and is therefore increasing with n. It converges to xϕ(∞).\nThe second term becomes n/integraltextξ+1/n\nξ(a/t)min(x,t)dt. This is a shrinking aver-\nage of the decreasing function amin(x/t,1) and is therefore increasing with n. It\nconverges to amin(x/ξ,1).\nLet the third term of (2.2) be ¯ℓn(t). Fory >0,\n/integraldisplay∞\ny¯ℓn(t)dt=1\nlog(n+1\nn)lim\nM→∞/parenleftbigg/integraldisplayM\nyϕ(t)dt\nt−/integraldisplayM\nyϕ(tn+1\nn)dt\nt/parenrightbigg\n=1\nlog(n+1\nn)lim\nM→∞/parenleftbigg/integraldisplayM\nyϕ(t)dt\nt−/integraldisplayMn+1\nn\nyn+1\nnϕ(t)dt\nt/parenrightbigg\n=/integraltextyn+1\nn\nyϕ(t)dt\nt\n/integraltextyn+1\nn\nydt\nt−lim\nM→∞/integraltextMn+1\nn\nMϕ(t)dt\nt\n/integraltextMn+1\nn\nMdt\nt\n=/integraltextyn+1\nn\nyϕ(t)dt\nt\n/integraltextyn+1\nn\nydt\nt−ϕ(∞).\nSinceϕis decreasing and right continuous, its shrinking average on ( y,yn+1\nn)\nincreases with nand converges to ϕ(y). Thus the last expression increases to\nϕ(y)−ϕ(∞) asn→ ∞. The identity,\nK0,1\nξh(x) =/integraldisplay∞\nξmin(x,t)h(t)dt=/integraldisplayx\n0/integraldisplay∞\nmax(y,ξ)h(t)dtdy,FOURIER SERIES IN WEIGHTED LORENTZ SPACES 7\nshows that K0,1\nξ¯ℓnalso increases with nand, by the Monotone Convergence Theo-\nrem,\nlim\nn→∞K0,1\nξ¯ℓn(x) = lim\nn→∞/integraldisplayx\n0/integraldisplay∞\nmax(y,ξ)¯ℓn(t)dtdy\n=/integraldisplayx\n0ϕ(max(y,ξ))−ϕ(∞)dy\n=\n\nxϕ(ξ)−xϕ(∞), 0< x < ξ;\nξϕ(ξ)+ ˜g(x)−˜g(ξ)−xϕ(∞),0< ξ≤x;\n˜g(x)−˜g(0+)−xϕ(∞), 0 =ξ < x.\nCombining the three terms of (2.2), we conclude that K0,1\nξℓnincreases with n.\nWhen 0< x < ξ the limit is,\nxϕ(∞)+a(x/ξ)+xϕ(ξ)−xϕ(∞) =x˜g(ξ)/ξ= ˜g(x)\nby Lemma 2.3. When 0 < ξ≤xthe limit is\nxϕ(∞)+a+ξϕ(ξ)+ ˜g(x)−˜g(ξ)−xϕ(∞) = ˜g(x),\nand when 0 = ξ < xit is\nxϕ(∞)+a+ ˜g(x)−˜g(0+)−xϕ(∞) = ˜g(x).\nThis completes the proof. /square\nThisapproximationofquasi-concavefunctionscanbeusedtogivea similarresult\nfor functions in Pβ\nξ∩Ωα,β. They can be realized as increasing limits of functions\nof typeKα,β\nξh, up to equivalence. This result extends Lemma 5 of [19].\nLemma 2.5. Supposeξ≥0, andα,β∈Rsatisfyα+β >0. Iff∈Pβ\nξ∩Ωα,β, then\nthere exists ˜f∈L+and a sequence of functions {hn}inL+such that f≤˜f≤2f\nandKα,β\nξhnincreases to ˜fpointwise.\nProof.Within this proof let y=xα+βands=tα+β. Fixf∈Pβ\nξ∩Ωα,βand define\ngby setting g(y) =xαf(x). This ensures that g∈P1\nξα+β∩Ω0,1. Let ˜gbe the least\nconcave majorant of the quasi-concave function gand apply Lemma 2.4, with ξ\nreplaced by ξα+β, to obtain functions ℓn∈L+such that K0,1\nξα+βℓnincreases to ˜ gas\nn→ ∞.\nSet˜f(x) =x−α˜g(y). Sinceg≤˜g≤2gwe also have f≤˜f≤2f. (Note that ˜fis\nnot the least concave majorant of fin general.) Then define hnby requiring that\nt−αhn(t)dt=ℓn(s)ds. Evidently, hn∈L+. Also,Kα,β\nξhn(x) is equal to\n/integraldisplay∞\nξmin(xβt−α,x−αtβ)hn(t)dt=x−α/integraldisplay∞\nξα+βmin(y,s)ℓn(s)ds=x−αK0,1\nξα+βℓn(y).\nTherefore, Kα,β\nξhn(x) is increasing with nand converges to x−α˜g(y) =˜f(x). This\ncompletes the proof. /square\nNow we have all the machinery to prove the main result of this section .8 JAVAD RASTEGARI AND GORD SINNAMON\nProposition 2.6. Suppose ξ≥0, andα,β∈Rsatisfyα+β >0. Let0< p≤1≤\nq <∞andu,v∈L+. IfA∈ A, then\n(2.3) sup\nt>ξ/bardblAkα,β\nt/bardblq,u\n/bardblkα,β\nt/bardblp,v≤sup\nf∈Pβ\nξ∩Ωα,β/bardblAf/bardblq,u\n/bardblf/bardblp,v≤2sup\nt>ξ/bardblAkα,β\nt/bardblq,u\n/bardblkα,β\nt/bardblp,v.\nProof.The definition, kα,β\nt(x) = min( xβt−α,x−αtβ), ensures that if t > ξthen\nkα,β\nt∈Pβ\nξ∩Ωα,β. This proves the first inequality of (2.3). For the other, let\nD= sup\nt>ξ/bardblAkα,β\nt/bardblq,u\n/bardblkα,β\nt/bardblp,v.\nWe apply Lemma 2.2 taking µandνto be Lebesgue measure on (0 ,∞). Note\nthat each A∈ Ais its own formal adjoint. Let λbe Lebesgue measure on ( ξ,∞),\nK=Kα,β\nξandk(x,t) =kα,β(x,t). The conclusion is that for all h∈L+,\n/bardblAKα,β\nξh/bardblq,u≤D/bardblKα,β\nξh/bardblp,v.\nNow fixf∈Pβ\nξ∩Ωα,βand apply Lemma 2.5 to get an increasing sequence hn∈L+\nsuch that f≤limn→∞Kα,β\nξhn≤2f. The averaging operator Apreserves order,\nso, by the Monotone Convergence Theorem,\n/bardblAf/bardblq,u≤lim\nn→∞/bardblAKα,β\nξhn/bardblq,u≤Dlim\nn→∞/bardblKα,β\nξhn/bardblp,v≤2D/bardblf/bardblp,v.\nThis proves the second inequality of (2.3). /square\nOneconsequenceofProposition2.6isthefollowingextensionofTheo rem1in[11]\n(see also Theorem 3 in [10]) from the range 1 ≤p≤q <∞to 0< p≤q <∞. It is\npossible to get such an extension directly from Maligranda’s Theorem 1 beginning\nwith the case p= 1 and making the substitution f/ma√sto→fp. By this method one\nobtains the constant 21/pin place of the smaller 21/qthat appears below.\nProposition 2.7. Supposeξ≥0, andα,β∈Rsatisfyα+β >0. If0< p≤q <∞\nandu,v∈L+, then\n(2.4) sup\nz>ξ/bardblkα,β\nz/bardblq,u\n/bardblkα,β\nz/bardblp,v≤sup\nf∈Pβ\nξ∩Ωα,β/bardblf/bardblq,u\n/bardblf/bardblp,v≤21/qsup\nz>ξ/bardblkα,β\nz/bardblq,u\n/bardblkα,β\nz/bardblp,v\nProof.Takingg=fq, it is routine to verify that,\nsup\nf∈Pβ\nξ∩Ωα,β/bardblf/bardblq,u\n/bardblf/bardblp,v=/parenleftbigg\nsup\ng∈Pqβ\nξ∩Ωqα,qβ/bardblg/bardbl1,u\n/bardblg/bardblp/q,v/parenrightbigg1/q\n.\nBut, (kα,β\nt(x))q=kqα,qβ\nt(x), so we also have,\nsup\nt>ξ/bardblkα,β\nt/bardblq,u\n/bardblkα,β\nt/bardblp,v=/parenleftbigg\nsup\nt>ξ/bardblkqα,qβ\nt/bardbl1,u\n/bardblkqα,qβ\nt/bardblp/q,v/parenrightbigg1/q\n.\nThe result now follows from Proposition 2.6, with indices p/qand 1, by taking A\nto be the identity. /square\nWe end this section by stating the special cases of Propositions 2.6 a nd 2.7 that\nwill be used for our results in inequalities for Fourier series. Recall th atP=P0\n1\nandωz(x) = min( z−2,x−2) =k2,0\nz(x).FOURIER SERIES IN WEIGHTED LORENTZ SPACES 9\nCorollary 2.8. Let0< p≤2≤q <∞andu,v∈L+. IfA∈ A, then\nsup\nz>1/bardblAωz/bardblq/2,u\n/bardblωz/bardblp/2,v≤sup\nf∈P∩Ω2,0/bardblAf/bardblq/2,u\n/bardblf/bardblp/2,v≤2sup\nz>1/bardblAωz/bardblq/2,u\n/bardblωz/bardblp/2,v.\nCorollary 2.9. Let0< p≤q <∞andu,v∈L+. Then\nsup\nz>1/bardblωz/bardblq/2,u\n/bardblωz/bardblp/2,v≤sup\nf∈P∩Ω2,0/bardblf/bardblq/2,u\n/bardblf/bardblp/2,v≤22/qsup\nz>1/bardblωz/bardblq/2,u\n/bardblωz/bardblp/2,v.\n3.Necessary Conditions\nHere we construct the test functions that produce our necessa ry condition for\nthe Fourier inequality\n(3.1) /bardblˆf/bardblΛq(u)≤C/bardblf/bardblΓp(w), f∈L1(T).\nThe condition we obtain is automatically necessary for the stronger inequality (1.1)\nas well. The method is similar to the construction given for the Fourier transform\nin [16], in that one test function is constructed for each averaging o perator in the\nclassA, see (1.3), and each value of a positive real parameter z. The details of\nconstruction in the Fourier series case are quite different, howeve r, because of the\nfinite measure on Tand the atomic measure on Z.\nThe idea is to take advantage of the large class of functions gwhose rearrange-\nments coincide with f∗, for a given f. It turns out that there is enough freedom\nwithin this class to ensure that the rearrangement ˆ g∗, of the Fourier series of g,\npossesses the properties we require. The first four lemmas are ne eded to give the\nmain construction in Lemma 3.5. The general necessary condition is p roved in\nTheorem 3.7.\nThroughout this section we use µto denote counting measure on Z.\nThe Fourier seriesofthe characteristicfunction ofan interval is e asyto calculate.\nThe first lemma gives an estimate of its rearrangement.\nLemma 3.1. Suppose z≥3and letf(x) =χ(0,1/z)(x), viewed as a function on T.\nThenˆf∗(y)≥1/(3πy+9πz).\nProof.The Fourier coefficients of fmay be computed directly. If k/\\e}atio\\slash= 0, then\nˆf(k) =/integraldisplay1\n0e−2πikxf(x)dx=e−ikπ/zsin(kπ/z)\nkπ.\nForα >0, let\nEα={k∈Z:|ˆf(k)|> α} ⊇ {k∈Z\\{0}:|sin(kπ/z)|> α|k|π}.\nTo estimate µ(Eα), the number of elements in Eα, note that any real interval of\nlengthLcontains at least L−1 integers. We will also make use of the following\nsimple estimate based on the convexity of the sine function: If nis an integer and\n|x−(2n−1)| ≤1, then|sin(πx/2)| ≥1−|x−(2n−1)|.\nLetNbe the greatest integer less than 1 /(απz), fix a positive integer n≤N\nand suppose kis an integer in the open interval of length z(1−απzn) centred at\n(z/2)(2n−1). Then,\n|k−(z/2)(2n−1)|<(z/2)(1−απzn)< z/2\nand hence ( n−1)z < k < nz . Also,\n|2k/z−(2n−1)|<1−απzn≤1,10 JAVAD RASTEGARI AND GORD SINNAMON\nso the sine function estimate gives,\n|sin(kπ/z)| ≥1−|2k/z−(2n−1)|> απzn > απk.\nIf follows that k∈((n−1)z,nz)∩Eα. Thus, there are at least z(1−απzn)−1\npositive integers in (( n−1)z,nz)∩Eα. Summing these from n= 1...Nshows\nthatµ((0,∞)∩Eα) is not less than,\n(z−1)N−z2απN(N+1)/2≥(z−1)/parenleftbigg1\nαπz−1/parenrightbigg\n−z\n2/parenleftbigg1\nαπz+1/parenrightbigg\n.\nEvidently, k∈Eαif and only if −k∈Eα. So, using z≥3,\nµ(Eα)≥2µ((0,∞)∩Eα)≥1−2/z\nαπ+2−3z≥1\n3απ−3z.\nThe definition of the rearrangement ensures that when α=ˆf∗(y),µ(Eα)≤y,\nso\n1\n3ˆf∗(y)π−3z≤y\nand we have\nˆf∗(y)≥1\n3πy+9πz.\n/square\nThe rearrangement of a characteristic function depends on the m easure of the\nunderlying set, but not on its geometry. On the other hand, the Fo urier series of\na characteristic function is profoundly affected by the geometry o f the underlying\nset. Here we take advantage of this fact to get a dilation-like result that behaves\noppositely to what we expect from a Fourier dilation.\nLemma 3.2. Letkbe a positive integer and z >1. Letf(x) =χ[0,1/(kz))(x). Then\nfor anyε >0there exists a function g∈L1(T)such that\ng∗(s) =f∗(s/k)andˆg∗(y)≥ˆf∗(y/k)−ε\nfor0≤s <1andy >0.\nProof.We show that for a sufficiently large integer M,\ng(x) =k−1/summationdisplay\nj=0e2πijMxf(x−j/(kz))\nwill be the desired function. First notice that the translates of fin the sum above\nare supported on disjoint subsets of [0 ,1). Thus,\n|g(x)|=k−1/summationdisplay\nj=0|e2πijMxf(x−j/(kz))|=k−1/summationdisplay\nj=0f(x−j/(kz)) =χ[0,1/z)(x).\nFurthermore |g|andfare both decreasing so, for 0 ≤s <1,\ng∗(s) =|g(s)|=χ[0,1/z)(s) =f(s/k) =f∗(s/k).\nThis shows that gsatisfies the first conclusion of the lemma no matter what Mis\nchosen.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 11\nTo establish the second conclusion we make use of the properties (1 .2) to get,\nˆg(n) =k−1/summationdisplay\nj=0e−2πi(n−jM)j/(kz)ˆf(n−jM).\nFixε >0 and choose M= 2k/(πε). For all nsatisfying |n|> M/2,\n|ˆf(n)|=/vextendsingle/vextendsingle/vextendsingle−einπ/(kz)\nnπsin(nπ/(kz))/vextendsingle/vextendsingle/vextendsingle≤1\nnπ<ε\nk.\nSo if|n−jM|< M/2 for some jthen, for every l/\\e}atio\\slash=j,|n−lM|> M/2 and hence\n|ˆf(n−lM)|< ε/k. It follows that if |n−jM|< M/2, then\n|ˆg(n)| ≥ |ˆf(n−jM)|−(k−1)ε/k≥ |ˆf(n−jM)|−ε.\nNow we can estimate the distribution function of ˆ g. Forα >0,\nµˆg(α) =µ{n∈Z:|ˆg(n)|> α}\n≥k−1/summationdisplay\nj=0µ{n∈(jM−M/2,jM+M/2) :|ˆg(n)|> α}\n≥k−1/summationdisplay\nj=0µ{n∈(jM−M/2,jM+M/2) :|ˆf(n−jM)|−ε > α}\n=kµ{n∈(−M/2,M/2) :|ˆf(n)|> α+ε}\nSince|ˆf(n)|< εwhenn /∈(−M/2,M/2),\nµˆg(α)≥kµ{n∈Z:|ˆf(n)|> α+ε}=kµˆf(α+ε).\nNow for y >0,\ny≥µˆg(ˆg∗(y))≥kµˆf(ˆg∗(y)+ε)\nand hence\nˆf∗(y/k)≤ˆf∗/parenleftbig\nµˆf/parenleftbig\nˆg(y)+ε/parenrightbig/parenrightbig\n≤ˆg∗(y)+ε,\nas required. /square\nThe last two results combine to give a useful one-parameter family o f estimates\nfor the rearrangement of the Fourier series of a characteristic f unction with an\nunderlying set of fixed measure.\nLemma 3.3. Forz≥3,r >0andε >0there exists a function g∈L1(T)such\nthat\ng∗=χ[0,1/z)andˆg∗(y)≥1\n3πy/r+9π(r+1)z−ε\nProof.Letkbe the integer satisfying r≤k < r+1 and let f=χ[0,1/kz). Then by\nLemma 3.2 there exists a gsuch that,\ng∗(s) =f∗(s/k) =χ[0,1/z)andˆg∗(y)≥ˆf∗(y/k)−ε\nLemma 3.1 yields\nˆg∗(y)≥1\n3πy/k+9πkz−ε≥1\n3πy/r+9π(r+1)z−ε\n/square12 JAVAD RASTEGARI AND GORD SINNAMON\nThe next lemma is a variation of the construction in Lemma 3.2. This time the\ncharacteristic function is subdivided into infinitely many parts of diffe ring sizes.\nThe increase in generality is balanced by the coarser estimate obtain ed for the\nrearrangement of the Fourier series.\nLemma3.4. Let{pj}be a sequence of non-negative real numberssatisfying/summationtext∞\nj=1pj=\np0≤1. For each pjletfj=χ[0,pj)be a function on the unit circle. Then for any\nε >0there exists a function g∈L1(T)such that\ng∗=χ[0,p0)andˆg∗(y)≥ˆf∗\nj(y)−ε, j= 1,2,....\nProof.LetX1= 0 and Xj=/summationtextj−1\nl=1plforj≥2. Define gby,\ng(x) =∞/summationdisplay\nj=1e2πiMjxfj(x−Xj),\nwhere the Mj,j= 1,2,...are to be chosen later. The definitions of X1,X2,...\nensure that the translates fj(x−Xj) have disjoint supports, and that,\n|g(x)|=∞/summationdisplay\nj=1|e2πiMjxfj(x−Xj)|=χ[0,p0).\nSince|g|is decreasing, g∗=|g|=χ[0,p0), the first conclusion of the lemma.\nThe Fourier coefficients of gare given by,\nˆg(n) =∞/summationdisplay\nj=1e−2πi(n−Mj)Xjˆfj(n−Mj).\n(Note that since the series defining gconverges in L1(T), the series defining ˆ g\nconverges in L∞(Z) and hence pointwise.) We choose M1,M2,...so that the\nintervals, Ij= (Mj−2j/(πε),Mj+ 2j/(πε)) are disjoint for j= 1,2,.... This\nimplies that if n∈Ij, thenn/\\e}atio\\slash∈Ilforl/\\e}atio\\slash=jso,\n|ˆfl(n−Ml)|=/vextendsingle/vextendsingle/vextendsingle/vextendsinglesin((n−Ml)πpl)\n(n−Ml)π/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n|n−Ml|π<ε\n2l.\nThus, for n∈Ij,\n|ˆg(n)| ≥ |ˆfj(n−Mj)|−/summationdisplay\nl/negationslash=jε\n2l≥ˆfj(n−Mj)−ε.\nFor anyjand any α >0,\nµˆg(α)≥µ{n∈Ij:|ˆg(n)|> α} ≥µ{n∈Ij:|ˆfj(n−Mj)|> α+ε}.\nBut|ˆfj(n−Mj)|< εforn /∈Ij, so\nµˆg(α)≥µ{n∈Z:|ˆfj(n−Mj)|> α+ε}=µˆfj(α+ε).\nAsinLemma3.2thisestimateforthedistributionfunctionsgives, ˆ g∗(y)≥ˆf∗\nj(y)−ε,\nthe desired estimate for the rearrangements. /square\nLemma 3.5. Letz≥3andA∈ A. For each ε >0there exists a function\nf∈L1(T)such that\nf∗≤χ[0,1/z)and(Aωz)1/2≤c1(ˆf∗+ε)\nwithc1= 183.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 13\nProof.Fixε >0 and let {(ai,bi)}be the intervals associated with the averaging\noperator A. We will build the function fin pieces and assemble them using Lemma\n3.4. The first piece, f0=χ[0,1/4z), satisfies f∗\n0=χ[0,1/4z)so, by Lemma 3.1,\nˆf∗\n0(y)≥(3πy+9π(4z))−1≥(39πmax(y,z))−1=ωz(y)1/2/(39π).\nIfysatisfiesAωz(y)≤2ωz(y), then\n(3.2) Aωz(y)1/2≤39√\n2πˆf∗\n0(y)≤c1ˆf∗\n0(y).\nThe second piece is needed only when zis contained in one of the intervals of A. If\nthere is one, call it ( a0,b0). By Lemma 3.3, with r=/radicalbig\nb0/(8z) andzreplaced by\n8z/3, there exists a function g0such that g∗\n0=χ[0,3/(8z))and\nˆg0∗(y)+ε/2≥/parenleftBig\n3πy/radicalbig\n8z/b0+9π/parenleftBig/radicalbig\nb0/(8z)+1/parenrightBig\n(8z/3)/parenrightBig−1\n=/parenleftBig\n6√\n2π(y/b0)+6√\n2π+24π(z/b0)1/2/parenrightBig−1\n(b0z)−1/2.\nIfy∈(a0,b0) then both y/b0andz/b0are less than 1 so,\nˆg0∗(y)+ε/2≥(12√\n2π+24π)−1(b0z)−1/2.\nAlso, ify∈(a0,b0), the monotonicity of ωzimplies that,\nAωz(y) =1\nb0−a0/integraldisplayb0\na0ωz(t)dt≤1\nb0/integraldisplayb0\n0ωz(t)dt≤1\nb0/integraldisplay∞\n0ωz(t)dt=2\nb0z.\nThus,\n(3.3) Aωz(y)1/2≤(12√\n2π+24π)√\n2(ˆg0∗(y)+ε/2)≤c1(ˆg0∗(y)+ε/2).\nThe remaining pieces of fare indexed by certain intervals of A. Let\nJ={j/\\e}atio\\slash= 0 :z≤aj≤bj/2}.\nFor each j∈J, apply Lemma 3.3, with r=/radicalbig\nbj/(16aj) andzreplaced by 16 aj/3,\nto produce a function gjsuch that g∗\nj=χ[0,3/(16aj))and\nˆgj∗(y)+ε/2≥/parenleftbigg\n3πy/radicalBig\n16aj/bj+9π/parenleftbigg/radicalBig\nbj/(16aj)+1/parenrightbigg\n(16aj/3)/parenrightbigg−1\n=/parenleftBig\n12π(y/bj)+12π+24√\n2π(2aj/bj)1/2/parenrightBig−1\n(ajbj)−1/2.\nIfy∈(aj,bj) then both y/bjand 2aj/bjare less than 1 so,\nˆgj∗(y)+ε/2≥(24π+24√\n2π)−1(ajbj)−1/2.\nBut fory∈(aj,bj),Aωz(y) = 1/(ajbj) so\n(3.4) Aωz(y)1/2≤(24π+24√\n2π)(ˆgj∗(y)+ε/2)≤c1(ˆgj∗(y)+ε/2).\nTo apply Lemma 3.4 to the functions f0,g0, andgjforj∈Jwe need to estimate\nthe sums of the lengths of the intervals involved. For each j∈J, letmjbe the\nsmallest integer such that 2mjz≤aj. Sincez≤aj, eachmj≥0. To see that\nmj/\\e}atio\\slash=mkfor distinct j,k∈J, suppose aj≤ak. Since the intervals of Aare14 JAVAD RASTEGARI AND GORD SINNAMON\ndisjoint, bj≤akand we see that, mj> mkbecause, 2mj+1z≤2aj< bj≤ak.\nSince the mjare all different,\n/summationdisplay\nj∈J1\naj≤1\nz/summationdisplay\nj∈J2−mj≤1\nz∞/summationdisplay\nm=02−m=2\nz.\nTherefore,\n1\n4z+3\n8z+/summationdisplay\nj∈J3\n16aj≤1\nz≤1\nand Lemma 3.4 guarantees the existence of a function fsuch that,\nf∗≤χ[0,1/z),ˆf∗≥ˆf0∗−ε/2,ˆf∗≥ˆg0∗−ε/2,andˆf∗≥ˆgj∗−ε/2 forj∈J.\nTo see that ( Aωz)1/2≤c1(ˆf∗+ε), lety >0. IfAωz(y)≤2ωz(y), then (3.2) shows\nthat\n(Aωz)(y)1/2≤c1(ˆf0∗(y)+ε/2)≤c1(ˆf∗(y)+ε).\nIfyandzare in the same interval of A, then (3.3) shows that\n(Aωz)(y)1/2≤c1(ˆg0∗(y)+ε/2)≤c1(ˆf∗(y)+ε).\nAny other ysatisfiesAωz(y)>2ωz(y) and is not in an interval of Awithz. Since\nAωz(y)/\\e}atio\\slash=ωz(y),yis in some interval ( aj,bj) on which ωzis not constant. Thus\nz < bj. Butzis not in the interval that contains ysoz≤aj. Therefore,\n1\najbj=Aωz(y)>2ωz(y) =2\ny2≥2\nb2\nj\nand we see that aj< bj/2 soj∈J. Now (3.4) yields,\n(Aωz)(y)1/2≤c1(ˆgj∗(y)+ε/2)≤c1(ˆf∗(y)+ε)\nto complete the proof. /square\nThe restriction z≥3 in the last lemma is a technical one and can be removed.\nProposition 3.6. Letz≥1andA∈ A. For each ε >0there exists a function\nf∈L1(T)such that\nf∗≤χ[0,1/z)and(Aωz)1/2≤c(ˆf∗+ε)\nwithc= 3c1= 549.\nProof.Ifz≥3 then Lemma 3.5 implies the existence of the desired function f,\nbecausec1≤c.\nIf 1≤z <3 then we set z= 3 in Lemma 3.5 to get a function fsuch that\nf∗≤χ[0,1/3)and (Aω3)1/2≤c1(ˆf∗+ε). Clearly, f∗≤χ[0,1/z). We also have\nωz≤9ω3which implies Aωz≤9Aω3and completes the proof. /square\nThe main result of the section follows. It uses the test functions ju st constructed\nto give a necessary condition for the Fourier series inequality (3.5). As we will see\nin the next section, for a large range of indices, the condition is also s ufficient.\nTheorem 3.7. Suppose 0< p <∞,0< q≤ ∞, and for some C >0,u,w∈L+\nsatisfy\n(3.5) /bardblˆf/bardblΛq(u)≤C/bardblf/bardblΓp(w), f∈L1(T).FOURIER SERIES IN WEIGHTED LORENTZ SPACES 15\nfor allf∈L1(T). Then\nsup\nz>1sup\nA∈A/bardblAωz/bardblq/2,u\n/bardblωz/bardblp/2,v≤c2C2\nwherecis the constant in Proposition 3.6. Here v(t) =tp−2w(1/t).\nProof.Makingthe changeofvariable t/ma√sto→1/ton the right-handside, (3.5) becomes,\n(3.6) /bardblˆf∗/bardblq,u≤C/parenleftbigg/integraldisplay∞\n0/parenleftbigg/integraldisplay1/t\n0f∗/parenrightbiggp\nv(t)dt/parenrightbigg1/p\n.\nFixA∈ A,z >1, andε∈(0,1). LetY >0 and use Proposition 3.6 to choose a\nfunction f:T→Csuch that f∗≤χ[0,1/z)and\n(Aωz)1/2≤c(ˆf∗+(ε/c)Aωz(Y)1/2).\nSinceωzis decreasing, so is Aωz. Thus, for y∈[0,Y),\nAωz(y)1/2≤cˆf∗(y)+εAωz(Y)1/2≤cˆf∗(y)+εAωz(y)1/2,\nso (1−ε)Aωz(y)1/2≤cˆf∗(y). Therefore,\n(1−ε)2/bardbl(Aωz)χ[0,Y)/bardblq/2,u≤c2/bardblˆf∗χ[0,Y)/bardbl2\nq,u≤c2/bardblˆf∗/bardbl2\nq,u.\nBut, for all y,\n/integraldisplay1/y\n0f∗(t)dt≤/integraldisplay1/y\n0χ[0,1/z)(t)dt=ωz(y)1/2,\nso, using (3.6),\n/bardblˆf∗/bardblq,u≤C/bardblω1/2\nz/bardblp,v=C/bardblωz/bardbl1/2\np/2,v.\nWe conclude that\n(1−ε)2/bardbl(Aωz)χ[0,Y)/bardblq/2,u≤c2C2/bardblωz/bardblp/2,v.\nLettingY→ ∞, and then ε→0, gives,\n/bardblAωz/bardblq/2,u≤c2C2/bardblωz/bardblp/2,v\nand completes the proof. /square\nA slight simplification of the above proof gives the corresponding res ult for the\nFourierseriesinequality between Λ-spaces. Notice that in this case both the Fourier\ninequality and the weight condition depend only on the values of w(t) for 0< t <1.\nCorollary 3.8. Suppose 0< p <∞,0< q <∞, and for some C >0,u,w∈L+\nsatisfy\n/bardblˆf/bardblΛq(u)≤C/bardblf/bardblΛp(w), f∈L1(T).\nfor allf∈L1(T). Then\nsup\nz>1sup\nA∈A/bardblAωz/bardblq/2,u\n/bardblχ(0,1/z)/bardblp/2,w≤c2C2\nwherecis the constant in Proposition 3.6.16 JAVAD RASTEGARI AND GORD SINNAMON\n4.Main Results\nIn this section we present weight conditions that ensure the bound edness of\nthe Fourier coefficient map between Lorentz spaces. For a large ra nge of indices\nthese coincide with the necessary conditions obtained in the previou s section to\ngive a characterization of exactly those weights for which the map is bounded. The\nfocus is on the inequality (1.1), which expresses the boundedness o f the Fourier\ncoefficient map from Γ p(w) to Γ q(u) but we will see that exactly the same weight\nconditions give boundedness from Γ p(w) to Λq(u). Under mild conditions on wthe\nboundedness from Λ p(w) to Λ q(u) is also equivalent.\nWhile the most interesting results involve weights wthat are supported on [0 ,1],\notherweightsarepermitted. The interested readermayverify th at both the Fourier\ninequalities and the various weight conditions depend on wχ(1,∞)only through the\nvalue of/integraltext∞\n1w(t)dt\ntp. Similarly, any weight uis permitted, but the most interesting\ncases involve weights uthat are constant on [ n−1,n) forn= 1,2.... See Theorem\n4.7(iii) for an indication that only the values/integraltextn\nn−1u(t)dtforn= 1,2,..., are of\nsignificance.\nFor sufficiency of the weight conditions we actually prove the bounde dness result\nfor a large class of operators that includes the Fourier coefficient m ap. Let ( X,µ)\nand (Y,ν) beσ-finite measure spaces with µ(X) = 1, and let Tbe a sublinear\noperator from L1\nµ+L2\nµtoL2\nν+L∞\nν. We say that Tis of type (1 ,∞) and (2,2)\nprovided Tis a bounded map both from L1\nµtoL∞\nνand from L2\nµtoL2\nν. The Fourier\ncoefficient map is one such operator; in this case µis Lebesgue measure on [0 ,1]\nandνis counting measure on Z.\nLetTdenote the collection of all sublinear operators Tof type (1 ,∞) and (2,2),\nover all probability measures µandσ-finite measures ν. Propositions 4.2 and 4.4,\nbelow, give several weight conditions that are sufficient for the ineq uality,\n(4.1) /bardblTf/bardblΓq(u)≤C/bardblf/bardblΓp(w), f∈L1\nµ,\nto hold for all T∈ T. Recalling that v(t) =tp−2w(1/t) it is easy to rewrite this as,\n(4.2)/parenleftbigg/integraldisplay∞\n0(Tf)∗∗(t)qu(t)dt/parenrightbigg1/q\n≤C/parenleftbigg/integraldisplay∞\n0/parenleftbigg/integraldisplay1\nt\n0f∗(s)ds/parenrightbiggp\nv(t)dt/parenrightbigg1/p\n, f∈L1\nµ.\n(Note that since µis a finite measure, L1\nµ+L2\nµ=L1\nµ.)\nThe resultsofthis sectionarebasedonthe followingcorollaryofare arrangement\nestimate from [9].\nProposition 4.1. Suppose (X,µ)and(Y,ν)areσ-finite measure spaces and let T\nbe a sublinear operator from L1\nµ+L2\nµtoL2\nν+L∞\nν. ThenT∈ Tif and only if there\nexists a constant DTsuch that\n(4.3)/integraldisplayz\n0(Tf)∗∗(t)2dt≤DT/integraldisplayz\n0/parenleftbigg/integraldisplay1\nt\n0f∗(s)ds/parenrightbigg2\ndt\nfor allz >0andf∈L1\nµ+L2\nµ.\nThis result appears in [9] with ( Tf)∗instead of the larger ( Tf)∗∗. But Hardy’s\ninequality shows if T∈ T, then so is the map f/ma√sto→(Tf)∗∗. Since (( TF)∗∗)∗=\n(TF)∗∗, we obtain the statement above. In the case that Tis the Fourier coefficient\nmap, we may take DT= 8.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 17\nThe next two theorems give sufficient conditions for the Fourier ineq uality (1.1).\nRecallthat afunction h: (0,∞)→[0,∞)isinP∩Ω2,0provided t2h(t) isincreasing,\nh(t) is decreasing, and h(t) is constant on (0 ,1).\nProposition 4.2. Suppose 0< p <∞,0< q <∞, andu,v∈L+. Let\nCΘ= sup\nh∈P∩Ω2,0/bardblh/bardblΘq/2(u)\n/bardblh/bardblp/2,v.\nThen for each T∈ Tthe inequality (4.1) holds with C=√DTCΘ. In particular, the\nFourier inequalities (1.1) and (3.5) hold with C=√8CΘ. Herew(t) =tp−2v(1/t).\nProof.LetT∈ Tand let ( X,µ) be its associated probability space. Fix f∈L1\nµ.\nLethfandϕfbe defined by\nhf(t) =/parenleftbigg/integraldisplay1/t\n0f∗(s)ds/parenrightbigg2\nandϕf(t) = (1/DT)(Tf)∗∗(t)2,\nwhereDTis the constant from Proposition 4.1. Notice that hf(t) is decreasing and\nt2hf(t) =f∗∗(1/t)2is increasing. Also, since µ(X) = 1,f∗vanishes outside the\ninterval (0 ,1) and therefore hfis constant on (0 ,1). It follows that hf∈P∩Ω2,0.\nIn addition, ϕfis decreasingandProposition4.1implies that ϕ∗∗\nf≤h∗∗\nf. Therefore,\n/bardblϕf/bardblq/2,u≤ /bardblhf/bardblΘq/2(u)≤CΘ/bardblhf/bardblp/2,v.\nThis implies (4.2) with C=√DTCΘ.\nSincew(t) =tp−2v(1/t), (4.2) becomes (4.1). Taking Tto be the Fourier coeffi-\ncient map, and DT= 8, we obtain (1.1) with C=√8CΘ. The weaker inequality\n(3.5) is an immediate consequence. /square\nRemark 4.3. Suppose q=pandu=v∈Bp/2. Then Θp/2(u) = Λ p/2(u),\nwith equivalent norms, so for hdecreasing, /bardblh/bardblΘp/2(u)≈ /bardblh/bardblΛp/2(u)=/bardblh/bardblp/2,v. It\nfollows that CΘ<∞and we have, with w(t) =tp−2u(1/t),\n/bardblˆf/bardblΓp(u)≤C/bardblf/bardblΓp(w), f∈L1.\nIn particular, when 1< p <2andu(t) =v(t) =tp−2, we recover the well-known\nfact that the Fourier transform maps Lpinto the power-weighted Lorentz space ℓp′,p.\nRecall that,\n/bardblˆf/bardblℓp′,p=/parenleftbigg/integraldisplay∞\n0tp/p′−1(ˆf)∗(t)pdt/parenrightbigg1/p\n.\nDespite this example, CΘcanoften be difficult toestimate directly, sowe provide\na number of estimates in the next proposition. One that will figure pr ominently in\nour weight characterization is,\n(4.4) Cω= sup\nz>1/bardblωz/bardblΘq/2(u)\n/bardblωz/bardblp/2,v.\nRecall that ωz(t) = min( t−2,z−2). Also recall that uodenotes the level function\nofuwith respect to Lebesgue measure. Both will be needed in the state ment and\nproof of the next theorem. In view of Proposition 4.2, each of the f ollowing upper\nbounds for CΘgives a sufficient condition for (4.1) and hence for (1.1).\nProposition 4.4. Suppose 0< p <∞,0< q <∞, andu,v∈L+.18 JAVAD RASTEGARI AND GORD SINNAMON\n(i)For any pandq,\nCΘ≤sup\nh∈P∩Ω2,0/parenleftbigg/integraldisplay∞\n0/parenleftbigg1\nt/integraldisplayt\n0h(s)ds/parenrightbiggq/2\nu(t)dt/parenrightbigg2/q/parenleftbigg/integraldisplay∞\n0h(t)p/2v(t)dt/parenrightbigg−2/p\n.\n(ii)Ifq≥2, then\nCΘ= sup\nh∈P∩Ω2,0sup\nA∈A/bardblAh/bardblq/2,u\n/bardblh/bardblp/2,vand\nCΘ≤sup\nh∈P∩Ω2,0/parenleftbigg/integraldisplay∞\n0h(t)q/2uo(t)dt/parenrightbigg2/q/parenleftbigg/integraldisplay∞\n0h(t)p/2v(t)dt/parenrightbigg−2/p\n.\n(iii)Ifp≤qandq≥2then\nCΘ≤(4q′)2/qsup\n1<1/x<y/parenleftbigg1\nxy/integraldisplayy\n0u(t)dt/parenrightbigg2/q/parenleftbigg/integraldisplay∞\n0ωx(t)p/2w(t)dt/parenrightbigg−2/p\n.\n(iv)Ifp≤2≤qthenCΘ≤2Cω.\nProof.Inequality (1.4) shows that for any decreasing h,\n/bardblh/bardblΘq/2(u)≤ /bardblh/bardblΓq/2(u)=/parenleftbigg/integraldisplay∞\n0/parenleftbigg1\nt/integraldisplayt\n0h(s)ds/parenrightbiggq/2\nu(t)dt/parenrightbigg2/q\n,\nwhich proves (i).\nWhenq≥2, Corollary 2.4 of [16] shows that for each decreasing h,\n(4.5) /bardblh/bardblΘq/2(u)= sup\nA∈A/bardblAh/bardblq/2,u≤ /bardblh/bardblq/2,uo.\nThis gives both the equation and the upper bound in (ii). It also shows that when\nq≥2,\n(4.6) Cω= sup\nz>1sup\nA∈A/bardblAωz/bardblq/2,u\n/bardblωz/bardblp/2,v,\nwhich will be useful later.\nFor (iii) we begin by applying Corollary 2.9 to the upper bound from (ii). S ince\np≤q,\nCΘ≤sup\nh∈P∩Ω2,0/bardblh/bardblq/2,uo\n/bardblh/bardblp/2,v≤22/qsup\nz>1/bardblωz/bardblq/2,uo\n/bardblωz/bardblp/2,v.\nLetx= 1/z, and make the change of variable t/ma√sto→1/tin the denominator to get,\n/bardblωz/bardblp/2,v=/parenleftbigg/integraldisplay∞\n0min(tp,xp)t−pw(t)dt/parenrightbigg2/p\n=x2/parenleftbigg/integraldisplay∞\n0ωx(t)p/2w(t)dt/parenrightbigg2/p\n.\nTo estimate the numerator, observe that since uois decreasing,\n/integraldisplay∞\nzuo(t)dt\ntq≤uo(z)/integraldisplay∞\nzdt\ntq=uo(z)z−q\nq−1/integraldisplayz\n0dt≤z−q\nq−1/integraldisplayz\n0uo(t)dt.\nThus,\n/bardblωz/bardblq/2,uo=/parenleftbigg\nz−q/integraldisplayz\n0uo(t)dt+/integraldisplay∞\nzuo(t)dt\ntq/parenrightbigg2/q\n≤/parenleftbigg\nq′z−q/integraldisplayz\n0uo(t)dt/parenrightbigg2/q\n.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 19\nBut Lemma 2.5 of [16] shows that\n1\nz/integraldisplayz\n0uo(t)dt≤2sup\ny≥z1\ny/integraldisplayy\n0u(t)dt,\nso,\n/bardblωz/bardblq/2,uo≤/parenleftbigg\n2q′z1−qsup\ny≥z1\ny/integraldisplayy\n0u(t)dt/parenrightbigg2/q\n=x2/parenleftbigg\n2q′sup\n1/x≤y1\nxy/integraldisplayy\n0u(t)dt/parenrightbigg2/q\n.\nThis estimate gives,\nCΘ≤22/qsup\n1/x>1x2/parenleftbigg\n2q′sup\n1/x≤y1\nxy/integraldisplayy\n0u(t)dt/parenrightbigg2/q\nx−2/parenleftbigg/integraldisplay∞\n0ωx(t)p/2w(t)dt/parenrightbigg−2/p\n,\nwhich simplifies to the conclusion of (iii).\nTo prove (iv), apply Corollary 2.8 to the equation from (ii) to get\nCΘ= sup\nh∈P∩Ω2,0sup\nA∈A/bardblAh/bardblq/2,u\n/bardblh/bardblp/2,v≤2sup\nz>1sup\nA∈A/bardblAωz/bardblq/2,u\n/bardblωz/bardblp/2,v= 2Cω,\nwhere the last equality is (4.6). /square\nNext we combine the sufficiency results above with the necessary co nditions\nobtained in Section 3 to obtain a necessary and sufficient condition fo r the bound-\nedness of the Fourier coefficient map between weighted Lorentz sp aces. Recall that\nv(t) =tp−2w(1/t) and refer to expressions (4.4) and (4.6) for the constant Cω.\nTheorem 4.5. Let0< p≤2≤q <∞andu,w∈L+. The Fourier inequality,\n/bardblˆf/bardblΓq(u)≤C/bardblf/bardblΓp(w), f∈L1(T),\nholds if and only if Cω<∞. Moreover, for the best constant C,\n√\nCω\n549≤C≤4/radicalbig\nCω.\nProof.LetCbe the least constant, finite or infinite, in the aboveFourier inequalit y.\nPropositions 4.2 and 4.4(iv) show that for any f∈L1(T),\n/bardblˆf/bardblΓq(u)≤/radicalbig\n8CΘ/bardblf/bardblΓp(w)≤4/radicalbig\nCω/bardblf/bardblΓp(w).\nThus,C≤4√Cω.\nOn the other hand, inequality (1.4) shows that we also have,\n/bardblˆf/bardblΛq(u)≤C/bardblf/bardblΓp(w), f∈L1(T),\nso (4.6) and Theorem 3.7 give,\nCω= sup\nz>1sup\nA∈A/bardblAωz/bardblq/2,u\n/bardblωz/bardblp/2,v≤(549C)2.\nThis completes the proof. /square\nIn the case q= 2 the necessary and sufficient condition Cω<∞can be put in\na form that is especially simple to estimate.20 JAVAD RASTEGARI AND GORD SINNAMON\nTheorem 4.6. Let0< p≤2andu,w∈L+. IfCis the best constant in the\nFourier inequality,\n/bardblˆf/bardblΓ2(u)≤C/bardblf/bardblΓp(w), f∈L1(T),\nthen\nCxy/549≤C≤8Cxy,\nwhere\nCxy= sup\n1<1/x<y/parenleftbigg1\nxy/integraldisplayy\n0u(t)dt/parenrightbigg1/2/parenleftbigg\nx−p/integraldisplayx\n0w(t)dt+/integraldisplay∞\nxw(t)dt\ntp/parenrightbigg−1/p\n.\nProof.Withq= 2, Proposition 4.4(iii) shows CΘ≤8C2\nxygivingC≤√8CΘ≤\n8Cxy.\nIt remainstoshowthat Cxy/549≤C. Since1\ny/integraltexty\n0uo(t)dtisadecreasingfunction\nthat majorizes1\ny/integraltexty\n0u(t)dt, we have\n(4.7) sup\n1/x<y1\ny/integraldisplayy\n0u(t)dt≤x/integraldisplay1/x\n0uo(t)dt.\nSo, taking z= 1/xand applying Lemma 2.2 of [16],\nsup\n1/x<y1\nxy/integraldisplayy\n0u(t)dt≤/integraldisplayz\n0uo(t)dt≤z2/bardblωz/bardbl1,uo=z2sup\nA∈A/bardblAωz/bardbl1,u.\nAlso,\nx−p/integraldisplayx\n0w(t)dt+/integraldisplay∞\nxw(t)dt\ntp=zp/bardblωz/bardblp/2\np/2,v.\nTherefore, by (4.6) and Theorem 4.5,\nC2\nxy≤sup\nz>1sup\nA∈A/bardblAωz/bardbl1,u\n/bardblωz/bardblp/2,v=Cω≤(549C)2.\nTaking square roots completes the proof. /square\nExamining the proofs of the last two theorems gives a more general result. We\nrecord it without tracking the estimates of the constants involved .\nTheorem 4.7. Let0< p≤2≤q <∞andu,w∈L+. The following are\nequivalent.\n(i)For each T∈ Tthere exists a finite constant Csuch that\n(4.8) /bardblTf/bardblΓq(u)≤C/bardblf/bardblΓp(w), f∈L1\nµ.(Hereµdepends on T.)\n(ii)There exists a finite constant Csuch that\n/bardblˆf/bardblΓq(u)≤C/bardblf/bardblΓp(w), f∈L1(T).\n(iii)There exists a finite constant Csuch that\n/bardblˆf/bardblΛq(u)≤C/bardblf/bardblΓp(w), f∈L1(T).\n(iv)Cω<∞.\nWhenq= 2,Cxy<∞is also equivalent.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 21\nProof.Since the Fourier coefficient map is in T, (i) implies (ii). Inequality (1.4)\nshows that (ii) implies (iii). But if (iii) holds for some finite constant C, then the\nproof of Theorem 4.5 provides Cω≤(549C)2<∞and gives (iv). To complete the\ncircle apply Propositions 4.2 and 4.4(iv) to see that for each T∈ T, (4.8) holds\nwithC=√DTCΘ≤√2DTCω<∞.\nThe last statement of the theorem follows from Theorem 4.6. /square\nUnder a mild a priori condition on the weight u, the necessary and sufficient\ncondition simplifies and the above theorem extends to a larger range of indices.\nRecall that if uis decreasing, then u∈B1,∞andu∈Bq/2for every q >2.\nProposition 4.8. Let0< p≤q <∞,2≤qandu,w∈L+. Suppose that\nu∈Bq/2ifq >2, andu∈B1,∞ifq= 2. Then the Fourier inequality,\n/bardblˆf/bardblΓq(u)≤C/bardblf/bardblΓp(w), f∈L1(T).\nholds if and only if\n(4.9) sup\n0<x<1/parenleftbigg/integraldisplay1/x\n0u(t)dt/parenrightbigg1/q/parenleftbigg\nx−p/integraldisplayx\n0w(t)dt+/integraldisplay∞\nxw(t)dt\ntp/parenrightbigg−1/p\nis finite. In fact, under the above hypothesis on u, statements (i), (ii) and (iii) of\nTheorem 4.7 are all equivalent to the finiteness of (4.9).\nProof.First suppose q= 2. The B1,∞condition on ushows that there exists a\nconstant b1(u) such that for each x,\n/integraldisplay1/x\n0u(t)dt≤sup\n1<xy1\nxy/integraldisplayy\n0u(t)dt≤b1(u)/integraldisplay1/x\n0u(t)dt\nsoCxy<∞if and only if (4.9) is finite. Since p≤q= 2, Theorem 4.7 completes\nthe proof.\nNow consider the case q >2 and suppose that (4.9) is finite. Since u∈Bq/2\nthere exists a constant csuch that inequality (1.5) holds. This and (1.4) give, for\nany decreasing h,\n/bardblh/bardblΘq/2(u)≤ /bardblh/bardblΓq/2(u)≤c/bardblh/bardblΛq/2(u)=c/bardblh/bardblq/2,u.\nTherefore\nCΘ≤csup\nh∈P∩Ω2,0/bardblh/bardblq/2,u\n/bardblh/bardblp/2,v≤c22/qsup\nz>1/bardblωz/bardblq/2,u\n/bardblωz/bardblp/2,v,\nwhere the second inequality is from Corollary 2.9.\nBut/integraldisplay∞\nzu(t)dt\ntq≤1\nzq/2/integraldisplay∞\nzu(t)dt\ntq/2≤bq/2(u)\nzq/integraldisplayz\n0u(t)dt,\nso\n/bardblωz/bardblq/2,u=/parenleftbigg1\nzq/integraldisplayz\n0u(t)dt+/integraldisplay∞\nzu(t)dt\ntq/parenrightbiggq/2\n≤(1+bq/2(u))2/q/parenleftbigg1\nzq/integraldisplayz\n0u(t)dt/parenrightbigg2/q\n.\nTherefore, CΘis bounded above by a multiple of,\nsup\nz>1/parenleftbigg1\nzq/integraldisplayz\n0u(t)dt/parenrightbigg2/q/parenleftbigg1\nzp/integraldisplayz\n0v(t)dt+/integraldisplay∞\nzv(t)dt\ntp/parenrightbigg−2/p\n.22 JAVAD RASTEGARI AND GORD SINNAMON\nUsingw(t) =tp−2v(1/t) and letting z= 1/xwe see that the last expression is equal\nto the square of (4.9) and therefore CΘ<∞. Proposition 4.2 shows that Part (i)\nof Theorem 4.7 holds. For any pandq, Part (i) implies Part (ii).\nFor the converse, still in the case q >2, suppose the Fourier inequality,\n/bardblˆf/bardblΓq(u)≤C/bardblf/bardblΓp(w), f∈L1(T).\nholds. That is, Part (ii) of Theorem 4.7 holds. Then (1.4) shows that P art (iii) also\nholds, so we may apply Theorem 3.7. Since the identity operator is in A, we have,\nsup\nz>1/bardblz−2χ(0,z)/bardblq/2,u\n/bardblωz/bardblp/2,v≤sup\nz>1/bardblωz/bardblq/2,u\n/bardblωz/bardblp/2,v≤sup\nz>1sup\nA∈A/bardblAωz/bardblq/2,u\n/bardblωz/bardblp/2,v<∞.\nUsingw(t) =tp−2v(1/t) and letting z= 1/x, we see that (4.9) is finite. This\ncompletes the proof. /square\nOne consequence of this theorem is an analogue for the Fourier coe fficient map\nof Theorem 2 in [3], a result for the Fourier transform. In the origina l result, u\nwas assumed to be decreasing. In this analogue we have weakened t his condition;\na decreasing function is in both Bq/2andB1,∞.\nTheorem 4.9. Letuandwbe weight functions on (0,∞).\n(i)Suppose 1< p≤q <∞,q≥2, andw∈Bp. Also suppose that u∈Bq/2\nifq >2, andu∈B1,∞ifq= 2. If\n(4.10) sup\n0<x<1x/parenleftbigg/integraldisplay1/x\n0u(t)dt/parenrightbigg1/q/parenleftbigg/integraldisplayx\n0w(t)dt/parenrightbigg−1/p\n<∞\nthen there is a C >0such that\n(4.11) /bardblˆf/bardblΛq(u)≤C/bardblf/bardblΛp(w), f∈L1(T).\n(ii)Conversely, if (4.11) is satisfied for any weight functions uandwon(0,∞)\nand for1< p,q < ∞, then (4.10) holds.\nProof.As always, v(t) =tp−2w(1/t). For Part (i), since w∈Bp,/bardbl · /bardblΛp(w)and\n/bardbl·/bardblΓp(w)are equivalent norms. Therefore (4.11) is equivalent to Theorem 4.7 (iii).\nA routine calculation using w∈Bpshows that (4.10) is equivalent to the finiteness\nof (4.9). Now Proposition 4.8 completes the proof of Part (i).\nFor part (ii), apply Corollary 3.8, taking Ato be the identity operator, to get,\nsup\nz>1/bardblωz/bardblq/2,u\n/bardblχ(0,1/z)/bardblp/2,w<∞.\nSincez−2χ(0,z)≤ωz, this implies\nsup\nz>1/parenleftbigg\nz−q/integraldisplayz\n0u(t)dt/parenrightbiggq/2/parenleftBigg/integraldisplay1/z\n0w(t)dt/parenrightBigg−2/p\n<∞.\nReplacing zby 1/xand taking square roots proves (4.10). /square\nAs afinal result for this section we showthe sufficiency, for the Fou rier coefficient\nmap on Lorentz spaces, of a weight condition analogous to one used in Theorem 1\nof [3], a result for the Fourier transform on Lebesgue spaces.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 23\nTheorem 4.10. Let1< p≤q <∞and2≤q. Assume uandware weight\nfunctions on (0,∞). If\nsup\n0<x<1/parenleftbigg/integraldisplay1/x\n0uo(t)dt/parenrightbigg1/q/parenleftbigg/integraldisplayx\n0w(t)1−p′dt/parenrightbigg1/p′\n<∞\nthen there exists C >0such that\n/bardblˆf/bardblΛq(u)≤C/bardblf/bardblΛp(w)\nfor allf∈L1(T).\nProof.Letσ(t) =tq−2uo(1/t) so that\n/integraldisplay1/x\n0uo(t)dt=/integraldisplay∞\nxσ(t)dt\ntq\nand hence,\nsup\n0<x<1/parenleftbigg/integraldisplay1/x\n0uo(t)dt/parenrightbigg1/q/parenleftbigg\nx−q/integraldisplayx\n0σ(t)dt+/integraldisplay∞\nxσ(t)dt\ntq/parenrightbigg−1/q\n≤1.\nIn view of (4.7) we may apply Propositions 4.4(iii) and 4.2, in the case p=qand\nw=σ, to see that\n/bardblˆf/bardblΓq(u)≤¯C/bardblf/bardblΓq(σ), f∈L1(T),\nfor some ¯C <∞.\nLetBbe the supremum in the hypothesis of the theorem. Then,\nsup\n0<x<1/parenleftbigg/integraldisplay1\nxσ(t)dt\ntq/parenrightbigg1/q/parenleftbigg/integraldisplayx\n0w(t)1−p′dt/parenrightbigg1/p′\n≤B <∞.\nBy Theorem 1 in [7], there exists a finite constant csuch that the weighted Hardy\ninequality,\n/parenleftbigg/integraldisplay1\n0/parenleftbigg1\nt/integraldisplayt\n0g(s)ds/parenrightbiggq\nσ(t)dt/parenrightbigg1/q\n≤c/parenleftbigg/integraldisplay1\n0g(t)pw(t)dt/parenrightbigg1/p\n,\nholds for all g≥0.\nIff∈L1(T) thenf∗is zero on (1 ,∞). Therefore,\n/bardblf/bardblq\nΓq(σ)=/integraldisplay1\n0/parenleftbigg1\nt/integraldisplayt\n0f∗(s)ds/parenrightbiggq\nσ(t)dt+/parenleftbigg/integraldisplay1\n0f∗(s)ds/parenrightbiggq/integraldisplay∞\n1σ(t)dt\ntq.\nThe weighted Hardy inequality above, with g=f∗, shows that the first term is\nbounded above by,\ncq/parenleftbigg/integraldisplay1\n0f∗(t)pw(t)dt/parenrightbiggq/p\n=cq/bardblf/bardblq\nΛp(w).\nH¨ older’s inequality shows that the second term is bounded above by\n/parenleftbigg/integraldisplay1\n0f∗(t)pw(t)dt/parenrightbiggq/p/parenleftbigg/integraldisplay1\n0w(t)1−p′dt/parenrightbiggq/p′/integraldisplay∞\n1σ(t)dt\ntq≤Bq/bardblf/bardblq\nΛp(w).\nPutting these together, we have,\n/bardblˆf/bardblΛq(u)≤ /bardblˆf/bardblΓq(u)≤¯C/bardblf/bardblΓq(σ)≤(cq+Bq)1/q¯C/bardblf/bardblΛq(w).\nThis completes the proof. /square24 JAVAD RASTEGARI AND GORD SINNAMON\n5.Applications\nThe space LlogLconsists of functions fdefined on [0 ,1] for which the integral\nof|f|log(2+|f|) is finite. It is well known that the space coincides with the Lorentz\nspace Λ 1(1−log(t)) = Γ 1(1). See, for example, Corollary 10.2 in [5].\nTakingp= 1 and w= 1 in Proposition 4.4(iii) and Theorem 4.6 gives a descrip-\ntion of spaces that contain the Fourier series of all functions in LlogL.\nTheorem 5.1. Suppose 2≤q <∞anduis a weight. If\n(5.1) sup\nz>1z\n(1+logz)qsup\ny>z1\ny/integraldisplayy\n0u(t)dt <∞\nthenF:LlogL→Γq(u)and hence F:LlogL→Λq(u). When q= 2, condition\n(5.1) is also necessary.\nRemark 5.2. Ifuis decreasing, or satisfies the weaker condition u∈B1,∞then\nsupy>z1\ny/integraltexty\n0u(t)dtmay be replaced by1\nz/integraltextz\n0u(t)dtin the previous theorem.\nButLlogLis only one of a large class of Lorentz spaces known as Lorentz-\nZygmund spaces. One can define Lorentz-Zygmund spaces for fu nctions on any\nσ-finite measure space by letting,\n/bardblf/bardblLr,p(logL)α=/braceleftBigg/parenleftbig/integraltext∞\n0/bracketleftbig\nt1/r(1+|logt|)αf∗(t)/bracketrightbigpdt\nt/parenrightbig1/p,0< p <∞,\nsup0<t<∞t1/r(1+|logt|)αf∗(t), p =∞,\nand setting Lr,p(logL)α={f:/bardblf/bardblLr,p(logL)α<∞}. When the underlying mea-\nsure is Lebesgue measure on [0 ,1] we write Lr,p(logL)αand when the underlying\nmeasure is counting measure on Zwe write ℓr,p(logℓ)α.\nThe Lorentz-Zygmund spaces were introduced and studied in [5]. T hey are\nspecialcasesofthe LorentzΛ p(w)-spaces. Specifically, Lr,p(logL)α= Λp(w), where\nw(t) =tp/r−1(1−logt)αpχ(0,1)(t),\nandℓr,p(logℓ)α= Λp(u), where\nu(t) =tp/r−1(1+|logt|)αpχ(0,∞)(t).\nInterpolationtheoryprovidesapowerfulapproachtofindingcon ditionsonLorentz-\nZygmund indices that are sufficient to imply boundedness of the Four ier series map\nbetween Lorentz-Zygmund spaces. This is done in [5] and elsewher e. For example,\nif 1< p <∞,β∈R, and 1< r <2, the method of Remark 4.3 may be applied\nwithu(t) =tp/r′−1(1+|log(t)|)qβto show that the Fourier transform is bounded\nfromLr,p(logL)βtoℓr′,p(logℓ)β. We omit the details.\nOur next result complements these by providing necessary conditio ns.\nObserve that Lr,p(logL)α/\\e}atio\\slash={0}if and only if tp/r−1(1−logt)αis integrable\nnear zero. To avoid the trivial case we assume that\n(5.2) r <∞,or/braceleftbig\nr=∞, p <∞, αp <−1/bracerightbig\n,or/braceleftbig\nr=∞, p=∞, α≤0/bracerightbig\n.\nTheorem 5.3. Suppose p,q∈(0,∞),r,s∈(0,∞],α,β∈(−∞,∞)and (5.2)\nholds. If F:Lr,p(logL)α→ℓs,q(logℓ)βthen either s >2, ors= 2andβ≤0.\nAlso, either 1/r+1/s <1, or1/r+1/s= 1andβ≤α.FOURIER SERIES IN WEIGHTED LORENTZ SPACES 25\nProof.The hypothesis may be stated as, /bardblˆf/bardblΛq(u)≤C/bardblf/bardblΛp(w)forf∈L1(T),\nwhere\nw(t) =tp/r−1(1−logt)αpχ(0,1)(t) and u(t) =tq/s−1(1+|logt|)βqχ(0,∞)(t).\nSo Corollary 3.8 implies,\n(5.3) sup\nz>1supA∈A/bardblAωz/bardblq/2,u/parenleftBig/integraltext1/z\n0w(t)dt/parenrightBig2/p<∞.\nSince (??holds, the denominator is finite for each z >1 and it follows that the\nnumerator is finite as well. In particular, lim y→∞/bardblAyωz/bardblq/2,u<∞where, for\ny > z,Ayis the averaging operator based on the single interval (0 ,y). For this\noperator,\nAyωz(t) =1\ny/parenleftbigg2\nz−1\ny/parenrightbigg\nχ(0,y)(t)+1\nt2χ(y,∞)(t)≥1\nyzχ(0,y)(t),\nso\nlim\ny→∞/parenleftbigg1\nyq/2/integraldisplayy\n0tq/s−1(1+|logt|)βqdt/parenrightbigg2/q\n<∞.\nFrom this we conclude that either s >2, ors= 2 and β≤0. This proves the first\nconclusion of the theorem.\nFor the second conclusion, take Ato be the identity operator in (5.3). Since\nz−2χ(0,z)(t)≤ωz(t),\nsup\nz>11\nz2/parenleftbigg/integraldisplayz\n0tq/s−1(1+|logt|)βqdt/parenrightbigg2/q/parenleftBigg/integraldisplay1/z\n0tp/r−1(1−logt)αpdt/parenrightBigg−2/p\n<∞.\nBut an easy estimate gives,\nd\ndt(tq/s(1+|logt|)βq)≤tq/s−1(1+|logt|)βq(q/s+|β|q)\nso\n(q/s+|β|q)/integraldisplayz\n0tq/s−1(1+|logt|)βqdt≥zq/s(1+|logz|)βq.\nIt follows that\nlim\nz→∞z−2/parenleftbig\nzq/s(1+logz)βq/parenrightbig2/q\n/parenleftBig/integraltext1/z\n0tp/r−1(1−logt)αpdt/parenrightBig2/p<∞.\nTaking the 2 /pexponent outside the limit and applying L’Hospital’s rule, gives,\nlim\nz→∞z−p+p/s−1(1+logz)βp(−p+p/s+βp/(1+logz))\n−z−2z1−p/r(1+logz)αp<∞.\nNow it follows that either 1 /r+1/s <1, or 1/r+1/s= 1 and β≤α. /square26 JAVAD RASTEGARI AND GORD SINNAMON\nReferences\n[1] M. Ari˜ no and B. Muckenhoupt, Maximal functions on class ical Lorentz spaces and Hardy’s\ninequality with weights for non-increasing functions. Tra ns. Amer. Math. Soc., 320(1990),\n727–735.\n[2] J. J. Benedetto and H. P. Heinig, Weighted Hardy spaces an d the Laplace transform, Har-\nmonic analysis (Cortona, 1982), 240–277, Lecture Notes in M ath.,992, Springer, Berlin,\n1983.\n[3] J. J. Benedetto and H. P. Heinig, Weighted Fourier inequa lities: new proofs and generaliza-\ntions. J. Fourier Anal. Appl., 9(2003), 1–37.\n[4] J. J. Benedetto, H. P. Heinig and R. Johnson, Weighted Har dy spaces and the Laplace\ntransform II, Math. Nachr. 132(1987), 29–55.\n[5] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Di ssertationes Math. (Rozprawy\nMat.),175(1980), 1–67.\n[6] C. Bennett and R. Sharpley, Interpolation of operators . Pure and Applied Mathematics 129,\nAcademic Press Inc., Boston, 1988.\n[7] J. S. Bradley, Hardy inequalities with mixed norms, Cana d. Math. Bull. 21(1978), 405–408.\n[8] M.Carro, A.Garc´ ıa del Amoand J. Soria, Weak-Type Weigh ts and Normable Lorentz Spaces.\nProc. Amer. Math. Soc., 124(1996), 849–857.\n[9] M.Jodeit, Jr., and A.Torchinsky, Inequalities forFour iertransforms.Studia Math., 37(1971),\n245–276.\n[10] L. Maligranda, Weighted inequalities for monotone fun ctions. Collect. Math., 48(1997), 687–\n700.\n[11] L. Maligranda, Weighted inequalities for quasi-monot one functions. J. London Math. Soc.,\n57(1998), 363–370.\n[12] M. Mastylo and G. Sinnamon, A Calder´ on couple of down sp aces. J. Funct. Anal., 240(2006),\n192–225.\n[13] E. Sawyer, Boundedness of classical operators on class ical Lorentz spaces. Studia Math.,\n96(1990), 145–158.\n[14] G. Sinnamon, The level function in rearrangement invar iant spaces. Publ. Mat., 45(2001),\n175–198.\n[15] G. Sinnamon, Embeddings of concave functions and duals of Lorentz spaces. Publ. Mat.,\n46(2002), 489–515.\n[16] G. Sinnamon, The Fourier transform in weighted Lorentz spaces. Publ. Mat., 47(2003), 3–29.\n[17] G. Sinnamon, Transferring monotonicity in weighted no rm inequalities. Collect. Math.,\n54(2003), 181–216.\n[18] G. Sinnamon, Monotonicity in Banach function spaces. N AFSA 8Nonlinear analysis, function\nspaces and applications. Vol. 8, 204–240, Czech. Acad. Sci. , Prague, 2007.\n[19] G. Sinnamon, Fourier inequalities and a new Lorentz spa ce. Banach and function spaces II,\n145–155, Yokohama Publ., Yokohama, 2008.\nDepartment of Mathematics, University of Western Ontario, L ondon, Canada\nE-mail address :jrastega@uwo.ca\nDepartment of Mathematics, University of Western Ontario, L ondon, Canada\nE-mail address :sinnamon@uwo.ca"    },    {        "title": "2108.04702v2.Superradiant_instability_of_the_Kerr_like_black_hole_in_Einstein_bumblebee_gravity.pdf",        "content": "Superradiant instability of the Kerr-like black hole in\nEinstein-bumblebee gravity\nRui Jiang,1Rui-Hui Lin,1,\u0003and Xiang-Hua Zhai1,y\n1Division of Mathematics and Theoretical Physics,\nShanghai Normal University, 100 Guilin Road, Shanghai 200234, China\nAbstract\nAn exact Kerr-like solution has been obtained recently in Einstein-bumblebee gravity model\nwhere Lorentz symmetry is spontaneously broken. In this paper, we investigate the superradiant\ninstability of the Kerr-like black hole under the perturbation of a massive scalar \feld. We \fnd the\nLorentz breaking parameter Ldoes not a\u000bect the superradiance regime or the regime of the bound\nstates. However, since Lappears in the metric and its e\u000bect cannot be erased by rede\fning the\nrotation parameter ~ a=p\n1 +La, it indeed a\u000bects the bound state spectrum and the superradiance.\nWe calculate the bound state spectrum via the continued-fraction method and show the in\ruence of\nLon the maximum binding energy and the damping rate. The superradiant instability could occur\nsince the superradiance condition and the bound state condition could be both satis\fed. Compared\nwith Kerr black hole, the nature of the superradiant instability of this black hole depends non-\nmonotonously not only on the rotation parameter of the black hole ~ aand the product of the\nblack hole mass Mand the \feld mass \u0016, but also on the Lorentz breaking parameter L. Through\nthe Monte Carlo method, we \fnd that for l=m= 1 state the most unstable mode occurs at\nL=\u00000:79637, ~a=M = 0:99884 and M\u0016= 0:43920, with the maximum growth rate of the \feld\n!IM= 1:676\u000210\u00006, which is about 10 times of that in Kerr black hole.\n\u0003linrh@shnu.edu.cn\nyzhaixh@shnu.edu.cn\n1arXiv:2108.04702v2  [gr-qc]  5 Nov 2021I. INTRODUCTION\nLorentz invariance is one of the most important symmetries in General Relativity (GR)\nthat works well in describing gravitation at the classical level. However, Lorentz invariance\nmay not be an exact symmetry at all energy scales[1]. It is shown that Lorentz invariance\nwill be strongly violated at the Planck scale ( \u00181019GeV) in some theories of quantum\ngravity(QG)[2]. Therefore, Lorentz violation in gravitation theories is worth studying in\nanticipation of a deeper understanding of nature. And hence, various theories involving\nLorentz violation, such as ghost condensation[3], warped brane world, and Einstein-aether\ntheory[4{6], have been proposed and investigated. Moreover, the possibility of spontaneous\nLorentz symmetry breaking(LSB) was considered. In 1989, Kosteleck\u0013 y and Samuel pre-\nsented a potential mechanism for the Lorentz breaking that may be generic in many string\ntheories[7, 8]. The main idea is to \fnd a model containing the essential features of the\ne\u000bective action that would arise in a string theory with tensor-induced breaking. One of\nthe simplest ways to implement this idea is the so-called Einstein-bumblebee gravity. In\nthis theory, a vector \feld ruled by a potential acquires a non-zero vacuum expectation value\n(VEV). The vector \feld is then frozen at its VEV, which chooses a preferred spacetime\ndirection in the local frames and spontaneously breaks the Lorentz symmetry.\nSince bumblebee gravity can be viewed as an endeavor to explore QG, it is then important\nto search for black hole solutions in this theory in that the strong gravitation environment\naround a black hole may provide information about QG. Casana et al obtained an exact\nSchwarzschild-like solution in 2018[9]. Then, the light de\rection[10{12] and quasinormal\nmodes[13] of this black hole have been addressed. Moreover, spherically symmetric black hole\nsolutions with cosmological constant[14], global monopole[15], or Einstein-Gauss-Bonnet\nterm[16] have also been found. As a more practical scenario, axial symmetry in Einstein-\nbumblebee gravity is investigated and an exact Kerr-like black hole is obtained by Ding et\nal[17]. The studies related to this solution have been extended to the e\u000bects of the matter\nand light around it[18{20]. Furthermore, a Kerr-Sen-like black hole has also been found[21].\nOn the other hand, the study of black hole stability dates back to 1957. Regge and\nWheeler proved that the Schwarzschild black hole is stable under small perturbations of the\nmetric[22, 23]. In the following studies, the propagation of the Klein-Gordon \feld around\na black hole becomes an important tool for investigating the stability of the corresponding\n2black hole. For a rotating black hole, the perturbing bosons may extract rotational energy\nfrom the black hole through the superradiance mechanism[24{26]. For this to occur, it is\nrequired that the frequency !of the wave is smaller than a critical value determined by\nthe azimuthal number mof the perturbation and the angular velocity \n Hof the black hole\nhorizon\n!<!c\u0011m\nH: (1)\nIf this superradiant condition is satis\fed, the scattered wave will be ampli\fed, where the\nadditional energy comes from the black hole. Furthermore, if the superradiant process\noccurs repeatedly, the rotational energy of the black hole may be extracted for multiple\ntimes, triggering the superradiant instability of the black hole. This can be ful\flled with\nthe \\black hole bomb\" idea postulated by Press and Teukolsky[27], by arranging a special\nmirror surrounding the black hole to re\rect the wave between the mirror and the black\nhole back and forth. Superradiant instability triggered by a mirror-like boundary condition\nhas been studied extensively[28{33]. Besides these arranged walls to re\rect the scattered\nwaves, a natural wall may exist if the spacetime is asymptotically non-\rat. For example, an\nanti-de Sitter spacetime is able to provide such re\rection and help trigger the superradiant\ninstability[34{41]. Apart from being re\rected by walls, scattered waves may be pulled back\nto the black hole if the perturbing boson has nonzero rest mass. The massive waves can nat-\nurally form bound states in the gravitational system and hence may trigger the superradiant\ninstability[42{48].\nIn this paper, we would like to investigate the stability of the Kerr-like black hole in the\nbumblebee gravity. We intend to study the bound states of the massive scalar \feld in the\nbackground of the Kerr-like black hole and hence the superradiant instability of the black\nhole. We obtain the frequency regime of the bound states and compute the bound state\nspectrum. When superradiant instability occurs, the scalar \feld will grow over time. In\nthis case, there may be such a speci\fc frequency of the bound state that this process will\nprogress most rapidly. For Kerr spacetime, the maximum growth rate of the scalar \feld was\nfound to be !IM= 1:72\u000210\u00007[49]. On this basis, we will look into the in\ruence of the\nLSB parameter Lon the maximum growth rate.\nThe paper is organized as follows. In Sec.II, we brie\ry review the Kerr-like black hole\nsolution in the bumblebee gravity. Then, we give the necessary conditions for the bound\nstate in Sec.III. The bound state spectrum is computed by using the continued-fraction\n3method in Sec.IV. In Sec.V, we discuss the superradiant instability and further study the\nin\ruence of the LSB parameter Lon the maximum growth rate. The last section is devoted\nto conclusion and discussions.\nThroughout the paper, we follow the metric convention ( \u0000;+;+;+) and the units G=\n~=c= 1.\nII. THE KERR-LIKE BLACK HOLE SOLUTION IN THE BUMBLEBEE GRAV-\nITY\nUnder the framework of the bumblebee gravity theory, the spontaneous LSB is induced\nby a vector B\u0016that has a non-zero VEV. The dynamics of the gravitational \feld will also be\na\u000bected if it couples to B\u0016. Therefore, the action describing the LSB gravitation includes the\nfree parts of both the gravitational \feld and the bumblebee \feld, the potential of B\u0016that\nleads to a non-zero VEV, and the coupling between B\u0016and the geometry of the spacetime.\nBased on the Riemannian description of the spacetime manifold, such an action can be\nwritten as[50]\nS=Z\nd4p\u0000g\u00021\n16\u0019(R+\u0018B\u0016B\u0017R\u0016\u0017)\u00001\n4B\u0016\u0017B\u0016\u0017\u0000V\u0000\nB\u0016B\u0016\u0006b2\u0001\u0003\n; (2)\nwhere\u0018is the coupling constant and the bumblebee \feld strength B\u0016\u0017is de\fned as\nB\u0016\u0017=@\u0016B\u0017\u0000@\u0017B\u0016: (3)\nThe potential Vtakes a minimum at B\u0016B\u0016\u0006b2= 0 with a real positive b2, and therefore\ntriggers the LSB of the bumblebee \feld. With V= 0 andV0= 0, the \feld B\u0016is considered\nto be frozen at its VEV, i.e. hB\u0016i=b\u0016, whereb\u0016b\u0016=b2. And the \feld equation is then\n0 =R\u0016\u0017\u0000\u0014b\u0016\u000bb\u000b\n\u0017+\u0014\n4g\u0016\u0017b\u000b\fb\u000b\f+\u0018b\u0016b\u000bR\u000b\u0017\n+\u0018b\u0017b\u000bR\u000b\u0016\u0000\u0018\n2g\u0016\u0017b\u000bb\fR\u000b\f\u0000\u0018\n2r\u000br\u0016(b\u000bb\u0017)\n\u0000\u0018\n2r\u000br\u0017(b\u000bb\u0016) +\u0018\n2r2(b\u0016b\u0017);(4)\nwhere\u0014= 8\u0019.\nFor the axially symmetric scenario, by assuming the bumblebee \feld to be a purely\nradial form of b\u0016= (0;b(r;\u0012);0;0), a Kerr-like black hole solution is found in Ref.[17]. In\n4the standard Boyer-Lindquist coordinates, this solution is written as\nds2=\u0000\u0012\n1\u00002Mr\n\u001a2\u0013\ndt2+\u001a2\n\u0001dr2+\u001a2d\u00122\n+Asin2\u0012\n\u001a2d'2\u00004Mrap\n1 +Lsin2\u0012\n\u001a2dtd';(5)\nwhere\n\u001a2=r2+ (1 +L)a2cos2\u0012;\n\u0001 =r2\u00002Mr\n1 +L+a2;\nA=\u0002\nr2+ (1 +L)a2\u00032\u0000\u0001 (1 +L)2a2sin2\u0012:(6)\nHereL=\u0018b2,Mis the ADM mass and ais the Boyer-Lindquist parameter. Eqs.(5) and(6)\ncan be written in a form very resemblant to the Kerr solution\nds2=\u0000\u0012\n1\u00002Mr\n~\u001a2\u0013\ndt2+~\u001a2\n~\u0001dr2+ ~\u001a2d\u00122\n+~Asin2\u0012\n~\u001a2d'2\u00004Mr~asin2\u0012\n~\u001a2dtd';(7)\nwhere\n~a=p\n1 +La;\n~\u001a2=r2+ ~a2cos2\u0012;\n~\u0001 =r2\u00002Mr+ ~a2\n1 +L;\n~A=\u0000\nr2+ ~a2\u00012\u0000(r2\u00002Mr+ ~a2)~a2sin2\u0012:(8)\nNote that Eq.(7) now looks exactly the same as the Kerr metric in form. The cost,\nhowever, is that ~\u0001 still involves explicitly the LSB parameter Lthat cannot be absorbed in\nthe new spin parameter ~ a. So, the e\u000bect of Lvia~\u0001 cannot be erased by rede\fning ~ a. Hence\nthe metric written in the form of Eqs.(7) and (8) is essentially di\u000berent from the Kerr metric\nas long as the purely radial LSB parameter Lis nonzero. The event horizons locate at\nr\u0006=M\u0006p\nM2\u0000(1 +L)a2=M\u0006p\nM2\u0000~a2; (9)\nwherer+andr\u0000correspond to outer and inner horizons, respectively. The condition for the\nblack hole to exist is then given by\np\n1 +La= ~a\u0014M: (10)\n5And the angular velocity of the outer horizon is\n\nH=ap\n1 +L\nr2\n++ (1 +L)a2=~a\nr2\n++ ~a2: (11)\nSince ~aplays a similar role to the Boyer-Lindquist parameter ain the Kerr metric, these\nphysical quantities (9)-(11) and the superradiant condition(1) also have the same forms via\n~aas the Kerr black hole, and do not rely on aorLseparately. In this sense, the e\u000bect of\nthe LSB parameter Lvia these measurable quantities is not trackable. However, as is seen\nin Eq.(8), the e\u000bect of Lon the metric via ~\u0001 cannot be erased. Therefore, we will proceed\nwith ~ainstead of the original Boyer-Lindquist parameter a, and will see in the following\nsections that the LSB parameter Lindeed a\u000bects the superradiance.\nIII. FREQUENCY REGIME OF THE BOUND STATES\nWe consider a scalar \feld \t with mass \u0016propagating in the background of the Kerr-like\nblack hole (5), which is described by the Klein-Gordon equation\n1p\u0000g@\u0016\u0000\ng\u0016\u0017p\u0000g@\u0017\t\u0001\n=\u00162\t: (12)\nWith the separation of \t,\n\t =Rlm(r)Slm(\u0012)eim\u001ee\u0000i!t; (13)\nEq.(12) can be decomposed into a radial part\n~\u0001d\ndr\u0012\n~\u0001dRlm\ndr\u0013\n+URlm= 0 (14)\nwith\nU=[!(r2+ ~a2)\u0000~am]2\n1 +L+~\u0001(2m~a!\u0000Alm\u0000~a2!2\u0000\u00162r2); (15)\nand an angular part\n1\nsin\u0012d\nd\u0012\u0012\nsin\u0012dSlm\nd\u0012\u0013\n+\u0014\nAlm+ ~a2\u0000\n!2\u0000\u00162\u0001\ncos2\u0012\u0000m2\nsin2\u0012\u0015\nSlm= 0; (16)\nwhereRlm's are radial functions, Slm's are spheroidal harmonic functions[51], and Alm's are\nthe angular eigenvalues that may be expanded as a power series\nAlm=l(l+ 1) +1X\nk=1fk\u0002\n~a2\u0000\n\u00162\u0000!2\u0001\u0003k(17)\n6if ~a2(\u00162\u0000!2)\u0014m2. In addition, eigenvalue Almcan also be computed numerically with\ncontinued-fraction method[52].\nThe radial equation(14) can be further written in the form\nd2eR\ndx2+eUeR= 0; (18)\nwhere\neR=p\nr2+ ~a2Rlm; (19)\nand the tortoise coordinate xis de\fned by\ndx\ndr=r2+ ~a2\n~\u0001(1 +L): (20)\nThe e\u000bective potential eUin Eq.(18) reads\neU=U(1 +L)2\n(r2+ ~a2)2\u0000r2~\u00012(1 +L)2\n(r2+ ~a2)4\u0000~\u0001(1 +L)\nr2+ ~a2\"\nr~\u0001(1 +L)\n(r2+ ~a2)2#0\n; (21)\nwhere the prime represents the derivative with respect to r.eUtends to (!2\u0000\u00162) (1 +L) at\nthe spatial in\fnity and ( !\u0000m\nH)2(1 +L) at the outer horizon. Therefore, Eq.(18) can be\ntreated asymptotically as a wave equation at the horizon or the spatial in\fnity. Since we are\ninterested in the bound state solutions, the wave towards the spatial in\fnity should decay\nexponentially and the wave at the horizon should be purely ingoing. Hence, the asymptotic\nstates of the wave are\nRlm\u00188\n><\n>:e\u0000i(!\u0000!c)p1+Lx; x!\u00001;\n1\nxe\u0000p\n(\u00162\u0000!2)(1+L)x; x!+1;(22)\nand the frequency !must satisfy\n!2<\u00162: (23)\nBesides, a trapping potential well outside the black hole is also required for bound states[53].\nWe then proceed to consider the existence of the trapping potential well. With a new radial\nfunction de\fned by\n \u0011~\u00011=2Rlm; (24)\nEq.(14) can be rewritten in the form of a Schr odinger-like wave equation\nd2 \ndr2+\u0002\n(1 +L)!2\u0000V\u0003\n= 0; (25)\n7where\n(1 +L)!2\u0000V=U(1 +L)2+M2\u0000~a2\n~\u00012(1 +L)2: (26)\nFor larger, the e\u000bective potential and its derivative with respect to rare\nV=\u00162(1 +L)\u00002(1 +L)M(2!2\u0000\u00162)\nr+O(1\nr2) (27)\nand\nV0=2(1 +L)M(2!2\u0000\u00162)\nr2+O(1\nr3); (28)\nrespectively. A trapping well exists if V0!0+asr!1 , which means that\n2!2\u0000\u00162>0: (29)\nTherefore, the frequency regime for the bound states is independent of LSB parameter L\nand is given by\n\u0016p\n2<!<\u0016: (30)\nIt is worth noting that in Ref.[54] the author obtained a di\u000berent result about the fre-\nquency regime due to an equation decomposition di\u000berent from Eq.(14) of this work. By\ncareful checking, we make sure our equation decomposition (Eqs.(14)-(16)) is in accordance\nwith the results in mathematical handbooks and can also be con\frmed by the recent work\nabout the quasinormal modes of this Kerr-like black hole[20]. We argue that the decompo-\nsition of the \feld equation (Eq.(3.3)) in Ref.[54] is erroneous in that it apparently lacks the\nhighly nontrivial angular eigenvalues Alm. Consequently, the incorrect equation decompo-\nsition in Ref.[54] leads to an incorrect asymptotic behavior of the radial function, which is\ninconsistent with Eq.(22) in this work and with other related work[55].\nIV. BOUND STATE SPECTRUM\nFor a given parameter Land a scalar \feld with mass \u0016, the boundary condition(22)\nsingles out a discrete set of bound state frequencies f!ng. In this section, we compute the\nbound state frequencies using the continued-fraction method. To this end, the asymptotic\nbehavior of the radial function is written as\nRlm(r!r+)\u0018(r\u0000r+)\u0000i\u001b(31)\n8and\nRlm(r!1 )\u0018r\u001f\u00001ekr; (32)\nwhere\n\u001b=r2\n++ ~a2\nr+\u0000r\u0000(!\u0000!c)p\n1 +L; (33)\n\u001f=M(\u00162\u00002!2)(1 +L)\nk; (34)\nand\nk=\u0000p\n(\u00162\u0000!2)(1 +L): (35)\nThe solution of radial equation can be expressed as\nRlm= (r\u0000r+)\u0000i\u001b(r\u0000r\u0000)\u001f\u00001+i\u001bekr1X\nn=0an(r\u0000r+\nr\u0000r\u0000)n(36)\nso that both Eqs.(31) and(32) can be accommodated. Substituting (36) into Eq.(14), one\nobtains a three-term recurrence relation for the coe\u000ecients an,\n\u000b0a1+\f0a0= 0; (37)\n\u000bnan+1+\fnan+\rnan\u00001= 0;n= 1;2;3\u0001\u0001\u0001; (38)\nwhere\n\u000bn=n2+ (c0+ 1)n+c0;\n\fn=\u00002n2+c1+c2;\n\rn=n2+c3n+c4:(39)\n9The constants c0;c1;c2;c3andc4are given by\nc0=1\u00002Mi!p\n1 +L\u0000i\n\u0011\u0010\n2M2!p\n1 +L\u0000~amp\n1 +L\u0011\n;\nc1=\u00002c0+ 2(M+ 2\u0011)k\u00002M!2(1 +L)\nk;\nc2=\u0000Alm(1 +L)\u0000(M+\u0011)2k2+ 2(M+\u0011)M!2(1 +L)\n\u00002Mi!p\n1 +L\u0000c0\u0014\n\u0000(M+\u0011)k+M!2(1 +L)\nk\n+ 1\u00002Mi!p\n1 +L\u0015\n;\nc3=\u00002Mi!p\n1 +L\u0000i\n\u0011\u0012\n2M2!p\n1 +L\u0000~amp\n1 +L\u0013\n\u00002M(\u00162\u00002!2)(1 +L)\nk;\nc4=(k+i!p\n1 +L)2M\u0014i\n\u0011\u0012\n2M2!p\n1 +L\u0000~amp\n1 +L\u0013\n+M(\u00162\u00002!2)(1 +L)\nk\u0015\n=k;(40)\nwhere\n\u0011=p\nM2\u0000~a2: (41)\nOne can see that the factor (1 + L) orp\n1 +Lappears many times in Eq.(40) and cannot\nbe absorbed in ~ a. This again roots in the fact that ~ aand ~\u0001 given in Eq.(8) have a di\u000berent\nrelation from their counterparts in the Kerr case and ~\u0001 involves explicitly the LSB parameter\nL. At last, the radial equation(14) involving ~\u0001 results in the multiple presence of Lin\nEq.(40), which shows that Lwill play a signi\fcant role in the bound state spectrum. It is\neasy to check that when L= 0, the expressions for \u000bn;\fnand\rnreduce to Kerr case given\nin Ref.[56].\nThe ratio of successive coe\u000ecients anis given by an in\fnite generalized continued fraction\nan+1\nan=\u0000\rn+1\n\fn+1\u0000\u000bn+1\rn+2\n\fn+2\u0000\u000bn+2\rn+3\n\fn+3\u0000\u0001\u0001\u0001: (42)\nSubstituting n= 0 into the above expression and comparing with Eq.(37), one can obtain\nthe characteristic equation\n\f0\u0000\u000b0\r1\n\f1\u0000\u000b1\r2\n\f2\u0000\u000b2\r3\n\f3\u0000\u0001\u0001\u0001= 0: (43)\nThis equation is only satis\fed for particular values of !corresponding to the frequencies of\nthe bound state modes. We compute these frequencies numerically by minimizing the abso-\nlute value of the left side of Eq.(43). For a given bound state mode, the resulting frequency !\n100.2 0.4 0.6 0.8 1.0 1.2 1.40.900.951.00ωR\n/μ\nL=-0.2\nL=-0.1\nL=0\nL=0.1\nL=0.2\n0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.35-0.30-0.25-0.20-0.15-0.10-0.050.00\nMμωI/μFIG. 1. Bound state frequencies versus the mass coupling M\u0016for di\u000berent parameter L, where we\nhave setl=m= 1 and ~a=M = 0:9.\nis generally a complex number. The real part of it ( !R) represents the oscillating frequency\nof the \feld, while the imaginary part ( !I) indicates the evolution of the wave amplitude.\n!Ibeing negative or positive means a decaying or growing scalar \feld, respectively. And\nthe magnitude of | !I| represents the corresponding damping or growth rate. Among the\ngrowing modes of the \feld, if exist, the ones with l=m= 1 grow most rapidly. Therefore,\nwe will focus on the behavior of these modes in the following discussion.\nFig.1 shows how the bound state frequency of the state with n= 0;l=m= 1 and\n~a=M = 0:9 vary with the mass coupling M\u0016 and LSB parameter L. The upper panel\nillustrates the real part of the frequency ( !R). One immediately observes that !R=\u0016has a\nminimum at some M\u0016, which corresponds to the maximum binding energy. It is obvious\nthat the maximum binding energy increases with L. For example, when L= 0 (Kerr case),\nthe maximum binding energy is around 13 :1% of the rest mass energy. It becomes \u001811:97%\nforL=\u00000:2 and\u001814:08% forL= 0:2. The lower panel illustrates the imaginary part\nof the frequency ( !I). It shows that the damping rate increases with L. Moreover, !Iis\n11actually positive at low couplings, which we cannot discern in this \fgure but is just the\nregime of the superradiant instability we will analyze in the following section.\nV. SUPERRADIANT INSTABILITY\nIn this section, we focus on unstable modes in which the imaginary part of the frequency\nis positive ( !I>0). This happens when Eqs.(1) and (30) are both satis\fed.\nFig.2 and Fig.3 show the detailed behaviors of the positive !Iversus the mass coupling\nM\u0016for di\u000berent Land ~a. In Fig.2, we take ~ a=M = 0:9 and the curves correspond to di\u000berent\nL. And in Fig.3, we take L=\u00000:5 and the curves correspond to di\u000berent ~ a. We also plot\nthe!Rcompared to the critical frequency !cof superradiance in this regime. Recall that\nsuperradiance occurs if and only if ! < !c. It is obvious from the two plots that !Iis\npositive when !R< !c. It is then con\frmed that the growth of the \feld comes from the\nsuperradiance mechanism.\nL=-0.9\nL=-0.8\nL=-0.7\nL=-0.5\nL=0\n05.×10-81.×10-71.5×10-72.×10-7\u0001IM\n0.15 0.20 0.25 0.300\n-0.05\n-0.1\n-0.15\nM\u0002(\n\u0001R-\n\u0001c)M\nFIG. 2. Bound state spectrum of a scalar \feld ( l=m= 1) for di\u000berent values of Lwhen ~a=M = 0:9.\nOne can also see from the two plots that each curve of !Ihas a peak and there is a highest\npeak of!Icorresponding to L=\u00000:8 in Fig.2 and ~ a=M = 0:998 in Fig.3. That is, for a given\n12a/M=0.9\na/M=0.95\na/M=0.99\na/M=0.998\na/M=0.99902.×10-74.×10-76.×10-78.×10-7ωIM\n0.20 0.25 0.30 0.35 0.40 0.45 0.500-0.05\n-0.1\n-0.15\n-0.2\n-0.25\n-0.3\nMμ(ωR-ωc)MFIG. 3. Bound state spectrum of a scalar \feld ( l=m= 1) for di\u000berent values of ~ awhenL=\u00000:5.\n~aorL, there may be a certain set of parameters that leads to a maximum growth rate !\u0003\nI.\nThen, taking di\u000berent L, we search for the !\u0003\nI, as well as the corresponding set of parameters\n~a\u0003andM\u0016\u0003by Monte Carlo method. The results are shown in Table I where the parameters\nwith an asterisk represent their values corresponding to the maximum growth rate. It is\nclear that the result for Kerr black hole, the maximum growth rate !\u0003\nIM= 1:72440\u000210\u00007\nata\u0003=M= 0:99663 and M\u0016\u0003= 0:448901[49], is recovered when L= 0. One can see that\nasLincreases, the rotation speed ~ a\u0003required to reach the maximum growth rate decreases,\nwhile the mass coupling M\u0016\u0003increases \frst and then decreases although it does not change\nsigni\fcantly. Most importantly, the maximum growth rate !\u0003\nIdoes not vary monotonously.\nIt is suspected that there may exist an overall maximum growth rate when all three of the\nparameters of L, ~aandM\u0016are taken into account, which corresponds to the most unstable\nmode of the \feld. Using the Monte Carlo method, we \fnd the most unstable mode occurs\natL=\u00000:79637, ~a=M = 0:99884 andM\u0016= 0:43920, with !IM= 1:676\u000210\u00006. It is about\n10 times of the maximum growth rate in Kerr black hole.\n13TABLE I. Maximum instability growth rate of di\u000berent Lwithl=m= 1.\nL ~a\u0003=MM\u0016\u0003!\u0003\nIM\n-0.95 0.99962 0.425919 5.28603\u000210\u00007\n-0.9 0.99932 0.432026 1.20933\u000210\u00006\n-0.8 0.99886 0.439019 1.67536\u000210\u00006\n-0.7 0.99848 0.442775 1.48849\u000210\u00006\n-0.5 0.99786 0.446469 8.58337\u000210\u00007\n-0.4 0.99759 0.447413 6.23684\u000210\u00007\n-0.3 0.99733 0.448040 4.50565\u000210\u00007\n-0.2 0.99708 0.448460 3.25704\u000210\u00007\n-0.1 0.99685 0.448736 2.36348\u000210\u00007\n00.99663 0.448901 1.72440\u000210\u00007\n0.10.99642 0.448995 1.26593\u000210\u00007\n0.20.99622 0.449046 9.35381\u000210\u00008\n0.30.99603 0.449049 6.95651\u000210\u00008\n0.40.99584 0.449008 5.20677\u000210\u00008\n0.50.99567 0.448966 3.92135\u000210\u00008\nVI. CONCLUSION AND DISCUSSIONS\nIn this paper, we have looked into the instability of the Kerr-like black hole in the Einstein-\nbumblebee gravity by considering the perturbation of a massive scalar \feld. Using a suitable\nrede\fnition of the spin parameter ~ a=p\n1 +La, we rewrite the metric into a form most\nresembling to the Kerr solution in GR. In this form, ~ aplays a similar role to the Boyer-\nLindquist parameter ain Kerr metric. The e\u000bect of the LSB parameter Lvia the physical\nquantities such as the horizon radii or angular velocity cannot be separated from ~ aand\nhence is not trackable. It follows that the frequency regimes of superradiance and bound\nstates are una\u000bected by L. However, the e\u000bect of Lon the metric cannot be fully erased\nby rede\fning ~ ain thatLstill appears explicitly in ~\u0001. Our analysis shows that the bound\nstate spectrum and superradiance are indeed a\u000bected by L. In particular, we calculate the\nbound state spectrum via the continued-fraction method and show the in\ruence of LSB\n14parameter Lon the maximum binding energy and the damping rate. The superradiant\ninstability could occur for this black hole since the superradiance condition and the bound\nstate condition could be both satis\fed. We \fnd the growth rate of the \feld does not depend\nmonotonously on L. In view of a parameter space spanned by the LSB parameter L, the\nrotation parameter ~ aand the mass coupling M\u0016, there exists an overall maximum growth\nrate!I, which corresponds to the most unstable mode of the \feld. By Monte Carlo method,\nwe have found that the most unstable mode occurs at L=\u00000:79637, ~a=M = 0:99884 and\nM\u0016= 0:43920 forl=m= 1 state, with !IM= 1:676\u000210\u00006. It is about 10 times of that\nin Kerr black hole.\nRecent studies show that when nonlinear e\u000bects are taken into account, the instability\nof Kerr black hole will result in a rotating black hole embedded in a massive bosonic \feld\nthat orbits around the black hole[57{59]. 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Pretorius, Physical Review Letters 119 (2017), 10.1103/phys-\nrevlett.119.041101.\n18"    },    {        "title": "1902.10241v3.A_fully_implicit__scalable__conservative_nonlinear_relativistic_Fokker_Planck_0D_2P_solver_for_runaway_electron.pdf",        "content": "A fully implicit, scalable, conservative nonlinear relativistic\nFokker-Planck 0D-2P solver for runaway electrons\nDon Daniel\u0003, William T. Taitano, Luis Chacón\nLos Alamos National Laboratory, New Mexico, USA\nAbstract\nUpon application of a sufficiently strong electric field, electrons break away from thermal equilibrium\nand approach relativistic speeds. These highly energetic ‘runaway’ electrons ( \u0018MeV) play a significant\nrole in tokamak disruption physics, and therefore their accurate understanding is essential to develop\nreliable mitigation strategies. For this purpose, we have developed a fully implicit solver for the 0D-2P\n(i.e., including two momenta coordinates) relativistic nonlinear Fokker-Planck equation (rFP). As in\nearlier implicit rFP studies (NORSE, CQL3D), electron-ion interactions are modeled using the Lorentz\noperator, and synchrotron damping using the Abraham-Lorentz-Dirac reaction term. However, our\nimplementation improves on these earlier studies by 1) ensuring exact conservation properties for electron\ncollisions, 2) strictly preserving positivity, and 3) being scalable algorithmically and in parallel. Key to\nour proposed approach is an efficient multigrid preconditioner for the linearized rFP equation, a multigrid\nelliptic solver for the Braams-Karney potentials [Braams and Karney, Phys. Rev. Lett. 59, 16 (1987)],\nand a novel adaptive technique to determine the associated boundary values. We verify the accuracy\nand efficiency of the proposed scheme with numerical results ranging from small electric-field electrical\nconductivity measurements to the accurate reproduction of runaway tail dynamics when strong electric\nfields are applied.\n1 Introduction\nRelativistic Coulomb collisions are modeled using an extended version of the Landau-Fokker-Planck collision\noperator [1]. Similarly to its non-relativistic counterpart [2], the operator assumes small-angle collisions,\nis well-posed, and features strict conservation of total particle number, total momentum, and total energy\n[3]. However, its accurate numerical solution is difficult because of its integro-differential formulation, which\nintroduces scalability and discretization challenges. In this study, we propose a finite-difference-based conser-\nvative, parallel, fully implicit solver for the 0D-2P relativistic Fokker-Planck (rFP) electron-electron collision\noperator. This work builds and improves on earlier rFP algorithms as implemented in the NORSE [4] and\nCQL3D [5, 6] codes.\nThe solver proposed in this study is primarily designed to simulate runaway electrons produced by a\nlarge loop voltage. In tokamaks, the loop voltage induced during disruptions can produce a large amount\nof runaway electrons, which may severely damage plasma facing materials [7, 8]. The generated runaway\ncurrent is also affected by secondary mechanisms such as energy transfers from the primary runaway electron\ncurrent to the thermal electrons through knock-on (large-angle) collisions. Understanding these nonlinear\nmechanisms may be essential to develop either avoidance or mitigation strategies for runaway electrons in\ntokamaks.\nA solver designed to capture runaway-electron dynamics benefits from certain features. For example,\ncapturing small-amplitude tails necessitates strict positivity preservation. Runaway-electron generation time\nmay be large: a sizable runaway tail length may take hundreds of electron-electron thermal collision-time\nscales to develop. Therefore, an implicit solver that can step over stiff thermal collision-time scales is\n\u0003corresponding author email: dond@lanl.gov\n1arXiv:1902.10241v3  [physics.comp-ph]  1 Sep 2019desirable. The ability to use large time steps also demands that the scheme be asymptotic preserving, which\nin turn requires enforcing strict conservation properties [9]. It is also essential that the solver be optimal\nand scales with the number of mesh points, as resolving small-scale features may require fine grids while\nfitting long tails may require large domains. The relativistic Fokker-Planck operator can be expressed either\nin integral form [1] or in differential form [10]. We employ the differential form, in which the collisional\ncoefficients are expressed in terms of relativistic potentials. This form is more conducive to an optimal\nO(N)solver, where Nis the number of grid points, as the integral form produces an O(N2)scaling when\nsymmetry is preserved in the integral operators (which is required to achieve strict conservation properties\n[11, 12]). To obtain a nearly optimal scaling ( O(N\u000blogN), with\u000b&1) with the differential formulation, we\npropose a multigrid-preconditioned GMRES [13] solver for the potentials along with an adaptive treatment\nforevaluatingpotentialfar-fieldboundaryconditions[whichgreatlydecreasestheircomputationalcomplexity\nfromO(N3=2)toO(N1:1)]. We note that it is in principle possible to improve the scaling resulting from the\nintegral form of the collision operator by the use of optimal integral methods such as fast multipole methods\n[14]. However, such methods can break the numerical symmetry of the integrals, resulting in the loss of\nstrict conservation properties. In this regard, fast integral methods would not improve on the proposed\noptimal algorithm for the differential formulation, and would require similar strategies to recover the strict\nconservation properties of the collision operator.\nWith regard to time-stepping, we propose a conservative, fully implicit nonlinear scheme, which as a\nresult is asymptotic preserving (i.e., it captures the Maxwell-Jüttner distribution as a steady-state solution\nto our system). Earlier algorithmic approaches proposed for this system are either linearly implicit (e.g.,\nNORSE [4]), or lack strict conservation properties (e.g., CQL3D [5, 6]). The implicit solver proposed in this\nstudy satisfies discrete conservation properties, is preconditioned for optimal algorithmic performance, and is\nscalable in parallel. As a result, our algorithm scales as O(N1:1logN):Our conservation and preconditioning\nstrategies follow closely those proposed in Ref. [9].\nThe rest of the paper is organized as follows. In x2, we discuss the full relativistic electron-electron\noperator, the Lorentz operator for electron-ion interactions, and the Abraham-Lorentz-Dirac reaction term\nfor modeling losses due to synchrotron damping. Then, in x3, we discuss the algorithmic aspects with regard\nto the discrete conservation strategy, positivity preservation, and our near-optimal strategy for determination\nof the potentials. In x4, we briefly describe our fully implicit nonlinear solver using an Anderson Acceleration\nscheme. Inx5, we discuss the numerical results that demonstrate the correctness of our implementation, and\nfinally inx6 we list the conclusions and scope for future work.\n2 Formulation\nWe model a homogeneous quasi-neutral plasma. We evolve the electron species with the relativistic Fokker-\nPlanck equation for the electron distribution function, fe, in the presence of background species \f,\n@tfe+@~ p\u0001h\n(~E+~FS)fei\n=X\n\f=i;eC(f\f;fe); (1)\nwheretis time normalized with the relativistic electron collision time,\n\u001crelativistic\nee =4\u0019\u000f2\n0m2\nec3\nq4eneln \u0003ee;\n~ pis the momentum vector normalized with mec,meis the electron mass, cis the speed of light, qeis the\nelectron charge, ~Eis the electric field normalized with the critical value for runaway electron generation\n[8],Ec=neq3\neln \u0003ee=4\u0019\u000f2\n0mec2,neis the electron number density, \u000f0is the electrical permittivity, ln \u0003eeis\nthe Coulomb logarithm, ~FSrefers to the electron friction coefficients associated with synchrotron radiation\ndamping effects (defined in detail later), and Cis the collision operator given by,\nC(f\f;fe) =@~ p\u0001\u0014\nD\f\u0001@~ pfe\u0000me\nm\f~F\ffe\u0015\n; (2)\nwhereD\frepresents the collisional diffusion tensor coefficients and ~F\frepresents the collisional friction\nvector coefficients (computed based on the appropriate background species f\f). Though Eq. (1) in principle\n2of the potentials is intractable because it results in poor scalingO(N2)whereNis the number of points. The collisionalcoefficients areDe=\u0000\u0000\u00001[L+I+~p~pm2e]h1+4\u0000\u00001[L\u0000I\u0000~p~pm2↵]h2(3)~Fe=\u0000\u0000\u00001~K(g0\u00002g1)(4)where the operatorL(.)=\u0000\u00002@2(.)@~v@~v\u0000~v@(.)@~v\u0000@(.)@~v~vK(.)=\u0000\u00001@(.)@~v.Note that velocity is normalized with the speed of light,~v=~p/\u0000me,\u0000=p1+~p.~p/m2e,a n d@/@~v=(\u0000/me)(m2e+~p~p)·@/@~p.To obtain the transport coefficients, we first solve for thehpotentials using the differential equations,[L+ 1]h0=fe[L\u00003]h1=h0(5)[L\u00003]h2=h1and thegpotentials usingLg0=feLg1=g0(6)where the operatorL(.)=(m2eI+~p~p):@2(.)@~p@~p+3~p.@(.)@~p.For solving the linear potential equations, we require values at the boundaries, this is determined via the integral Green’sfunction formulations [1],h0=\u000014⇡Z(r2\u00001)\u00001/2f\u0000(~p0)\u00000d3˜p0,h1=\u000018⇡Zpr2\u00001f\u0000(~p0)\u00000d3˜p0,h2=\u0000132⇡Z(rcosh\u00001r\u0000pr2\u00001)f\u0000(~p0)\u00000d3˜p0,(7)g0=\u000014⇡Zr(r2\u00001)\u00001/2f\u0000(~p0)\u00000d3˜p0,g1=\u000018⇡Zcosh\u00001rf\u0000(~p0)\u00000d3˜p0,wherer=\u0000\u00000\u0000~p·~p0/m2e.2.2 Modeling external effects2.2.1 Electron-ion scattering operatorThe relativistic Lorentz operator for modelling electron-ion scattering operations which is a valid approximation consideringthe ions are more massive than the underlying electrons,mi>>\u0000me.The diffusion coeffients areDkk=Zeff2⌫p2?p2,Dk?=D?k=\u0000Zeff2⌫p?pkp2,D??=Zeff2⌫p2kkp2,wherep2=p2?+p2kand⌫is the velocity magnitude. The diffusion coefficients in the azimuthal direction is zero becausethe distribution function is axisymmetric about the magnetic field.3work [6] and involves a redistribution of the discretization errors so as to satisfy conservation properties in the discretelimit.We do not enforce conservation symmetries,hpk,C(fe,fi)+C(fi,fe)i=0h\u0000,C(fe,fi)+C(fi,fe)i=0for the electron-ion collisions. Because, we do not consider the evolution of ion species,fi,t h i si sn o te n f o r c e di nt h ecode. Ions are assumed to be cold and massive in comparison to the electrons, and the electron-ions are modelled by theelectron-ion scattering operator.3.3 Positivity-preserving treatmentsWe describe numerical treatments done to preserve positivity of the distribution function, accurate positivity-preservingschemes are essential in capturing small amplitude runaway tails.3.3.1 Anisotropic diffusion tensor terms.The divergence of the diagonal terms in the diffusion tensor,Drpfe|kkandDrpfe|??,s a t i s f yt h ed i s c r e t em a x i m u mprincple. The divergence of the off-diagonal diffusion tensor terms violate the maximum principle resulting in loss ofboundedness. To avoid this, we reformulate the off-diagonal components [7] as,Drpfe|k?=Dk?@p?fe=Dk?@p?lnfe|{z}FeffectivefeDrpfe|?k=D?k@pklnfe|{z}FeffectivefeOnce formulated as an advective term, we then use flux-limiting schemes such as SMART when calculating thedivergence to ensure the electron distribution function is bounded.3.3.2 Advective terms.The Sharp and Monotonic Algorithm for Realistic Transport (SMART) is used for flux limiting in the advective terms.Please see appendix.3.4 Resolving singular integrands in azimuthal directionThe boundary conditions for the relativistic potential equationsh, gare found using the integral formulations (7). Becausethe distribution is axisymmetric, the 3D integration can be rewritten as a 2D momentum space integration and a 1Dazimuthal angle integration. Forh1,h2,g1relativistic potentials, we use trapezoidal numerical integration with 24 discretepoints in the\u0000angle. However, the integrandsHandIingo,hobecomes singular in the limit ofp0!p=)r!1making numerical integration inaccurateh\u0000,0=\u000014⇡Zf\u0000(p0k,p0?)\u00000dp0kdp0?Z\u00001(r2\u00001)1/2d\u0000|{z}I=\u0000I4⇡Zf\u0000(p0k,p0?)\u00000dp0kdp0?,g\u0000,0=\u000014⇡Zf\u0000(p0k,p0?)\u00000dp0kdp0?Z\u0000r(r2\u00001)1/2d\u0000|{z}H=\u0000H4⇡Zf\u0000(p0k,p0?)\u00000dp0kdp0?,To remove possible errors, we express the azimuthal integration in terms of complete integrals of the first and third kind,I=Z2⇡0d\u0000pr2\u00001=4K(m)d(t+\u0000t\u0000)pa2\u0000a1\u0000=4K(m)(t+\u0000t\u0000)pa2+a1\u0000,(8)H=(a2\u0000b2t+)I+4b2pa2+a1\u0000⇧(e2,m)(9)5xxxxxxxxxof the potentials is intractable because it results in poor scalingO(N2)whereNis the number of points. The collisionalcoefficients areDe=\u0000\u0000\u00001[L+I+~p~pm2e]h1+4\u0000\u00001[L\u0000I\u0000~p~pm2↵]h2(3)~Fe=\u0000\u0000\u00001~K(g0\u00002g1)(4)where the operatorL(.)=\u0000\u00002@2(.)@~v@~v\u0000~v@(.)@~v\u0000@(.)@~v~vK(.)=\u0000\u00001@(.)@~v.Note that velocity is normalized with the speed of light,~v=~p/\u0000me,\u0000=p1+~p.~p/m2e,a n d@/@~v=(\u0000/me)(m2e+~p~p)·@/@~p.To obtain the transport coefficients, we first solve for thehpotentials using the differential equations,[L+ 1]h0=fe[L\u00003]h1=h0(5)[L\u00003]h2=h1and thegpotentials usingLg0=feLg1=g0(6)where the operatorL(.)=(m2eI+~p~p):@2(.)@~p@~p+3~p.@(.)@~p.For solving the linear potential equations, we require values at the boundaries, this is determined via the integral Green’sfunction formulations [1],h0=\u000014⇡Z(r2\u00001)\u00001/2f\u0000(~p0)\u00000d3˜p0,h1=\u000018⇡Zpr2\u00001f\u0000(~p0)\u00000d3˜p0,h2=\u0000132⇡Z(rcosh\u00001r\u0000pr2\u00001)f\u0000(~p0)\u00000d3˜p0,(7)g0=\u000014⇡Zr(r2\u00001)\u00001/2f\u0000(~p0)\u00000d3˜p0,g1=\u000018⇡Zcosh\u00001rf\u0000(~p0)\u00000d3˜p0,wherer=\u0000\u00000\u0000~p·~p0/m2e.2.2 Modeling external effects2.2.1 Electron-ion scattering operatorThe relativistic Lorentz operator for modelling electron-ion scattering operations which is a valid approximation consideringthe ions are more massive than the underlying electrons,mi>>\u0000me.The diffusion coeffients areDkk=Zeff2⌫p2?p2,Dk?=D?k=\u0000Zeff2⌫p?pkp2,D??=Zeff2⌫p2kkp2,wherep2=p2?+p2kand⌫is the velocity magnitude. The diffusion coefficients in the azimuthal direction is zero becausethe distribution function is axisymmetric about the magnetic field.3\noooooooooooo(j,k)(j+1/2, k)(j, k+1/2)x(j+1, k)(j+3/2, k)(j+1, k+1/2)x(j, k-1)x(j+1/2, k-1)(j, k-1/2)CYLINDRICAL DOMAINMOMENTUM SPACEINDEXINGFigure 1: We consider a cylindrical geometry representation (pk;p?)with azimuthal symmetry (left). Fol-\nlowing a finite volume formulation, we define the distribution function on cell centers (crosses) and fluxes\non edges (arrows). The ghost cells (circles) are exterior to domain boundaries. A typical stencil is shown on\nthe right. The discrete volume for cell (j;k)is computed as \u0001Vj;k= 2\u0019p?;k\u0001pk;j\u0001p?;k;where \u0001pk;jand\n\u0001p?;kare the discrete momentum space cell sizes in the parallel and perpendicular directions.\nmay be used for multiple species, here we only consider the evolution of electrons interacting with themselves,\nions and external electric fields.\nThe distribution function is described in a two-dimensional cylindrical domain (pk;p?), with the sub-\nscriptskand?referring to directions parallel and perpendicular to the magnetic field, respectively (see\nFig. 1). The azimuthal direction is ignored because the distribution is axisymmetric. The electron-electron\ninteractions are described using the full form of the collision operator, while the electron-ion interaction is\nmodeled with the Lorentz operator (which assumes the ions to be cold and infinitely massive, mi>>\rme\nwith\r=p\n1 +p2the Lorentz factor).\n2.1 Electron-electron collisions\nThe collisional coefficients, D\fand~F\f, for electrons in Eq. (2) are expressed in terms of the Braams-Karney\npotentials [10]. These potentials are obtained by inverting a set of elliptic equations. In this study, the\nelliptic solves are performed optimally with parallel multigrid-preconditioned GMRES techniques, with a\nscaling ofO(NlogN). The collisional coefficients are given by [10]:\nDe=\u00004\u0019\nn\f\r\u00001[L+P]h1+ 4\r\u00001[L\u0000P]h2; (3)\n~Fe=\u00004\u0019\nn\f\r\u00001~K(g0\u00002g1); (4)\nwhere the operators L,~K, andPare defined as:\nL =P\u0001@2 \n@~ p@~ p\u0001P\u0000P\u0012\n~ p\u0001@ \n@~ p\u0013\n;\n~K =P\u0001@ \n@~ p;\nP=I+~ p~ p:\nTo obtain the transport coefficients, we first compute the hpotentials by solving the partial differential\nequations,\n[L+ 1]h0=fe;\n[L\u00003]h1=h0; (5)\n[L\u00003]h2=h1;\nand then the gpotentials by solving:\n3Lg0=fe;\nLg1=g0: (6)\nHere, the operator Lis defined as:\nL =P:@2 \n@~ p@~ p+ 3~ p\u0001@ \n@~ p: (7)\nTo solve these linear potential equations, we require far-field boundary conditions. They are determined\nfrom the Green’s function solution of the elliptic equations, Eqs. (5,6) [1]:\nh0=\u00001\n4\u0019Z\n(r2\u00001)\u00001=2f\f(~p0)\n\r0d3~p0;\nh1=\u00001\n8\u0019Zp\nr2\u00001f\f(~p0)\n\r0d3~p0;\nh2=\u00001\n32\u0019Z\n(rcosh\u00001r\u0000p\nr2\u00001)f\f(~p0)\n\r0d3~p0; (8)\ng0=\u00001\n4\u0019Z\nr(r2\u00001)\u00001=2f\f(~p0)\n\r0d3~p0;\ng1=\u00001\n8\u0019Z\ncosh\u00001rf\f(~p0)\n\r0d3~p0;\nwherer=\r\r0\u0000~ p\u0001~p0. Note that the integral kernels of h0andg0are singular when r!1 (~ p!~p0);which\nrequire a specialized numerical treatment in terms of elliptic integrals for accuracy and efficiency (see x3.5\nand App. B). Also, we have devised an efficient adaptive algorithm to fill ghost cells that prevents these\nboundary integrals from leading to an O(N3=2)scaling of the computational complexity (see also x3.5).\n2.2 Modeling external effects\nWe consider several external effects, including an imposed electric field, ~E= (Ek;0), ions, and synchrotron\nradiation,~FS.\nElectron-ion scattering is modeled with the Lorentz or pitch-angle scattering operator [3, 15, 4], which\nassumes ions are cold and infinitely massive. The operator causes scattering of the electrons in the pitch\nangle (arccos[pk=p])and, in this simplified form, it preserves kinetic energy. It has finite diffusion coefficients\nand zero friction coefficients, given by:\nDi;kk=Ze\u000b\n2vp2\n?\np2; Di;k?=Di;?k=\u0000Ze\u000b\n2vp?pk\np2; Di;??=Ze\u000b\n2vp2\nk\np2;~Fi=~0; (9)\nwherep2=p2\n?+p2\nk,vis the velocity magnitude (normalized with c), andZe\u000b=PniZ2\ni=PniZiis\nthe effective ion-charge state ( niandZirefers to ion densities and charges). For a quasi-neutral plasma,PniZi=ne:Note that the electron-ion collision operator becomes singular at the origin v!0. We mollify\nthis singularity by reformulating the singular part, as:\n1\nv\u00191p\nv2+v2\ncut;\nwherevcut=pcut=p\n1 +p2\ncutis the velocity cut-off, with pcut= 2\u0001p. Note this approximation of the singular\nterm in the cylindrical space introduces a finite but small amount of heating as p!0.\nFinally, we consider synchrotron radiation, which results in loss of momentum for the electrons. We\nmodel this with the Abraham-Lorentz-Dirac reaction term [15, 16]. The reaction term has finite friction\ncoefficients, given by:\n4FS;?=\u0000Sp?\n\r(1 +p2\n?); FS;k=\u0000Spk\n\rp2\n?; (10)\nwhereS=\u001crelativistic\nee=\u001crrelates the time scale of the synchrotron-radiation damping, \u001cr, to that of electron-\nelectron relativistic collisions, \u001crelativistic\nee .\n3 Algorithm\n3.1 General discretization strategy\nWe employ a conservative finite-difference scheme. The distribution is evaluated at cell centers, while the\nfriction and diffusion fluxes are evaluated at cell faces. Recall the electron-electron collision operator is the\ndivergence of a collisional flux,\nC(fe;fe) =@~ p\u0001(Derpfe\u0000~Fefe)\u0019\u000e~ p\u0001(~RD\u0000~RF) =\u000e~ p\u0001~R; (11)\nwhere~RDand~RFare the diffusion and friction fluxes, respectively, and \u000e~ p\u0001denotes the discrete form of the\ndivergence operator, which in cylindrical-momentum space is written as:\n\u0010\n\u000e~ p\u0001~R\u0011\nj;k=\u0012Rk;j+1=2;k\u0000Rk;j\u00001=2;k\n\u0001pk;j+p?;k+1=2R?;j;k+1=2\u0000p?;k\u00001=2R?;j;k\u00001=2\np?;k\u0001p?;k\u0013\n:(12)\nFluxes at cell faces are given by:\nRD;k;j+1\n2;k=\u0000\nDkk@pkfe+Dk?@p?fe\u0001\nj+1\n2;k; RD;?;j;k+1\n2=\u0000\nD?k@pkfe+D??@p?fe\u0001\nj;k+1\n2;\nRF;k;j+1\n2;k=Fk;j+1\n2;kfe;j+1\n2;k; RF;?;j;k+1\n2=F?;j+1\n2;kfe;j;k+1\n2:\nThe potential operator L(Eq. 7) is discretized using central differences (see App. A1 for details). The\npotentials are evaluated at cell centers and their boundary conditions are specified at ghost cells. For the\npotential Eqs. (5)-(6), we apply far-field Dirichlet boundary conditions using Eqs. (8) as discussed in x3.5.\nThe collisional coefficients, D\fand~F\f, are evaluated at cell centers using the computed potentials (see\nApp. A2 for discretization details). The collisional coefficients at the ghost cells are evaluated by linearly\nextrapolating the values from adjacent cell-centered values. Note that the ghost cells also store distribution\nand coefficient data for cross-processor communication using an MPI framework.\nThe coefficients for external effects (such as scattering due to ion interactions, Eq. (9), synchrotron\ndamping effects, Eq. (10), and electric field acceleration terms) are evaluated at cell-centers. Where needed,\nvaluesatcellfacesarefoundbyaveragingtwoadjacentcell-centeredvalueswithinthecomputationaldomain.\nAs is typical in kinetic simulations, the outer domain boundaries are selected such that the distribution is\nsufficiently small there.\n3.2 Discrete conservation strategy for the e-e collision operator\nThe relativistic electron-electron collision operator conserves the total particle number, momentum, ~ p=\r~ v,\nand energy E=\r, as the moments of the collision operator satisfy:\nh1;C(fe;fe)ip= 0; (13)\nhpk;C(fe;fe)ip= 0; (14)\nh\r;C(fe;fe)ip= 0; (15)\nwhereha;bip=R\npab2\u0019p?dpkdp?. Discretely, these inner products may be approximated by a mid-point\nquadrature rule as:\n5hA;BiD\np\u00192\u0019NkX\nj=1N?X\nk=1Aj;kBj;kp?;k\u0001pk;j\u0001p?;k;\nwhere the superscript Drefers to the discrete representation of the summation operator, and \u0001pk;jand\n\u0001p?;kare the width and height of a rectangular cell located at (j;k):\nIn general, the relationships shown in Eqs. (13-15) will not be satisfied due to numerical errors. Discrete\nparticle number conservation (Eq. 13) is trivially satisfied by setting the normal component of diffusion\nand friction fluxes to zero at the boundary. However, enforcing Eqs. (14,15) is more challenging. A recent\nstudy [9] enforced these conservation properties discretely by redistributing the numerical errors via discrete\nnonlinear constraints. We employ a similar methodology here. Firstly, we multiply the diffusion flux by a\nfactor\n\u0011= 1 +\u00110+\u00111(pk\u0000\u0016pk);\nwherethemagnitudesof \u00110and\u00111areexpectedtobeoftheorderoftruncationerror,and \u0016pk=hfe;pkip=h1;feip\nis the mean momentum. Thus, the discrete collision operator is of the form,\nCD(fe;fe) =\u000e~ p\u0001(~\u0011RD\u0000~RF);\nIntegrating over the cylindrical-momentum domain,\nhpk;CD(fe;fe)iD\np= 0;\nh\r;CD(fe;fe)iD\np= 0; (16)\nwe obtain a system of two equations,\n\"\nh\r\u000e~ p\u0001~RiD\nph\r(pk\u0000\u0016pk)\u000e~ p\u0001~RDiD\np\nhpk\u000e~ p\u0001~RDiD\nphpk(pk\u0000\u0016pk)\u000e~ p\u0001~RDiD\np#\u0014\u00110\n\u00111\u0015\n=\"\nh\r\u000e~ p\u0001(~RF\u0000~RD)iD\np\nhpk\u000e~ p\u0001(~RF\u0000~RD)iD\np#\n; (17)\nfor unknowns [\u00110; \u00111], which can be inverted straightforwardly. This strategy conserves momentum and\nenergy at the discrete level for electron-electron collisions. Note that, because we assume the ions to be\ncold and infinitely massive, there are no conservation properties associated with electron-ion collisions (i.e.\ndiscrete representation may result in a net energy and momentum loss).\n3.3 Time stepping strategy\nA huge separation in time scales exists in runaway-electron dynamics. Long time-scales are of the order\nof the relativistic collision times, O(\u001crelativistic\nee ). For typical bulk temperatures \u0002 =T=mec2\u001810\u00004, this\nimplies a time-scale separation of six orders of magnitude between thermal and relativistic time scales (as\n\u001cthermal\nee = \u00023=2\u001crelativistic\nee ). Stepping over fast time scales demands a fully implicit temporal scheme with\nstrict conservation and positivity preservation properties. We describe our approach next.\nThe discrete system of equations representing the effects of electron-electron collisional interactions C\nand external effects, E, on electron evolution can be written as:\n\u000etfn\ne=CD(fn\ne;fn\ne) +~C(fn\ni;fn\ne)\u0000\u000e~ p:h\n(~FS+~E)fn\nei\n|{z }\nE(fne); (18)\nwhere the superscript Drepresents the appropriate discrete form defined in x3.2, and ~Crepresents the\nLorentz operator (see Eq. 9). For a general implicit backward time discretization scheme at time step n, we\nhave,\n\u000etfn\ne=X\ni=0;1;2:::bifn\u0000i\ne\n\u0001t;\n6where constants, bi, satisfyP\nibi= 0:We use both first-order (Euler, BDF1) and second-order (BDF2)\nschemes for time advancing. For BDF1, b0=\u00001andb1= 1and for BDF2 with constant time steps,\nbo= 3=2; b1=\u00002,b2= 1=2. The coefficients can be generalized for non-uniform time steps.\nMultiplying Eq. (18) with ~ c= (1;pk;\r), and averaging over the momentum space, we obtain:\nX\ni=0;1;2:::bih~ c;fn\u0000i\neiD\np\n\u0001t=h~ c;CD(fn\ne;fn\ne)iD\np+h~ c;E(fn\ne)iD\np:\nBecause of the discrete conservation properties of the electron-electron collision operator, the first term in\nthe right hand side vanishes. Therefore, any change in the total momentum or energy of electrons can only\nbe due to external effects such as ion-electron collisions, synchrotron radiation, and electric field acceleration.\n3.4 Positivity-preserving strategy\nPositivity-preserving schemes are essential to capture small-amplitude runaway tails. Our strategy is to\nleveragethestructureofthedifferentialoperators(advection-diffusion), anduseexistingpositivity-preserving\ndiscretizations for these terms.\nFor all advective terms in the relativistic kinetic equation, we use the positivity-preserving SMART flux\nlimiter [17] to construct the associated fluxes. For the diagonal components of the tensor diffusion term,\nD\u0001rpfejkkandD\u0001rpfej??, we employ a standard second-order discretization:\n(D\u0001rpfe)kk;j+1\n2;k=\u0000\nDkk@pkfe\u0001\nj+1\n2;k\u0019Dkk;j+1;k+Dkk;j;k\n2fe;j+1;k\u0000fe;j;k\n\u0001pk;j+1\n2;\n(D\u0001rpfe)??;j;k+1\n2= (D??@p?fe)j;k+1\n2\u0019D??;j;k+1+D??;j;k\n2fe;j;k+1\u0000fe;j;k\n\u0001p?;k+1\n2;\nwhich is numerically well-posed (does not feature a null space and features a maximum principle). However,\nunless care is taken, the off-diagonal diffusion tensor terms do not feature a discrete maximum principle,\nresulting in loss of boundedness. To address this issue, we reformulate the off-diagonal components as\neffective friction forces as proposed in Ref. [18]:\n(D\u0001rpfe)k? =Dk?@p?fe=feDk?@p?lnfe|{z}\nFeff\nk=feFe\u000b\nk=Re\u000b\nF;k;\n(D\u0001rpfe)?k =feD?k@pklnfe|{z}\nFeff\n?=feFe\u000b\n?=Re\u000b\nF;?: (19)\nOnce formulated as advective terms, we use flux-limiting advective schemes (similar to the collisional friction\nterms) to calculate the effective flux. Discretization details can be found in App. A3.\n3.5 Strategy for evaluating boundary conditions of collision potentials\nThe boundary conditions for the relativistic potential equations for handgare found using the integral\nformulations in Eq. (8). For the h1;h2;g1relativistic potentials, we use a trapezoidal-rule numerical inte-\ngration with 24 discrete points in the \u001eangle. However, the kernels in g0;h0become singular in the limit of\np0!p=)r!1;complicating a direct numerical integration. These complexities can be eliminated by\nreformulating these integrals in terms of complete elliptic integrals. We begin by noting that, because the\ndistribution is axisymmetric, the 3D momentum-space integration of the Green’s function can be rewritten\nas a 2D momentum space integration over the PDF and a 1D azimuthal angle integration as:\n7Algorithm 1 Adaptive spline based potential boundary treatment.\n1. Initialize a set of knots.\n2. Evaluate potential integrals and create spline (cubic or higher-order).\n3. Bisect original knots to create new knots.\n4. Evaluate potential integrals at each knot and check error using Eq. (20) : j\u001eI\u0000\u001eSj:\n5. Where error is small, stop local bisection. Where error is large, go to step 3.\nh\f;0=\u00001\n4\u0019Zf\f(p0\nk;p0\n?)\n\r0p?dp0\nkdp0\n?Z\n\u001e1\n(r2\u00001)1=2d\u001e\n|{z}\nI=\u00001\n4\u0019Zf\f(p0\nk;p0\n?)\n\r0I(pk;p?;p0\nk;p0\n?)p?dp0\nkdp0\n?;\ng\f;0=\u00001\n4\u0019Zf\f(p0\nk;p0\n?)\n\r0p?dp0\nkdp0\n?Z\n\u001er\n(r2\u00001)1=2d\u001e\n|{z}\nH=\u00001\n4\u0019Zf\f(p0\nk;p0\n?)\n\r0H(pk;p?;p0\nk;p0\n?)p?dp0\nkdp0\n?;\nThe segregated integrals IandHare then written in terms of complete integrals of the first and third kind\n(see App. B).\nHowever, even after these reductions, evaluating potentials at all ghost points in the boundary remains\nexpensive. Thereareapproximately O(N1=2)ghost-cellboundarypoints, eachpointrequiring O(N)integrals\nwhen using Eqs. (8). This makes the potential boundary evaluations scale poorly with the number of mesh\npoints,N[i.e.,O(N3=2)]. To ameliorate the scaling for the boundary-condition treatment, we adaptively\nselect a small number of boundary points for the potential evaluations to match a given accuracy, with the\nremaining ghost points found by interpolation using a high-order spline. The adaptive algorithm to find the\nminimum number of spline knots needed for a given tolerance is outlined in Algorithm 1, and illustrated in\nFig. 2. We begin with a set of uniformly distributed ghost points at the boundary, for example, four points\n(black crosses in the first row), where we evaluate the values of the potential integral. We fit a cubic (or\nhigher order) spline through these values (blue crosses in the second row). New knots are then created by\nbisection (black crosses in the third row), where integrals are again evaluated. The absolute error is then\ncomputed as the difference between the value given by the spline interpolation, \u001eS, and the actual value of\nthe potential integral at the targeted points \u001eI:\nab=j\u001eI\u0000\u001eSj: (20)\nIntervals delimited by the set of knots that do not satisfy the prescribed tolerance (e.g., red knot in the fourth\nrow) are bisected further. This process is continued until a spline fit of the desired accuracy is obtained.\nTo ensure the spline error is commensurate with other sources of error in the algorithm, in practice the\nabsolute tolerance criterion is chosen to be a function of the momentum mesh spacing as:\nab\u00180:05\u0001p?\u0001pk: (21)\nFigure 3 illustrates the adaptive knots generated with the adaptive spline algorithm for the g0potential\nfor a Maxwell-Jüttner distribution of \u0002 = 10\u00002in a mesh of Nk= 2048andN?= 512:The red dots along\nthe left, top, and right boundaries point to the location of the spline knots generated using Algorithm 1.\nThe contour of g0is also illustrated to demonstrate the variation of g0at points close and far away from the\ndistribution. The algorithm generates more spline knots where the function varies significantly. At the far\nright boundary, the points are few and equally spaced, as the function variation is small. A clear benefit of\nthe adaptive spline approach can be seen at the top boundary, where it is determined that only 41 functional\nevaluations are needed for an accurate estimate of the potential along the entire boundary, which spans a\ntotal of 2048mesh points.\n8Figure 2: Illustration of adaptive spline technique.\ndesirable. The ability to use large time steps also demands that the scheme be asymptotic preserving, whichin turn requires enforcing strict conservation properties [22]. It is also essential that the solver be optimaland scales with the number of mesh points, as resolving small-scale features may require fine grids whilefitting long tails may require large domains. The relativistic Fokker-Planck operator can be expressed eitherin integral form [4] or in differential form [5]. We employ the differential form, in which the collisionalcoefficients are expressed in terms of relativistic potentials. This form is more conducive to an optimalO(N)solver, whereNis the number of grid points, as the integral form produces anO(N2)scaling when symmetryis preserved in the integral operators (which is required to achieve strict conservation properies [14, 20]).To obtain an optimalO(N)scaling with the differential formulation, we propose a multigrid-preconditionedGMRES [18] solver for the potentials along with an adaptive treatment for evaluating potential boundaryconditions. It is possible to improve the scaling resulting from the integral form by the use of optimal integralmethods such as fast multipole methods [3]. However, such methods can break the numerical symmetry ofthe integrals, leading to the loss of strict conservation properties. In this regard, fast integral methodsresemble the proposed optimal solve of the differential formulation, and will require similar strategies torecover the strict conservation properties of the collision operator.With regard to time-stepping, we propose a conservative, fully implicit nonlinear scheme, which as a resultis asymptotic preserving (i.e., it captures the Maxwell-Jüttner distribution as a steady-state solution to oursystem). Earlier algorithmic approaches proposed for this system are either linearly implicit (e.g., NORSE[21]), or lack strict conservation properties (e.g., CQL3D [13, 16]). The implicit solver proposed in this studysatisfies discrete conservation properties, is preconditioned for optimalO(N)algorithmic performance, andis scalable in parallel. Our conservation and preconditioning strategies follow closely Ref. [22].The rest of the paper is organized as follows. In§2, we discuss the full relativistic electron-electronoperator, the Lorentz operator for electron-ion interactions, and the Abraham-Lorentz-Dirac reaction termfor modeling losses due to synchrotron damping. Then, in§3, we discuss the algorithmic aspects with regardto the discrete conservation strategy, positivity preservation, and our optimal strategy for determination ofthe potentials. In§4, we briefly describe our fully implicit nonlinear solver using an Anderson Accelerationscheme. In§5, we discuss the numerical results that demonstrate the correctness of our implementation, andfinally in§6w el i s tt h ec o n c l u s i o n sa n ds c o p ef o rf u t u r ew o r k .2 FormulationWe model a homogeneous quasi-neutral plasma. We evolve the electron species with the relativistic Fokker-Planck equation for the electron distribution function,fe,i nt h ep r e s e n c eo fb a c k g r o u n ds p e c i e s\u0000,@tfe+@~p·h(~E+~FS)fei=X\u0000=i,eC(f\u0000,fe),(1)wheretis time normalized with the relativistic electron collision time,⌧relativisticee=4⇡✏20m2ec3q4eneln⇤ee,~pis the momentum vector normalized withmec,meis the electron mass,cis the speed of light,qeis theelectron charge,~Eis the electric field normalized with the critical value for runaway electron generation[8],Ec=neq3eln⇤ee/4⇡✏20mec2,neis the electron number density,✏0is the electrical permittivity,ln⇤eeisthe Coulomb logarithm,~FSrefers to the electron friction coefficients associated with synchrotron radiationdamping effects (defined in detail later), andCis the collision operator. Though Eq. 1 in principle may beused for multiple species, here we only consider the evolution of electrons interacting with themselves, ionsand external electric fields.The distribution function is described in a two-dimensional cylindrical domain(pk,p?), with the sub-scriptskand?referring to directions parallel and perpendicular to the magnetic field, respectively (seeFig. 1). The azimuthal direction is ignored because the distribution is axisymmetric. The electron-electroninteractions are described using the full form of the collision operator, while the electron-ion interaction ismodeled with the Lorentz operator (which assumes the ions to be cold and infinitely massive,mi>>\u0000mewith\u0000=p1+p2the Lorentz factor).2\nFigure 3: Adaptive spline knots (fourth-order spline) in a uniform computational domain with Nk= 2048\nandN?= 512:The figure illustrates the g0potential for a Maxwell-Jüttner distribution of \u0002 = 10\u00002. The\nred dots represent the location of the spline knots, comprising a total of 41 knots at the top boundary, and\n13 and 9 knots at the left and right boundaries, respectively.\n9Assuming equi-spaced knots, we can get an estimate of how the number of splining knots, Np, scales with\nthe total degrees of freedom, N, by comparing the spline error with the tolerance in Eq. 21:\n1\nN\u0018\u00121\nNp\u0013ns+1\n=)Np/N1=(ns+1): (22)\nHere,nsis the order of the spine. For instance, for a fourth-order spline, this result predicts Np\u0018N0:2.\nHowever, we expect this estimate to be very conservative, and it does not take into the account the adaptive\ndistribution of the knots. Numerical experiments in Sec. x5.2.1 show that Np\u0018O(N0:13)for a fourth-order\nspline.\n4 Nonlinear solver\nThe spatial and temporal discretization techniques prescribed in x3 lead to a coupled nonlinear system of\nequations, which requires an iterative nonlinear solver for the distribution function. We use an Anderson\nAcceleration scheme [19] to converge iteratively the system, which we briefly summarize next.\nGiven a fixed point map based Picard iteration,\nfk+1=G(fk);\nwhere the superscript kdenotes the iteration step, Anderson Acceleration scheme [20] accelerates the con-\nvergence of the Picard iteration by using the history of past nonlinear solutions via:\nfk+1=mkX\ni=0\u000bk\niG(fk\u0000mk+i)|{z}\nfk\u0000mk+i+1; (23)\nwhere in this study mk= min(5;k):The coefficients \u000bk\niare determined via an optimization procedure that\nminimizes,\n\r\r\r\r\rmkX\ni=0\u000bk\ni\u0000\nG(fk\u0000mk+i)\u0000fk\u0000mk+i\u0001\r\r\r\r\r;\nsubject toPmk\ni=0\u000bk\ni= 1.\nTo enable preconditioning of the Anderson iteration, our fixed map is based on a quasi-Newton iteration,\nwhere:\nfk+1=G(fk) =fk+\u000efk=fk\u0000\u0000\nPk\u0001\u00001Rk; (24)\nwithPkthe preconditioner, \u000efkthe nonlinear increment, and Rkthe nonlinear residual. Given an electron\ndistribution, fe, the residual for the nonlinear system is evaluated as outlined in Algorithm 2. Note that if\nPis the Jacobian, i.e. Pk= (@R=@fe)k, then Eq. (24) becomes a Newton iteration.\nThe residual contribution from electron-electron collisions requires the solutions of five potentials, which\nrequire inversions of the linear equations in Eqs. (5,6). These are inverted for each nonlinear iteration\nat flux-assembly time along with the computation of conservation constraints \u00110and\u00111, see Eq. 17. The\nnonlineareliminationoftheresidualsassociatedwiththepotentialsandconservationconstraintsfollowsfrom\nprevious studies [21, 9], and enables a conservative, optimal O(NlogN)solver when the Poisson operators\nare inverted optimally and scalably. Here, the linear potential equations are solved using a multigrid-\npreconditioned GMRES [22] solver. The multigrid preconditioner features 1 V cycle with 4 passes of damped\nJacobi (damping factor of 0.7), along with agglomeration for restriction and a second-order prolongation. At\nthe beginning of the solve, the five potentials are solved using a tighter relative tolerance criteria of 10\u00008and\nthen followed by a looser relative-tolerance criteria of 10\u00005\u000010\u00007during each nonlinear solve, depending\non the problem.\nThe preconditioner in Eq. (24) is obtained by Picard linearization of the potentials and subsequent\ndiscretization of the full system,\n10Algorithm 2 Evaluating nonlinear residual, R.\n1. Compute \u000etfeand boundary conditions for potentials.\n2. Invert potential equations for h0;h1;h2;g0;g1using Eqs. (5-6) and evaluate collisional coefficients.\n3. Compute collision operator, C(fe;fe), and enforce conservation symmetries.\n4. Compute external physics: electron-ion scattering operator, ~C(fi;fe), synchrotron damping radiation\nand parallel electric field acceleration ; \u000e~ p\u0001h\n(~FS+~E)fei\n:\n5. Assemble nonlinear residual:\nR(fe) =\u000etfe\u0000C(fe;fe)\u0000~C(fi;fe) +\u000e~ p\u0001h\n(~FS+~E)fei\n:\nPk\u000ef=\u000et\u000ef\u0000C(fk\u00001\ne;\u000ef)\u0000E(\u000ef);\nwhereEis a linear operator representing the net external effects on electrons, see Eq. (18). The transport\ncoefficients in the electron-electron collision operator, C, are Picard-linearized and computed at the previous\nnonlinear iteration, k. All advective terms in the preconditioner are discretized using a linear upwinding\nscheme. During each nonlinear step k, the linear system Pk\u000efk=\u0000Rkis solved with one multigrid V-cycle\nand 3 passes of damped Jacobi (with damping constant 0.7). We use agglomeration for restriction and\nsecond-order prolongation.\nThenonlineariterationendswhenthedesiredrelativenonlinearresidualconvergenceratio rNLisreached,\nrNL=kRkk\nkRk=0k:\nCases with large disparities in signal amplitudes, for example a Maxwellian thermal bulk along with runaway\ntail, may require a tighter convergence ratio, rNL= 10\u00007, to capture accurately the small-amplitude tail.\nIn contrast, a single deforming electron thermal bulk can use a significantly looser nonlinear convergence,\nrNL= 10\u00004, for accurate results.\n5 Results\nWe begin this section with some verification studies, and finish it with scalability and accuracy studies to\nassess the performance of the algorithm.\n5.1 Verification\nWe first verify conservation properties of the equilibrium Maxwell-Jüttner distributions either at rest or\nmoving with a mean momentum, pb:Note that all computations are performed in the stationary reference\nframe. Then, we verify conservation properties during collisional relaxation, and also benchmark the calcu-\nlation of electrical conductivity under the action of both weak and strong electric fields with previous studies\n[3, 23, 4]. Finally, we verify our algorithm with recent calculations of runaway dynamics in the t!1limit\n[8, 15, 16]. Verification results were obtained using the second-order BDF2 time-stepping scheme except\nwhen explicitly stated (see x3.3 for details on the time-stepping scheme).\n5.1.1 Preservation of stationary and boosted Maxwell-Jüttner distributions.\nThe computational domain is uniform with Nk= 256andN?= 128:The nonlinear residual, rNL, is\nconverged to a relative tolerance of 10\u00004unless otherwise specified. The discrete conservation properties\nare satisfied to nonlinear tolerance and are independent of the time step used. The electron number density\n11(a) ( b)\nFigure 4: Preservation of a stationary Maxwell-Jüttner distribution, fMJ\ne, for \u0002 = 1,ne= 1;Nk= 256;\nandN?= 128, see Eq. (25). (a)Log contour of electron distribution fe. The distribution remains\nunchanged as a function of time (not shown). (b)Time evolution of relative errors (Eq. 26) of number\ndensity (blue), relativistic momentum (red) and relativistic energy (green). Note time is normalized with\n\u001crelativistic\nee ,\u001crelativistic\nee =\u001cthermal\neeas\u0002 = 1.\nis normalized, ne= 1. The domain is chosen such that the distribution function is sufficiently small at\nboundaries. The entire domain is shown in the figures illustrating the distribution function. Fig. 4 (a)\nillustrates a static Maxwell-Jüttner (MJ):\nfMJ\ne=ne\n4\u0019\u0002K2(1=\u0002)exp\u0014\n\u0000\r(p)\n\u0002\u0015\n; (25)\nin log scale with normalized temperature \u0002 =T=mec2= 1. In Eq. 25, K2is the modified Bessel function of\nthesecondkind. Wehaveconfirmedthatthedistributionretainsitsinitialshapeforthewholesimulation. To\nillustrate this, Fig. 4 (b)demonstrates the evolution of the relative errors in the number density, momentum\nand energy for 200 \u001cee. The relative errors are measured as,\nrelative error =jg(t)\u0000g(0)j\ng(0): (26)\nwheregis either the number density, momentum or energy. The figure shows that the relative errors in\nnumber density are one part in 1011, while errors in relativistic momentum and energy remain smaller than\none part in 108.\nFor a boosted (translated) MJ, the equilibrium distribution appears deformed and is given by:\nfBMJ\ne =ne\n4\u0019\u0002b\rbK2(1=\u0002b)exp\u0014\n\u0000\rb\r\u0000pbpk\n\u0002b\u0015\n; (27)\nwhere the subscript bdenotes the values in the boosted frame. Fig. 5 illustrates a boosted MJ equilibrium\ndistribution with \u0002b= 0:15, in a frame boosted by pb= 2and with\rb=p\n1 +p2\nb:The relative errors are\nshown for 10\u001crelativistic\nee\u0018100\u001cthermal\nee, demonstrating identical behavior as in the stationary MJ case.\n5.1.2 Conservation properties during collisional relaxation dynamics\nTo explore collisional relaxation dynamics, we consider two cases, one which features an initial configuration\nof two boosted MJ distributions, and the other which features a randomized initial distribution. Simulations\nhave been run till the distributions have relaxed to a single Maxwell-Jüttner.\nFigure 6 (a)illustrates the collisional relaxation of two MJ distributions boosted by 2 units in opposite\ndirections. Fig. 6 (b)depicts the relative errors in number density, momentum, and energy. After an initial\ntransient stage t2(0;300), the relative errors in momentum and number density remain small and bounded\nin time.\n12(a) ( b)\nFigure 5: Preservation of a boosted Maxwell-Jüttner distribution, fBMJ\ne, forne= 1;\u0002b= 0:15;pb= 2;Nk=\n256;andN?= 128, see Eq. 27. (a)Log contour of electron distribution, fe.(b)Time evolution of relative\nerrors in number density, momentum, and energy.\n(a) ( b)\nFigure 6: Collisional relaxation of two boosted Maxwell-Jüttner distributions, fBMJ\ne, withne= 1,pb=\n\u00002;2,\u0002b= 0:15;Nk= 256;andN?= 128, see Eq. 27. (a)Evolution of electron distribution contours from\ntwo distinct distributions at initial time (top) to a single Maxwell-Jüttner at final time (bottom) (b)Time\nevolution of relative errors measured during the collisional relaxation process.\n13(a) ( b) ( c)\nFigure 7: Thermal relaxation of a random perturbed Maxwell-Jüttner distribution, frand\ne, forNk= 256and\nN?= 128, see Eq. (28). (a)Initial random electron distribution, see Eq. (28). (b)\u0000(c)Evolution of relative\nerrors in discrete conservation properties for a nonlinear relative tolerance of 10\u00004in(b)and10\u00006in(c):\nTo demonstrate that the discrete conservation strategy works in more complicated cases, we consider the\nthermalization of a random distribution, of the form:\nfrand\ne =P\nhP;1ip; whereP(pk;p?) =J(pk;p?)\n4\u0019\u0002K2(1=\u0002)exp\u0014\n\u0000\r(p)\n\u0002\u0015\n; (28)\nwhere \u0002 = 1andJis a random number function with a uniform distribution in the range [0;1], see Fig.\n7. The presence of large gradients in the distribution and small tails makes this an excellent problem to\ntest discrete conservation errors and positivity preservation. For a nonlinear relative tolerance of 10\u00004, the\nrelativeerrorsinmomentumarelargerthaninpreviouscases, 1partin 104. Tighteningtherelativenonlinear\ntolerance to 10\u00006results in a commensurate decrease of the errors to 1 part in 106.\n5.1.3 Electrical conductivity in weak and strong electric fields\nWe consider next the case where collisional friction balances an externally imposed electric field, leading\nto finite electrical conductivity. We verify the code for a wide range of initial temperatures with electrical\nconductivity results provided by Braams and Karney [3]. To measure conductivity, we apply a small electric\nfield, ^Ek= 10\u00003EDwhere ^Ekis the parallel electric field, ED=Ec=\u0002is the Dreicer field, and Ecthe\nConnor-Hastie critical electric field [8]. The electron distribution is initialized using the Maxwell-Jüttner\ndistribution at various temperatures \u0002and the normalized electrical conductivity is computed as:\n\u0016\u001b=Ze\u000b\nneqe\u00023=2^j\nEk; ^j=\u0000neqevk; (29)\nwherevk=pk=\r. A small electric field deforms the Maxwellian slightly to produce a net electron flow in\nthe positive pkdirection. To prevent numerical overflow, for \u0002>10\u00003the initial distribution is defined\nusing a non-relativistic Maxwellian. Fig. 8 (a)depicts \u0016\u001b=Z e\u000bat various temperatures. The numerical\nsimulation results (circles) are in excellent agreement with the analytical results (lines) from Ref. [3]. The\nelectrical conductivity measurements are made after the simulation reaches a quasi-steady-state after an\ninitial transient. Because of the applied electric field, the plasma slowly heats up, and the quasi-steady-state\ntemperatures, \u0002 =hfe(~ p;t)p2=2\rip, are larger (but close) to their initial value. The electrical conductivity,\n\u0016\u001b, in Eq. (29) is computed using the quasi-steady-state temperature.\nFigure 8 (b)illustrates the time evolution of electrical conductivity for various electric-field strengths in\nthe non-relativistic limit. Results of NORSE [4] (circles) and Weng et al. [23] (squares) are also shown. The\nelectron distribution is initialized with a Maxwellian corresponding to an initial temperature of \u0002 = 10\u00004\nandne= 1. Note that the Weng et al. study uses the nonrelativistic Fokker-Planck operator. For all\nvalues of electric field, we have good agreement with earlier studies. For the case of ^Ek= 0:01ED;we have\nbetter agreement with NORSE than Weng et al.. Ref. [4] hypothesizes that the observed deviations between\n14(a) ( b)\n012345610−1100101\nE∥t/√Θσ\n  \nˆE∥/ED=1ˆE∥/ED=0.1ˆE∥/ED=0.01NORSEWeng2008\nFigure 8: Verification under weak and strong electric fields. The computational domain is uniform with\nNk= 512andN?= 256:(a)Normalized electrical conductivity vs. \u0002for various effective ion charges,\nZe\u000b2[1;10], and a weak electric field of ^Ek=ED= 0:001. The momentum domain sizes vary with a\nminimum of p?2(0;0:12)andpk2(\u00000:12;0:12)for\u0002 = 10\u00004and a maximum of p?2(0;40)and\npk2(\u000040;40)for\u0002 = 5.(b)Time evolution of electrical conductivity for various strengths of the electric\nfield for an effective ion charge of Ze\u000b= 1. The electron distribution was initialized with a Maxwellian for\n\u0002 = 10\u00004andne= 1. The momentum domains are p?2(0;0:3)andpk2(\u00000:3;0:3)for strong electric\nfield ( ^Ek=ED= 1),p?2(0;0:2)andpk2(\u00000:2;0:2)for intermediate electric field ( ^Ek=ED= 0:1), and\np?2(0;0:12)andpk2(\u00000:12;0:12)for weak electric field ( ^Ek=ED= 0:01).\nNORSE and Weng et al. in the small ^E=EDlimit may be due to numerical heating in Weng et al.. Our\nresults also suggest the same.\n5.1.4 Reproducing runaway-electron tail dynamics\nTo verify runaway dynamics with existing linear test-particle studies [8, 16], we performed two linearized\nnumerical simulations where we keep the collisional cofficients fixed in time to those of a Maxwell-Jüttner\ndistribution with ne= 1and\u0002 = 0:01. In the first simulation, we applied an electric field 2.25 times the\ncritical value, i.e Ek= 2:25:This causes some electrons to overcome the frictional force and accelerate to\nhigh speeds. Fig. 9 demonstrates the evolution of the runaway tail at 42000\u001cthermal\neecollision times. The\nasymptotic slope of the runaway tail as predicted by Connor-Hastie [8] is:\nftail\ne/1\npkexp\u0012\n\u0000(Ek+ 1)p2\n?\n2(1 +Ze\u000b)pk\u0013\n:\nAs can be seen in the figure, the runaway tail produced by the algorithm is in excellent agreement with the\nasymptotic theoretical results.\nIn the second simulation, we verify the runaway electron dynamics in the presence of the synchrotron\nradiation damping term in Eq. (10). Because we are interested in the steady-state as t!1, time marching\nis performed efficiently with a BDF1 time stepping scheme (rather than BDF2). Fig. 10 shows the electron\ndistribution function at t= 777(i.e.,\u0019777;000\u001cthermal\nee) with a damping coefficient of S= 0:1. Other\nparameters are Ek= 2:25;\u0002 = 0:01;andZe\u000b= 1. Ref. [15] performed a linearized initial value problem and\ndescribes the evolution of runaway electrons in the momentum space as a two-step phenomenon, beginning\nwith the relative fast formation of a long runaway tail and a much slower rise of the bump to a steady-state\nsolution. We find similar behavior here with the electrons accumulating in the momentum space around\np0\u001918, to form a second maximum. The location of the second maximum is in good agreement with Ref.\n[16]. Note that, when collisional coefficients are evolved nonlinearly, we see heating of the bulk (not shown)\n150 10 20 30 40 50 6010−1210−1010−810−610−410−2100\np/bardblf\n  \nInitial\n7000 e-e\n14000 e-e\n21000 e-e\n28000 e-e\n35000 e-e\n42000 e-e\ntimeConnor-Hastie\nprediction of runaway tail slopeFigure 9: Verification of runaway tail dynamics. Electron distribution function vs. pkforp?= \u0001p?=2; Ek=\n2:25;\u0002 = 0:01andZe\u000b= 1. The computational domain is uniform with Nk= 2048andN?= 512with\np?2(0;10)andpk2(\u000010;60). The initial Maxwellian is represented by the blue line concentrated at the\norigin,pk= 0:At 7000\u001cthermal\nee(green line), we see a finite tail develop from the Maxwellian. This tail\ngrows steadily as time increases. Time step \u0001t= 0:01i.e. 10\u001cthermal\nee, and a nonlinear relative convergence\ntolerance of rNL= 10\u00006.\nleading to slide-away effects as previously reported in Ref. [4].\n5.2 Solver performance\n5.2.1 Algorithmic and parallel scalability.\nTable 1 lists weak parallel scalability results for the thermalization of a random electron distribution, see\nfrand\nein Eq. (28), with \u0002 = 1:For simplicity, we employ the BDF1 time-stepping scheme for scalability\ntests. We report the wall clock time (WCT) per time step, the number of nonlinear iterations (NLI) per\nimplicit time step \u0001t, the ratio WCT/NLI (which is an indirect measure of communication costs), and the\nratio between implicit and explicit time steps (which is a measure of numerical stiffness). The potential\nlinear iterations (PLI) is the average number of GMRES iterations required for each potential solve. The\nabsolute tolerances are set to 10\u00008. For the initial solve, the potentials are converged tightly to a relative\nSecond maximum\nFigure 10: Verification of synchrotron radiation physics. Electron distribution function in the (pk;p?)space\nforEk= 2:25;\u0002 = 0:01;Ze\u000b= 1and synchrotron damping factor S= 0:1. The second maximum is located\natp\u001918:The computational domain is uniform with Nk= 2048andN?= 512:The contour is plotted\nat timet= 777:Time step, \u0001t= 0:5i.e. 500\u001cthermal\nee, and a nonlinear relative-convergence tolerance of\nrNL= 10\u00006.\n16Figure 11: Left: Weak parallel scaling test with a domain size per core of 64\u000232grid points. The figure\ndepicts the wall clock time (WCT) per nonlinear iteration (NLI) as a function of the number of cores, and\ndemonstrates that the scaling is consistent with an O(N0:1logN) scaling. Right: Cumulative number of\nadaptive spline knots (for all potentials) Npvs. the total mesh points, N;demonstrating an overall scaling\nofNp\u0018O(N0:13).\ntolerance of 10\u00008, with a looser tolerance of 10\u00006used for subsequent nonlinear iterations. The adaptive\nspline tolerance is defined by Eq. (21) and a fourth-order spline is used for piece-wise interpolation. The\nexplicit time step is calculated as:\n\u0001texplicit = 0:25 min\n\f=i;e \n\u0001p2\nk\nmax(D\f;kk);\u0001p2\n?\nmax(D\f;??);\u0001p?\nmax(A\f;?);\u0001pk\nmax(A\f;k)!\n:\nTable 1 demonstrates excellent parallel scalability (WCT/NLI) up to 4096 processors. When increasing the\nnumber of processors while keeping the problem size per processor constant, we are effectively increasing the\nresolution of the problem, thus making the problem harder to solve for a fixed time step (as evidenced by\nthe increasing implicit-to-explicit timestep ratio). This manifests in a very mild growth of the number of\nnonlinear iterations (NLI) as we increase the number of cores (the iteration count increases only by a factor\nof 3 over a three-order-of-magnitude increase of the problem size).Table 2 lists parallel and algorithmic\nscalability results for the case of collisional relaxation of two Maxwell-Jüttner distributions, boosted by one\nmomentum unit in opposite directions and a normalized temperature of 10\u00001in their frames of reference,\nseefBMJ\nein Eq. (27). We observe good parallel (WCT/NLI) and algorithmic scalability (NLI) up to 4096\nprocessors with \u0001t=\u0001texplicit\u0018460for the high resolution case of 4096\u00022048.\nFig. 11 (a)illustrates weak scaling results of the wall clock time per nonlinear iteration (WCT/NLI) vs.\nthe number of cores for the random electron thermalization (red line, Table 1) and the boosted MJ relaxation\n(blue line, Table 2). We observe excellent parallel scalability in both cases. The expected scaling for parallel\nmultigrid-based solvers is WCT/NLI \u0018O(logN), resulting in an overall algorithmic scaling of O(NlogN).\nInstead, we find WCT/NLI \u0018O(N0:1logN). The additional factor of N0:1originates in the growth of the\nnumber of spline knots with N, as predicted by Eq. 22 and demonstrated in Fig. 11 (b). In the figure, we\nshow that the cumulative number of spline knots (for all potentials) increases as O(N0:13)for a fourth-order\nspline, which is more benign than the predicted one in Eq. 22 (which assumed uniformly spaced knots),\nand much more efficient than the naive scaling of O(N3=2). We have also confirmed that the scaling of\nnumber of spline knots with Ndepends on the spline order, with lower orders resulting in a larger exponent.\nFor instance, a cubic spline for the same tests results in Npscaling asO(N0:17)(results not shown). It is\ninteresting to note that, from the figure, the O(N0:1)scaling seems to disappear at large core count (large\nproblem sizes), which we speculate is due to the problem becoming too well resolved by the mesh.\n17NkN?npNLI per \u0001tWCT (sec) per \u0001tWCT/NLI \u0001t=\u0001texplicit PLI\n128644 4.5 4.85 1.07 82 7.8\n25612816 4 6.0 1.5 329 9.14\n51225664 4 7.4 1.85 1318 9.475\n1024512256 5 11.1 2.2 5270 9.088\n204810241024 6.5 15.8 2.43 21083 8.969\n409620484096 12 30.6 2.55 84331 9.0\nTable 1: Parallel and algorithmic scaling tests: Thermalization of a random distribution in domain with\npk2(\u000010;10),p?2(0;10), and \u0001t= 1:The results are averaged over 2 time steps with nonlinear relative\nconvergence tolerance of rNL= 10\u00005(considering more time steps is not useful, as the solution has already\nsettled into a MJ distribution).\nNkN?npNLI per \u0001tWCT (sec) per \u0001tWCT/NLI \u0001t=\u0001texplicit PLI\n128644 3.2 3.42 1.07 0.45 8.5\n25612816 3.3 4.63 1.4 1.8 8.6\n51225664 3 5.85 1.95 7.2 10.8\n1024512256 3 6.9 2.3 29 11.6\n204810241024 3 8.93 2.98 115 12.4\n409620484096 4 13.4 3.35 460 13.55\nTable 2: Parallel and algorithmic scaling tests: Collisional relaxation of two boosted Maxwell-Jüttner dis-\ntributions in domain with \u0002b= 0:1,pb=\u00001;1; pk2(\u000015;15),p?2(0;15), and \u0001t= 0:01. Note that the\ntime step chosen is comparable to the thermal collision time in boosted frame, i.e. \u001cthermal\nee;b\u00190:04\u001crelativistic\nee:\nThe results are averaged over 10 time steps with rNL= 10\u00005.\n(a) ( b)\nFigure 12: Spatial and temporal accuracy measurement of the proposed scheme using the two boosted MJ\nconfiguration.\n185.2.2 Spatial and temporal accuracy\nFigure12 (a)illustratesthespatialaccuracyoftheproposedschemewiththeboostedMJcollisionalrelaxation\nproblem described in x5.2.1. The ‘exact’ electron distribution fexact\neis obtained at t= 0:15(before steady-\nstate is reached) with Nk= 1024,N?= 512and\u0001t= 0:01, while the three data points (solid dots)\ncorrespond to coarser grids of 128\u000264,256\u0002128and512\u0002256. The blue dashed line corresponds to a\nsecond-ordererrorscaling. The `2\u0000normoftheerrorbetweenthe‘exact’andnumericalelectrondistributions\nis computed as:\nkfe\u0000fexact\nek2=0\n@NkX\nj=1N?X\nk=1(fe;j;k\u0000fexact\ne;j;k )22\u0019p?;k\u0001pk;j\u0001p?;k1\nA0:5\n:\nWe confirm that the proposed implementation is second-order accurate in space.\nFigure 12 (b) illustrates the temporal accuracy of the implementation. In this case, the ’exact’ electron\ndistribution fexact\neis obtained in a 256\u0002128grid using the BDF2 time advancement scheme with \u0001t=\n5\u000210\u00004, see description in x3.3. The four data points correspond to larger time steps of \u0001t= 10\u00004;2\u0002\n10\u00004;4\u000210\u00004;and10\u00003. The proposed implementation is confirmed to be second-order accurate in time.\n6 Conclusions\nWe have developed a fully implicit, nearly optimal, relativistic nonlinear Fokker-Plank algorithm with strict\nconservation properties.. We consider a 0D2P cylindrical momentum-space representation. The solver em-\nploys the differential form of the Fokker-Planck equation, which requires the solution of five relativistic\npotentials in momentum space to obtain the collisional coefficients. Singularities in the potential integral\nformulations are resolved by expressing them in terms of complete elliptic integrals of the second and third\nkind. To ensure a benign scaling of the potential solves with the total number of mesh points N, we em-\nploy a multigrid-preconditioned GMRES solver, and have developed an adaptive spline technique for finding\nfar-field boundary conditions for the potentials. The adaptive spline technique results in a small additional\nexponent in the algorithmic scaling of O(N0:1). Positivity of the distribution function is ensured using\na continuum-based reformulation approach [18] combined with robust positivity-preserving discretizations\nschemes [17]. Using an Anderson Acceleration fixed-point iteration scheme for our nonlinear solves, also\npreconditioned with multigrid techniques, we obtain an algorithm that overall scales as O(N1:1logN). The\nlogNcontribution is due to the parallel multigrid techniques employed, and the N0:1contribution is from\nthe proposed adaptive spline technique. We have demonstrated second-order accuracy in both space and\ntime, and characterized the performance of our parallel implementation. We have verified our solver by\ncomparing with previous results for electrical conductivity measurements in the weak and strong electric\nfield limits. We have demonstrated the accuracy of conserved quantities in electron-electron collisions, with\nsmall relative errors, in number density, relativistic momentum, and energy. In addition, we have exam-\nined runaway dynamics and verified it by comparing to known results [8, 15, 16]. In future work, we will\nextend this method to study inhomogeneous plasmas by considering the spatial dependence of the electron\ndistribution function.\nAcknowledgements\nThe authors thank Z. Guo for help in verifying the algorithm and E. Hirvijoki for insightful discussions\non the properties of the relativistic operator. The authors also thank C. McDevitt and X. Tang for useful\ninputs during the course of this project. This work was supported by the US Department of Energy through\nthe Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National\nSecurity, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract\nNo. 89233218CNA000001). This research used resources provided by the Los Alamos National Laboratory\nInstitutional Computing Program.\n19A Discretization of operators in potential equations and collisional\ncoefficients\nA1. Linear potential equations.\nThe potential operator Lconsists of Hessian and advective terms,\nL = (I+~ p~ p) :@2 \n@~ p@~ p+ 3~ p:@ \n@~ p:\nThe terms are discretized using central differencing. The Laplacian piece is computed as:\n\u0012\nI:@2 \n@~ p@~ p\u0013\nj;k= \n@2 \n@p2\nk+@2 \n@p2\n?!\nj;k=Xj+1\n2;k\u0000Xj\u00001\n2;k\n\u0001pk;j+p?;k+1\n2Yj;k+1\n2\u0000p?;k\u00001\n2Yj;k\u00001\n2\np?;k\u0001p?;k;\nwhere,\nXj+1\n2;k= j+1;k\u0000 j;k\n\u0001pk;j+1\n2; Yj;k+1\n2=( j;k+1\u0000 j;k)\n\u0001p?;k+1\n2:\nThe remaining Hessian piece is computed as:\n\u0012\n~ p~ p:@2 \n@~ p@~ p\u0013\nj;k=\u0014pk;jpk;jp?;kpk;j\npk;jp?;kp?;kp?;k\u0015\n:2\n4@2 \n@p2\nk@2 \n@pk@p?\n@2 \n@p?@pk@2 \n@p2\n?3\n5\nj;k\n=pk;jpk;jQj+1\n2;k\u0000Qj\u00001\n2;k\n\u0001pk;j+p?;kp?;kRj;k+1\n2\u0000Rj;k\u00001\n2\n\u0001p?;k\n+ 2p?;kpk;jTj;k+1\n2\u0000Tj;k\u00001\n2\n\u0001p?;k;\nwhere\nQj+1\n2;k= j+1;k\u0000 j;k\n\u0001pk;j+1\n2; Rj;k+1\n2= j;k+1\u0000 j;k\n\u0001p?;k+1\n2;\nTj;k+1\n2=1\n2\u0012 j+1\n2;k+1\u0000 j\u00001\n2;k+1\n\u0001pk;j+ j+1\n2;k\u0000 j\u00001\n2;k\n\u0001pk;j\u0013\n:\nThe advective piece is computed as:\n\u0012\n3~ p:@ \n@~ p\u0013\nj;k= 3\u0012\npk;j j+1\n2;k\u0000 j\u00001\n2;k\n\u0001pk;j+p?;k j;k+1\n2\u0000 j;k\u00001\n2\n\u0001p?;k\u0013\n:\nNotethecellfacedvaluesof  arefoundbylinearaveragingacrosscellcenteredvalues, forexample  j+1=2;k=\n0:5( j;k+ j+1;k)and j;k+1=2= 0:5( j;k+ j;k+1).\nA2. Collisional coefficients.\nOnce the potentials are determined, the friction coefficients are evaluated using Eq. (4). The components\nof~K at the cell center are defined as:\n(K )j;k=\u0010\nI+~ p~ p\u0011\nj;k\u0001\u0012@ \n@~ p\u0013\nj;k\n=2\n664(1 +pk;jpk;j)\u0012 j+1\n2;k\u0000 j\u00001\n2;k\n\u0001pk;j\u0013\n+pk;jp?;k\u0012 j;k+1\n2\u0000 j;k\u00001\n2\n\u0001p?;k\u0013\n(1 +p?;kp?;k)\u0012 j;k+1\n2\u0000 j;k\u00001\n2\n\u0001p?;k\u0013\n+pk;jp?;k\u0012 j+1\n2;k\u0000 j\u00001\n2;k\n\u0001pk;j\u00133\n775\n20The cell face values of  are found by taking the average of cell-centered values. A similar discretization\napproach is used when evaluating the diffusion coefficient, Eq. (3) .\nA3. Reformulated off-diagonal tensor diffusion terms (effective friction coeffi-\ncients).\nTheoff-diagonaldiffusioncoefficientsareexpressedaseffectivefrictioncoefficientsoftheform Dk?@lnf=@p?\nandD?k@lnf=@pk, see Eq (19). The momentum-space derivatives of lnfat cell centers are evaluated by\naveraging the cell vertex values, for example:\n\u0012@lnf\n@p?\u0013\nj;k=1\n4 \u0012@lnf\n@p?\u0013\nj+1\n2;k+1\n2+\u0012@lnf\n@p?\u0013\nj\u00001\n2;k+1\n2+\u0012@lnf\n@p?\u0013\nj\u00001\n2;k\u00001\n2+\u0012@lnf\n@p?\u0013\nj+1\n2;k\u00001\n2!\n;\nwhere the cell vertex value is obtained by averaging over adjacent face-centered values:\n\u0012@lnf\n@p?\u0013\nj+1\n2;k+1\n2=1\n2 \nln(jfj;k+1j+\u000fl)\u0000ln (jfj;kj+\u000fl)\n\u0001p?;k+1\n2+ln (jfj+1;k+1j+\u000fl)\u0000ln (jfj+1;kj+\u000fl)\n\u0001p?;k+1\n2!\n;\n\u0012@lnf\n@p?\u0013\nj+1\n2;k\u00001\n2=1\n2 \nln(jfj;kj+\u000fl)\u0000ln (jfj;k\u00001j+\u000fl)\n\u0001p?;k\u00001\n2+ln (jfj+1;kj+\u000fl)\u0000ln (jfj+1;k\u00001j+\u000fl)\n\u0001p?;k\u00001\n2!\n;\n\u0012@lnf\n@p?\u0013\nj\u00001\n2;k+1\n2=1\n2 \nln(jfj\u00001;k+1j+\u000fl)\u0000ln (jfj\u00001;kj+\u000fl)\n\u0001p?;k+1\n2+ln (jfj;k+1j+\u000fl)\u0000ln (jfj;kj+\u000fl)\n\u0001p?;k+1\n2!\n;\n\u0012@lnf\n@p?\u0013\nj\u00001\n2;k\u00001\n2=1\n2 \nln(jfj\u00001;kj+\u000fl)\u0000ln (jfj\u00001;k\u00001j+\u000fl)\n\u0001p?;k\u00001\n2+ln (jfj;kj+\u000fl)\u0000ln (jfj;k\u00001j+\u000fl)\n\u0001p?;k\u00001\n2!\n;\nwhere\u000fl= 10\u000030is added to mollify singularities. Once computed at the cell centers, the friction coefficients\nat the cell faces are found by linear averaging.\nB Solution of singular integrals in relativistic potentials\nB1. Solution of first singular integral\nWe seek a solution of the integral:\nI=Z2\u0019\n0d\u001ep\nr2\u00001; (30)\nwith:\nr=p\n(1 +p2)(1 + (p0)2)\u0000p\u0001p0=p\n(1 +p2)(1 + (p0)2)\u0000pkp0\nk|{z }\na2\u0000p?p0\n?|{z}\nb2cos\u001e=a2\u0000b2cos \b:\nWe consider the case of b2>0. Noter2\u00001 = (r+ 1)(r\u00001). Sincer\u00151, it follows that:\na2\u0015b2+ 1: (31)\nTo begin, we consider the change of variable t= cos\u001e. We consider the following cases:\nt= cos\u001e ; \u001e2[0;\u0019\n2];\u001e2[3\u0019\n2;2\u0019] ;d\u001e=\u0000dtp\n1\u0000t2\nt=\u0000cos\u001e ; \u001e2[\u0019\n2;3\u0019\n2] ;d\u001e=dtp\n1\u0000t2:\n21This gives:\nI= 2Z1\n0dtp\n(1\u0000t2)(a2+ 1\u0000b2t)(a2\u00001\u0000b2t)| {z }\nI1+2Z1\n0dtp\n(1\u0000t2)(a2+ 1 +b2t)(a2\u00001 +b2t)| {z }\nI2:\nSolution of I2integral\nWe begin with the integral I2. We follow Abramowitz & Stegun [24], and consider the polynomials:\nQ1= 1\u0000t2;\nQ2= (a2+ 1 +b2t)(a2\u00001 +b2t):\nThese polynomials have real roots \u00061,\u0000a2+1\nb2,\u0000a2\u00001\nb2. Because of Eq. (31), it is apparent that the last\ntwo roots are\u0014\u00001, and hence Q1andQ2do not have nested roots. In this case, one can consider the\ntransformation to the canonical forms of the elliptic integrals by constructing the polynomial:\nQ1\u0000\u0015Q2=\u0000(1 +\u0015b4)t2\u00002b2a2\u0015t+ 1\u0000\u0015(a4\u00001): (32)\nSeeking a zero discriminant for the quadratic form in tgives the following value for \u0015:\nb4a4\u00152= (\u0015(a4\u00001)\u00001)(1 +\u0015b4))\u00152b4\u0000\u0015(a4\u0000b4\u00001) + 1 = 0\n)\u0015\u0006=(a4\u0000b4\u00001)\u0006p\n(a4\u0000b4\u00001)2\u00004b4\n2b4: (33)\nNote that these roots are real and semi-positive, since, by Eq. (31):\n(a4\u0000b4\u00001)\u00152b2:\nAlso, it is clear that\n\u0015+>\u0015\u0000>0; (34)\nand that:\n\u0015+\u0015\u0000=1\nb4: (35)\nSince the discriminant for Eq. (32) vanishes, it follows that the roots of Q1\u0000\u0015Q2are perfect squares and\nare given by:\nt=\u0000t\u0006;t\u0006=\u0015\u0006b2a2\n1 +\u0015\u0006b4: (36)\nTherefore:\nQ1\u0000\u0015+Q2=\u0000(1 +\u0015b4)(t+t+)2; (37)\nQ1\u0000\u0015\u0000Q2=\u0000(1 +\u0015b4)(t+t\u0000)2: (38)\nAt this point, it is useful to point out a few properties of the roots t\u0006in Eq. (36). Firstly, from Eq. (34)\nit follows that:\nt+>t\u0000: (39)\nSecondly, from the polynomial in Eq. (32) and the properties of the quadratic equations, we can write:\nt2\n\u0006=\u0015\u0006(a4\u00001)\u00001\n1 +\u0015\u0006b4; (40)\nwhich can be used to prove that:\nt\u0000\u00141 (41)\n22(needed for later) as follows:\nt2\n\u0000\u00141,\u0015\u0000(a4\u0000b4\u00001)|{z}\n\u00152b2<2,\u0015\u0000\u00141=b2; (42)\nwhich can be shown to be true when noting that:\n(a4\u0000b4\u00001)2\u00004b4= (a4\u0000b4\u00001\u00002b2)(a4\u0000b4\u00001 + 2b2)\u0015(a4\u0000b4\u00001\u00002b2)2:\nThe inequality follows from Eq. (33). Finally, using Eq. (42) and the inequality above, we can also readily\nprove that:\nt+=\u0015+b2a2\n1 +\u0015+b4=a2\nb21\n\u0015\u0000+ 1\u0015a2\nb2+ 1\u00151; (43)\nwhich will be important later.\nEqs (37, 38) can be solved for Q1andQ2as follows:\nQ2=a2+(t+t+)2\u0000a2\u0000(t+t\u0000)2;\nQ1=a1+(t+t+)2\u0000a1\u0000(t+t\u0000)2:\nHere:\na2\u0006=1 +\u0015\u0006b4\n\u0015+\u0000\u0015\u0000;a1\u0006=\u0015\u0007(1 +\u0015\u0006b4)\n\u0015+\u0000\u0015\u0000:\nNote that:\n\u000fa2+>a2\u0000(from Eq. 34).\n\u000fa1+=a1\u0000=t\u0000=t+<1(from Eq. 39).\nFrom the expressions of Q1,Q2, one can write:\nQ1Q2= (t+t+)4\u0014\na2+\u0000a2\u0000(t+t\u0000)2\n(t+t+)2\u0015\u0014\na1+\u0000a1\u0000(t+t\u0000)2\n(t+t+)2\u0015\n:\nFollowing Ref. [24], we postulate the change of variables:\nw=(t+t\u0000)\n(t+t+))dw=t+\u0000t\u0000\n(t++t)2dt:\nHence:\nI2=Z1\n0dtpQ1Q2=1\nt+\u0000t\u0000Zw1\nw0dwp\n[a2+\u0000a2\u0000w2] [a1+\u0000a1\u0000w2]: (44)\nHere:\nw0=t\u0000\nt+<1 ;w1=1 +t\u0000\n1 +t+;w0<w 1<1:\nThe result in Eq. (44) can be written as a canonical elliptic integral by considering:\na1+\na1\u0000=t\u0000\nt+=w0=e2<1 ;a2+\na2\u0000=1 +\u0015+b4\n1 +\u0015\u0000b4=d2>1; (45)\nto find:\nI2=Z1\n0dtpQ1Q2=1\n(t+\u0000t\u0000)pa2\u0000a1\u0000Ze\ne2dwp\n[d2\u0000w2] [e2\u0000w2]:\nHere, we have used the surprising property that:\nw2\n1=\u00121 +t\u0000\n1 +t+\u00132\n=t\u0000\nt+=e2)w1=e;\nwhich can be demonstrated by using the definition of t\u0006(Eq. 36) and t2\n\u0006(Eq. 40).\n23Solution of I1integral\nThe solution of the integral I1follows a similar development, except now:\nQ1= 1\u0000t2;\nQ2= (a2+ 1\u0000b2t)(a2\u00001\u0000b2t):\nWith these definitions, it can be shown that the discriminant of the combination Q1\u0000\u0015Q2is exactly the\nsame, and therefore so are the solutions \u0015\u0006. However the roots in tnow have opposite signs:\nt=t\u0006;t\u0006=\u0015\u0006b2a2\n1 +\u0015\u0006b4; (46)\nand the factorization of Q1;2reads:\nQ2=a2+(t\u0000t+)2\u0000a2\u0000(t\u0000t\u0000)2;\nQ1=a1+(t\u0000t+)2\u0000a1\u0000(t\u0000t\u0000)2:\nFrom the expressions of Q1,Q2, one can write:\nQ1Q2= (t\u0000t+)4\u0014\na2+\u0000a2\u0000(t\u0000t\u0000)2\n(t\u0000t+)2\u0015\u0014\na1+\u0000a1\u0000(t\u0000t\u0000)2\n(t\u0000t+)2\u0015\n:\nFollowing Ref. [24], we postulate the change of variables:\nw=(t\u0000t\u0000)\n(t+\u0000t))dw=t+\u0000t\u0000\n(t+\u0000t)2dt:\nWhen postulating this change of variables, we have taken into account the fact that t\u00141< t+(Eq. 43),\nand thatt\u0000<1(Eq. 41). It follows that:\nI1=Z1\n0dtpQ1Q2=1\nt+\u0000t\u0000Zw1\nw0dwp\n[a2+\u0000a2\u0000w2] [a1+\u0000a1\u0000w2]; (47)\nwhere:\nw0=\u0000t\u0000\nt+=\u0000e2<0 ;w1=1\u0000t\u0000\nt+\u00001>0:\nAs before, one can readily prove that:\nw2\n1=\u00121\u0000t\u0000\nt+\u00001\u00132\n=t\u0000\nt+=e2;\nand therefore w1=e. There results:\nI1=Z1\n0dtpQ1Q2=1\n(t+\u0000t\u0000)pa2\u0000a1\u0000Ze\n\u0000e2dwp\n[d2\u0000w2] [e2\u0000w2]: (48)\nSolution of total integral I\nWhen combining these two solutions, we find:\nI= 2(I1+I2) =2\n(t+\u0000t\u0000)pa2\u0000a1\u0000\"Ze\ne2dwp\n[d2\u0000w2] [e2\u0000w2]+Ze\n\u0000e2dwp\n[d2\u0000w2] [e2\u0000w2]#\n=2\n(t+\u0000t\u0000)pa2\u0000a1\u0000\u0014Ze\ne2+Ze\n0+Z0\n\u0000e2\u0015\n=2\n(t+\u0000t\u0000)pa2\u0000a1\u0000\"Ze\ne2+Ze\n0+Ze2\n0#\n=4\n(t+\u0000t\u0000)pa2\u0000a1\u0000Ze\n0dwp\n[d2\u0000w2] [e2\u0000w2];\n24which can be written in terms of the complete elliptic integral of the second kind as [24]:\nI=4K(m)\nd(t+\u0000t\u0000)pa2\u0000a1\u0000=4K(m)\n(t+\u0000t\u0000)pa2+a1\u0000;\nwhere in the last step we have used the definition of d(Eq. 45), and where:\nm=e2=d2:\nB2. Solution of second singular integral\nIn the previous section, we determined the root structure of the radicand and removed the odd terms in\nthe radicand. We employ this approach and also use ideas from Ref. [25] to express the following elliptic\nintegral,\nH=Z2\u0019\nord\u001ep\nr2\u00001;\nin terms of Legendre’s elliptic functions. Recall that\nr=a2\u0000b2cos\u001e:\nThe integral can thus be expressed as,\nH= 2Z1\n0(a2\u0000b2t)dtp\n(1\u0000t2)(a2+ 1\u0000b2t)(a2\u00001\u0000b2t)+ 2Z1\n0(a2+b2t)dtp\n(1\u0000t2)(a2+ 1 +b2t)(a2\u00001 +b2t):\nWe can regroup as we know from xB1 the solution when the numerator is unity,\nH=a2I+ 2Z1\n0\u0000b2tdtp\n(1\u0000t2)(a2+ 1\u0000b2t)(a2\u00001\u0000b2t)| {z }\nH1+2Z1\n0b2tdtp\n(1\u0000t2)(a2+ 1 +b2t)(a2\u00001 +b2t)| {z }\nH2:(49)\nRemoving the odd terms in the radicand, we obtain,\nH1=Z1\n0\u0000b2tdtpQ1Q2=\u0000b2\n(t+\u0000t\u0000)pa2\u0000a1\u0000Ze\n\u0000e2(wt++t\u0000)=(1 +w)dwp\n[d2\u0000w2] [e2\u0000w2]\n=\u0000b2\n(t+\u0000t\u0000)pa2\u0000a1\u0000Ze\n\u0000e2R1(w)dwp\n[d2\u0000w2] [e2\u0000w2];\nand\nH2=Z1\n0b2tdtpQ1Q2=b2\n(t+\u0000t\u0000)pa2\u0000a1\u0000Ze\ne2(\u0000wt++t\u0000)=(\u00001 +w)dwp\n[d2\u0000w2] [e2\u0000w2]\n=b2\n(t+\u0000t\u0000)pa2\u0000a1\u0000Ze\ne2R2(w)dwp\n[d2\u0000w2] [e2\u0000w2]:\nThe rational functions of w,R1andR2, can be expressed in terms of odd and even functions. This is because\nthe odd term can be simplified into elementary functions via trigonometric substitutions, see Ref. [25].\nR1(w) =w\n1\u0000w2(t+\u0000t\u0000) +t\u0000\u0000w2t+\n1\u0000w2\nR2(w) =w\n1\u0000w2(t+\u0000t\u0000)\u0000t\u0000\u0000w2t+\n1\u0000w2\n25However, in our case we observe that these terms cancel each other. Examining the odd terms in R1and\nR2, and adding together their contribution to H, we find:\nHodd\n1+Hodd\n2=\u0000Ze\n\u0000e2+Ze\ne2=\u0000Ze2\n\u0000e2= 0;\nas the odd function is asymmetric about the origin. The even term can be further factorized into:\nt\u0000\u0000w2t+\n1\u0000w2=t+\u0000t+\u0000t\u0000\n1\u0000w2:\nPutting the above expression into H1andH2, the contributions of the even terms may be expressed as,\nH1+H2=\u0000b2t+\n(t+\u0000t\u0000)pa2\u0000a1\u0000 Ze\n\u0000e2dwp\n[d2\u0000w2] [e2\u0000w2]+Ze\ne2dwp\n[d2\u0000w2] [e2\u0000w2]!\n+b2\npa2\u0000a1\u0000 Ze\n\u0000e2dw\n(1\u0000w2)p\n[d2\u0000w2] [e2\u0000w2]+Ze\ne2dw\n(1\u0000w2)p\n[d2\u0000w2] [e2\u0000w2]!\nSimplifying and regrouping, we obtain the expression for H, see Eq. (49) as,\nH= (a2\u0000b2t+)I+4b2\npa2+a1\u0000\u0005(e2;m) (50)\nwhere \u0005is the complete elliptic integral of the third kind, with e2<1. This formula has been verified\nnumerically. Also in the above, we made use of the following step which was described earlier in the previous\nsection. It is as follows,\nZe\n\u0000e2+Ze\ne2=Z0\n\u0000e2+2Ze\n0\u0000Ze2\n0\nThen substituting t=w=eto get the final form (50). The complete elliptic integral of the third kind is given\nby,\n\u0005(e2;m) =Z1\n0dt\n(1\u0000e2t)p\n(1\u0000t2)(1\u0000mt2):\nReferences\n[1] ST Beliaev and GI Budker. The relativistic kinetic equation. In Soviet Physics Doklady , volume 1, page\n218, 1956.\n[2] Marshall N Rosenbluth, William M MacDonald, and David L Judd. Fokker-planck equation for an\ninverse-square force. Physical Review , 107(1):1, 1957.\n[3] Bastiaan J Braams and Charles FF Karney. Conductivity of a relativistic plasma. Physics of Fluids B:\nPlasma Physics , 1(7):1355–1368, 1989.\n[4] Adam Stahl, Matt Landreman, O Embréus, and Tünde Fülöp. Norse: A solver for the relativistic\nnon-linear fokker–planck equation for electrons in a homogeneous plasma. Computer Physics Commu-\nnications , 212:269–279, 2017.\n[5] RW Harvey, VS Chan, SC Chiu, TE Evans, MN Rosenbluth, and DG Whyte. Runaway electron pro-\nduction in diii-d killer pellet experiments, calculated with the cql3d/kprad model. Physics of Plasmas ,\n7(11):4590–4599, 2000.\n[6] Y Petrov and R W Harvey. Benchmarking the fully relativistic collision operator in CQL3D . CompX\nreport CompX-2009-1, 2009.\n26[7] Leslie Colin Woods. Theory of tokamak transport: new aspects for nuclear fusion reactor design . John\nWiley & Sons, 2006.\n[8] JW Connor and RJ Hastie. Relativistic limitations on runaway electrons. Nuclear fusion , 15(3):415,\n1975.\n[9] William T Taitano, Luis Chacón, AN Simakov, and K Molvig. A mass, momentum, and energy conserv-\ning, fully implicit, scalable algorithm for the multi-dimensional, multi-species rosenbluth–fokker–planck\nequation. Journal of Computational Physics , 297:357–380, 2015.\n[10] Bastiaan J Braams and Charles FF Karney. Differential form of the collision integral for a relativistic\nplasma.Physical review letters , 59(16):1817, 1987.\n[11] Eero Hirvijoki. Conservative finite-element method for the relativistic coulomb collision operator. arXiv\npreprint arXiv:1903.07403 , 2019.\n[12] Takashi Shiroto and Yasuhiko Sentoku. Structure-preserving strategy for conservative simulation of\nrelativistic nonlinear landau–fokker–planck equation. arXiv preprint arXiv:1902.07866 , 2019.\n[13] Youcef Saad and Martin H Schultz. Gmres: A generalized minimal residual algorithm for solving\nnonsymmetric linear systems. SIAM Journal on scientific and statistical computing , 7(3):856–869, 1986.\n[14] Rick Beatson and Leslie Greengard. A short course on fast multipole methods. Wavelets,multilevel\nmethods and elliptic PDEs , 1:1–37, 1997.\n[15] J Decker, E Hirvijoki, O Embreus, Y Peysson, A Stahl, I Pusztai, and T Fülöp. Numerical char-\nacterization of bump formation in the runaway electron tail. Plasma Physics and Controlled Fusion ,\n58(2):025016, 2016.\n[16] Zehua Guo, Christopher J McDevitt, and Xian-Zhu Tang. Phase-space dynamics of runaway electrons\nin magnetic fields. Plasma Physics and Controlled Fusion , 59(4):044003, 2017.\n[17] PH Gaskell and AKC Lau. Curvature-compensated convective transport: Smart, a new boundedness-\npreserving transport algorithm. International Journal for numerical methods in fluids , 8(6):617–641,\n1988.\n[18] Erasmus J Du Toit, Martin R O‘Brien, and Roddy GL Vann. Positivity-preserving scheme for two-\ndimensional advection–diffusion equations including mixed derivatives. Computer Physics Communica-\ntions, 228:61–68, 2018.\n[19] DonaldGAnderson. Iterativeproceduresfornonlinearintegralequations. Journal of the ACM (JACM) ,\n12(4):547–560, 1965.\n[20] Homer F Walker and Peng Ni. Anderson acceleration for fixed-point iterations. SIAM Journal on\nNumerical Analysis , 49(4):1715–1735, 2011.\n[21] L Chacón, DC Barnes, DA Knoll, and GH Miley. An implicit energy-conservative 2d fokker–planck\nalgorithm: Ii. jacobian-free newton–krylov solver. Journal of Computational Physics , 157(2):654–682,\n2000.\n[22] Youcef Saad and Martin H Schultz. Gmres: A generalized minimal residual algorithm for solving\nnonsymmetric linear systems. SIAM Journal on scientific and statistical computing , 7(3):856–869, 1986.\n[23] SM Weng, Zheng-Ming Sheng, MQ He, J Zhang, PA Norreys, M Sherlock, and APL Robinson. Plasma\ncurrents and electron distribution functions under a dc electric field of arbitrary strength. Physical\nreview letters , 100(18):185001, 2008.\n[24] Milton Abramowitz and Irene A Stegun. Handbook of mathematical functions: with formulas, graphs,\nand mathematical tables , volume 55. Courier Corporation, 1965.\n[25] George Labahn and Mark Mutrie. Reduction of elliptic integrals to legendre normal form . University of\nWaterloo, Computer Science Department, 1997.\n27"    },    {        "title": "1401.2829v1.NLSE_for_quantum_plasmas_with_the_radiation_damping.pdf",        "content": "arXiv:1401.2829v1  [physics.plasm-ph]  13 Jan 2014NLSE for quantum plasmas with the radiation damping\nPavel A. Andreev∗\nM. V. Lomonosov Moscow State University, Moscow, Russian Fe deration.\n(Dated: February 17, 2018)\nWe consider contribution of the radiation damping in the qua ntum hydrodynamic equations for\nspinless particles. We discuss possibility of obtaining of corresponding non-linear Schrodinger equa-\ntion (NLSE) for the macroscopic wave function. We compare co ntribution of the radiation damping\nwith weakly (or semi-) relativistic effects appearing in the second order by v/c. The radiation\ndamping appears in the third order by v/c. So it might be small er than weakly relativistic effects,\nbut it gives damping of the Langmuir waves which can be consid erable.\nPACS numbers: 52.35.-g, 52.30.Ex, 52.27.Ny\nKeywords: Langmuir waves, quantum hydrodynamics, radiati on damping, semi-relativistic effects\nI. INTRODUCTION\nSome relativistic effects havebeen consideredin classic\nandquantumplasmas[1]-[10]. Inmanycasesone-particle\nSchrodinger, Pauli, Klein-Gordon, and Dirac equations\nhave been used to derive set of quantum hydrodynamic\nequations [10], [11], [12], [13]. Some collective effects\nin classic relativistic plasmas with the radiation damp-\ning were considered in Ref. [14]. A method of rigorous\nderivation of non-relativistic and semi-relativistic hydro-\ndynamicequationsfrommany-particleSchrodingerequa-\ntion was suggested in Ref. [15] and developed in Refs.\n[8], [9], [16], [17]. Semi-relativistic effects appears in the\nsecond order by the parameter v/cshowing ratio of the\nparticlevelocity vto thespeed oflight c. In thispaperwe\nare interested in contribution of the radiation damping\nin the evolution of quantum plasmas working in terms of\nquantum hydrodynamics (QHD). This effect appears in\nthe third order by v/c, when electromagnetic radiation\nof particles arises in theory. So system can not be de-\nscribed in terms of Hamiltonian of particles, the electro-\nmagnetic field have to be explicitly accounted. However\nthe method of classic hydrodynamic derivation suggested\nin Refs. [18], [19] does not apply Hamiltonian descrip-\ntion using the Newton equations. Recent applications\nand discussions of this method can be found in Refs. [4]\nand [20]. Hence this method gives possibility to derive\nhydrodynamics with the radiation damping. Comparing\nfinal equations with similar QHD equation we make gen-\neralization of obtained equation on quantum plasmas.\nHydrodynamic equations appears as natural represen-\ntation of classic and quantum dynamics of many-particle\nsystems. In some cases the set of continuity and Euler\nequations including the quantum Bohm potential can be\nrepresented as non-linear Schrodinger equation (NLSE)\nformacroscopicwavefunctiondefinedintermsofthepar-\nticle concentration and the potential of velocity field [15],\n[21]. Famous examples of NLSE are the Gross-Pitaevskii\nequation for Bose-Einstein condensates of neutral atoms\n∗Electronic address: andreevpa@physics.msu.ruandGinsburg-Landauequationforsuperconductors. Dif-\nferent methods for dealing with NLSEs have been de-\nveloped, hence they can be used for weakly relativistic\nquantum plasmas with the radiation damping as well.\nWe gave an old example of studying of the radiation\ndamping in plasmas [14]. However it is a topic under\ndiscussion in recent papers as well. For instance laser-\nplasma interactions in ultra-relativistic regime were con-\nsidered in Ref. [22] to calculate the nonlinear dielectric\npermittivity, ponderomotive and dissipative forces act-\ning in plasmas. The motion of a cold electron fluid ac-\ncounting for the radiation reaction force in the Lorentz-\nAbraham-Dirac form was used (see formula 3). Some\nradiative and quantum electrodynamics effects were nu-\nmerically modeled for ultra-relativistic laser-plasma in-\nteractions as well [23]. The scalar and spinor quantum\nelectrodynamics in the presence of strong laser fields in\nplasmas were considered in Ref. [24].\nII. CONSTRUCTION OF MACROSCOPIC\nEQUATIONS\nMicroscopic classic motion of each particle in plasmas\nobeys the Newton equation\nmi˙vi=Fi, (1)\nwhereiis the number of particle and force acting on the\nparticle includes the radiation damping fialong with the\nLorentz force\nFi=qiEi+qi\nc[vi,Bi]+fi, (2)\nwhereEiandBiareelectricandmagneticfieldsactingon\nith particle and creating by other particles of the system,\nmiare masses of particles, qiare charges of particles, vi\nis the velocity of particles, and cis the speed of light.\nNon-relativistic force for the radiation damping ap-\npears as\nfi≈2q2\ni\n3c3¨vi≈2q3\ni\n3mic3˙Ei+2q4\ni\n3m2\nic4[Ei,Bi] (3)2\n(see Ref. [25] section 9, paragraph 75, formula 75.8).\nThe second identity in formula (3) has been made taking\nderivative of the acceleration with respect to time and\nneglecting ˙f. In the second term of the Lorentz force\nwe have used ˙v=eE, and we have not considered time\nderivative of the magnetic field. Doing these approxi-\nmations we keep considering terms in the third order on\nv/c.\nAstheresultofmanipulationsdescribedabovewehave\nan approximate equations for classic motion of electrons\nwith the radiation damping. We can use it as framework\nto derive classic hydrodynamic equations describing col-\nlective evolution of considering system. To this end we\nuse method suggested in Ref. [18] and briefly reviewed\nin Ref. [20]. This method gives the following continuity\nand Euler equations\n∂tn+∇(nv) = 0 (4)\nand\nmn(∂t+v∇)v+∇p=qnE\n+q\ncn[v,B]+2q3\n3mc3˙E+2q4\n3m2c4[E,B] (5)\nwritten in the self-consistent field approximation allow-\ning to truncate the chain of HD equations at using of an\nequation of state for the thermal pressure. In equations\n(4) and (5) we have next physical quantities nis the par-\nticle concentration, vis the velocity field, ∂tand∇are\nthe time and spatial derivatives, and pis the thermal\npressure. Different terms in equations (4) and (5) have\nfollowing meaning. The continuity equation (4) shows\nconservation of the particles number and gives time evo-\nlution of the particle concentration. The Euler equation\n(5) is the equation of particles current (particle flux) evo-\nlution. In non-relativistic systems it coincides with the\nequation for the momentum current (or the momentum\nflux). In fact we should present all weakly relativistic\nterms in equation (5), but we do not want to present\nrather huge equation, for the weakly relativistic effects in\nquantum hydrodynamics see Refs. [8], [9]. These equa-\ntions are coupled with the equations of field\n∇E(r,t) = 4π/summationdisplay\naqana(r,t), (6)\n∇×E(r,t) = 0, (7)\n∇B(r,t) = 0, (8)\nand\n∇×B(r,t) =4π\nc/summationdisplay\naqana(r,t)va(r,t).(9)\nWe do not have time derivatives of field here, since only\ntrace of radiation in the approximation under considera-\ntion is the radiation damping.We see a force field caused by the radiation damping.\nComparingclassicandquantumhydrodynamicequations\nwe can get contribution of the radiation damping in the\nset of QHD equations. The quantum Euler equation dif-\nfers from the classic one by the quantum Bohm potential\nand has following form\nmn(∂t+v∇)v+∇p\n−¯h2\n2m∇/parenleftBigg\n△n\nn−(∇n)2\n2n2/parenrightBigg\n=qnE\n+q\ncn[v,B]+2q3\n3mc3˙E+2q4\n3m2c4[E,B].(10)\nConsidering quantum plasmas of charged Fermi particles\nwe should consider the Fermi pressure instead of thermal\npressure.\nHaving equations (4) and (10) we can derive a non-\nlinear Schrodinger equation for the macroscopic wave\nfunction [17] neglecting by the magnetic field and assum-\ning that the electric field is potential E=−∇ϕ\nΦ =√nexp/parenleftbiggı\n¯hmφ/parenrightbigg\n, (11)\nwhereφis the potential of the velocity field v=∇φ.\nConsidering the potential electric field we can intro-\nduce an effective potential electric field including contri-\nbution of the radiation damping\n/tildewideE=−∇/tildewideϕ=−/parenleftbigg\n1+2e2\n3mc3∂t/parenrightbigg\n∇ϕ. (12)\nDifferentiating the macroscopic wave function (11) with\nrespect to time we obtain a non-linear Schrodinger equa-\ntion\nı¯h∂tΦ(r,t) =/parenleftbigg\n−¯h2∇2\n2m+(3π2)2\n3¯h2\n2mn2\n3(r,t)+q/tildewideϕ/parenrightbigg\nΦ(r,t)\n(13)\nwithn=|Φ|2.\nWriting explicit form of the potential of effective elec-\ntric field we represent the NLSE as follows\nı¯h∂tΦ(r,t) =/parenleftbigg\n−¯h2∇2\n2m\n+(3π2)2\n3¯h2\n2mn2\n3(r,t)+qϕ+2q3\n3mc3∂tϕ/parenrightbigg\nΦ(r,t),(14)\nwhere the potential of electric field depends on the parti-\ncle concentration via the Maxwell equations (8), (9), and\nconsequently it depends on the macroscopic wave func-\ntion. Hence we have closed set of equations consisting of\nthe NLSE and the Maxwell equations.3\nThe dispersion dependence of semi-relativistic Lang-\nmuir waves in absence of the radiation damping was ob-\ntained in Ref. [8]. It has the following form\nω2\nSR(k) =ω2\nLe/parenleftbigg\n1−ς¯h2k2\n4m2c2−5T\n2mc2/parenrightbigg\n+¯h2k4\n4m2−¯h4k6\n8m4c2+γT0\nmk2, (15)\nwhere we included shifts of the Langmuir frequency by\nboth the semi-relativistic part of kinetic energy and the\nDarwin interaction presented by term proportional to ς.\nSimultaneous account of both effects gives ς= 0, for\ndetailsseeRef. [8]. Thelastpartofthefirsttermappears\nasconsequenceofthethermal -semi-relativisticforcefield.\nOtherterms havefollowingmeaning: the quantum Bohm\npotential, thesemi-relativisticpartofthequantumBohm\npotential, and the contribution of the thermal pressure.\nIn linear approximation on small perturbations of hy-\ndrodynamic variables δn=Nexp(−iωt+ikr) andδv=\nUexp(−iωt+ikr)wecangetadispersionequationgiving\nω(k)\nω2+ıλω−Ω2= 0. (16)\nDispersion dependence ω(k) arises as\nω=1\n2/parenleftbigg\n−ıλ±2Ω/radicalbigg\n1−λ2\n4Ω2/parenrightbigg\n, (17)\nincluding weakly-relativistic nature of the radiation\ndamping we can give approximate formula for the dis-\npersion dependence\nω≈Ω/bracketleftbigg\n1−λ2\n8Ω2−ıλ\n2Ω/bracketrightbigg\n(18)\ncontaining\nλ=8πe4n0\n3m2c3=2\n3ω2\nLere\nc, (19)\nwherere≡e2/(mc2) is the classic radius of electron, and\nωLeis the Langmuir frequency, and\nΩ2=1\nm∂p\n∂nk2+¯h2k4\n4m2+ω2\nLe (20)with∂p\n∂n=γT0, andγ= 3.\nIn formula (18) we can see imaginary part of the fre-\nquency, which is negative and gives to damping of the\nsemi-relativistic quantum Langmuir waves.\nPresence of the semi-relativistic effects given by for-\nmula (15) changes Ω, but it does not affect structure of\nsolution (17).\nProper consideration based on full spinless semi-\nrelativistic theory [8] gives ω2\nSR(k) (15) instead of Ω. Fi-\nnally we can rewrite formula (18) as follows\nReω=ωSR(k)−1\n18r2\ne\nc2ω4\nLe/radicalBig\nω2\nLe+3T0\nmk2+¯h2k4\n4m2,(21)\nand\nImω=−1\n3ω2\nLere\nc. (22)\nFormula shows that the damping of Langmuir waves\ncaused by the radiation damping do not contain contri-\nbution of quantum effects.\nIII. CONCLUSION\nWe have considered weakly relativistic evolution of\nquantum plasmas including radiation damping. We have\nderived corresponding equations of quantum hydrody-\nnamics. Solving these equations we obtained spectrum\nof the Langmuir waves. We have found that the radia-\ntion damping gives a shift of the real part of frequency\nof the Langmuir waves, but it also leads to damping of\nthe waves. We should mention that this damping is not\naffected by the quantum effects in considered approxima-\ntion.\nWe have shown that the QHD equations with the ra-\ndiation damping can be represented in form of a NLSE\nfor the wave function in medium. This equation may\nget its place in list of the Ginsburg-Landau and Gross-\nPitaevskii equations. Each of them has been very useful\nin their own fields. Therefore the NLSE derived in this\npaper opens similar possibilities for quantum plasmas.\nObtained in this paper equations allows to study dif-\nferent effects in the weakly relativistic quantum plasmas\nwith the radiation damping.\n[1] F. A. Asenjo, F. A. Borotto, Abraham C.-L. Chian, V.\nMunoz, J. A. Valdivia, E. L. Rempel, Phys. Rev. E 85,\n046406 (2012).\n[2] R. A. Lopez, F. A. Asenjo, V. Munoz, A. C.-L. Chian,\nand J. A. Valdivia, Phys. Rev. E 88, 023105 (2013).\n[3] C. Ruyer, L. Gremillet, D. Benisti, and G. Bonnaud,\nPhys. Plasmas 20, 112104 (2013).\n[4] P. A. Andreev, arXiv:1208.0998.[5] S. M. Mahajan, Phys. Rev. Lett. 90, 035001 (2003).\n[6] S. M. Mahajan, Z. Yoshida, Phys. Plasm. 18, 055701\n(2011).\n[7] F. A. Asenjo, J. Zamanian, M. Marklund, G. Brodin, and\nP. Johansson, New J. Phys. 14, 073042 (2012).\n[8] A. Yu. Ivanov, P. A. Andreev, L. S. Kuz’menkov, arXiv:\n1209.6124.\n[9] A. Yu. Ivanov and P. A. Andreev, Russ. Phys. J. 56, 3254\n(2013).\n[10] F. Haas, B. Eliasson, P. K. Shukla, Phys. Rev. E 85,\n056411 (2012).\n[11] T. Takabayashi, Progr. Theor. Phys. 14, 283 (1955).\n[12] P. K. Shukla, B. Eliasson, Rev. Mod. Phys. 83, 885\n(2011).\n[13] F. A. Asenjo, V. Munoz, J. A. Valdivia, S. M. Mahajan,\nPhys. Plasm. 18, 012107 (2011).\n[14] L. S. Kuz’menkov and P. A. Polyakov, Sov. Phys. JETP\n55, 82 (1982).\n[15] L. S. Kuzmenkov, S. G. Maksimov, Theor. Math. Phys.\n118, N20, 227 (1999).\n[16] P. A. Andreev, L. S. Kuz’menkov, Rus. J. Phys. 50, N12,\n1251 (2007).\n[17] P. A. Andreev, L. S. Kuzmenkov, M. I. Trukhanova,\nPhys. Rev. B 84, 245401 (2011).[18] M. A. Drofa, L. S. Kuzmenkov, Theor. Math. Phys. 108,\nN1, 849 (1996).\n[19] L.S.Kuz’menkov,Theoretical andMathematical Physic s\n86, 159 (1991).\n[20] P. A. Andreev and L. S. Kuzmenkov, PIERS Proceed-\nings, Moscow, Russia, August 19-23, p. 158 (2012).\n[21] P. A. Andreev, L. S. Kuz’menkov, Phys. Rev. A 78,\n053624 (2008).\n[22] A. V. Bashinov and A. V. Kim, arXiv:1309.5811.\n[23] M. Lobet, E. d’Humleres, M. Grech, C. Ruyer, X.\nDavoine, L. Gremillet, arXiv:1311.1107.\n[24] E. Raicher, S. Eliezer, and A. Zigler, arXiv:1312.3088\n[25] L. D. Landau and E. M. Lifshitz, The Classical Theory\nof Fields (Butterworth-Heinemann, 1975)."    },    {        "title": "1907.04499v1.Determination_of_the_damping_co_efficient_of_electrons_in_optically_transparent_glasses_at_the_true_resonance_frequency_in_the_ultraviolet_from_an_analysis_of_the_Lorentz_Maxwell_model_of_dispersion.pdf",        "content": "Determination of the damping coe\u000ecient of\nelectrons in optically transparent glasses at the\ntrue resonance frequency in the ultraviolet from\nan analysis of the Lorentz-Maxwell model of\ndispersion\nSurajit Chakrabarti\n(Ramakrishna Mission Vidyamandira)\nHowrah, India\nThe Lorentz-Maxwell model of dispersion of light has been analyzed in this paper\nto determine the true resonance frequency in the ultraviolet for the electrons in\noptically transparent glasses and the damping coe\u000ecient at this frequency. For\nthis we needed the refractive indices of glass in the optical frequency range. We\nargue that the true resonance condition in the absorption region prevails when\nthe frequency at which the absorption coe\u000ecient is maximum is the same as the\nfrequency at which the average energy per cycle of the electrons is also a max-\nimum. We have simultaneously solved the two equations obtained from the two\nmaxima conditions numerically to arrive at a unique solution for the true resonance\nfrequency and the damping coe\u000ecient at this frequency. Assuming the damping\ncoe\u000ecient to be constant over a small frequency range in the absorption region,\nwe have determined the frequencies at which the extinction coe\u000ecient and the re-\n\rectance are maxima. These frequencies match very well with the published data\nfor silica glasses available from the literature.\n1arXiv:1907.04499v1  [physics.optics]  10 Jul 20191 Introduction\nThe Lorentz-Maxwell model of dispersion of electromagnetic waves in matter is\nvery successful in describing the properties of matter under the action of electro-\nmagnetic waves over its whole spectrum where the wavelength is large compared to\nthe interatomic distances. The model is generally studied in the optical frequency\nrange where only the oscillation of electrons bound to atoms and molecules is rel-\nevant for the study of dispersion. Two important parameters of the model namely\nthe natural oscillation frequency and the plasma frequency of the electrons in a\ndielectric medium like glass can be easily determined from the refractive indices\nof a glass prism measured in the optical band [1] where glass is transparent. In\na condensed system like glass one has to include the e\u000bect of the local \feld on\nthe electrons apart from the \feld of the incident wave. This leads to another fre-\nquency which is conventionally known as the resonance frequency and is related to\nthe plasma and the natural oscillation frequencies of the electron [2]. Though it is\ncalled the resonance frequency, there is no proof that the absorption coe\u000ecient is\nmaximum at this frequency.\nIn order to study the absorption of EM waves in matter, a phenomenologi-\ncal variable called the damping coe\u000ecient is introduced in the Lorentz-Maxwell\nmodel. Glass is opaque in the ultraviolet indicating that it has a strong absorption\nthere. In scienti\fc literature, there are innumerable experimental works which have\nstudied the interaction of silica glasses with electromagnetic waves over its whole\nspectrum. A summary of these works can be found in Kitamura et al. [3]. From\nthe experimental data on the extinction coe\u000ecient for silica glass in the ultraviolet,\nwe can \fnd the frequency at which this coe\u000ecient is maximum. However, as far as\nwe are aware, there has been no theoretical study so far which has determined this\nfrequency by an analysis of the Lorentz-Maxwell model of dispersion. The main\nproblem with the theoretical analysis is the fact that it has not been possible so\nfar to determine the value of the damping coe\u000ecient theoretically.\n2In this work we have determined the damping coe\u000ecient at the true resonance\nfrequency which we de\fne to be the frequency at which the absorption coe\u000ecient\nfor the energy of the electromagnetic \feld in the medium is maximum. We have\ndone this theoretically by taking the natural oscillation frequency and the plasma\nfrequency determined from the refractive indices of glass in the optical region as two\nknown parameters of the Lorentz-Maxwell model. We have formed two algebraic\nequations containing the true resonance frequency and the damping coe\u000ecient\nas two unknown variables. We have solved these two equations simultaneously by\nnumerical method to \fnd a unique solution for the two variables. With the value of\nthe damping coe\u000ecient known, we have explored the anomalous dispersion region\nin the ultraviolet for glass.\nIt is well known that the Kramers-Kronig relations [3] allow us to determine\nthe imaginary part of the dielectric constant from an integral of the real part over\nthe whole range of frequencies and vice versa. The theory is based on a very\ngeneral causality argument and a linear response of the medium to an external\nperturbation. We have, on the other hand, determined the damping coe\u000ecient of\nthe Lorentz-Maxwell model of dispersion starting from the refractive indices in the\noptical region corresponding to the real value of the dielectric constant. From this\nwe have extracted the information about the absorptive region in the ultraviolet\ncorresponding to the imaginary part of the dielectric constant.\nIn section 2 we give the outline of the Lorentz-Maxwell model. In section 3\nwe o\u000ber our physical argument for the method adopted to determine the damping\ncoe\u000ecient and the true resonance frequency. The next four sections are just an\nexecution of these ideas. We conclude with a summary of the work.\n32 Lorentz-Maxwell model of dispersion\nIn the Lorentz model [4] of dispersion of light in a dense medium like a solid or\nliquid, electrons execute forced simple harmonic oscillations with damping in the\ncombined \feld of the incident electromagnetic wave of frequency !and the local\n\feld. The local \feld arises as a result of the interaction of the electron with the\n\felds of other atoms close by. Without any loss of generality we can assume that\nthe direction in which the electron is oscillating is the ydirection. We can write\nthe equation of motion as\ny+\r_y+!2\n0y=qE0\n0\nme\u0000i!t: (1)\nwhereE0\n0is the amplitude of the e\u000bective \feld acting on the electrons. Here !0is\nthe natural oscillation frequency and \ris the damping coe\u000ecient of the electron.\nIn the steady state the electron will oscillate at a frequency !of the incident wave\nthough shifted in phase. E0\n0is related to the amplitude of the \feld ( Ei0) outside\nfrom where it is incident on the medium as\nE0\n0=1 +\u001f\n3\n1 +D\u001fEi0: (2)\nThe\u001f\n3term in equation (2) arises as a result of the e\u000bect of local \feld in the\nLorentz-Lorenz theory of dielectric polarizability valid for an isotropic medium [5]\nwhere\u001fis the electric susceptibility. Dis the depolarisation factor, a dimensionless\nnumber of the order of unity [6]. The dielectric function of the medium is given by\n\u000f= 1 +\u001f: (3)\nUsing Maxwell's phenomenological relation \u000f=n2\ncwherencis the complex refrac-\ntive index and the Lorentz-Lorenz equation [5], we arrive at the following equation\nfor a number of resonance regions [7,8].\nn2\nc\u00001\nn2\nc+ 2=Nq2\n3\u000f0mX\njfj\n!2\n0j\u0000!2\u0000i\rj!: (4)\n4Herefjis the fraction of electrons that have a natural oscillation frequency !0j\nand damping constant \rjwith \u0006fj= 1.Nis the density of electrons taking part\nin dispersion. It is a common practice to assume a single dominant absorption\nfrequency which is true in many practical cases and which makes the analysis\nsimpler [9]. With this assumption fj= 1 and equation (4) can be written as\nn2\nc= 1 +!2\np\n!2\nn\u0000!2\u0000i\r!(5)\nwhere the plasma frequency !pis given by\n!2\np=Nq2\n\u000f0m(6)\nand we de\fne\n!2\nn=!2\n0\u0000!2\np\n3: (7)\nIn scienti\fc literature [2], !0is known as the natural oscillation frequency of the\nelectrons and !nis known conventionally as the resonance frequency. So far, au-\nthors have used some chosen values of the damping coe\u000ecient \rand the plasma\nfrequency which mimic the absorptive properties of dielectric materials, in order\nto carry out model analysis [9]. We have actually determined the damping coe\u000e-\ncient from a prior knowledge of the natural oscillation frequency and the plasma\nfrequency of a glass medium.\nIn the optical limit where the absorption in glass is negligible we take \r= 0.\nIn this limit the refractive index is real and equation (5) reduces to\nn2= 1 +!2\np\n!2\nn\u0000!2: (8)\nwhich is essentially the Sellmeier's formula [7] for dispersion in the frequency do-\nmain with one absorption band. If we have a set of measurements of refractive\nindices of a glass prism for several optical wavelengths, we can determine !nand\n!pusing equation (8) [1].The resonance wavelength which falls in the ultraviolet\nregion, has been determined in a similar work [7]. Once !nand!pare known, !0\ncan be determined using equation (7).\n5In the absorptive region the dielectric function picks up an imaginary part\ngiven by\nn2\nc=\u000f=\u000f1+i\u000f2: (9)\nThe refractive index ( n) and the extinction coe\u000ecient ( \u0014) known as optical con-\nstants are written as\nnc=n+i\u0014 (10)\nwhere\u0014represents the attenuation factor of the amplitude of the electromagnetic\nwave in an absorptive medium. Using the last two equations and equation (5) we\nobtain for the real and imaginary parts of the complex dielectric function [10],\n\u000f1=n2\u0000\u00142= 1 +!2\np(!2\nn\u0000!2)\n(!2\nn\u0000!2)2+\r2!2(11)\nand\n\u000f2= 2n\u0014=!2\np\r!\n(!2\nn\u0000!2)2+\r2!2: (12)\nWe can express n2and\u00142as functions of frequency !using equations (11) and\n(12). The details and the \fnal expressions have been shown in Appendix A.\nThe absorption coe\u000ecient of the incident EM wave \u000bis given by [7,10],\n\u000b=2!\nc\u0014 (13)\nwherecis the speed of light in vacuum. \u000bgives the attenuation coe\u000ecient of\nthe intensity of the incident wave. Intensity is the rate of \row of energy per\nunit area normal to a surface. \u000bwill be a maximum at the frequency at which\nthe absorption of energy by the electrons from the EM \feld is maximum. This\ngives the condition of resonance. It is a general practice to consider !nde\fned in\nequation (7) as the resonance frequency though there is no proof that the energy\nabsorption is maximum at this frequency. So we do not assume a priori !nto be\nthe resonance frequency. In the next section we will describe our strategy to \fnd\nthe true resonance frequency and in the results section we will see that the true\nresonance frequency is di\u000berent from both !0and!nand lies between them. There\n6is no real reason to call !nthe resonance frequency. We treat the true resonance\nfrequency as an unknown variable to be found from our analysis.\nThe damping coe\u000ecient \ris introduced in the Lorentz model to explain absorp-\ntion. We model \rsuch that it is zero in the optical band and upto the frequency\n!n. In the absorption band we assume that \ris constant from frequency !nto\n!0. Above!0,\rfalls down and rises again to another constant value of \rin the\nnext resonance region if the material under study has one. With this model for \r\nin mind we can extrapolate equation (8) to \fnd !nand!p. In the next section we\nwill explain how to get the constant value of \rin the absorption region and the\ntrue resonance frequency.\nEven if the system under study may have several absorption bands, we can\nstudy it with the assumption of a single resonance region. The optical waves os-\ncillate the outermost electrons of atoms and molecules having the lowest natural\nfrequencies and as a result we get the phenomenon of refraction. With an anal-\nysis of the refractive indices in the optical region under this assumption of single\nresonance, we are most likely to \fnd information about the absorption band with\nthe lowest natural oscillation frequency in the ultraviolet closest to the optical\nband. This will of course depend on the strength of the resonance. The justi\fca-\ntion of the single resonance calculations with the chosen model for \rcan be found\nfrom the results of our theoretical calculations which will be found to match the\nexperimental results very well.\n73 Physical argument for the method adopted to\ndetermine the damping coe\u000ecient at the true\nresonance frequency\nFrom various experiments on the absorption of EM waves in matter, we know\nthat the absorption coe\u000ecient ( \u000b) attains a maximum value at a characteristic\nfrequency. We try to \fnd this frequency where \u000bis maximum. We di\u000berentiate\n\u000bwith respect to frequency and equating the derivative to zero get one equation.\nHowever, we have two unknown variables in the theory - the damping coe\u000ecient\nand the true resonance frequency. We look for a second equation.\nThe incident EM wave interacts with the electrons bound to the atoms and\nmolecules.The electrons execute a forced simple harmonic oscillation with damp-\ning. The total energy of the electron is time dependent, as the electron is being\nperturbed by a time dependent harmonic force.The average energy of the electron\nper cycle can be worked out easily [11]. We \fnd the frequency at which this av-\nerage energy per cycle is maximum. This leads to another equation involving the\ntwo unknown variables. When the frequency at which \u000bis maximum is the same\nas the frequency at which the average energy per cycle of the electron is also a\nmaximum, the electromagnetic wave will share its energy most with the electrons\nand will be attenuated most. This will constitute the true condition of resonance.\nBy solving the two equations simultaneously using numerical method, we \fnd both\nthe variables. We call the characteristic frequency, the true resonance frequency\n!tand the damping coe\u000ecient at the true resonance frequency \rt.\nHeitler [12] has proposed a quantum theory of the phenomenon of damping.\nAccording to this theory the damping coe\u000ecient is dependent on frequency though\nof a very slowly varying nature near resonances. This gives support to our earlier\nassumption that the damping coe\u000ecient is a constant within a small frequency\nrange about the resonance frequency. However, it can be taken as zero in the\n8optical band where glass is transparent and absorption is negligible.\n4 Condition for the maximum of the absorption\ncoe\u000ecient as a function of frequency\nOur aim in this section is to \fnd the frequency at which \u000bis maximum. We\n\frst di\u000berentiate \u000bwith respect to !assuming\rconstant. In order to \fnd the\nderivative of \u000bwe \frst di\u000berentiate equations (11) and (12) with respect to !. We\n\fnd two algebraic equations involvingdn\nd!andd\u0014\nd!. By eliminatingdn\nd!from the two\nequations, we get the expression ford\u0014\nd!and henced\u000b\nd!using equation (13). We have\nshown the di\u000berentiations in Appendix B. Eliminatingdn\nd!between equations (B.2)\nand (B.3) we get\n2d\u0014\nd!(n+\u00142\nn) =A\u0000B\nC(14)\nwhere\nA=!2\np(!2\nn\u0000!2)2[\r\u00002\u0014\nn!] (15)\nand\nB=!2\np!\r[!\r2\u00004(!2\nn\u0000!2)!\u00002\u0014\nn!2\nn\r] (16)\nand\nC= [(!2\nn\u0000!2)2+\r2!2]2: (17)\nFrom this we get\nd\u000b\nd!=\u0014\nc[2 +n\n\u0014!\nn2+\u00142A\u0000B\nC]: (18)\nIf\u000bis maximum thend\u000b\nd!should be zero. So we write at the maximum\n!(A\u0000B)\nC=\u00002\u0014\nn(n2+\u00142): (19)\nIt is to be noted that two sides of equation (19) are dimensionless and they will be\ncompared later numerically to \fnd the solution for the true resonance frequency\nand the damping coe\u000ecient.\n95 Condition for the maximum of the average en-\nergy per cycle of the electron as a function of\nfrequency\nIn the steady state the electron will oscillate at a frequency !as given by the\nsteady state solution of equation (1) and the total energy of the system averaged\nover a period is given by [11],\nE(!) =1\n4(qE0\n0)2\nm(!2+!2\n0)\n[(!2\n0\u0000!2)2+ (!\r)2]=1\n4(qE0\n0)2\nmg(!) (20)\nwhere\ng(!) =(!2+!2\n0)\n[(!2\n0\u0000!2)2+ (!\r)2]: (21)\nEquation (2) shows the relationship between the incident electric \feld and the \feld\nacting on an electron. With the variation of frequency in the ultraviolet we can\nimagine that the amplitude of the incident \feld is kept constant. However, the\namplitudeE0\n0is dependent on \u001fwhich is frequency dependent. Lorentz theory is\nbased on the assumption that the response \u001fof the medium to the external \feld\nis small [13]. In equation (2), \u001fappears both in the numerator as well as in the\ndenominator. With the depolarization factor Dpositive, any variation of \u001fin the\nnumerator will be o\u000bset to some extent by the variation in \u001fin the denominator.\nSo we neglect the variation of the term E0\n0with frequency and assume it to be\nconstant. To \fnd the derivative of the average energy per cycle E(!), it is su\u000ecient\nto \fnd the derivative of the function g(!) given by equation (21) with respect to\n!. Equating the derivative to zero, we \fnd the condition at which the average\nenergy per cycle is maximum. It turns out that the frequency is given by\n!=!0[r\n4\u0000(\r\n!0)2\u00001]1\n2: (22)\nIf the incident electromagnetic wave can oscillate the bound electrons steadily at\nfrequency!given by the last equation, then the wave has to deliver maximum\nenergy per cycle and its absorption will be maximum.\n10It is clear from equation (22) that for real values of !we should have the ratio\nf=\r\n!0<p\n3. By trial we take several positive values of fupto its maximum of\np\n3. For each value of fwe \fnd the values of \rand!using equation (22) with the\nknown value of !0. With these values we determine the refractive index nand the\nextinction coe\u000ecient \u0014in the resonance region using equations (A.3) and (A.4)\nrespectively. We put these values in equation (19) and try to see for which value\noffthis equation is satis\fed. From fwhich satis\fes equation (19) we calculate \rt\nand!tusing equation (22).\n6 Determination of the damping coe\u000ecient and\nthe true resonance frequency\nThe results of an experiment performed with a prism made of \rint glass have been\nreported by Chakrabarti [1]. In this experiment1\nn2\u00001has been plotted against the\ninverse wavelength squared at optical range following equation (8). From this plot\nwe have determined the values of !n,!p:The value of !0has been estimated using\nequation (7). The errors in these frequencies are less than 1% :Refractive indices as\na function of wavelengths have been shown in table 1 and the parameters needed\nfor this work have been shown in table 2.\nIt has been shown in the discussion following equation (22) that the maximum\nvalue off=\r\n!0isp\n3. So we take trial values of flike 0.1, 0.2 upto 1.7. For all these\nvalues offwe determine \rand then!using equation (22). We then determine n\nand\u0014using equations (A.3) and (A.4) respectively and calculate A,B,Caccording\nto equations (15),(16),(17) respectively. With these values we try to see whether\nthey satisfy equation (19) or not. The algorithm for this is as follows: suppose\nwe call the left side of equation (19), Y1 and the right side, Y2. Now we form\na parameter, Y=Y1\u0000Y2. For all values of ffrom 0.1 to 1.7 we calculate Y\nand \fnd between which two values of f, the sign of Ychanges. In our case there\n11was only one sign change in Ybetweenfvalues 0.6 and 0.7. Now we check this\nregion more closely, that is between 0.60 to 0.70 at an interval of 0.01. We \fnd\nthat forf= 0:65,Y=\u00000:001 and for f= 0:66,Y= 0:053. So there is a zero\ncrossing between fvalues 0.65 and 0.66. Proceeding similarly we \fnally \fnd that\nforf= 0:6501,Y=\u00000:0005 and for f= 0:6502,Y= 0:000055. So the solution\nlies between 0.6501 and 0.6502. We take the solution as f= 0:65015 with an error\nof 0:00005 which is just \u00060:008%. From this value of fwe get\rtsince we know\nthe value of !0. We get the value of !tby putting the value of fin equation (22).\nThese values are shown in table 3. The values of the damping coe\u000ecient and the\ntrue resonance frequency turn out to be\n\rt= 11:6\u00021015rad/s\nwith an error of only 0 :76% and\n!t= 16:8\u00021015rad/s\nwith an error similar to the error in !0.!tlies between !nand!0. This solution\nis unique.\nWe \fnd that !tdi\u000bers signi\fcantly from !n. We have presented the parameters\nat true resonance frequency in table 4. The maximum value of the absorption\ncoe\u000ecient\u000bat the true resonance frequency has come out to be 8 :18\u0002107m\u00001\ncorresponding to an attenuation length 12 :2 nm.\nOnce we determine the damping coe\u000ecient, we can determine the absorption\ncoe\u000b\fcient \u000bin a small frequency range about !twhere we assume \ris constant\nand equal to \rt. In \fgure 1 we plot \u000bas a function of x=!\n!0betweenx=\n!n\n!0= 0:81 to 1. We assume that within the frequency range !nto!0,\rremains\nconstant. We clearly \fnd that \u000battains a maximum value at x=!t\n!0= 0:94:\nJackson [14] has given the frequency range for absorption coe\u000ecient \u000bin the\nultraviolet as well as the plasma frequency of water. In the ultraviolet, we are\nconcerned with electronic oscillations as the most important component responsible\n12for dispersion. So the properties of glass will not be too di\u000berent from water at\nthese frequencies. This work [14] shows that the maximum value of \u000bis around\n108m\u00001which is approximately the same as the maximum value that we have\nobtained. The frequency at which the maximum occurs is also the same in order\nof magnitude as !t:The full width at half maximum of the absorption curve can\nbe read o\u000b approximately. It is determined to be approximately 15 \u00021015rad/s\nwhich is of the same order of magnitude that we have obtained for the value of \rt.\nThe damping coe\u000ecient determined in this work is rather large.This broad ab-\nsorption in the ultraviolet is due to the outer electrons in the atoms and molecules\nof the solid which take part in the process of dispersion. The outer electrons are\na\u000bected by the collisions and the electric \felds of the neighbouring atoms. Con-\nsequently an extensive region of continuous absorption is obtained in solids and\nliquids [8,15]. So this large value of \rtis expected. This value of \rtis valid only in\nthe resonance region.\n7 Some other results\nWe assume the damping coe\u000b\fcient \rtis constant within the small frequency range\nfrom!nto!0in which!tlies. We now \fnd the frequency !\u0014at which the extinction\ncoe\u000ecient\u0014is maximum, assuming !\u0014is close to!t. For\u0014to be maximum,d\u0014\nd!\nshould be zero and from equation (14) this should occur when A=B:Here we scan\na parameter!\n!0from 0.1 to 1.0. For each value of !and known \rtwe calculate\nnand\u0014using equations (A.3) and (A.4) respectively. We then check whether\nthe condition A=Bis satis\fed. Once again we \fnd a unique solution with\n!\u0014\n!0= 0:84843:This corresponds to an angular frequency !\u0014= 15:1\u00021015rad/s.\nFrom this we get\n\u0017\u0014= 2:40\u00021015Hz\n13corresponding to a wavelength\n\u0015\u0014= 0:125\u0016m:\nIn \fgure 2 we have plotted \u0014determined by our theoretical analysis, as a function of\n!\n!0:This \fgure clearly shows that \u0014has attained a maximum value. The maximum\nvalue of\u0014that we have obtained at !=!\u0014is 0.769. This is of the same order\nof magnitude as the maximum value shown in \fgure 2 of Kitamura et al.[3] which\ngives the maximum of \u0014in the ultraviolet at a wavelength very near to 0 :12\u0016m.\nThis is very close to the wavelength \u0015\u0014that we have obtained. In the last column\nof table 3 we show the value of !\u0014.\u0015\u0014that we have determined may be seen from\n\fgure 1 of [3] to be the nearest to the optical wavelengths on long wavelength side\nas we claimed in last paragraph of section 2. The data in \fgure 1 of [3] may be\nshowing other resonance/s at shorter wavelengths or larger frequencies.\nIn \fgure 3 we show a plot of re\rectance Ras a function of frequency in the\nabsorption region where Ris the normal re\rectivity de\fned in [3] as\nR=(n\u00001)2+\u00142\n(n+ 1)2+\u00142: (23)\nWe \fnd that this function has a peak at!\n!0= 0:908 with the maximum value of R=\n0:118. The peak position corresponds to != 16:2\u00021015rad/s:The re\rectance peak\nposition corresponds to an energy 10.6 eV. Experimentally, silica glasses show the\nlowest frequency re\rectance peak in the ultraviolet region at 10.2 eV [16] which is\nvery close to what we have found theoretically. This is once again as we claimed in\nthe last paragraph of section 2. Sigel has shown in \fgure 3 [16] that the re\rectance\npeak position is the same for two glassy materials though their re\rectance values\nat this frequency are di\u000berent. Similarly the actual values of refractive indices and\nextinction coe\u000ecients at the same frequency can vary signi\fcantly due to glass\nmanufacturing process [3]. Data for these coe\u000ecients in the absorption region for\n\rint glasses were not available in the literature. However, the order of magnitude\nof these coe\u000ecients is similar to that of other silica glasses. Since we are getting\n14the peak positions of the extinction coe\u000ecient and the re\rectance data very close\nto the published data for silica glasses, we conclude that the value of the damping\ncoe\u000ecient obtained by us is correct.\nWe have shown in table 5 the values of n,\u0014,\u000b, the absorption length1\n\u000band\ng(!) at frequencies close to !t. They have been calculated at three frequencies !0,\n!\u0014and!nfor the same \rt. Comparing tables 4 and 5 we \fnd that \u000bandg(!) are\nindeed maxima at !t. The anomalous nature of variation of the refractive index is\nevident from the values of nat frequencies around !t.\nIn section 4 we determined the condition for \u000bto be maximum. In a similar\nway we can \fnd the condition for the refractive index nto be an extremum by\nequating its derivative to zero. This derivative can be obtained by eliminatingd\u0014\nd!\nfrom equations (B.2) and (B.3). Interestingly, this condition gives two solutions\nfor the frequency. The refractive index is maximum at one and minimum at the\nother frequency. We \fnd that the Lorentz-Maxwell model of dispersion reproduces\nall the features of anomalous dispersion in the absorption region as observed in\nactual experiments [3].\n8 Conclusions\nThe Lorentz-Maxwell model of dispersion of electromagnetic waves in matter has\nbeen studied in this paper with an analysis of the phenomenon of absorption in\nthe ultraviolet in dielectrics like \rint glass. We have shown that if we know the\nrefractive indices of glass fairly accurately in the optical frequencies, we can explore\nthe anomalous dispersion region in the ultraviolet quantitatively. The key \fnding\nof this work is that the damping coe\u000ecient of the model can be determined by\na simple argument. We also determine the frequency at which the absorption\ncoe\u000ecient is maximum. We call this the true resonance frequency. In the optical\nregion where glass is transparent, the damping coe\u000ecient can be assumed to be\n15zero. In the absorptive part the damping coe\u000ecient has been taken to be a constant\nwithin a short range of frequencies. The value of the damping coe\u000ecient matches\nin order of magnitude with the experimental width of the absorption coe\u000ecient\ndata for water available from literature.\nOnce the damping coe\u000ecient is determined, we can \fnd the frequencies at\nwhich the extinction coe\u000ecient and the re\rectance are maxima. These frequencies\nmatch very well with the experimental data available in the literature for silica\nglasses. This indirectly shows that the value of \rdetermined by us is correct. Our\nassumption of a single resonance should give us the information of the absorption\nband closest to the optical frequencies. We actually observe this by comparing the\npeak positions of the extinction coe\u000ecient and re\rectance data obtained by us with\nthat found from literature. Refractive indices estimated at di\u000berent frequencies\nclose to the true resonance frequency in the absorption region reveal the anomalous\nnature of dispersion. All the features of dispersion by a dielectric like glass in the\nultraviolet absorption region have been reproduced from our theoretical analysis\nof the Lorentz-Maxwell model.\n9 Acknowledgment\nThe author would like to thank Prof. Jayanta Kumar Bhattacharjee for some\nhelpful discussions. Thanks are due to Prof. Debashis Mukherjee for making some\nhelpful comments on the paper.\n16Appendix A\nEquations (11) and (12) can be easily inverted and we get [2],\nn2=1\n2[(\u000f2\n1+\u000f2\n2)1\n2+\u000f1] (A.1)\nand\n\u00142=1\n2[(\u000f2\n1+\u000f2\n2)1\n2\u0000\u000f1]: (A.2)\nWe express \u000f1and\u000f2as functions of frequency using equations (11) and (12)\nrespectively. After a fairly straightforward algebra we arrive at the \fnal expressions\nforn2and\u00142as functions of frequency !.\nn2=[!4\np\r2!2+ (!04+\r2!2+!2\np!02)2]1\n2+ (!04+\r2!2+!2\np!02)\n2(!04+\r2!2)(A.3)\n\u00142=[!4\np\r2!2+ (!04+\r2!2+!2\np!02)2]1\n2\u0000(!04+\r2!2+!2\np!02)\n2(!04+\r2!2)(A.4)\nwhere\n!02=!2\nn\u0000!2: (A.5)\nThese equations are exact and will be used for determining nand\u0014in the resonance\nregion in the ultraviolet.\nIt can be easily checked that in the limit \rtending to zero, \u0014becomes zero\nat all frequencies and vice versa. Thus the medium is transparent in the optical\nfrequencies as it should. In this limit the refractive index nsatis\fes a relation\nwhich has been used in the \frst place to get the parameters !nand!p[1].\nAppendix B\nWe \frst \fnd the derivative of1\n(!2n\u0000!2)2+\r2!2with respect to !and get\nd\nd![1\n(!2\nn\u0000!2)2+\r2!2] =2![2(!2\nn\u0000!2)\u0000\r2]\n[(!2\nn\u0000!2)2+\r2!2]2(B.1)\n17Di\u000berentiating equation (11) with respect to !we get\nndn\nd!\u0000\u0014d\u0014\nd!=\u0000!!2\np[(!2\nn\u0000!2)2+\r2!2] +!!2\np(!2\nn\u0000!2)[2(!2\nn\u0000!2)\u0000\r2]\n[(!2\nn\u0000!2)2+\r2!2]2\n=!!2\np[(!2\nn\u0000!2)2\u0000!2\nn\r2]\n[(!2\nn\u0000!2)2+\r2!2]2(B.2)\nSimilarly di\u000berentiating equation (12) with respect to !we get\n2\u0014dn\nd!+ 2nd\u0014\nd!=!2\np\r[(!2\nn\u0000!2)2+\r2!2] + 2!2!2\np\r[2(!2\nn\u0000!2)\u0000\r2]\n[(!2\nn\u0000!2)2+\r2!2]2\n=!2\np\r(!2\nn\u0000!2)[(!2\nn\u0000!2) + 4!2]\u0000!2!2\np\r3\n[(!2\nn\u0000!2)2+\r2!2]2:(B.3)\n18REFERENCES\n1. Chakrabarti S 2006 Phys. Educ. 23167- 175 (New Delhi, India: South\nAsian Publishers PVT LTD)\n2. Christy R W 1972 Am.J.Phys. 401403-1419\n3. Kitamura R, Pilon L and Jonasz M 2007 Applied Optics 46(33) 8118-8133\n4. Almog I F, Bradley M S and Bulovic V 2011 The Lorentz Oscillator and its\nApplications ( MIT OpenCourseWare, MIT6-007S11/lorentz)\n5. Born M, Wolf E 1980 Principles of Optics ( New York:Pergamon Press,\nSixth.Edn.) p 85\n6. Kittel C 1976 Introduction To Solid State Physics (New Delhi:WileyEastern\nLimited, 5th Edn.) p 405\n7. Hecht E 2002 Optics ( Delhi,India:Pearson Education, 4th Edn.) p 71,\n85,70,128\n8. Feynman R P, Leighton R B, Sands M 2003 The Feynman Lectures on\nPhysics, 2nd Volume (New Delhi,India:Narosa Publishing House) p 1211,1210\n9. Oughstun K E, Cartwright N A 2003 Optics Express 11(13) 1541-1546\n10. Tanner D B 2013 Optical e\u000bects in solids (www.phys.u\r.edu/tanner/notes.pdf\n)\n11. Kleppner D, Kolenkow R J 1973 An Introduction To Mechanics (New Delhi\n:Tata Mcgraw Hill ) p 426\n12. Heitler W 1954 The quantum theory of radiation (Oxford University Press,\n3rd. Edn.) p 163\n1913. Seitz F 1940 The Modern Theory of Solids (McGraw-Hill ,International Series\nIn Pure And Applied Physics) p 629\n14. Jackson J D 1999 Classical Electrodynamics 3rd edn. (John Wiley & Sons,\nInc) p 314\n15. Jenkins F A , White H E 1957 Fundamentals of Optics (Mcgraw-Hill book\ncompany,Inc, 3rd Edn.) p 486\n16. Sigel G H Jr 1973/74 Journal of Non-Crystalline Solids 13372-398\n20Table 1: Refractive indices as a function of wavelengths for the \rint glass\nprism [1]\nwavelength refractive index\n\u0015(nm) n\n706.544 1.6087\n667.815 1.6108\n587.574 1.6167\n504.774 1.6259\n501.567 1.6264\n492.193 1.6277\n471.314 1.6311\n447.148 1.6358\n438.793 1.6377\n21Table 2: Parameters obtained from \ftting of data of refractive indices to\nLorentz model[1]\n!n N ! p !0\nrad/s m\u00003rad/s rad/s\n14:5\u000210151:02\u0002102918:0\u0002101517:8\u00021015\nTable 3: Table for \rt,!tand!\u0014\nf=\rt\n!0\rt !t !\u0014\nrad/s rad/s rad/s\n0.65015 11 :6\u0002101516:8\u0002101515:1\u00021015\nTable 4: Parameters at the true resonance frequency\nfrequency n \u0014 \u000b1\n\u000bg(!) =4mE(!)\n(qE0\n0)2\n! m\u00001\u0016m s2=rad2\n!t 0.995 0.729 8 :18\u00021070.0122 0:153\u000210\u000031\nTable 5: Values of some parameters in the ultraviolet region\nfrequency n \u0014 \u000b1\n\u000bg(!) =4mE(!)\n(qE0\n0)2\n! m\u00001\u0016m s2=rad2\n!0 0.906 0.678 8 :06\u00021070.0124 0:149\u000210\u000031\n!\u0014 1.18 0.769 7 :76\u00021070.0129 0:141\u000210\u000031\n!n 1.26 0.763 7 :39\u00021070.0135 0:133\u000210\u000031\n22 7.2 7.4 7.6 7.8 8 8.2 8.4\n 0.8  0.85  0.9  0.95  1  1.05α(107 m-1)\nω/ω0Figure 1: Distribution of the absorption coe\u000ecient \u000bin the absorption region\n23 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78\n 0.75  0.8  0.85  0.9  0.95  1κ\nω/ω0Figure 2: Distribution of the extinction coe\u000ecient \u0014in the absorption region\n24 0.115 0.1155 0.116 0.1165 0.117 0.1175 0.118 0.1185\n 0.8 0.82 0.84 0.86 0.88  0.9 0.92 0.94 0.96 0.98  1 1.02R\nω/ω0Figure 3: Distribution of Re\rectance Rin the absorption region\n25"    },    {        "title": "2311.08121v3.Enhanced_classical_radiation_damping_of_electronic_cyclotron_motion_in_the_vicinity_of_the_Van_Hove_singularity_in_a_waveguide.pdf",        "content": "arXiv:2311.08121v3  [physics.class-ph]  6 Feb 2024Enhanced classical radiation damping of electronic cyclot ron motion in the vicinity of\nthe Van Hove singularity in a waveguide\nYuki Goto∗\nNational Institute for Fusion Science, National Institute s of Natural Sciences, Toki 509-5292, Japan\nSavannah Garmon\nDepartment of Physics, Osaka Metropolitan University, Sak ai 599-8531, Japan and\nInstitute of Industrial Science, The University of Tokyo, K ashiwa, 277-8574, Japan\nTomio Petrosky\nCenter for Complex Quantum Systems, The University of Texas at Austin, Austin, 78705, USA and\nInstitute of Industrial Science, The University of Tokyo, K ashiwa, 277-8574, Japan\nWe study the damping process of electron cyclotron motion an d the resulting emission in a\nwaveguide using the classical Friedrichs model without rel ying on perturbation analysis such as\nFermi’s golden rule. A Van Hove singularity appears at the lo wer bound (or cut-off frequency) of the\ndispersionassociated witheachoftheelectromagnetic fiel dmodesinthewaveguide. Inthevicinityof\nthe Van Hove singularity, we found that not only is the decay p rocess associated with the resonance\npole enhanced (amplification factor ∼104) but the branch-point effect is also comparably enhanced.\nAs a result, the timescale on which most of the decay occurs is dramatically shortened. Further,\nthis suggests that the non-Markovian branch point effect sho uld be experimentally observable in\nthe vicinity of the Van Hove singularity. Our treatment yiel ds a physically-acceptable solution\nwithout the problematic runaway solution that is well known to appear in the traditional treatment\nof classical radiation damping based on the Abraham-Lorent z equation.\nI. INTRODUCTION\nCyclotron motion appears in many contexts in nature,\nsuch as auroras in Earth’s magnetosphere, neutron stars\nin space, free electrons within metals, and magnetic con-\nfinement in fusion plasmas. Cyclotron motion plays a\ncrucial role in astrophysics, condensed matter physics,\nand fusion science. As is well known, cyclotron motion\nis a typical example of classical radiation damping. The\nphenomenological Abraham-Lorentz equation has been\nobtained to describe radiation damping in classical me-\nchanics [1].\nHowever, the usefulness of the Abraham-Lorentzequa-\ntion is limited because it leads to an unphysical solu-\ntion, i.e., the so-called runaway solution , due to the force\nthat is proportional to the third-order time derivative\nfor the position of the charged particle. Dirac proposed\na class of initial conditions that eliminate the runaway\nsolution, however his proposal introduces non-causality\n[1]. The inconsistency in the Abraham-Lorentz equation\nwith the basic laws of physics arises from the fact that\nthe field emitted in this process is calculated based on\nthe Li´ enard–Wiechert potentials which ignore the effects\nof the back-reaction of the self-generated field resulting\nfrom the motion of charged particles.\nWe have two main objectives in this paper. First, we\ndemonstrate that by using the normal mode description\nfortheclassicalradiationfield, asintroducedinourprevi-\nouspaper[2], wecandescribeclassicalradiationdamping\nin a manner that eliminates the inconsistency mentioned\n∗goto.yuki@nifs.ac.jpabove. This is achieved through the application of a for-\nmalism parallel to that introduced by Friedrichs, which\nis commonly used to describe radiation damping dynam-\nics in quantum mechanics [3–6]. In quantum mechanics,\na decaying solution exists without any runaway behav-\nior. Secondly, we determine the decay evolution of clas-\nsical cyclotron motion and the intensity of the emitted\nclassical fields without relying on perturbation analysis\nwithin the framework of the classical Friedrichs model\nintroduced in this paper. This enables explicit evalua-\ntion of the decay rate of the cyclotron motion and the\nintensity of the emitted classical fields, even in scenarios\nwhere Fermi’s golden rule may not be applicable, such\nas when the cyclotron frequency approaches the cut-off\nfrequency of the waveguide.\nIn this paper, we will consider the classical system\nwhere an electron inside a rectangular electromagnetic\nwaveguide exhibits cyclotron motion with variable fre-\nquency under the influence of a (static, uniform) external\nmagnetic field that is applied in parallel with the length\nof the waveguide. It is well known that the dispersion\nrelation of light in a waveguide has cut-off frequencies\ndepending on each mode. For each mode, the continuous\nfrequencyspectrum ofthe wavehasabranchpoint singu-\nlarity at the lower bound of the spectrum in the complex\nfrequency plane. This singular point corresponds to the\nVan Hove singularity that appears in solid state physics\nin quantum mechanics [7, 8]. In our approach, we ap-\nply the method developed in quantum mechanics as a\nsecond quantization formalism to our study [4], but now\nwe reinterpret the annihilation and creation operators of\nthe photon as normal modes of classical light that obey\nthe Poisson bracket instead of the commutation relation.\nSince the Poisson bracket obeys the same algebra as the2\ncommutator, wecanobtainanexactsolutionoftheequa-\ntion of motion for classical radiation damping. In other\nwords, our method can be said to be a classicalization of\nquantum mechanics. As a result, physically meaningful\nsolutions are obtained that avoid runaway solutions and\nthe non-causal behavior associated with the Abraham-\nLorentz equation mentioned above.\nIn early studies of cavity quantum electrodynamics,\nKleppner showed that the spontaneous emission rate of\nan atom in a waveguide would experience a dramatic\nspike when the transition frequency is tuned to the cut-\noff(continuum band edge) ofone of the waveguidemodes\n[9]. This spike occurs as a result of the Van Hove singu-\nlarity at the cut-off mode. However, Kleppner’s analysis\ncould only provide a qualitative assessment since it re-\nlies on Fermi’s golden rule, which estimates the transi-\ntion rate in proportion to the density of states, which di-\nverges at the Van Hove singularity. Over a decade later,\nstudies of emission rates in photonic band gap materi-\nals revealed more precisely the enhancement factor due\nto the singularity [10–12], which was later independently\ndiscoveredby two of the present authorsworking directly\non the original problem of transmission rates of atoms in\nwaveguides [13]. Relying on non-perturbative methods,\nit was shown in [13] that the spontaneous emission rate\nγis modified near the cut-off such that it can be ex-\npanded in powers of λ4/3, in which λ≪1 is the dimen-\nsionless coupling constant between the dipole moment of\nthe atom and the field mode in the waveguide. In the\ncontext of the present work, λis of the order 10−6. On\nthe other hand, when the transition frequency is far de-\ntunedfromthecut-offfrequencysuchthatFermi’sgolden\nrule can be applied (we call this the Fermi region ),γis\nproportional to λ2. Hence, the enhancement in the Van\nHove singularity region compared to the Fermi region is\nλ4/3/λ2=λ−2/3≃104times larger. This spontaneous\nemission process can be attributed to the appearance of\nthe resonance pole in the complex energy plane, which is\nassociated with Markovian (irreversible) dynamics.\nHowever, in addition to the resonance pole effect, one\nshould expect that the non-Markovian branch-point ef-\nfect is also enhanced near the cut-off frequency. It is\nwell established that the branch point effect is non-\nnegligible at extremely long time scales and very short\ntime scales. Under typical circumstances, the branch-\npoint effect leads to an inverse power-law decay that\nonly appears after many lifetimes of the exponential de-\ncay have passed [14–19]. On the other hand, on short\ntime scales, the pole effect by itself would lead to pure\nexponential decay, which is inconsistent with the short\ntimescale behavior, including the initial conditions. This\nis because there is a non-negligible deviation from ex-\nponential decay on short time scales. Thus the branch-\npointandthepoleeffectmustbetakentogethertoobtain\nphysically sensible results. With the exception of the two\nexperiments in Refs. [16, 19], the initial state is usually\nso depleted by the time the branch-point effect appears\non the long time scale that it is extremely difficult to ob-\nservein experiment. But this effect canbe enhancednear\nFIG. 1. Physical model and coordinate system. The (static,\nuniform) external magnetic field is applied in the positive z\ndirection along the rectangular waveguide, and the electro n\nexperiences cyclotron motion in the waveguide.\nthe cut-off frequency (or band edge energy) [18, 20–24].\nIn the present work, we extend our previous analysis\nto the case of cyclotronic motion inside a waveguide and\nfurther we include the branch-point effect that was pre-\nviously neglected. Our results suggest that the branch-\npoint effect indeed should be experimentally accessible\nin the vicinity of the Van Hove singularity at the cut-off\nfrequency.\nThis paper is composed of six sections. In section 2,\ntheHamiltonian basedonourphysicalmodel isprovided,\nand the time evolution of normal modes is discussed. In\nsection3, the decayrate ofthe electronis derived. In sec-\ntion 4, the expressions for the field evolution associated\nwith the pole effect and branch point effect are derived.\nIn Section 5, their numerical calculation are discussed.\nThe conclusion of the paper is provided in section 6.\nII. MODEL AND FORMALISM\nLet us consider an electron in cyclotron motion in the\nrectangularwaveguide(seeFig.1). Weassumethewaveg-\nuide walls are perfect conductors. The electron couples\nto the electromagnetic field in the waveguide, which is\ninfinitely long in both directions along the zaxis. The\nelectron is located at the origin of the zaxis. The static\nuniform external magnetic field is applied in the positive\nzdirection. Thewidthofthewaveguideinthe xdirection\nisxLand the height in the ydirection is yL. We assume\nyL> xL. Also, we choose the origin for xandyat a\ncorner of the rectangular cross section of the waveguide,\nanduse aright-handedcoordinatesystem. Thecyclotron\nfrequency can be adjusted by changing the magnitude of\nthe magnetic field. For our numerical simulations below,\nwe choose ( xL,yL) = (0.1 m,0.2 m), for which the mag-3\nnitude of the magnetic field is of order 10−2T, and the\nLarmorradiusis oforder10−3m. Hence the width ofthe\nwaveguide is much larger than the Larmor radius, thus\nwe can neglect the forces acting on the electron from the\nwalls of the waveguide. These conditions induce the cy-\nclotronemissionwith the emmited light in the microwave\nfrequency regime. Under these conditions, we can write\nthe Hamiltonian of the system for the electron as\nH=He+Hf, (1)\nin which we have defined the Hamiltonian of the electron\nHeand Hamiltonian of the field Hfas follows:\nHe≡/integraldisplay\ndr1\n2me{p+e[Aex(r)+Ain(r)]}2δ(r−re),\n(2)\nHf≡/integraldisplay\ndr/bracketleftBigg\nε0\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂Ain(r)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+1\n2µ0|∇×Ain(r)|2/bracketrightBigg\n,\n(3)\nwheree,me,ε0, andµ0are the elementary charge, the\nmass of the electron, the vacuum permittivity, and the\nvacuum permeability, respectively. Furthermore, ris the\ncoordinate variable of the waveguide, and rerepresents\nthe position variable of the electron. The quantity pis\nthe momentum vector of the electron. We assume pis\nnonzero only in the xandydirections because motion in\nthezdirection does not affect the field emission. Thus,\nthe electron rotates in the x-yplane, and the center of\nthe cyclotron motion is ( xc,yc). Also,Aex(r) is the vec-\ntor potential for the (static, uniform) external magnetic\nfield, whereas Ain(r) is the vector potential for the prop-\nagation modes and dressing modes around the electron\nin the waveguide. In the waveguide, there are two kind\nof propagation modes because of the boundary condi-\ntions, which are called transverse electric (TE) modes\nand transverse magnetic (TM) modes. For the TE (TM)\nmodes, there is no electric (magnetic) field in the direc-\ntion of propagation. Thus, the vector potential Ain(r) is\nrepresented by the linear combination\nAin(r) =ATE\nin(r)+ATM\nin(r). (4)\nBy solving the sourceless Maxwell equations inside the\nrectangular waveguidewith the boundary conditions and\nchoosing the Coulomb gauge ∇ ·Ain(r) = 0, we obtain\nthe explicit form of the vector potentials\nATE\nin(r) =/summationdisplay\nm,n/integraldisplay\nkNA\nχm,n√ωk/bracketleftbiggnπ\nyLFCS\nm,n(x,y)ex\n−mπ\nxLFSC\nm,n(x,y)ey/bracketrightbigg\nqTE\nkeikz+c.c.,(5)\nATM\nin(r) =/summationdisplay\nm,n/integraldisplay\nkNAck\nχm,n/radicalbig\nω3\nk/bracketleftbiggmπ\nxLFCS\nm,n(x,y)ex\n+nπ\nyLFSC\nm,n(x,y)ey−iχ2\nm,n\nkFSS\nm,n(x,y)ez/bracketrightBigg\n×qTM\nkeikz+c.c.,(6)where\nNA=−1√πε0xLyL, (7)\nχm,n=/radicalBigg/parenleftbiggmπ\nxL/parenrightbigg2\n+/parenleftbiggnπ\nyL/parenrightbigg2\n, (8)\nand\nFCS\nm,n(x,y) = cos/parenleftbiggmπ\nxLx/parenrightbigg\nsin/parenleftbiggnπ\nyLy/parenrightbigg\n,(9)\nFSC\nm,n(x,y) = sin/parenleftbiggmπ\nxLx/parenrightbigg\ncos/parenleftbiggnπ\nyLy/parenrightbigg\n,(10)\nFSS\nm,n(x,y) = sin/parenleftbiggmπ\nxLx/parenrightbigg\nsin/parenleftbiggnπ\nyLy/parenrightbigg\n.(11)\nThe abbreviation c.c.represents complex conjugate. The\ndispersion relation of light in the waveguide is given by\nωk=c/radicalBig\nχ2m,n+k2, (12)\nwherec,ωk, andkarethe speed oflight, the frequency of\nthefieldradiation,andthe zcomponentofthewavenum-\nber, respectively. The dispersion relation is a nonlinear\nfunction in kandhascut-offfrequencies cχm,ndepending\non each mode (see Fig.2). The quantities qTE\nkandqTM\nk\nare the normal modes of the field. The symbol/summationtext\nm,n/integraltext\nk\nis an abbreviation for/summationtext∞\nm=0/summationtext∞\nn=0/integraltext∞\n−∞dk. The wave\nvectork= (mπ/xL,nπ/y L,k) is discrete in the xandy\ndirections due to the confinement of the waveguide. The\ndiscretepair( m,n)rangesoverallpairsofintegersfrom0\nto∞, excluding (0 ,0), while kis continuous from −∞to\n∞. We adopt a symmetric gauge as the external vector\npotential Aex(r), which gives a static uniform magnetic\nfieldB(B >0) in the zdirection:\nAex(r) =/parenleftbigg\n−By\n2,Bx\n2,0/parenrightbigg\n. (13)\nIn this study we only consider the TE 01mode, which\ncan be well separated from the other modes (see Fig.2).\nThen, the Hamiltonian can be approximated as follows\n(see Appendix A for detailed calculation):\nH≃ω1q∗\n1q1+/integraldisplay∞\n−∞dkωkq∗\nkqk\n+λ/integraldisplay∞\n−∞dk(q1−q∗\n1)(Vkqk−V∗\nkq∗\nk),(14)\nwhereω1is the cyclotron frequency of the electron asso-\nciated with field mode q1,λis the dimensionless coupling\nconstant of order 10−6andVkis the interaction form fac-\ntor, each given as follows:\nω1≡eB\nme, (15)\nλ≡/radicalBigg\ne2ωc\n2meε0c3, (16)\nVk≡i√\nc3\n√πxLyL/radicalbiggω1\nωcFCS\n0,1(xc,yc)\n√ωkeikzc.(17)4\nFIG. 2. Mode structure of the dispersion relation of light\nin the rectangular waveguide from the lowest mode to the\nfourth mode. The wave-number kand frequency ωkare non-\ndimensionalized by using the cut-off frequency ωc≡cχ0,1at\nthe lowest mode of the waveguide as the unit of frequency\nandc/ωcas the unit of length. As a specific example in the\ndiscussion, ( xL,yL) = (0.1 m,0.2 m) has been chosen, hence\nωc≃4.7 GHz.\nWe putzc= 0, assuming the electron is rotating at the\norigin of coordinate z. Since we consider only the TE 01\nmode, the bold subscript in ωkandqkhas been replaced\nwith scalar k. This Hamiltonian has been obtained by\nthe following two approximations, in addition to the low-\nest mode approximation with the TE 01mode. One is\nto neglect the field-field interaction term proportional to\nthe square of Ain(r) (see Eq.(A4) in Appendix). The\nother is the dipole approximation (see Eq.(A50) in Ap-\npendix). We note that virtual process terms such as q1qk\nandq∗\n1q∗\nkare retained, i.e. the rotating wave approxi-\nmation is not imposed. As a result, the field treatment\nis consistent with relativity, and the emission process\nsatisfies causality. We call this Hamiltonian the classi-\ncal Friedrichs model because if second quantization were\nperformed on the normal modes of this Hamiltonian, it\nwould become the well-known Friedrichs model in quan-\ntum mechanics [4]. However, we do not perform second\nquantization but instead treat the normal modes as clas-\nsical quantities. For the arbitrary functions fandgwith\nthese classical quantities as the arguments, we define the\nfollowing Poisson bracket:\n{f,g} ≡ −i/summationdisplay\nα/parenleftbigg∂f\n∂qα∂g\n∂q∗α−∂f\n∂q∗α∂g\n∂qα/parenrightbigg\n,(18)\nwhereαis 1 orkand we have used a conventional dis-\ncrete notation for the case of the continuous variable, i.e.\nα=kfor the continuous variable, in which case the sum-\nmation symbol is replaced by an integral. This definition\nis consistent with the definition of the Poisson brackets\nin terms of canonical variables (see Eq.( ??) in AppendixA). The normal modes satisfy the Poisson brackets,\n{q1,q∗\n1}=−i, (19)\n{qk,q∗\nk′}=−iδ(k−k′), (20)\n{qα,qβ}= 0, (21)\nwhereβis 1 orkwithα/ne}ationslash=β. Since the Poisson brack-\nets satisfy the same algebra as the commutation rela-\ntion, all algebraic calculations performed for our classical\nFriedrichs model lead to essentially the same results as\nthe quantum case.\nHere, we perform the non-dimensionalization of the\nHamiltonian (14). At first, we recognize that eB/χ2\nm,n\nhas the physical dimensions of an action variable written\nintermsoftheHamiltonianparameters. Inthisstudy, we\nspecifically focus on the vicinity of the Van Hove singu-\nlarity associated with the lowest mode of the waveguide.\nConsequently, we carry out the non-dimensionalization\nof the Hamiltonian by using unit J0≡eB/χ2\n0,1. More-\nover, we introduce dimensionlessquantities by measuring\nfrequency in the unit of the cut-off frequency ωc≡cχ0,1\nof the lowest waveguide mode, in which cis the speed\nof light. Then, representing the dimensionless quantities\nwith an overbar, we have\n¯H=w1¯q∗\n1¯q1+/integraldisplay∞\n−∞dκwκ¯q∗\nκ¯qκ\n+λ/integraldisplay∞\n−∞dκ(¯q1−¯q∗\n1)/parenleftbig¯Vκ¯qκ−¯V∗\nκ¯q∗\nκ/parenrightbig\n,(22)\nwhere¯H≡H/(ωcJ0), ¯q1≡q1/√J0, ¯qκ≡qk//radicalbig\ncJ0/ωc,\n¯Vκ≡Vk/√cωc,w1≡ω1/ωc, andwκ≡ωk/ωc. The\ntimetis also replaced by the dimensionless quantity τ≡\nωct. In addition, the dispersion relation of light in the\nwaveguide can be written as\nwκ=/radicalbig\n1+κ2, (23)\nwhereκ≡(c/ωc)k.\nSince Eq.(22) is a bilinear Hamiltonian, we can exactly\n“diagonalize” it as\n¯H= ˜w1¯Q∗\n1¯Q1+/integraldisplay∞\n−∞dκwκ¯Q∗\nκ¯Qκ,(24)\nwhere¯Qαaretherenormalizednormalmodes1. Thevari-\nables (qα,q∗\nα) and (¯Qα,¯Q∗\nα) are connected by a linear\n1We note that if the cyclotron motion were to occur in free spac e\nwithout boundaries such as the walls of the waveguide, the re nor-\nmalized normal mode of the cyclotron motion ¯Q1would not ex-\nist and the transformed Hamiltonian would be given by Eq.(24 )\nwithout the first term in the right-hand side. In this case, th e\nBogoliubov transformation consists of Eqs.(26), (30), and (31)\nwith¯Q1= 0, which was an amazing discovery by Friedrichs in\n[3]. We found that the non-vanishing mode ¯Q1is recovered in\nthis case due to the Van Hove singularity in the waveguide.5\ntransformation,\n¯Q1=¯N1/bracketleftbiggw1+ ˜w1\n2w1¯q1+w1−˜w1\n2w1¯q∗\n1\n+λ/integraldisplay∞\n−∞dκ/parenleftbigg¯Vκ\nwκ−˜w1¯qκ+¯V∗\nκ\nwκ+ ˜w1¯q∗\nκ/parenrightbigg/bracketrightbigg\n,(25)\n¯Qκ= ¯qk−2λw1¯V∗\nκ\n¯ξ+(wκ)/bracketleftbiggwκ+w1\n2w1¯q1−wκ−w1\n2w1¯q∗\n1\n+λ/integraldisplay∞\n−∞dκ′/parenleftbigg¯Vκ′\nwκ′−wκ−iε¯qκ′+¯V∗\nκ′\nwκ′+wκ¯q∗\nκ′/parenrightbigg/bracketrightbigg\n.\n(26)\nwhereεis a positive infinitesimal. Hereafter, we leave\nthe limit ε→0 implicit to avoid heavy notation. The\nfunction ¯ξ±is defined by\n¯ξ±(ζ)≡ζ2−w2\n1−4λ2w1/integraldisplay∞\n−∞dκwκ|¯Vκ|2\n[ζ2−w2κ]±.(27)\nHere, the ±symbol in the integrandfactor 1 //bracketleftbig\nζ2−w2\nκ/bracketrightbig±\nmeans that the denominator is evaluated on a Riemann\nsheet that is analytically continued. The plus symbol in-\ndicatesanalyticcontinuationfromtheupperhalfplaneto\nthe lower half plane, and vice versa for the minus sym-\nbol. Also, ˜ w1is the renormalized frequency associated\nwith the stable mode ¯Q1, which is given by\n¯ξ(˜w1) = 0. (28)\nThe normalization factor ¯N1associated with this mode\nis given by\n¯N1≡/bracketleftBigg\n1\n2w1d¯ξ(ζ)\ndζ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nζ=˜w1/bracketrightBigg−1\n2\n. (29)\nWe refer to the above transformation as the classical Bo-\ngoliubovtransformation(seeAppendixBfordetailedcal-\nculation). The inverse transformation is given by\n¯q1=¯N1/parenleftbigg˜w1+w1\n2w1¯Q1−˜w1−w1\n2w1¯Q∗\n1/parenrightbigg\n−λ/integraldisplay∞\n−∞dκ/bracketleftbigg(wκ+w1)¯Vκ\n¯ξ−(wκ)¯Qκ+(wκ−w1)¯V∗\nκ\n¯ξ+(wκ)¯Q∗\nκ/bracketrightbigg\n,\n(30)\n¯qκ=¯Qκ−λ¯N1¯V∗\nκ/parenleftbigg¯Q1\n˜w1−wκ+¯Q∗\n1\n˜w1+wκ/parenrightbigg\n+2λ2w1¯V∗\nκ/integraldisplay∞\n−∞dκ′/bracketleftbigg¯Vκ′\n¯ξ−(wκ′)(wκ′−wκ−iε)¯Qκ′\n+¯V∗\nκ′\n¯ξ+(wκ′)(wκ′+wκ)¯Q∗\nκ′/bracketrightbigg\n.(31)\nCorrespondingtoEq.(19)throughEq.(21), therenormal-\nized normal modes ( ¯Qα,¯Q∗\nα) satisfy the Poissonbrackets,\n{¯Q1,¯Q∗\n1}=−i, (32)\n{¯Qκ,¯Q∗\nκ′}=−iδ(κ−κ′), (33)\n{¯Qα,¯Qβ}= 0. (34)where the Poisson bracket can be expressed by replacing\nqαby¯Qαin Eq.(18).\nPerforming the integral over κin Eq.(27), we find\n¯ξ±(ζ) =ζ2−w2\n1±λ2G2w1/radicalbig\n1−ζ2, (35)\nwhere we have defined the dimensionless constant\nG2≡4c2\nxLyLω2csin2/parenleftbiggπ\nyLyc/parenrightbigg\n, (36)\nfor which G∼1. Furthermore, the normalization factor\n(29) can be evaluated by using Eq.(35), which gives\n¯N1=/bracketleftBigg\n˜w1\nw1+λ2G2˜w1\n2(1−˜w2\n1)3/2/bracketrightBigg−1\n2\n, (37)\nwhere¯N1→0 in the limit w1→ ∞, because ˜ w1→1 in\nthis limit (see Fig.3(a)).\nSince the Hamiltonian (22) was diagonalized as\nEq.(24), we can obtain the time evolution of the renor-\nmalized normal mode ¯Qαby Hamilton’s equation of mo-\ntion. Hamilton’s equation of motion for ¯Qαis\nid¯Qα\ndτ=−LH¯Qα, (38)\nwhereLHis the Liouville operator (Liouvillian), which is\ndefined by the Poisson bracket with the Hamiltonian and\nisageneratorofthetimeevolutioninclassicalmechanics,\nLHf≡i{H,f}. (39)\nThe imaginary number iin front of the right-hand side\nof Eq.(39) is introduced so the Liouvillian LHis a Her-\nmitian operator in the Hilbert space. Substituting the\nHamiltonian (24) into Eq.(39), one can see that ¯Qαis an\neigenfunction of LH,\n−LH¯Qα= ˜wα¯Qα, (40)\nwhere ˜w1/ne}ationslash=w1is the renormalized frequency of the\nrenormalized normal mode of the cyclotron motion, and\n˜wκ=wκis the frequency of the the renormalized field\nmode. By Eqs.(38) and (40), the time evolution of ¯Qα\nbecomes\n¯Qα=¯Qα(0)e−i˜wατ, (41)\nAssuming ¯ qκ= 0 atτ= 0, the initial conditions ¯Q1(0)\nand¯Qκ(0) are given through Eqs.(25) and (26) by\n¯Q1(0) =¯N1/bracketleftbiggw1+ ˜w1\n2w1¯q1(0)+w1−˜w1\n2w1¯q∗\n1(0)/bracketrightbigg\n,(42)\n¯Qκ(0) =−2λw1¯V∗\nκ\n¯ξ+(wκ)/bracketleftbiggwκ+w1\n2w1¯q1(0)−wκ−w1\n2w1¯q∗\n1(0)/bracketrightbigg\n.\n(43)6\nFIG. 3. Pole locations in the vicinity of the Van Hove sin-\ngularity. (a) is the real part, (b) is the imaginary part. The\nsolid lines are for ζ0, the dashed lines are for ζ1andζ2, re-\nspectively. The imaginary part of the ζ1in (b) provides the\ndecay rate of the cyclotron motion. The maximum decay rate\non the Van Hove singularity ( w1−1 = 0) is 104times larger\ncompared to the Fermi region ( w1≫1) which is located to\nthe far right, outside of the graph.\nIII. DECAY RATE OF THE CYCLOTRON\nMOTION\nThe electron mode ¯ q1in Eq.(30) consists of two terms,\n¯q1(τ) = ¯q1,s(τ)+ ¯q1,d(τ), (44)\nwhich are the discrete component associated with ¯Q1,\ngiven by\n¯q1,s(τ)≡¯N1/bracketleftbigg˜w1+w1\n2w1¯Q1(0)e−i˜w1τ\n−˜w1−w1\n2w1¯Q∗\n1(0)ei˜w1τ/bracketrightbigg\n,(45)and the field component\n¯q1,d(τ)≡ −λ/integraldisplay∞\n−∞dκ/bracketleftbigg(wκ+w1)¯Vκ\n¯ξ−(wκ)¯Qκ(0)e−iwκτ\n+(wκ−w1)¯V∗\nκ\n¯ξ+(wκ)¯Q∗\nκ(0)eiwκτ/bracketrightbigg\n.(46)\nAs can be seen when substituting ¯Q1(0) into Eq.(45), ¯ q1,s\nsimply gives a stable oscillation without decay. There-\nfore, the cyclotron motion inside the waveguide never\nstops even as τ→+∞. This means the electron experi-\nences perpetual acceleration without emitting radiation\neven asτ→+∞. This is possible because of the Van\nHove singularity.\nTo evaluate ¯ q1,d, we substitute ¯Qκ(0) into Eq.(46) and\nchange the integration variable from κtowκ, to obtain\n¯q1,d(τ) = 2λ2/integraldisplay∞\n1dwκwκ/radicalbig\nw2κ−1/vextendsingle/vextendsingle¯Vκ/vextendsingle/vextendsingle2\n/vextendsingle/vextendsingle¯ξ+(wκ)/vextendsingle/vextendsingle2\n×/bracketleftbig\n(wκ+w1)h(wκ)e−iwκτ\n+(wκ−w1)h∗(wκ)eiwκτ/bracketrightbig\n,(47)\nwhere\nh(wκ)≡w1Re[¯q1(0)]+iwκIm[¯q1(0)],(48)\nand\n/vextendsingle/vextendsingle¯ξ+(wκ)/vextendsingle/vextendsingle2=¯ξ+(wκ)¯ξ−(wκ). (49)\nHere, the factor dκ/dw κ=wκ//radicalbig\nw2κ−1 is the density of\nstates, which diverges at the lower limit of the integral.\nThis divergence is the Van Hove singularity.\nAs can be seen, ¯ q1,dgives the time decay of electron\nmotion because the denominetor in Eq.(47) has poles in\nthe complex frequency plane. By putting the denomine-\ntor in Eq.(47) to zero, we obtain a double cubic equation\nas follows:\n/parenleftbig\nζ2\nn/parenrightbig3−(2w2\n1+1)/parenleftbig\nζ2\nn/parenrightbig2+(2w2\n1+w4\n1)ζ2\nn\n−w4\n1+λ4G4w2\n1= 0,(50)\nwhere we have introduced the dimensionless variable ζ2\nn.\nThe general solutions of Eq.(50) are given by the stan-\ndard method to solve the cubic equation as\nζ2\nn=e2\n3niπα1\n3w1++e−2\n3niπα1\n3w1−+dw1,(51)\nwhere\nαw1±≡−qw1±/radicalbigq2w1+4p3w1\n2, (52)\nand\npw1≡ −1\n32(w2\n1−1)2, (53)\nqw1≡2\n33(w2\n1−1)3+w2\n1λ4G4, (54)\ndw1≡1\n3(2w2\n1+1). (55)7\nHeren= 0, 1 or 2. By taking the square root of eq.(51),\nwe obtain three solutions ζnwith Re[ ζn]>0. Addition-\nally, the three solutions with Re[ ζn]<0 are given by\nputting a minus sign to the solutions with Re[ ζn]>0.\nIn the following, we show the behaviour of the three so-\nlutions with Re[ ζn]>0. The solution of this equation\nlabeledζ0is real for all values of w1and corresponds to\nthe bound state discussed previously. The solutions ζ1\nandζ2can be real or complex-valued, depending on the\nvalue ofw1as described below.\nFig.3 shows the behavior of the solutions ζnin the\nvicinity of the cut-off frequency (i.e., in the vicinity of\nthe Van Hove singularity) in the waveguide. Fig.3(a)\nrepresents the real part of the solutions, while (b) repre-\nsents the imaginary part. In both figures, the solid line\ncorresponds to the bound state ζ0, while the dashed lines\nrepresent the two complex solutions ζ1andζ2. The com-\nplex solution with negative imaginary part corresponds\nto the resonance state that breaks time symmetry.\nMoreover, one can see in Fig.3(a) that there exists a\npointw0−1wheretworealsolutionscoalescebeforeturn-\ning into a resonance and its partner anti-resonance state\nwith complex eigenvalues. This point is known as an ex-\nceptional point [24]. The properties of the system near\nthe exceptional point are interesting, but we will focus\nin this paper more narrowly on the Van Hove singularity\nitself.\nExactly at the Van Hove singularity, we can obtain\nζ2\n1as the exact solution of the cubic equation (50) after\nputtingw1= 1,\nζ2\n1= 1+1\n2λ4\n3G4\n3−i√\n3\n2λ4\n3G4\n3, (56)\nwhereGis the dimensionless constant with typical order\n1 as shown in Eq.(36). The resonance state ζ1at the Van\nHove singularity is then given by taking the square root\nof Eq.(56), then we have\nζ1= 1+1\n4λ4\n3G4\n3−i√\n3\n4λ4\n3G4\n3+O/parenleftBig\nλ8\n3/parenrightBig\n.(57)\nThis expression can also be obtained by Puiseux expan-\nsion in terms of λ[24]. The imaginary part of this ex-\npression gives the decay rate at the Van Hove singular-\nity. From the form of Eq.(57), it is clear that this result\ncannot be obtained by the usual perturbation analysis\nwith the series expansion in terms of λas in the case\nof Fermi’s golden rule that gives the decay rate propor-\ntional to λ2. We can see that the decay rate in the vicin-\nity of the Van Hove singularity is magnified by a factor\nλ−2/3(=λ4/3/λ2) compared with the Fermi region. For\nthe electron cyclotron motion, this magnification factor\nis about 104.\nRegarding this discussion, we elaborate further on the\npoint that ˜ w1andwκrepresent the real eigenvalues in\nthe first Riemann sheet and the complex eigenvalues in\nthe secondRiemannsheet ofthe Liouvillian, respectively.\nRemarkably, ¯Q1does not experience time decay, whereas\n¯Qκexperiencestime decay. As brieflymentioned in Foot-\nnote 1, the existence of the Van Hove singularity allows\nFIG. 4. Pole locations on the complex frequency plane. ζ0is\nlocated in the first Riemann sheet. ζ1andζ2are located in\nthe second Riemann sheet.\nfor this non-decaying mode. Consequently, there is a sce-\nnario where cyclotron motion can persist without emit-\nting light. In other words, in classical systems, there\nare instances where an electron can undergo accelerated\nmotion without emitting light and continue its motion\nwithout decay.\nIV. TIME EVOLUTION OF THE FIELD MODE\nThe field modes ¯ qκin Eq.(31) consists of three terms,\n¯qκ(τ) = ¯qκ,p(τ)+ ¯qκ,s(τ)+ ¯qκ,d(τ),(58)\nwhere\n¯qκ,p(τ)≡¯Qκ(0)e−iwκτ, (59)\n¯qκ,s(τ)≡ −λ¯N1¯V∗\nκ/bracketleftbigg¯Q1(0)e−i˜w1τ\n˜w1−wκ+¯Q∗\n1(0)ei˜w1τ\n˜w1+wκ/bracketrightbigg\n,(60)\n¯qκ,d(τ)≡2λ2w1¯V∗\nκ/integraldisplay∞\n−∞dκ′/bracketleftbigg¯Vκ′¯Qκ′(0)e−iwκ′τ\n¯ξ−(wκ′)(wκ′−wκ−iε)\n+¯V∗\nκ′¯Q∗\nκ′(0)eiwκ′τ\n¯ξ+(wκ′)(wκ′+wκ)/bracketrightbigg\n.(61)\nThe term ¯ qκ,pdescribes the free propagation of the field\ninside the waveguide. On the other hand, ¯ qκ,sand ¯qκ,d\ndescribe the steady oscillation and the decaying compo-\nnents of the field around the electron, respectively.\nThe focus of our interest is a detailed analysis of the\npropagating wave emitted toward the far distance from\nthe electron in cyclotron motion, which is denoted as\n¯qκ,p. Therefore, we do not further discuss the bound\nfields around the electron ¯ qκ,sand ¯qκ,din the present\nwork.\nFor the TE 01mode which we consider here, the vector8\npotential (5) is approximated as\nAp(Z,τ)≃Ac/integraldisplay∞\n−∞dκ1√wκ¯qκ,p(τ)eiκZ+c.c.,(62)\nwhereZis the dimensionless coordinate variable on the z\ndirection defined by Z≡(ωc/c)zand we have introduced\nthe notation Ap(Z,τ) as an abbreviation for ATE01\nin,x(Z,τ).\nThe quantity Acis the amplitude with the dimension of\na vector potential:\nAc≡/radicalbigg\nJ0\ncFCS\n0,1(xc,yc)NA. (63)\nWe note that ( x,y) in Eq.(63) has been replaced with\nthe rotation center of the electron ( xc,yc), because of\nthe dipole approximation.\nIn the following part of this paper, we calculate the\ntime evolution at Z= 0. This is because, once the field\nis emitted from the cyclotron motion, the field can prop-\nagate freely inside the waveguide. Thus, if one wishes,\nit is possible to calculate the propagation of the emit-\nted field by using the linear wave equation with a given\nsource that is the time evolution of the field at Z= 0.\nBy substituting Eq.(59) with initial condition (43) into\nEq.(62), the electric field at Z= 0 is given by\nEp(0,τ) =−ωc∂Ap(0,τ)\n∂τ\n=λEc/integraldisplay∞\n−∞dκh(wκ)\n¯ξ+(wκ)e−iwκτ+c.c.,(64)\nwhereh(wκ) is given in Eq.(48) and the factor Ecin\nEq.(64) is a quantity with the dimension of the electric\nfield defined as\nEc≡−2√w1cFCS\n0,1(xc,yc)Ac√πxLyL\n=ω2\nc\n2π/radicalbigg\nw1J0\nc3ε0G2. (65)\nBy changing the integration variable from κtowκ, we\nhave\nEp(0,τ) =λEc/integraldisplay∞\n1dwκ˜g+(wκ)e−iwκτ+c.c..(66)\nwhere\n¯g+(wκ)≡wκ/radicalbig\nw2κ−1h(wκ)\n¯ξ+(wκ)\n=wκw1Re[¯q1(0)]+iw2\nκIm[¯q1(0)]/radicalbig\nw2κ−1(w2κ−w2\n1)−iλ2G2w1.(67)\nHere again, dκ/dw κ=wκ//radicalbig\nw2κ−1 is the density of\nstates, which diverges at the lower limit of the integral.\nWe note that the function ¯ g+does not diverge at the\nlowerlimit ofthe integration. This is because ¯ξ+given in\nthe denominator of Eq.(35) also has a divergence at the\nlower limit of integration, thus the product of the densityof states dκ/dw κand the function in the denominator ¯ξ+\nsuppresses the divergence of the integrand. However, the\nlower limit of the integral is still a singular point. This is\nbecause the denominator of ¯ g+includes the branch point\ndue to the density of states.\nPutting the denominator of ¯ g+to zero in Eq.(66), we\nobtain the same cubic equation in Eq.(50) because the\nintegrands of Eq.(47) and Eq.(66) have the same factor\n¯ξ+in the denominator. Thus, the integrand of Eq.(66)\nalso has one real pole (bound state ζ0) and two complex\npoles (resonance and anti-resonance states, ζ1andζ2) on\nthe complex frequency plane, which behave as shown in\nFig.3. One can perform the integration in Eq.(66) by\nanalytic continuation of the integration variable into the\ncomplex frequency plane. For this case, the complex fre-\nquency plane consists of two Riemann sheets as shown in\nFig.4. In Fig.4, the bound state exists on the real axis of\nthe first Riemann sheet of the complex frequency plane,\nwhereas the resonance and anti-resonance poles coexist\nascomplexconjugatesonthesecondRiemannsheet. The\nresonancestateappearsonthe lowerhalfplane, while the\nanti-resonance state appears on the upper half plane.\nForτ >0, we perform a contour deformation as shown\nin Fig.4. The integral contours C−andC+cancel out.\nMoreover, it is easy to show that the integral contribu-\ntionsCR1−andCR2−become zero in the limit R→ ∞.\nThen,Ep(0,τ) is composed of only two terms as\nEp(0,τ) =Ep,p(0,τ)+Ep,b(0,τ),(68)\nwhere\nEp,p(0,τ)≡λEc/integraldisplay\nCp−dwk¯g+(wκ)e−iwkτ+c.c.,(69)\nEp,b(0,τ)≡λEc/integraldisplay\nCb−dwk¯g+(wκ)e−iwkτ+c.c..(70)\nThis means that the pole effect Ep,p(0,τ) and the branch\npoint effect Ep,b(0,τ) are the only contribution to the\nintegration in Eq.(66).\nA. Pole effect\nSince we know the location of the pole ζ1, we can per-\nform the integral (69) using the residue theorem, thus\nEp,p(0,τ) =−2πiλEc/radicalbig\nζ2\n1−1h(ζ1)e−iζ1τ\n3ζ2\n1−w2\n1−2+c.c..(71)\nThe time evolution of the emitted field at the Van Hove\nsingularity is obtained by substituting Eqs.(56) and (57)\ninto Eq.(71) with w1= 1, is given by\nEp,p(0,τ) =4πλ1\n3Ec|h(ζ1)|\n3√\n3G2\n3\n×sin/bracketleftbigg/parenleftbigg\n1+1\n4λ4\n3G4\n3/parenrightbigg\nτ+π\n6+θ/bracketrightbigg\ne−√\n3\n4λ4\n3G4\n3τ.\n(72)9\nwhere\n|h(ζ1)|2={Re[¯q1(0)]}2+{Im[¯q1(0)]}2\n+1\n2λ4\n3G4\n3/parenleftBig√\n3Re[¯q1(0)]+Im[¯ q1(0)]/parenrightBig\nIm[¯q1(0)]\n+1\n4λ8\n3G8\n3{Im[¯q1(0)]}2(73)\nThe phase θsatisfies\nsinθ=Im[¯q1(0)]−√\n3Re[¯q1(0)]\n2|h(ζ1)|,(74)\ncosθ=√\n3Im[¯q1(0)]+Re[¯ q1(0)]\n2|h(ζ1)|.(75)B. Branch point effect\nNext, weevaluate the branchpoint effect. The integra-\ntion path Cb−in Eq.(70) gives the path from the branch\npoint 1 to 1 −iR(R→ ∞) in the complex frequency\nplane (see Fig.4),\nEp,b(0,τ) =λEclim\nR→∞/integraldisplay1−iR\n1dwκ¯g+(wκ)e−iwκτ+c.c..\n(76)\nWe change the integration variable from wκtosas\nwκ= 1−is\nτ, (77)\nthen, we have\nEp,b(0,τ) =−iλEc\nτe−iτ/integraldisplay∞\n0ds¯g+/parenleftBig\n1−is\nτ/parenrightBig\ne−s+c.c.,\n(78)\nwhere we have taken the limit R→ ∞and,\n¯g+/parenleftBig\n1−is\nτ/parenrightBig\n=w1τ2(τ−is)Re[¯q1(0)]+iτ(τ−is)2Im[¯q1(0)]√s√−i2τ−s[(1−w2\n1)τ2−i2τs−s2]−iλ2G2w1τ3. (79)\nSince the integrand in Eq.(78) is expressed by known\nfunctions, we can evaluate the value of the integral (78)\nin terms of quadrature by numerical calculations. The\nresults will be shown in the next section.\nV. DISCUSSION\nFig.5 shows the numerical results of the absolute value\nof the pole effect in Eq.(71) and the branch point effect\nin Eq.(78) as functions of the dimensionless cyclotron\nfrequency w1. Fig.5(a), (b) and (c) show the dynamics\nwhenw1is in the Fermi region, between the Fermi re-\ngion and the Van Hove singularity region (we call this\nthe middle region), and the Van Hove singularity region,\nrespectively. The solid lines represent the pole effect,\nwhile the dashed lines represent the branch point effect.\nIn all regions, the pole effect leads to exponential decay,\nwhile the branch point effect results in power law decay.\nWhenw1is in the Fermi region as shown in Fig.5(a),\nthe branch point effect rapidly dies out as the order of\nthe amplitudebecomesmuchsmallerthanthe poleeffect,\nmaking it difficult to observe experimentally. However,\nwhenw1is in the Van Hove singularity region as shown\nin Fig.5(c), the branch point effect persists up to the\nrelaxation time of the pole effect with about the same\nmagnitude as the pole effect. This result suggests that\nthe non-Markovian branch point effect is experimentally\nobservable in the Van Hove singularity region.In each case, the timescale on which most of the decay\noccurs depends on the resonance state. For the Fermi\nregion 1 ≪w1, the function ¯ξ±(ζ) in Eq.(35) is approxi-\nmated as\n¯ξ±(ζ)≃ζ2−w2\n1±λ2G2w1/radicalbig\n1−w2\n1, (80)\nwhere we expand the third term on the right-hand side\nof Eq.(35) in a Taylor series around w1and keep only the\nlowest order. Since ¯ξ±(ζ) = 0 gives the pole locations,\nwe can obtain ζ2\n1in the Fermi region,\nζ2\n1≃w2\n1−iλ2G2w1/radicalbig\nw2\n1−1. (81)\nThe resonance eigenvalue ζ1in the Fermi region is then\ngiven by taking the square root of Eq.(81), and again\nexpanding to obtain\nζ1≃w1−iλ2G2\n2/radicalbig\nw2\n1−1, (82)\nthus the lifetime of the exponential decay in the Fermi\nregion (Fig.5(a)) is estimated by 1 /(λ2G2/2/radicalbig\nw2\n1−1)≃\n9.9×1012, which is consistent with Fermi’s golden rule2.\n2We note that it is well known that the Abraham-Lorentz equa-\ntion derived by using Li´ enard–Wiechert potentials gives t he de-10\nFIG. 5. Time evolution of the absolute value of the pole and br anch point effects: (a), (b) and (c) are represented when w1is in\nthe Fermi region, between the Fermi region and the Van Hove si ngularity region, and on the Van Hove singularity, respecti vely.\nThese absolute values are normalized by |E1| ≡ |Ep,p(0,1)|.\nFor the Van Hove singularity region (Fig.5(c)), the life-\ntime of the exponential decay is estimated from Eq.(72)\nas 1/(√\n3λ4\n3G4\n3/4)≃6.0×108. Hence, the timescale is\ndramatically shortened as a result of the Van Hove sin-\ngularity.\nIn our previous work Ref.[13], our original motivation\nwas partly to find a method to enhance the Markovian\nexponential decay associated with the resonance pole.\nHowever, as revealed here the non-Markovianprocess as-\nsociated with the branch-point effect is also (compara-\nbly) enhanced in the Van Hove region. Returning to this\nprevious objective, we have found here that by appropri-\nately tuning the cyclotron frequency it is also possible\nto enhance the Markovian dynamics in absolute terms,\nwhile maintaining its relative dominance over the non-\nMarkovian dynamics. This is shown in Fig.5(b) in the\nmiddle region.\nVI. CONCLUDING REMARKS\nInthis research,weanalyzedthe emissionfromanelec-\ntron in cyclotron motion in a waveguide using the clas-\nsical Friedrichs model without relying on perturbation\ntechniques. When the cyclotron frequency is tuned in\nthe vicinity of the cut-off frequency of a waveguide (i.e.,\nin the vicinity of the Van Hove singularity), we found not\nonly that the pole effect is sharply amplified (104times)\nas compared with the case of the Fermi region but also\nthatthebranchpointeffectisamplifiedtoaboutsameor-\nder of magnitude. This suggests that the non-Markovian\nbranch point effect is experimentally observable in the\nVan Hove singularity region. Further, we have found an\nintermediate case for which the Markovian dynamics are\nsignificantly enhanced, but are still dominant compared\ncay rate of an electron proportional to λ2in addition to the prob-\nlematic runaway solution. Thus, the Abraham-Lorentz equat ion\ncan be used only in the Fermi region.to the non-Markovian dynamics (see Fig.5(b)). Hence,\nenhancement ofboth effects should be experimentallyac-\ncessible.\nFurther, we haveobtained an essentiallyself-consistent\ndescription of classical radiation damping for the spe-\ncific case of an electron undergoing cyclotronic motion\nin a waveguide. In particular, we have implicitly dealt\nwith the self-forces acting on the electron due to its own\nemitted radiation when we exactly evaluated the integral\nterm in Eq.(27), which is the equivalent of the self-energy\nfunction in the quantum formalism. That is to say, we\naccounted for the back-reaction on the electron at the\nequivalentofalllevelsofperturbationtheorythroughour\nexact diagonalization procedure. We accomplished this\nby classicalizing the second quantized Friedrichs model\nand replacing commutation relations with Poissonbrack-\netsfor(what turnedout tobe) an exactlysolvablemodel.\nThis procedure avoids the well-known problems with\nobtaining a self-consistent description of radiation damp-\ning in classical electrodynamics. In particular, we have\nnot employed time-averaging or other phenomenological\napproximations that are used to obtain the Abraham-\nLorentz equation for the self-force on the electron. We\nhave only employed two significant approximations, nei-\nther of which is particularly intrusive: that the interac-\ntion with the field is localized on the electron’s center of\nmotion and field-field interactions have been neglected.\nNeither of these approximations results in problems such\nas pre-acceleration or the runaway solution known to\nplague the Abraham-Lorentz description. Of course, our\nmodel only applies to a specific case; however, one could\nfind other cases where classicalization would be applica-\nble and from that point more detailed physical descrip-\ntionscouldbeobtainedbytheuseofperturbationtheory.\nFinally, in this paper, we have focused on the emission\nprocess near the lowest cut-off mode in a rectangular\nwaveguide. In future work, we will study the emission\nprocess including the modes with higher-frequency cut-\noff, which will introduce interesting new effects. Further,\nwe will present results for the emission process from cy-\nclotron motion inside a cylindrical waveguide in which11\npropagating field modes with non-zero angular momen-\ntumcanbeproduced. Thisistheso-calledopticalvortex.\nWe will show that by tuning the cyclotron frequency to\nthe appropriate waveguide mode, one can select to pro-\nduce the optical vortex with a range of angular momen-\ntum values.\nACKNOWLEDGMENTS\nWe thank Dr. Satoshi Tanaka, Dr. Hiroaki Nakamura\nand Dr. Naomichi Hatano for helpful discussions related\nto this work. Y.G. acknowledges support from the Japan\nSociety for the PromotionofScience (JSPS) under KAK-\nENHI Grant No. 22K14025 and from the Research En-\nhancement Strategy Office in the National Institute for\nFusion Science. S.G. acknowledges support from JSPS\nunder KAKENHI Grant No. 18K03466.\nAppendix A: DERIVATION OF THE CLASSICAL\nFRIEDRICHS MODEL\nIn this appendix we derive Eq.(14) from Eq.(1). By\nexpanding Eq.(2), we obtain\nHe=1\n2me[p+eAex(re)]2\n+e\nme[p+eAex(re)]·Ain(re)\n+e2\n2meAin(re)·Ain(re).(A1)\nHere, we will define the unperturbed part H0\neand per-\nturbed part H1\neas\nH0\ne≡1\n2me[p+eAex(re)]2, (A2)\nH1\ne≡e\nme[p+eAex(re)]·Ain(re),(A3)\nthus,\nHe≃H0\ne+H1\ne, (A4)\nwhere we have neglected the field-field interaction term\npropotional to the square of Ain(r), which is the third\nterm in the right-hand side of Eq.(A1).\nBy substituting the explicit form of the symmetric\ngaugeAex(r) in Eq.(13) into Eq.(A2), H0\neis given by\nH0\ne=1\n2me/braceleftbigg/parenleftBig\npy+meω1\n2xe/parenrightBig2\n+/parenleftBig\npx−meω1\n2ye/parenrightBig2/bracerightbigg\n,\n(A5)\nwhere we impose the restriction that the zelement of pis\nnotpresentbecauseitdoesnotaffecttheemissionoffield.\nSince cyclotronmotionis circular, it ispresumed that H0\ne\ncanbeexpressedbyaharmonicoscillatorwithonedegree\nof freedom. Under this presumption, we express H0\neas\nH0\ne=ω1\n2(P2\n1+Q2\n1), (A6)where we defined the new variables as\nP1≡1√meω1/parenleftBig\npy+meω1\n2xe/parenrightBig\n,(A7)\nQ1≡1√meω1/parenleftBig\npx−meω1\n2ye/parenrightBig\n. (A8)\nForthearbitraryfunctions fandgwiththesevariablesas\nthe arguments, we define the following Poisson bracket:\n{f,g}1≡/parenleftbigg∂f\n∂P1∂g\n∂Q1−∂f\n∂Q1∂g\n∂P1/parenrightbigg\n.(A9)\nThese variables P1andQ1satisfy the Poisson brackets\nbelow;\n{P1,Q1}= 1, (A10)\n{P1,P1}= 0, (A11)\n{Q1,Q1}= 0, (A12)\nthus,P1andQ1are canonical variables. And it is\nproven that our presumption that H0\necan be expressed\nas a harmonic oscillator with one degree of freedom is\ncorrect. We note since there were originally four vari-\nables (xe,ye,px,py), there should also be another pair of\ncanonical variables P2andQ2, in addition to P1andQ1.\nHowever, P2andQ2do not appear in this Hamiltonian,\nand hence they are constants of the motion in this sys-\ntem.\nHowever,weneedtoknow P2andQ2inordertoobtain\nthe inverse canonical transformation. Hence we assume\nP2andQ2are a linear combination of ( xe,ye,px,py) be-\ncauseP1andQ1are also a linear transformation includ-\ning any of the elements ( xe,ye,px,py) in Eqs(A7) and\n(A8). Namely, P2andQ2are assumed as follows:\nP2=c1xe+c2px+c3ye+c4py,(A13)\nQ2=d1xe+d2px+d3ye+d4py.(A14)\nwhere (c1,c2,c3,c4) and (d1,d2,d3,d4) are unknown co-\nefficients. These coefficients are determined so that P2\nandQ2become canonical variables by using the Poisson\nbrackets;\n{P2,Q2}2= 1, (A15)\n{P2,P2}2= 0, (A16)\n{Q2,Q2}2= 0, (A17)\nwhere we have defined the following Poisson bracket for\nthe arbitrary functions fandgwith these variables as\nthe arguments\n{f,g}2≡/parenleftbigg∂f\n∂P2∂g\n∂Q2−∂f\n∂Q2∂g\n∂P2/parenrightbigg\n.(A18)\nThus, we obtain the relations for the coefficients,\nc3=meω1\n2c2, c4=−2\nmeω1c1,(A19)\nd3=meω1\n2d2, d4=−2\nmeω1d1,(A20)12\nand\nc1d2−c2d1=1\n2. (A21)\nTherefore, P2andQ2are given by\nP2=−2\nmeω1c1/parenleftBig\npy−meω1\n2xe/parenrightBig\n+c2/parenleftBig\npx+meω1\n2ye/parenrightBig\n,\n(A22)\nQ2=−2\nmeω1d1/parenleftBig\npy−meω1\n2xe/parenrightBig\n+d2/parenleftBig\npx+meω1\n2ye/parenrightBig\n.\n(A23)\nThe inverse transformation is given by\nxe=1√meω1P1−c2Q2+d2P2, (A24)\nye=−1√meω1Q1+2\nmeω1c1Q2−2\nmeω1d1P2,(A25)\npx=√meω1\n2Q1+c1Q2−d1P2, (A26)\npy=√meω1\n2P1+meω1\n2c2Q2−meω1\n2d2P2.(A27)\nBy substituting the explicit form of Aex(r) in Eq.(13)\ninto Eq.(A3) and carrying out the canonical transforma-\ntion by using P1andQ1,H1\neis given by\nH1\ne=/radicalBigg\nω1e2\nme[Q1Ain,x(re)+P1Ain,y(re)].(A28)\nwhere again only xandycomponents are present.\nFurthermore, we introduce the discrete normal mode\nby\nq1≡1√\n2(P1+iQ1), (A29)\nq∗\n1≡1√\n2(P1−iQ1). (A30)\nThe inverse transformation is given by\nP1=1√\n2(q1+q∗\n1), (A31)\nQ1=−i√\n2(q1−q∗\n1). (A32)\nThus, theelectronpartoftheHamiltonian He=H0\ne+H1\ne\nis given by\nHe=ω1q∗\n1q1\n−i/radicalBigg\nω1e2\n2me[(q1−q∗\n1)Ain,x(re)+i(q1+q∗\n1)Ain,y(re)].\n(A33)\nAlso, by substituting the vector potentials (5) and (6)into Eq.(3), the field part of the Hamiltonian Hfis ob-\ntained by using the continuous normal mode,\nHf=/summationdisplay\no/summationdisplay\nm,n/integraldisplay\nkωkqo\nk∗qo\nk, (A34)\nwhere the index of summation omeans TE or TM. We\nnote that this Hamiltonian can also be expressed in the\nform of a harmonic oscillator as follows:\nHf=/summationdisplay\no/summationdisplay\nm,n/integraldisplay\nk/parenleftbigg1\n2ε0Po\nk+1\n2ε0ω2\nkQo\nk/parenrightbigg\n,(A35)\nwhere\nPo\nk≡/radicalbiggε0ωk\n2(qo\nk+qo\nk∗), (A36)\nQo\nk≡−i√2ε0ωk(qo\nk−qo\nk∗). (A37)\nForthearbitraryfunctions fandgwiththesevariablesas\nthe arguments, we define the following Poisson bracket:\n{f,g}k≡/summationdisplay\nk/parenleftbigg∂f\n∂Po\nk∂g\n∂Qo\nk−∂f\n∂Qo\nk∂g\n∂Po\nk/parenrightbigg\n.(A38)\nHere,Po\nkandQo\nkrepresent the generalized momentum\nand the generalized coordinates, respectively, satisfying\nthe Poisson bracket below:\n{Po\nk,Qo\nk′}k=δ(k−k′), (A39)\n{Po\nk,Po\nk′}k= 0, (A40)\n{Qo\nk,Qo\nk′}k= 0. (A41)\nThe continuous normal modes are given by\nqo\nk=1√2ε0ωk(Po\nk+iε0ωkQo\nk),(A42)\nqo\nk∗=1√2ε0ωk(Po\nk−iε0ωkQo\nk).(A43)\nForthesePoissonbracketswithrespecttothesecanonical\nvariables, we define the following new Poisson brackets:\n{f,g} ≡/summationdisplay\nα′/parenleftbigg∂f\n∂Pα′∂g\n∂Qα′−∂f\n∂Qα′∂g\n∂Pα′/parenrightbigg\n,(A44)\nwhereα′is1, 2,or k. Thus,introducingthenormalmode\nin this way as Eqs.(A29), (A30), (A42) and (A43) yields\nEq.(18). Therefore, the full Hamiltonian His given by\nusing normal modes,\nH=ω1q∗\n1q1+/summationdisplay\no/summationdisplay\nm,n/integraldisplay\nkωkqo\nk∗qo\nk\n+λ/summationdisplay\no/summationdisplay\nm,n/integraldisplay\nk/braceleftbig\n(q1−q∗\n1)/bracketleftbig\nVo\nk,x(re)qo\nk−Vo\nk,x∗(re)qo\nk∗/bracketrightbig\n+i(q1+q∗\n1)/bracketleftbig\nVo\nk,y(re)qo\nk−Vo\nk,y∗(re)qo\nk∗/bracketrightbig/bracerightbig\n,(A45)13\nwhere we have defined the form factors:\nVTE\nk,x(re)≡i√\nc3\n√πxLyL/radicalbiggω1\nωcnπ\nyLFCS\nm,n(xe,ye)\nχm,n√ωkeikze,(A46)\nVTE\nk,y(re)≡ −i√\nc3\n√πxLyL/radicalbiggω1\nωcmπ\nxLFSC\nm,n(xe,ye)\nχm,n√ωkeikze,\n(A47)\nVTM\nk,x(re)≡i√\nc5\n√πxLyL/radicalbiggω1\nωcmπ\nxLkFCS\nm,n(xe,ye)\nχm,n/radicalbig\nω3\nkeikze,\n(A48)\nVTM\nk,y(re)≡i√\nc5\n√πxLyL/radicalbiggω1\nωcnπ\nyLkFSC\nm,n(xe,ye)\nχm,n/radicalbig\nω3\nkeikze.\n(A49)\nIn this study, we imposed the restriction that the z\nelement of pis not present because it does not af-\nfect the emission of field. Thus the position variable\nzeshould be replaced by the constant value zc. The\nquantity zcdenotes the zcomponent of the center of\nthe cyclotron motion rc= (xc,yc,zc). Therefore, the\nargument of the above form factors should be repre-\nsented by Vo\nk,u(xe,ye,zc). Note that the subscript uof\nVo\nk,u(xe,ye,zc) stands for xandy.\nNow, we use the dipole approximation in the form fac-\ntors, because the radius of cyclotron motion re−rcis\nmuch smaller than the width of the waveguide in the\nbothxandydirections:\nVo\nk,u(xe,ye,zc) =Vo\nk,u(rc)\n+∞/summationdisplay\nχ=11\nχ!/braceleftbigg/bracketleftbigg\n(xe−xc)∂\n∂x′e/bracketrightbiggχ\nVo\nk,u(x′\ne,ye,zc)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx′e=xc\n+/bracketleftbigg\n(ye−yc)∂\n∂y′e/bracketrightbiggχ\nVo\nk,u(xe,y′\ne,zc)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny′e=yc/bracerightBigg\n=Vo\nk,u(rc)+O/parenleftbigg\nh/parenleftbiggxe−xc\nxL,ye−yc\nyL/parenrightbigg/parenrightbigg\n≃Vo\nk,u(rc), (A50)\nwhere we have imposed |(xe−xc)/xL| ≪1 and|(ye−\nyc)/yL| ≪1. Thus, the full Hamiltonian can be approxi-\nmated as if the interaction between the electron and the\nfield occurs at the rotation center. Therefore we obtain,\nH≃ω1q∗\n1q1+/summationdisplay\no/summationdisplay\nm,n/integraldisplay\nkωkqo\nk∗qo\nk\n+λ/summationdisplay\no/summationdisplay\nm,n/integraldisplay\nk/braceleftbig\n(q1−q∗\n1)/bracketleftbig\nVo\nk,x(rc)qo\nk−Vo\nk,x∗(rc)qo\nk∗/bracketrightbig\n+i(q1+q∗\n1)/bracketleftbig\nVo\nk,y(rc)qo\nk−Vo\nk,y∗(rc)qo\nk∗/bracketrightbig/bracerightbig\n.(A51)\nFurthermore, we only consider the TE 01mode, which is\nthe lowest energy mode. Therefore, the Hamiltonian can\nbe approximated by Eq.(14). We note we have replaced\nsome variables in order to avoid heavy notation such as\nω0,1,k≡ωk,qTE\n0,1,k≡qk, andVTE\n0,1,k,x(rc)≡Vkin Eq.(14).Appendix B: CLASSICAL BOGOLIUBOV\nTRANSFORMATION\nIn this appendix we derive Eqs.(24), (25) and (26).\nFirst, we impose the periodic boundary condition on the\nHamiltonian (22) using a dimensionless box of size L.\nIn this study, the TE 01mode is taken into consideration,\nand the Hamiltonian (22) is modeled as one-dimensional.\nTherefore, one-dimensional box normalization is intro-\nduced. Then the dimensionless wave-number of the field\nκis discrete, i.e., κ= 2πj/Lwith any integer j. Thus\nEq.(22) reduces to the discretized Hamiltonian ˜Has fol-\nlows:\n˜H=w1¯q∗\n1¯q1+∞/summationdisplay\nκ=−∞wκ˜q∗\nκ˜qκ\n+λ∞/summationdisplay\nκ=−∞(¯q1−¯q∗\n1)/parenleftBig\n˜Vκ˜qκ−˜V∗\nκ˜q∗\nκ/parenrightBig\n,(B1)\nwhere\n˜qκ≡/radicalbigg\n2π\nL¯qκ, (B2)\n˜Vκ≡/radicalbigg\n2π\nL¯Vκ. (B3)\nWe note ¯ qκand¯Vκhave the order of L0. To deal with\nthe continuous wave-number of the field, we will take the\nlimitL→ ∞in the appropriate stage of calculations. In\nthis limit we have\n2π\nL∞/summationdisplay\nκ=−∞→/integraldisplay∞\n−∞dκ,L\n2πδκ,κ′→δ(κ−κ′).(B4)\nSinceEq.(B1)isabilinearHamiltonian, wecanexactly\n“diagonalize” it as\n˜H=W1˜Q∗\n1˜Q1+/summationdisplay\nκWκ˜Q∗\nκ˜Qκ, (B5)\nwhere we have abbreviated the summation sign for sim-\nplification as\n/summationdisplay\nκ≡∞/summationdisplay\nκ=−∞. (B6)\nHere˜Q1and˜Qκrepresent the renormalized normal\nmodes, while W1andWκdenote the renormalized fre-\nquencies. The normal modes satisfy the Poisson brack-\nets,\n/braceleftBig\n˜Q1,˜Q∗\n1/bracerightBig\n=−i, (B7)\n/braceleftBig\n˜Qκ,˜Q∗\nκ′/bracerightBig\n=−iδκ,κ′, (B8)\n/braceleftBig\n˜Qα,˜Qβ/bracerightBig\n= 0. (B9)\nWe note that in Eq.(B8) here, it is the Kronecker delta.\nThe eigenvalue equation for the Liouvillian composed14\nof this diagonalized Hamiltonian (B5) is the following,\nwith the eigenvalue associated with the renormalized fre-\nquency of the Hamiltonian (B5):\n−LH˜Qα=Wα˜Qα. (B10)\nFor calculating ˜Q1, we assume a linear transformation\nas follows:\n˜Q1=C11¯q1+D11¯q∗\n1+/summationdisplay\nκ(C1κ˜qκ+D1κ˜q∗\nκ),(B11)\nwhereC11,D11,C1κandD1κare coefficients. By substi-\ntuting Eq.(B11) into Eq.(B10) then comparing the coef-\nficients of ¯ q1, ¯q∗\n1, ˜qκand ˜q∗\nκon both sides, we have\nw1C11−λ/summationdisplay\nκ/parenleftBig\n˜V∗\nκC1κ+˜VκD1κ/parenrightBig\n=W1C11,(B12)\n−w1D11+λ/summationdisplay\nκ/parenleftBig\n˜V∗\nκC1κ+˜VκD1κ/parenrightBig\n=W1D11,(B13)\nwκC1κ−λ˜Vκ(C11+D11) =W1C1κ,(B14)\n−wκD1κ+λ˜Vκ(C11+D11) =W1D1κ.(B15)\nFrom Eqs.(B12) and (B13), we obtain\nD11=w1−W1\nw1+W1C11. (B16)\nBy substituting Eq.(B16) into Eq.(B14) and Eq.(B15),\nwe have\nC1κ=2w1\nw1+W1λ˜Vκ\nwκ−W1C11,(B17)\nD1κ=2w1\nw1+W1λ˜V∗\nκ\nwκ+W1C11.(B18)\nAlso, by substituting the relationships from Eqs.(B16) to\n(B18) into Eq.(B11), we have\n˜Q1=2w1\nw1+W1C11/bracketleftbiggw1+W1\n2w1¯q1+w1−W1\n2w1¯q∗\n1\n+/summationdisplay\nκ/parenleftBigg\nλ˜Vκ\nwκ−W1˜qκ+λ˜V∗\nκ\nwκ+W1˜q∗\nκ/parenrightBigg/bracketrightBigg\n.(B19)\nFurthermore, by substituting Eqs.(B17) and (B18) into\nEq.(B12), we obtain a transcendental equation that pro-\nvides a specific representation of W1:\n˜ξ1(W1) = 0, (B20)\nwhere\n˜ξ1(W)≡W2−w2\n1−4λ2w1/summationdisplay\nκwκ|˜Vκ|2\nW2−w2κ.(B21)\nThe coefficient C11can be identified from Poisson brack-\nets. By substituting ˜Q1and its complex conjugate into\nEq.(B7), we obtain\nC11=w1+W1\n2w1˜N1, (B22)where\n˜N1≡\n1\n2w1d˜ξ1(ζ)\ndζ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nζ=W1\n−1\n2\n.(B23)\nHere we take the limit of L→ ∞. In Eq.(B21), |˜Vκ|2in\nEq.(B3) has the order of L−1, so the summation sign is\nreplaced by an integral symbol, then Eq.(B21) reduces\nto Eq.(27). This integral can be exactly performed, and\nfrom Eq.(B20), we obtain W1= ˜w1, which is a real\neigenvalue of the Liouvillian on the first Riemann sheet.\nThis means that there is a frequency shift for W1from\nw1. Additionally, ˜N1also reduces to ¯N1in Eq.(29), thus\nEq.(B19) reduces to Eq.(25) on the limit of L→ ∞.\nFor˜Qκ, we can derive it using the same algebraic cal-\nculations as when we derived ˜Q1. In that case, ˜Qκis\nobtained by replacing W1withWκin Eq.(B19). How-\never, for the coefficient of ˜ qκ, ifWκ=wκ, this point\nbecomes a divergence point under the κintegral. There-\nfore, this point is considered separately in advance. In\nfact, through the following discussion, we will find that\nthe frequency shift of Wκis given by Wκ=wκ+O(L−1),\nand it becomes evident that there is no frequency shift\nin the field as the limit of L→ ∞. Thus we assume such\na linear transformations as follows:\n˜Qκ=Cκ1¯q1+Dκ1¯q∗\n1+Cκκ˜qκ\n+/summationdisplay\nκ′′C′\nκκ′˜qκ′+/summationdisplay\nκ′′Dκκ′′˜q∗\nκ′′,(B24)\nwhereCκ1,Dκ1,Cκκ,C′\nκκ′andDκκ′′are coefficients and\nthe summation symbol with prime means that κ′=κis\neliminated as\n/summationdisplay\nκ′′\n≡/summationdisplay\nκ′\nκ′/negationslash=κ. (B25)\nBy substituting Eq.(B24) into Eq.(B10) then comparing\nthe coefficients of ¯ q1, ¯q∗\n1, ˜qκ, ˜qκ′and ˜q∗\nκon both sides, we\nhave\nw1Cκ1−λ/parenleftBigg\n˜V∗\nκCκκ+/summationdisplay\nκ′′˜V∗\nκ′C′\nκκ′+/summationdisplay\nκ′′˜Vκ′′Dκκ′′/parenrightBigg\n=WκCκ1,(B26)\n−w1Dκ1+λ/parenleftBigg\n˜V∗\nκCκκ+/summationdisplay\nκ′′˜V∗\nκ′C′\nκκ′+/summationdisplay\nκ′′˜Vκ′′Dκκ′′/parenrightBigg\n=WκDκ1,(B27)\nwκCκκ−λ˜Vκ(Cκ1+Dκ1) =WκCκκ,(B28)\n−wκ′′Dκκ′′+λ˜V∗\nκ′′(Cκ1+Dκ1) =WκDκκ′′,(B29)\nwκ′C′\nκκ′−λ˜Vκ′(Cκ1+Dκ1) =WκC′\nκκ′.(B30)\nFrom Eqs.(B26) and (B27), we obtain\nDκ1=w1−Wκ\nw1+WκCκ1. (B31)15\nBy substituting Dκκ′′of Eq.(B29) with Eq.(B31) into\nEq.(B26) and C′\nκκ′of Eq.(B30) with Eq.(B31) into\nEq.(B27), we have\nCκ1=−w1+Wκ\n˜ξ2(Wκ)λ˜V∗\nκCκκ, (B32)\nDκ1=−w1−Wκ\n˜ξ2(Wκ)λ˜V∗\nκCκκ, (B33)\nwhere\n˜ξ2(Wκ)≡W2\nκ−w2\n1+2w1λ2|˜Vκ|2\nW2κ−w2\nκ′−4λ2w1/summationdisplay\nκ′′wκ′|˜Vκ′|2\nW2κ−w2\nκ′.\n(B34)\nBy substituting Eq.(B32) with Eq.(B31) into Eq.(B29)\nand Eq.(B33) with Eq.(B31) into Eq.(B30), we have\nC′\nκκ′=−2w1λ˜V∗\nκ\n˜ξ2(Wκ)λ˜Vκ′\nwκ′−WκCκκ, (B35)\nDκκ′′=−2w1λ˜V∗\nκ\n˜ξ2(Wκ)λ˜V∗\nκ′′\nwκ′′+WκCκκ. (B36)\nAlso, by substituting these relationships Eqs.(B32),\n(B33), (B35) and (B36) into Eq.(B24), we have\n˜Qκ=Cκκ/bracketleftBigg\n˜qκ−2w1λ˜V∗\nκ\n˜ξ2(Wκ)/parenleftbiggWκ+w1\n2w1¯q1−Wκ−w1\n2w1¯q∗\n1\n+/summationdisplay\nκ′′λ˜Vκ′\nwκ′−Wκ˜qκ′+/summationdisplay\nκ′′λ˜V∗\nκ′′\nwκ′′+Wκ˜qκ′′/parenrightBigg/bracketrightBigg\n.\n(B37)\nThe coefficient Cκκcan be identified from Poisson brack-\nets. Substituting ˜Qκand its complex conjugate intoEq.(B8), we obtain\nCκκC∗\nκκ+O(L−1) = 1. (B38)\nHereagainwetakethe limit of L→ ∞. Then, the second\nterm on the left-hand side of Eq.(B38) vanishes due to\nthe order of L−1in this limit. Thus we put\nCκκ=C∗\nκκ= 1. (B39)\nFurthermore, by substituting Eqs.(B32) and (B33) into\nEq.(B28), we obtain\nWκ=wκ+2ω1λ2|˜Vκ|2\n˜ξ2(Wκ). (B40)\nIn Eq.(B40), under the limit of L→ ∞, it becomes\nWκ=wκbecause|˜Vκ|2in Eq.(B3) has the order of L−1.\nThis means that the renormalized field frequencies are\nunderstood to remain unchanged from the original field\nfrequencies. Finally, in Eq.(B34), |˜Vκ|2in Eq.(B3) has\nthe orderof L−1, sothe summation sign is replaced byan\nintegral symbol and the third term of the right-hand side\nvanished, then Eq.(B34) reduces to Eq.(27). 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