Phil Howard (2007) Resonance behaviour for classes of billiards on the Poincaré half-plane.
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The classical and quantum mechanics of two linked classes of open billiard systems on the Poincaré half-plane is studied. These billiard systems are presented as models of arithmetic scattering systems under deformation. An investigation is made of the classical phase space and the stability of a certain family of periodic orbits is investigated. The movement of the positions and widths of the resonances is followed, as the shape of Artin’s billiard is deformed. One deformation varies its lower boundary, interpolating between integrable and fully chaotic cases. The other deformation translates the right hand wall of the billiard, thus interpolating between several examples of Hecke triangle billiards. The variation of the statistical properties of the spectra are focussed on and the transitions in the statistics of the resonance positions and widths are mapped out in detail near particular values of the deformation parameters, where the billiard is a fundamental domain for some arithmetic group. Analytic solutions for the scattering matrix and the resonance positions in these particular systems are derived and numerical results are obtained which are in excellent agreement with the predictions. Away from the arithmetic systems, both generic behaviour according to the predictions of Random Matrix Theory, and non-generic behaviour is found, with deviations occurring particularly in the long range statistics. In the integrable case, semiclassical WKB theory is used to produce accurate wavefunctions and eigenvalues. For the general deformation a number of numerical methods are explored, such as the finite element method, complex absorbing potentials and collocation, in order to find an optimum method to locate the resonance positions.
This is a Published version This version's date is: 30/05/2007 This item is peer reviewed
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[ABB+99] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra,J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney,and D. Sorensen. LAPACK Users’ Guide. Society for Industrialand Applied Mathematics, third edition, 1999. 109
[ADG+99] H. Alt, C. Dembowski, H.-D. Gr¨af, R. Hofferbert, H. Rehfeld,A. Richter, and C. Schmit. Experimental versus numerical eigenvaluesof a Bunimovich stadium billiard: A comparison. Phys. Rev.E, 60:2851–2857, 1999. 106
[AGG+97] H. Alt, H.-D. Gr¨af, T. Guhr, H. L. Harney, R. Hofferbert, H. Rehfeld,A. Richter, and P. Schardt. Correlation-hole method forthe spectra of superconducting microwave billiards. Phys. Rev. E,55:6674–6683, 1997. 128
[ALST05] R. Aurich, S. Lustig, F. Steiner, and H. Then. Indications aboutthe shape of the universe from the Wilkinson microwave anisotropyprobe data. Phys. Rev. Letters, 94:021301, 2005. 21, 38
[And58] P. W. Anderson. Absence of diffusion in certain random lattices.Phys. Rev., 109:1492–1505, 1958. 20
[Arf85] G. Arfken. Mathematical Methods for Physicists, pages 560–562.Academic Press, Inc., 1985. 55
[Art24] E. Artin. Ein mechanisches System mit quasi-ergodischen Bahnen.Abh. Math. Sem. d. Hamburgischen Universit¨at, 3:170–175, 1924.22, 37
[AS65] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions.Dover, 1965. 40, 108, 111, 161, 162
[ASS88] R. Aurich, M. Sieber, and F. Steiner. Quantum chaos of theHadamard-Gutzwiller model. Phys. Rev. Lett., 61:483–487, 1988.23
[Ave02] H. Avelin. Research announcement on the deformation of cuspforms. Technical report, Uppsala Univ, 2002. UUDM report2002:26, Uppsala. 24
[Bac03] A. Backer. Numerical aspects of eigenvalue and eigenfunction computationsfor chaotic quantum systems. In M. Degli Esposti andS. Graffi, editors, The Mathematical Aspects of Quantum Maps.Springer, 2003. 109
[BB70] R. Balian and C. Bloch. Distribution of eigenfrequencies for thewave equation in a finite domain I: three-dimensional problem withsmooth boundary surface. Ann. Phys., 60:401–447, 1970. 61
[BB86] K. Bartschat and P. G. Burke. Resfit - a multichannel resonancefitting program. Computer Physics Communications, 41, 1986. 95
[BB97] M. Brack and R Bhaduri. Semiclassical Physics. Addison Wesley,1997. 20, 115, 116
[BD95] H. Beijeren and J. R. Dorfman. Lyapunov exponents andKolmogorov-sinai entropy for the Lorentz gas at low densities. Phys.Rev. Letters, 74:4412–4415, 1995. 87
[Ber89] M. Berry. Quantum chaology, not quantum chaos. Physica Scripta,40:335–336, 1989. 20
[BGGS92] E. B. Bogomolny, B. Georgeot, M.-J. Giannoni, and C. Schmit.Chaotic billiards generated by arithmetic groups. Phys. Rev. Lett.,69:1477–1480, 1992. 22, 145, 148
[BGS84] O. Bohigas, M. J. Giannoni, and C. Schmit. Characterization ofchaotic quantum spectra and universality of level fluctuation laws.Phys. Rev. Lett., 52, 1984. 21, 119
[BGS86] O. Bohigas, M.-J. Giannoni, and C. Schmit. Spectral fluctuationsof classically chaotic quantum systems. LNP Vol. 263: QuantumChaos and Statistical Nuclear Physics, 263:18–40, 1986. 22
[BH76] H. P. Baltes and E. R. Hilf. Spectra of Finite Systems. BibliographischesInstitut, Mannheim, 1976. 61
[BK99] M. V. Berry and J. P. Keating. The Riemann zeros and eigenvalueasymptotics. SIAM Review, 41:236–266, 1999. 24, 119
[Bog06] E. Bogomolny. Quantum and arithmetical chaos. In Frontiers inNumber Theory, Physics and Geometry, Proceedings of Les Houcheswinter school 2003. Springer-Verlag, 2006. 39, 64, 68, 70
[Bou01] N. Bouhamou. The use of NAG mesh generation and sparse solverroutines for solving partial differential equations. Technical report,NAG Ltd, 2001. Reference no. TR1/01 (NP3615). 104
[Bro73] T. A. Brody. A statistical measure for the repulsion of energy levels.Lettere Al Nuovo Cimento, 7:482, 1973. 125
[BSS92] J. Bolte, G. Steil, and F. Steiner. Arithmetical chaos and violationof universality in energy level statistics. Phys. Rev. Lett., 69:2188–2191, 1992. 22, 112
[BT77] M. V. Berry and M. Tabor. Level clustering in the regular spectrum.Proc. R. Soc. Lond. A, 356:375–394, 1977. 119, 125
[BTU93] O. Bohigas, S. Tomsovic, and D. Ullmo. Manifestations of classicalphase space structures in quantum mechanics. Physics Reports,223:43–133, 1993. 20
[BV86] B. V. Bal´azs and A. Voros. Chaos on the pseudosphere. Phys. Rep.,143:109–240, 1986. 12, 22, 28, 34, 38, 45, 64, 70
[BV98] E. Balslev and A. Venkov. The Weyl law for subgroups of themodular group. Geom. Funct. Anal., 8(3):437–465, 1998. 24
[CAM+04] P. Cvitanovi´c, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay.Chaos: Classical and Quantum, page 555. Niels Bohr Institute,Copenhagen 2005, 2004. Stable version 11. 61
[CE89] P. Cvitanovi´c and B. J. Eckhardt. Periodic-orbit quantization ofchaotic systems. Phys. Rev. Lett., 63(8):823–826, 1989. 62
[CGS91] A. Csord´as, R. Graham, and P. Sz´epfalusy. Level statistics of a noncompactcosmological billiard. Phys. Rev. A, 44:1491–1499, 1991.66, 161
[CGSV94] A. Csord´as, R. Graham, P. Sz´epfalusy, and G. Vattay. Transitionfrom Poissonian to Gaussian-orthogonal-ensemble level statistics ina modified Artin’s billiard. Phys. Rev. E, 49:325–333, 1994. 82, 83,84, 86, 92, 124, 125, 127, 148
[CM03] N. Chernov and R. Markarian. Introduction to the Ergodic Theoryof Chaotic Billiards. IMPA, Rio de Janeiro, Brasil, 2003. 87
[Cox65] H. S. M. Coxeter. Non-Euclidean Geometry. University of TorontoPress, 1965. 34
[Cre95] P. Crehan. Chaotic spectra of classically integrable systems. Journalof Physics A, 28:6389–6394, 1995. 48
[Ein17] A. Einstein. Zum Quantensatz von Sommerfeld und Epstein. Verhandlungender Deutschen Physikalischen Gesellschaft, 19:82–92,1917. 19
[EMOT53] A. Erd´elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi.Higher Trascendental Functions: Volume 1. McGraw-Hill, 1953.
[FH65] R. P. Feynman and A. R. Hibbs. Quantum Physics and Path Integrals.McGraw-Hill, 1965. 57, 58
[FK97] R. Fricke and F. Klein. Vorlesungen ¨uber die Theorie der automorphenFunktionen. Teubner, Leipzig, 1897. 47
[FKS05] Y. V. Fyodorov, T. Kottos, and H.-J. St¨ockmann, editors. Specialissue on trends in quantum chaotic scattering, volume 38. Instituteof Phyiscs publishing, 2005. 23
[For29] L. Ford. Automorphic Functions. McGraw-Hill, 1929. 47
[Fri98] H. Friedrich. Theoretical Atomic Physics. Springer, 1998. 24, 56
[FS97] Y. Fyodorov and H.-J. Sommers. Statistics of resonance poles, phaseshifts and time delays in quantum chaotic scattering: Random matrixapproach for systems with broken time-reversal invariance. J.Math. Phys., 38, 1997. 78, 95, 119, 121
[GHSV91] R. Graham, R. H¨ubner, P. Sz´epfalusy, and G. Vattay. Level statisticsof a noncompact integrable billiard. Phys. Rev. A, 44:7002–7015, 1991. 61, 106, 111, 124
[GR89] P. Gaspard and S. Rice. Semiclassical quantization of the scatteringfrom a classically chaotic repellor. J. Chem. Phys., 90:2242–2254,1989. 51
[Gut67] M. C. Gutzwiller. Phase-integral approximation in momentumspace and the bound states of an atom. J. Math. Phys., 8:1979–2000, 1967. 58
[Gut83] M. C. Gutzwiller. Stochastic behavior in quantum scattering. PhysicaD: Nonlinear Phenomena, 7:341–355, 1983. 50
[Gut90] M. Gutzwiller. Chaos in Classical and Quantum Mechanics.springer-verlag, 1990. 20, 38, 46, 58, 112
[Had98] J. Hadamard. Les surfaces `a courbures oppos´es et leurs lignesg´eod´esiques. J. Math. Pures et Appl., 4:27–73, 1898. 22, 45
[HB82] D. A. Hejhal and B. Berg. Some new results concerning eigenvaluesof the non-Euclidean Laplacian for PSL(2, Z). Technical report,University of Minnesota, 1982. 110, 111, 112
[Hej92a] D. A. Hejhal. Eigenvalues of the Laplacian for Hecke triangle groups.Memoirs of the American Mathematical Society, 97, 1992. 70, 112
[Hej92b] D. A. Hejhal. Eigenvalues of the Laplacian for Hecke triangle groups.Mem. Amer. Math. Soc., 469, 1992. 110, 112
[Hel84] E. J. Heller. Bound-state eigenfunctions of classically chaotic Hamiltoniansystems: Scars of periodic orbits. Phys. Rev. Lett., 53:1515–1518, 1984. 21
[HIL+92] F. Haake, F. Izrailev, N. Lehmann, D. Saher, and H.-J. Sommers.Statistics of complex levels of random matrices for decaying systems.Z. Phys. B, 88:359–370, 1992. 122
[HMFOU05] P. J. Howard, F. Mota-Furtado, P. F. O’Mahony, and V. Uski.Statistics of resonances for a class of billiards on the Poincar´e halfplane.J. Phys. A, 38:10829–10841, 2005. 106
[Kea92] J. P. Keating. Periodic orbit resummation and the quantization ofchaos. Proc. R. Soc. Lond. A, 436:99–108, 1992. 62
[Kub73] T. Kubota. Elementary Theory of Eisenstein Series. John Wileyand Sons, 1973. 51
[LLJP86] L. Leviandier, M. Lombardi, R. Jost, and J. P. Pique. Fouriertransform: A tool to measure statistical level properties in verycomplex spectra. Physical Review Letters, 56:2449–2452, 1986. 128
[LP76] P. D. Lax and R. S. Phillips. Scattering Theory for AutomorphicFunctions. Princeton University Press, 1976. 41, 51
[LRP06] M. Lyly, J. Ruokolainen, and A. Pursula. Elmer, 2006.http://www.csc.fi/elmer/index.phtml. 101, 103
[LS93] M. Lombardi and T. H. Seligman. Universal and nonuniversalstatistical properties of levels and intensities for chaotic Rydbergmolecules. Phys. Rev. A, 47:3571–3586, 1993. 129, 131
[LSZ03] W. T. Lu, S. Sridhar, and M. Zworski. FractalWeyl laws for chaoticopen systems. Phys. Rev. Letters, 91, 2003. 61
[Maj98] A. W. Majewski. Does quantum chaos exist? a quantum Lyapunovexponents approach. ArXiv Quantum Physics e-prints, 1998, quantph/9805068. 20
[MHB+04] S. M¨uller, S. Heusler, P. Braun, F. Haake, and A. Altland. Semiclassicalfoundation of universality in quantum chaos. Phys. Rev.Lett., 93:014103, 2004. 21
[Moi97] N. Moiseyev. Quantum theory of resonances: calculating energies,widths and cross-sections by complex scaling. Physics Reports,302:211–203, 1997. 96
[Mon73] H. Montgomery. The pair correlation of zeros of the zeta function.Analytic Number Theory (Proceedings of Symposia in Pure Mathematics),24:181–193, 1973. 23
[MS05] F. Mezzadri and N. C. Snaith. Recent Perspectives in Random MatrixTheory and Number Theory. Cambridge University Press, 2005.
[Odl] A. M. Odlyzko. The first 100, 000 zeros of the Riemannzeta function, accurate to within 3 ∗ 10−9).http://www.dtc.umn.edu/∼odlyzko/zeta tables/zeros1. 56,110, 111, 112
[Odl89] A. M. Odlyzko. The 1020th zero of the Riemannzeta function and 70 million of its neighbours.http://www.dtc.umn.edu/∼odlyzko/unpublished/, 1989. 112,125, 129
[Olv74] F.W. J. Olver. Asymptotics and Special Functions. Academic Press,1974. 159, 161
[Ott93] E. Ott. Chaos in Dynamical Systems. Cambridge University Press,1993. 20
[PHH06] O. Pironneau, F. Hecht, and A. Le Hyaric. FreeFEM++, 2006.http://www.freefem.org/ff++. 80, 103
[PRSB00] E. Persson, I. Rotter, H.-J. St¨ockmann, and M. Barth. Observationof resonance trapping in an open microwave cavity. Physical ReviewLetters, 85:2478–2481, 2000. 122[PS85] R. S. Phillips and P. Sarnak. TheWeyl theorem and the deformationof discrete groups. Comm. Pure Appl. Math., 38:853–866, 1985. 24
[PT56] C. E. Porter and R. G. Thomas. Fluctuations of nuclear reactionwidths. Physical Review, 104:483, 1956. 21, 121
[PTVF92] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery.Numerical Recipes in FORTRAN - The Art of Scienctfic Computing.Cambridge University Press, 1992. 110, 155, 162
[RM93] U. V. Riss and H.-D. Meyer. Calculation of resonance energies andwidths using the complex absorbing potential method. J. Phys. B,26:4503–4536, 1993. 96, 97, 98
[RM02] L. Ramdas Ram-Mohan. Finite Element and Boundary ElementApplications in Quantum Mechanics. Oxford University Press, 2002.
[Rob86] M. Robnik. A simple separable Hamiltonian having bound statesin the continuum. Journal of Physics A, 19:3845–3848, 1986. 56
[RV98] M. Robnik and G. Veble. On spectral statistics of classically integrablesystems. Journal of Physics A, 31:4669–4704, 1998. 125
[SC93] R. Schack and C. Caves. Hypersensitivity to perturbations in thequantum baker’s map. Phys. Rev. Lett., 71(4):525–528, 1993. 20
[Sch78] D. Schattschneider. The plane symmetry groups: Their recognitionand notation. Am. Math. Monthly, 85:439–450, 1978. 47
[Sch80a] C. Schmit. Quantum and classical properties of some billiards onthe hyperbolic plane. In Chaos et Physique Quantique — Chaos andQuantum Physics, Les Houches, ecole d’´ete de physique theorique1989, session LII. Elsevier Science Publishers B V, 1980. 106, 121
[Sch80b] B. Schutz. Geometrical Methods of Mathematical Physics. CambridgeUniversity Press, 1980. 30, 36
[Sch95] R. Schack. Comment on ‘exponential sensitivity and chaos in quantumsystems’. Phys. Rev. Lett., 75(3):581, 1995. 20
[Sel56] A. Selberg. Harmonic analysis and discontinuous groups in weaklysymmetric Riemannian spaces with applications to Dirichlet series.Journal of the Indian Mathematical Society, 20:47–87, 1956. 63
[SH00] S. Sahoo and Y. K. Ho. Determination of resonance energy andwidth using the method of complex absorbing potential. ChineseJournal Of Physics, 38:127–138, 2000. 96
[SSCL93] M Sieber, U Smilansky, S C Creagh, and R G Littlejohn. Nongenericspectral statistics in the quantized stadium billiard. Journalof Physics A: Mathematical and General, 26(22):6217–6230, 1993.
[SSS03] D. Savin, V. Sokolov, and H.-J. Sommers. Is the concept of a non-Hermitian effective Hamiltonian relevant in the case of potentialscattering? Phys. Rev. E, 67, 2003. 78
[Ste94] G. Steil. Eigenvalues of the Laplacian and of the Hecke operatorsfor PSL(2, Z). Technical report, DESY, 1994. 110, 111
[St¨o99] H.-J. St¨ockmann. Quantum Chaos: an Introduction. CambridgeUniversity Press, 1999. 38, 119, 125, 127
[SZ89] V. V. Sokolov and V. G. Zelevinsky. Dynamics and statistics ofunstable quantum states. Nuclear Physics A, 504:562–588, 1989.
[Tay72] J. Taylor. Scattering Theory. John Wiley and Sons, Inc., 1972. 41,
[Tem75] N. M. Temme. On the numerical evaluation of the modified Besselfunction of the third kind. J. Comput. Phys., 19:324–337, 1975. 162
[The05] H. Then. Maaßcusp forms for large eigenvalues. Math. Comp.,74:363–381, 2005. 111
[THM96] G. Tanner, K. T. Hansen, and J. Main. The semiclassical resonancespectrum of hydrogen in a constant magnetic field. Nonlinearity,9:1641–1670, 1996. 62
[Tit51] E. C. Titchmarsh. The Theory of the Riemann Zeta-Function. OxfordUniversity Press, 1951. 54
[Ven78] A. B. Venkov. Selberg’s trace formula for the Hecke operatorgenerated by an involution, and the eigenvalues of the Laplace-Beltrami operator on the fundamental domain of the modular groupPSL(2, Z). Math. USSR Izvestiya, 12:448–462, 1978. 61, 64, 65
[vV28] J. H. van Vleck. The correspondence principle in the statisticalinterpretation of quantum mechanics. Proceedings of the NationalAcademy of Sciences of the United States of America, 14:178–188,1928.
[Wat66] G. N. Watson. A Treatise on the Theory of Bessel Functions. CambridgeUniversity Press, 1966. 40
[Wey11] H. Weyl. ¨ Uber die asymptotische Verteilung der Eigenwerte.G¨ottinger Nachrichten, 110, 1911. 60
[WJ89] D. M. Wardlaw and W. Jaworski. Time delay, resonances, Riemannzeros and chaos in a model quantum scattering system. Journal ofPhysics A, 22:3561–3575, 1989. 38, 43, 112
[Wol05] Wolfram Research, Inc. Mathematica, 2005. 110, 114