Asymptotic rate of quantum ergodicity in chaotic euclidean billiards

Communications on Pure and Applied Mathematics

Volumn 59, Issue 10, 2006, Pages 1457-1488

  Barnett, A H ^a,b  

^a NEW YORK UNIVERSITY   (United States)

^b Dartmouth College   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33748301988     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20150     Document Type: Article

References (66)

1. Agam, O.; Fishman, S. Semiclassical criterion for scars in wave functions of chaotic systems. Phys. Rev. Lett. 73 (1994), no. 6, 806-809.
2. Aurich, R.; Taglieber, M. On the rate of quantum ergodicity on hyperbolic surfaces and for billiards. Phys. D118 (1998), no. 1-2, 84-102.
3. Austin, E. J.; Wilkinson, M. Distribution of matrix elements of a classically chaotic system. Europhysics Lett. 20 (1992), 589-593.
4. Bäcker, A.; Schubert, R.; Stifter, P. On the number of bouncing ball modes in billiards. J. Phys. A 30 (1997), no. 19, 6783-6795.
5. Bäcker, A. Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems. The mathematical aspects of quantum maps, 91-144. Lecture Notes in Physics, 618. Springer, Berlin, 2003.
6. Bäcker, A.; Schubert, R.; Stifter, P. Rate of quantum ergodicity in Euclidean billiards. Phys. Rev. E (3) 57 (1998), no. 5, part A, 5425-5447. Erratum: Rate of quantum ergodicity in Euclidean billiards. Phys. Rev. E (3) 58 (1998), no. 4, 5192.
7. Barnett, A. H. Quasi-orthogonality on the boundary for euclidean laplace eigenfunctions. arXiv: math-ph/0601006, 2006.
8. Barnett, A. H. The scaling method for the Dirichlet eigenproblem. In preparation.
9. Barnett, A. H. Dissipation in deforming chaotic billiards. Doctoral dissertation, Harvard University, 2000. Also available at http://www.math.dartmouth.edu/~ahb/thesis_html.
10. Barnett, A. H.; Cohen, D.; Heller, E. J. Deformations and dilations of chaotic billiards: dissipation rate, and quasi-orthogonality of the boundary wavefunctions. Phys. Rev. Lett. 85 (2000), 1412. arXiv:nlin.CD/0003018, 2000.
11. Barnett, A.; Cohen, D.; Heller, E. J. Rate of energy absorption for a driven chaotic cavity. J. Phys. A 34 (2001), no. 3, 413-438. arXiv:nlin.CD/0006041, 2000.
12. Berry, M. V. Regular and irregular semiclassical wavefunctions. J. Phys. A 10 (1977), no. 12, 2083-2091.
13. Berry, M. V. Quantum scars of classical closed orbits in phase space. Proc. Roy. Soc. London Ser. A 423 (1989), no. 1864, 219-231.
14. Berry, M. Some quantum-to-classical asymptotics. Chaos et physique quantique (Les Houches, 1989), 251-304. North-Holland, Amsterdam, 1991.
15. Betcke, T.; Trefethen, L. Reviving the method of particular solutions. SIAM Rev. 47 (2005), no. 3, 469-491.
16. Bogomolny, E. B. Smoothed wave functions of chaotic quantum systems. Phys. D 31 (1988), no. 2, 169-189.
17. Bogomolny, E. B.; Georgeot, B.; Giannoni, M.-J.; Schmit, C. Chaotic billiards generated by arithmetic groups. Phys. Rev. Lett. 69 (1992), no. 10, 1477-1480.
18. Bohigas, O. Random matrix theories and chaotic dynamics. Chaos et physique quantique (Les Hauchen, 1989), 87-199. North-Holland, Amsterdam, 1991.
19. Boose, D.; Main, J. Distributions of transition matrix elements in classically mixed quantum systems. Phys. Rev. E 60 ( 1999), 2813-2844.
20. Casati, G.; Prosen, T. The quantum mechanics of chaotic billiards. Classical chaos and its quantum manifestations (Toulouse, 1998). Phys. D 131 (1999), no. 1-4, 293-310.
21. Cohen, D. Chaos and energy spreading for time-dependent Hamiltonians, and the various regimes in the theory of quantum dissipation. Ann. Physics 283 (2000), no. 2, 175-231.
22. Colin de Verdière, Y. Ergodicité et fonctions propres du laplacien. Comm. Math. Phys. 102 (1985), no. 3, 497-502.
23. Combescure, M.; Robert, D. Semiclassical sum rules and generalized coherent states. J. Math. Phys. 36 (1995), no. 12, 6596-6610.
24. Donnelly, H. Quantum unique ergodicity. Proc. Amer. Math. Soc. 131 (2003), no. 9, 2945-2951.
25. Eckhardt, B.; Fishman, S.; Keating, J.; Agam, O.; Main, J.; Müller, K. Approach to ergodicity in quantum wave functions. Phys. Rev. E 52 (1995), 5893-5903.
26. Faure, F.; Nonnenmacher, S.; De Bièvre, S. Scarred eigenstates for quantum cat maps of minimal periods. Comm. Math. Phys. 239 (2003), no. 3, 449-492.
27. Feingold, M.; Peres, A. Distribution of matrix elements of chaotic systems. Phys. Rev. A (3) 34 (1986), no. 1, 591-595.
28. Gardiner, C. W. Handbook of stochastic methods for physics, chemistry and the natural sciences. 3rd ed. Springer Series in Synergetics, 13. Springer, Berlin, 2004.
29. Gutzwiller, M. C. Chaos in classical and quantum mechanics. Interdisciplinary Applied Mathematics, 1. Springer, New York, 1990.
30. Heller, E. J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53 (1984), no. 16, 1515-1518.
31. Heller, Eric J. Wavepacket dynamics and quantum chaology. Chaos et physique quantique (Les Hauches, 1989), 547-664. North-Holland, Amsterdam, 1991.
32. Hortikar, S.; Srednicki, M. Random matrix elements and eigenfunctions in chaotic systems. Phys. Rev. E (3) 57 (1998), no. 6, 7313-7316.
33. Kaplan, L.; Heller, E. J. Linear and nonlinear theory of eigenfunction scars. Ann. Physics 264 (1998), no. 2, 171-206.
34. Kaplan, L.; Heller, E. J. Measuring scars of periodic orbits. Phys. Rev. E (3) 59 (1999), no. 6, 6609-6628.
35. Kuttler, J. R.; Sigillito, V. G. Eigenvalues of the Laplacian in two dimensions. SIAM Rev. 26 (1984), no. 2, 163-193.
36. Li, B.; Hu, B. Statistical analysis of scars in stadium billiard. J. Phys. A 31 (1998), no. 2, 483-504.
37. Lindenstrauss, E. Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2) 163 (2006), no. 1, 165-219.
38. Luo, W.; Sarnak, P. Quantum variance for Hecke eigenforms. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 769-799.
39. Martinez, A. An introduction to semiclassical and microlocal analysis. Universitext. Springer, New York, 2002.
40. Prosen, T. Statistical properties of matrix elements in a Hamilton system between integrability and chaos. Ann. Physics 235 (1994), no. 1, 115-164.
41. Prosen, T. Quantization of a generic chaotic 3D billiard with smooth boundary. I. Energy level statistics. Phys. Lett. A 233 (1997), no. 4-6, 323-331.
42. Prosen, T. Quantization of generic chaotic 3D billiard with smooth boundary. II. Structure of high-lying eigenstates. Phys. Lett. A 233 (1997), no. 4-6, 332-342.
43. Rudnick, Z; Sarnak, P. The behaviour of eigenstates of arithmetic hyperbolic manifolds. Comm. Math. Phys. 161 (1994), no. 1, 195-213.
44. Sarnak, P. Arithmetic quantum chaos. The Schur lectures (1992) (Tel Aviv), 183-236. Israel Mathematical Conference Proceedings, 8. Bar-Ilan University, Ramat Gan, Israel, 1995.
45. Sarnak, P. Spectra and eigenfunctions of Laplacians. Partial differential equations and their applications (Toronto, ON, 1995), 261-276. CRM Proceedings & Lecture Notes, 12. American Mathematical Society, Providence, R.I., 1997.
46. Sarnak, P. Spectra of hyperbolic surfaces. Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 441-478.
47. Schanz, H.; Kottos, T. Scars on quantum networks ignore the lyapunov exponent, Phys. Rev. Lett. 90 (2003), 234101.
48. Šnirel'man, A. I. Ergodic properties of eigenfunctions. (Russian) Uspehi Mat. Nauk 29 (1974), no. 6 (180), 181-182.
49. Sinaǐ, Ya. G. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25 (1970), no. 2 (152), 141-192.
50. Sivia, D. S. Data analysis. A Bayesian tutorial. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.
51. Steiner, F. Quantum chaos. arXiv:chao-dyn/9402001, 1994.
52. Tanner, G. How chaotic is the stadium billiard? A semiclassical analysis. J. Phys. A 30 (1997), no. 8, 2863-2888.
53. Tate, T. Some remarks on the off-diagonal asymptotics in quantum ergodicity. Asymptot. Anal. 19 (1999), no. 3-4, 289-296.
54. Veble, G.; Robnik, M.; Liu, J. Study of regular and irregular states in generic systems. J. Phys. A 32 (1999), no. 36, 6423-6444.
55. Vergini, E. Estudio cuántico y semiclásico de billares clásicamente caóticos. Doctoral dissertation, Universidad de Buenos Aires, 1995.
56. Vergini, E.; Saraceno, M. Calculation by scaling of highly excited states of billiards. Phys. Rev. E 52 (1995), no. 3, 2204-2207.
57. Wilkinson, M. A semiclassical sum rule for matrix elements of classically chaotic systems. J. Phys. A 20 (1987), no. 9, 2415-2423.
58. Zelditch, S. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55 (1987), no. 4, 919-941.
59. Zelditch, S. Quantum transition amplitudes for ergodic and for completely integrable systems, J. Funct. Anal. 94 (1990), no. 2, 415-436.
60. Zelditch, S. Quantum ergodicity on the sphere. Comm. Math. Phys. 146 (1992), no. 1, 61-71.
61. Zelditch, S. On the rate of quantum ergodicity. I. Upper bounds. Comm. Math. Phys. 160 (1994), no. 1, 81-92.
62. Zelditch, S. On the rate of quantum ergodicity. II. Lower bounds. Comm. Partial Differential Equations 19 (1994), no. 9-10, 1565-1579.
63. Zelditch, S. Quantum mixing. J. Funct. Anal. 140 (1996), no. 1, 68-86.
64. Zelditch, S. A random matrix model for quantum mixing. Internat. Math. Res. Notices 1996, no. 3, 115-137.
65. Zelditch, S. Quantum ergodicity and mixing of eigenfunctions. Elsevier Encyclopedia of Mathematical Physics, to appear. arXiv:math-ph/0503026, 2005.
66. Zelditch, S.: Zworski, M. Ergodicity of eigenfunctions for ergodic billiards. Comm. Math. Phys. 175 (1996), no. 3, 673-682.