Some open questions in 'wave chaos'

Nonlinearity

Volumn 21, Issue 8, 2008, Pages

  Nonnenmacher, Stéphane ^a,b  

^a DIF   (France)

^b CNRS   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 47849123200     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/21/8/T01     Document Type: Article

References (45)

1. Anantharaman N 2006 Entropy and the localization of eigenfunctions Ann. Math. at press
2. Anantharaman N and Nonnenmacher S 2007 Half-delocalization of eigenfunctions of the laplacian on an Anosov manifold Ann. Inst. Fourier 57 2465-523
3. Bérard P 1977 On the wave equation on a compact Riemannian manifold without conjugate points Math. Z. 155 249-76
4. Berry M V 1977 Regular and irregular semiclassical wave functions J. Phys. A: Math. Gen. 10 2083-91
5. Berry M V 1985 Semiclassical theory of spectral rigidity Proc. R. Soc. Lond., Ser. A 400 229-51
6. Bochi J and Viana M 2004 Lyapunov exponents: how frequently are dynamical systems hyperbolic? Modern Dynamical Systems and Applications ed M Brin et al (Cambridge: Cambridge University Press)
7. Bogomolny E, Georgeot B, Giannoni M-J and Schmit C 1992 Chaotic billiards generated by arithmetic groups Phys. Rev. Lett. 69 1477-800
8. Bolte J, Steil G and Steiner F 1992 Arithmetical chaos and violation of universality in energy level statistics Phys. Rev. Lett. 69 2188-92
9. Bohigas O, Giannoni M J and Schmit C 1984 Characterization of chaotic quantum spectra and universality of level fluctuation laws Phys. Rev. Lett. 52 1
10. Colin de Verdière Y 1985 Ergodicité et fonctions propres du laplacien Commun. Math. Phys. 102 497-502
11. Zelditch S 1987 Uniform distribution of the eigenfunctions on compact hyperbolic surfaces Duke Math. J. 55 919-41
12. Donnelly H 2003 Quantum unique ergodicity Proc. Am. Math. Soc. 131 2945-51
13. Eckhardt B, Fishman S, Keating J, Agam O, Main J and Müller K 1995 Approach to ergodicity in quantum wave functions Phys. Rev. E 52 5893-903
14. Barnett A H 2006 Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards Commun. Pure Appl. Math. 59 1457-88
15. Faure F 2007 Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula Ann. Inst. Fourier 57 2525-99
16. Faure F, Nonnenmacher S and De Bièvre S 2003 Scarred eigenstates for quantum cat maps of minimal periods Commun. Math. Phys. 239 449-92
17. Feingold M and Peres A 1986 Distribution of matrix elements of chaotic systems Phys. Rev. A 34 591-5
18. Heusler S, Müller S, Altland A, Braun P and Haake F 2007 Periodic-orbit theory of level correlations Phys. Rev. Lett. 98 044103
19. Keating J P and Müller S 2007 Resummation and the semiclassical theory of spectral statistics Proc. R. Soc. London A 463 3241-50
20. ∞ norms of eigenfunctions of arithmetic surfaces Ann. Math. 141 301-20
21. Keating J P, Novaes M, Prado S D and Sieber M 2007 Semiclassical structure of chaotic resonance eigenfunctions Phys. Rev. Lett. 97 150406
22. Nonnenmacher S and Rubin M 2007 Resonant eigenstates for a quantized chaotic system Nonlinearity 20 1387-420
23. Jakobson D, Polterovich I and Toth J 2007 Lower bounds for the remainder in Weyl's law on negatively curved surfaces Int. Math. Res. Not. 2007 142
24. Kurberg P and Rudnick Z 2001 Value distribution for eigenfunctions of desymmetrized quantum maps Int. Math. Res. Not. 18 995-1002
25. Kurlberg P and Rudnick Z 2005 On the distribution of matrix elements for the quantum cat map Ann. Math. 161 489-507
26. Lin K 2002 Numerical study of quantum resonances in chaotic scattering J. Comput. Phys. 176 295-329
27. Lindenstrauss E 2006 Invariant measures and arithmetic quantum unique ergodicity Ann. Math. 163 165-219
28. Luo W and Sarnak P 2004 Quantum variance for Hecke eigenforms Ann. Sci. Ecole Norm. Sup. 37 769-99
29. Mehta M L 1991 Random Matrices 2nd edn revised and enlarged (New York: Academic)
30. Nonnenmacher S and Zworski M 2007 Distribution of resonances for open quantum maps Commun. Math. Phys. 269 311-65
31. Peter M 2002 The correlation between multiplicities of closed geodesics on the modular surface Commun. Math. Phys. 225 171-89
32. Pollicott M and Sharp R 2006 Correlations for pairs of closed geodesics Invent. Math. A 163 1-24
33. Rudnick Z and Sarnak P 1994 The behaviour of eigenstates of arithmetic hyperbolic manifolds Commun. Math. Phys. 161 195-213
34. Shnirelman A 1974 Ergodic properties of eigenfunctions Usp. Math. Nauk. 29 181-2
35. Schomerus H and Tworzydło J 2004 Quantum-to-classical crossover of quasi-bound states in open quantum systems Phys. Rev. Lett. 93 154102
36. Schubert R 2005 On the rate of quantum ergodicity for quantised maps Preprint available at http://www.maths.bris.ac.uk/∼marcvs/publications.html
37. Silberman L and Venkatesh A 2008 Entropy bounds for Hecke eigenfunctions on division algebras Geo. Funct. Anal. at press
38. Sieber M and Richter K 2001 Correlations between periodic orbits and their rle in spectral statistics Phys. Scr. T 90 128-33
39. Sjöstrand J 1990 Geometric bounds on the density of resonances for semiclassical problems Duke Math. J. 60 1-57
40. Sjöstrand J and Zworski M 2007 Fractal upper bounds on the density of semiclassical resonances Duke Math. J. 137 381-459
41. p norm of spectral clusters for second-order elliptic operators on compact manifolds J. Funct. Anal. 77 123-38
42. Voros A 1977 Asymptotic -expansions of stationary quantum states Ann. Inst. H Poincaré A 26 343-403
43. Wilkinson M 1988 Random matrix theory in semiclassical quantum mechanics of chaotic systems J. Phys. A: Math. Gen. 21 1173-90
44. Zelditch S 1994 On the rate of quantum ergodicity: I. Upper bounds Commun. Math. Phys. 160 81-92
45. Schubert R 2006 Upper bounds on the rate of quantum ergodicity Ann. H Poincaré 7 1085-98