Fractal Weyl laws in discrete models of chaotic scattering

Journal of Physics A: Mathematical and General

Volumn 38, Issue 49, 2005, Pages 10683-10702

  Nonnenmacher, Stéphane ^a   Zworski, MacIej ^b  

^a CEA SACLAY   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 28244438770     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/38/49/014     Document Type: Article

References (42)

1. Balazs N L and Voros A 1989 The quantized baker's transformation Ann. Phys., NY 190 1-31
2. Baranger H U and Mello P A 1994 Phys. Rev. Lett. 73 142
3. Jalabert R A, Pichard J-L and Beenakker C W J 1994 Universal quantum signatures of chaos in ballistic transport Europhys. Lett. 27 255-60
4. Beenakker C W J 1997 Random-matrix theory of quantum transport Phys. Rep. 69 731-808
5. Blanter Ya M and Büttiker M 2000 Shot noise in mesoscopic conductors Phys. Rep. 336 1-166
6. Bianucci P, Paz J P and Saraceno M 2002 Decoherence for classically chaotic quantum maps Phys. Rev. E 65 046226
7. Bony J-F 2001 Résonances dans des domaines de taille h Int. Math. Res. Not. 16 817-47
8. Petkov V and Zworski M 2001 Semiclassical estimates on the scattering determinant Ann. Henri Poincaré 2 675-711
9. Burq N and Zworski M 2001 Resonance expansions in semi-classical propagation Commun. Math. Phys. 232 1-12
10. Nakamura S, Stefanov P and Zworski M 2003 Resonance expansions of propagators in the presence of potential barriers J. Funct. Anal. 205 180-205
11. Gaspard P and Rice S A 1989 Semiclassical quantization of the scattering from a classical chaotic repeller J. Chem. Phys. 90 2242-54
12. Gaspard P, Alonso D and Burghardt I 1995 New ways of understanding semiclassical quantization Adv. Chem. Phys. 90 105-364
13. Wirzba A 1999 Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems Phys. Rep. 309 1-116
14. Guillopé L, Lin K and Zworski M 2004 The Selberg zeta function for convex co-compact Schottky groups Commun. Math. Phys. 245 149-76
15. Ivrii V 1998 Microlocal Analysis and Precise Spectral Asymptotics (Berlin: Springer)
16. Lifermann J 1979 Les méthods rapides de transformation du signal: Fourier, Walsh, Hadamard, Haar (Paris: Masson)
17. Lin K 2002 Numerical study of quantum resonances in chaotic scattering J. Comput. Phys. 176 295-329
18. Lin K and Zworski M 2002 Quantum resonances in chaotic scattering Chem. Phys. Lett. 355 201-5
19. Lu W, Sridhar S and Zworski M 2003 Fractal Weyl laws for chaotic open systems Phys. Rev. Lett. 91 154101
20. Strain J and Zworski M 2004 Growth of the zeta function for a quadratic map and the dimension of the Julia set Nonlinearity 17 1607-22
21. Łoziński A, Pakoński P and Życzkowski K 2002 Irreversible quantum baker map Phys. Rev. E 66 065201 (R)
22. Łoziński A, Życzkowski K and Słomczyński W 2003 Quantum iterated function systems Phys. Rev. E 68 046110
23. Meenakshisundaram N and Lakshminarayan A 2005 Multifractal eigenstates of quantum chaos and the Thue-Morse sequence Phys. Rev. E 71 065303 (R)
24. Lakshminarayan A 2005 Shuffling cards, factoring numbers, and the quantum baker's map Preprint nlin.CD/0505057
25. Morita T 1994 Periodic orbits of a dynamical system in a compound central field and a perturbed billiards system Ergod. Theor. Dynam. Syst. 14 599-619
26. Nonnenmacher S and Zworski M 2005 Distribution of resonances for open quantum maps Preprint math-ph/0505034
27. Pakoński P, Życzkowski K and Kuś M 2001 Classical 1D maps, quantum graphs and ensembles of unitary matrices J. Phys. A: Math. Gen. 34 9303-17
28. Saraceno M 1990 Classical structures in the quantized baker transformation Ann. Phys., NY 199 37-60
29. Saraceno M and Vallejos R O 1996 The quantized D-transformation Chaos 6 193-9
30. Saraceno M and Voros A 1994 Towards a semiclassical theory of the quantum baker's map Physica D 79 206-68
31. Schack R and Caves C M 2000 Shifts on a finite qubit string: a class of quantum baker's maps Appl. Algebra Eng. Commun. Comput. 10 305-10
32. Schanz H, Puhlmann M and Geisel T 2003 Shot noise in chaotic cavities from action correlations Phys. Rev. Lett. 91 134101
33. Schomerus H and Tworzydło J 2004 Quantum-to-classical crossover of quasi-bound states in open quantum systems Phys. Rev. Lett. 93 154102
34. Sjöstrand J 1990 Geometric bounds on the density of resonances for semiclassical problems Duke Math. J. 60 1-57
35. Sjöstrand J and Zworski M 2005 Geometric bounds on the density of semiclassical resonances in small domains Preprint math.SP/0506307
36. Tanner G 2000 Spectral statistics for unitary transfer matrices of binary graphs J. Phys. A: Math. Gen. 33 3567-85
37. Tworzydło J, Tajic A, Schomerus H and Beenakker C W 2003 Dynamical model for the quantum-to-classical crossover of shot noise Phys. Rev. B 68 115313
38. Jacquod Ph and Sukhorukov E V 2004 Breakdown of universality in quantum chaotic transport: the two-phase dynamical fluid model Phys. Rev. Lett. 92 116801
39. Whitney R S and Jacquod Ph 2005 Microscopic theory for the quantum to classical crossover in chaotic transport Phys. Rev. Lett. 94 116801
40. Zworski M 1999 Resonances in physics and geometry Not. AMS 43 319-28
41. Zworski M 2002 Quantum resonances and partial differential equations Proc. ICM 3 243-52
42. Życzkowski K and Sommers H-J 2000 Truncations of random unitary matrices J. Phys. A: Math. Gen. 33 2045-57