Crystal properties of eigenstates for quantum cat maps

Nonlinearity

Volumn 10, Issue 6, 1997, Pages 1569-1598

  Nonnenmacher, S ^a  

^a CEA SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001405315     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/10/6/008     Document Type: Article

References (37)

1. Bargmann V 1961 On a Hubert space of analytic functions and an associated integral transform Commun. Pure Appl. Math. XIV 187-214 Bargmann V 1967 Commun. Pure Appl. Math. XX 1-101
2. Bargmann V 1961 On a Hubert space of analytic functions and an associated integral transform Commun. Pure Appl. Math. XIV 187-214 Bargmann V 1967 Commun. Pure Appl. Math. XX 1-101
3. Husimi K 1940 Some formal properties of the density matrix Proc. Phys. Math. Soc. Japan 22 264-314
4. Schnirelman A 1974 Ergodic properties of the eigenfunctions Usp. Math. Nauk. 29 181-2
5. Leboeuf P and Voros A 1990 Chaos-revealing multiplicative representation of quantum eigenstates J. Phys. A: Math. Gen. 23 1765-74 Leboeuf P and Voros A 1995 Quantum nodal points as fingerprints of classical chaos Quantum Chaos: Between Order and Disorder ed G Casati and B V Chirikov (Cambridge: Cambridge University Press) pp 507-33
6. Leboeuf P and Voros A 1990 Chaos-revealing multiplicative representation of quantum eigenstates J. Phys. A: Math. Gen. 23 1765-74 Leboeuf P and Voros A 1995 Quantum nodal points as fingerprints of classical chaos Quantum Chaos: Between Order and Disorder ed G Casati and B V Chirikov (Cambridge: Cambridge University Press) pp 507-33
7. Leboeuf P and Shukla P 1996 Universal fluctuations of zeros of chaotic wavefunctions J. Phys. A: Math. Gen. 29 4827-55
8. Leboeuf P 1991 Phase space approach to quantum dynamics J. Phys. A: Math. Gen. 24 4575-86
9. Voros A 1989 WKB method in the Bargmann representation Phys. Rev. A 40 6814-25
10. Arnold V I and Avez A 1968 Ergodic Problems in Classical Mechanics (New York: Benjamin)
11. Hannay J H and Berry M V 1980 Quantisation of linear maps on a torus - Fresnel diffraction by a periodic grating Physica D 1 267-91
12. Whittaker E T and Watson G N 1965 A Course of Modern Analysis (Cambridge: Cambridge University Press)
13. Serre J P 1970 Cours d'Arithmétique (Presses Universitaires de France)
14. Hardy G H and Wright E M 1979 An Introduction to the Theory of Numbers (Oxford: Clarendon)
15. Mumford D 1983 Tala Lectures on Thela I (Progress in Mathematics 28) (Boston, MA: Birkhäuser)
16. Kač V and Peterson D H 1984 Infinite dimensional Lie algebras, theta functions and modular forms Adv. Math. 53 125-264
17. Lang S 1970 Algebra (Reading, MA: Addison-Wesley)
18. Rudin W 1970 Real and Complex Analysis (New York: McGraw-Hill)
19. Reed M and Simon B 1972 Methods of Modern Mathematical Physics, I: Functional Analysis (London: Academic)
20. Eckhardt B 1986 Exact eigenfunctions for a quantized map J. Phys. A: Math. Gen. 19 1823-31
21. DeBièvre S, Degli Esposti M and Giachetti R 1995 Quantization of a class of piecewise affine transformations on the torus Commun. Math. Phys. 176 73-94
22. Degli Esposti M 1993 Quantization of the orientation-preserving automorphisms of the torus Ann. Inst. Henri Poincaré (Physique théorique) 58 323-41
23. Degli Esposti M, Graffi S and Isola S 1995 Classical limit of the quantized hyperbolic toral automorphisms Commun. Math. Phys. 167 471-507
24. Bouzouina A and De Bièvre S 1996 Equipartition of the eigenfunctions of quantized ergodic maps on the torus Commun. Math. Phys. 178 83-105
25. Klimek S, Leśniewski A, Maitra N and Rubin R 1997 Ergodic properties of quantized toral automorphisms J. Math. Phys. 38 67-83
26. Percival I and Vivaldi F 1987 Arithmetical properties of strongly chaotic motions Physica D 25 105-30
27. Bartuccelli F and Vivaldi F 1989 Ideal orbits of toral automorphisms Physica D 39 194-204
28. Keating J 1991 Asymptotic properties of the periodic orbits of the cat map Nonlinearity 4 277-307
29. Keating J 1991 The cat maps: Quantum mechanics and classical motion Nonlinearity 4 309-41
30. Takahashi K 1986 Wigner and Husimi function in quantum mechanics J. Phys. Soc. Japan 55 762-79 Takahashi 1989 Distribution functions in classical and quantum mechanics Prog. Theor. Phys. Suppl. 98 109-56
31. Takahashi K 1986 Wigner and Husimi function in quantum mechanics J. Phys. Soc. Japan 55 762-79 Takahashi 1989 Distribution functions in classical and quantum mechanics Prog. Theor. Phys. Suppl. 98 109-56
32. Colin de Verdière Y 1985 Ergodicité et fonctions propres du laplacien Commun. Math. Phys. 102 497-502
33. Helffer B, Martinez A and Robert D 1987 Ergodicité et limite semiclassique Commun. Math. Phys. 109 313-26
34. Zelditch S 1987 Uniform distribution of eigenfunctions on compact hyperbolic surfaces Duke Math. J. 55 919-41 Gérard P and Leichtnam E 1993 Ergodic properties of eigenfunctions for the Dirichlet problem Duke Math. J. 71 559-607 Zelditch S and Zworski M 1996 Ergodicity of eigenfunctions for Ergodic Billiards Commun. Math. Phys. 175 673-82
35. Zelditch S 1987 Uniform distribution of eigenfunctions on compact hyperbolic surfaces Duke Math. J. 55 919-41 Gérard P and Leichtnam E 1993 Ergodic properties of eigenfunctions for the Dirichlet problem Duke Math. J. 71 559-607 Zelditch S and Zworski M 1996 Ergodicity of eigenfunctions for Ergodic Billiards Commun. Math. Phys. 175 673-82
36. Zelditch S 1987 Uniform distribution of eigenfunctions on compact hyperbolic surfaces Duke Math. J. 55 919-41 Gérard P and Leichtnam E 1993 Ergodic properties of eigenfunctions for the Dirichlet problem Duke Math. J. 71 559-607 Zelditch S and Zworski M 1996 Ergodicity of eigenfunctions for Ergodic Billiards Commun. Math. Phys. 175 673-82
37. Zelditch S 1995 Lectures on quantum dynamics from the semiclassical viewpoint Lectures Given at the Centre Emile Borel (Paris), Proc. Semester 'Chaos and Quantization' (Oxford: Oxford University Press) in preparation