Semiclassical transition from an elliptical to an oval billiard

Journal of Physics A: Mathematical and General

Volumn 30, Issue 13, 1997, Pages 4563-4596

  Sieber, Martin ^a,b  

^a 91405 Orsay Campus   (France)

^b UNIVERSITY OF ULM   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031558090     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/30/13/011     Document Type: Article

References (53)

1. Cvitanović P (ed) 1992 Periodic orbit theory - theme issue Chaos 2 1-158
2. Tél T and Ott E (ed) 1993 Chaotic scattering - theme issue Chaos 3 417-782.
3. Casati G and Chirikov B (ed) 1995 Quantum Chaos (Cambridge: Cambridge University Press)
4. Gutzwiller M C 1971 Periodic orbits and classical quantization conditions J. Math. Phys. 12 343-58
5. Balian R and Bloch C 1972 Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillations Ann. Phys. 69 76-160
6. Berry M V and Tabor M 1976 Closed orbits and the regular bound spectrum Proc. R. Soc. A 349 101-23
7. Berry M V and Tabor M 1977 Calculating the bound spectrum by path summation in action-angle variables J. Phys. A: Math. Gen. 10 371-9
8. Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer)
9. Creagh S C and Littlejohn R G 1991 Semiclassical trace formulas in the presence of continuous symmetries Phys. Rev. A 44 836-50
10. Creagh S C and Littlejohn R G 1992 Semiclassical trace formulae for systems with non-Abelian symmetry J. Phys. A: Math. Gen. 25 1643-69
11. Meyer K R 1970 Generic bifurcations of periodic points Trans. Am. Math. Soc. 149 95-107
12. Bruno A D 1970 Instability in a Hamiltonian system and the distribution of asteroids Math. USSR Sbornik 12 271-312
13. Bruno A D Research on the restricted three body problem. I: Periodic solutions of a Hamiltonian system Preprint Inst. Prikl. Mat. Akad. Nauk SSSR Moskva (in Russian) p 44
14. de Almeida A M O and Hannay J H 1987 Resonant periodic orbits and the semiclassical energy spectrum J. Phys. A: Math. Gen. 20 5873-83
15. Kuś M, Haake F and Delande D 1993 Prebifurcation periodic ghost orbits in semiclassical quantization Phys. Rev. Lett. 71 2167-71
16. Atkins K M and Ezra G S 1994 Semiclassical density of states at symmetric pitchfork bifurcations in coupled quartic oscillators Phys. Rev. A 50 93-7
17. Sieber M 1996 Uniform approximation for bifurcations of periodic orbits with high repetition numbers J. Phys. A: Math. Gen. 29 4715-32
18. de Almeida A M O 1986 Semiclassical energy spectrum of quasi-integrable systems Quantum Chaos and Statistical Nuclear Physics (Lecture Notes in Physics 263) ed T Seligman (New York: Springer) pp 197-211
19. Creagh S C 1996 Trace formula for broken symmetry Ann. Phys. 248 60-94
20. Tomsovic S, Grinberg M and Ullmo D 1995 Semiclassical trace formulas of near-integrable systems: Resonances Phys. Rev. Lett. 75 4346-9
21. Ullmo D, Grinberg M and Tomsovic S 1996 Near-integrable systems: Resonances and semiclassical trace formulas Phys. Rev. E 54 136-52
22. Richens P J 1982 On quantisation using periodic classical orbits J. Phys. A: Math. Gen. 15 2101-10
23. Ayant Y and Arvieu R 1987 Semiclassical study of particle motion in two-dimensional and three-dimensional elliptical boxes: I J. Phys. A: Math. Gen. 20 397-409
24. Arvieu R and Ayant Y 1987 Semiclassical study of particle motion in two-dimensional and three-dimensional elliptical boxes: II J. Phys. A: Math. Gen. 20 1115-36
25. Gradshteyn I S and Ryzhik I M 1980 Table of Integrals, Series, and Products (San Diego, CA: Academic) corrected and enlarged edition
26. Keller J B and Rubinow S I 1960 Asymptotic solution of eigenvalue problems Ann. Phys. 9 24-75 Keller J B and Rubinow S I 1960 Ann. Phys. 10 303-5 (Errata)
27. Keller J B and Rubinow S I 1960 Asymptotic solution of eigenvalue problems Ann. Phys. 9 24-75 Keller J B and Rubinow S I 1960 Ann. Phys. 10 303-5 (Errata)
28. Sinai Y G 1976 Introduction to Ergodic Theory (Princeton, NJ: Princeton University Press)
29. Chang S-J and Friedberg R 1988 Elliptical billiards and Poncelet's theorem J. Math. Phys. 29 1537-50
30. Crespi B, Chang S-J and Shi K-J 1993 Elliptical billiards and hyperelliptic functions J. Math. Phys. 34 2257-89
31. Berry M V 1981 Regularity and chaos in classical mechanics, illustrated by three deformations of a circular 'billiard' Eur. J. Phys. 2 91-102
32. Tabanov M B 1994 Billiard in an ellipse, geodesics on an ellipsoid, and new elliptic coordinates Russ. Acad. Sci. Dokl. Math. 48 438-44
33. Okai S, Nishioka H and Ohta M 1990 Periodic orbits in elliptic billiards Mem. Konan Univ., Sci. Ser. 37 29-45
34. Richter P H, Wittek A, Kharlamov M P and Kharlamov A P 1995 Action integrals for ellipsoidal billiards Z. Naturf. a 50 693-710
35. Morse P M and Feshbach H 1953 Methods of Theoretical Physics, Part I and II (New York: McGraw-Hill)
36. Abramowitz M and Stegun I A (ed) 1965 Handbook of Mathematical Functions (New York: Dover)
37. Traiber A J S, Fendrik A J and Bernath M 1989 Level crossings and commuting observables for the quantum elliptic billiard J. Phys. A: Math. Gen. 22 L365-70
38. Waalkens H, Wiersig J and Dullin H R 1996 Elliptic Quantum Billiard Preprint chao-dyn/9612020
39. Ussishkin I The quantization of billiards in the scattering approach beyond the semiclassical approximation MSc Thesis Weizmann Institute of Science
40. Tanner G 1997 How chaotic is the stadium billiard? A semiclassical analysis J. Phys. A: Math. Gen. 30 2863-88
41. Bogomolny E B 1988 Smoothed wave functions of chaotic quantum systems Physica 31D 169-89
42. Reimann S M, Brack M, Magner A G, Blaschke J and Murthy M V N 1996 Circular quantum billiard with a singular magnetic flux line Phys. Rev. A 53 39-48
43. Lichtenberg A J and Lieberman M A 1992 Regular and Chaotic Dynamics (New York: Springer) 2nd edn
44. de Almeida A M O 1988 Hamiltonian Systems: Chaos and Quantization (Cambridge: Cambridge University Press)
45. Arnold V I (ed) 1993 Dynamical Systems III. Mathematical Aspects of Classical and Celestial Mechanics (Berlin: Springer)
46. Rimmer R J 1982 Generic Bifurcations for Involutory Area Preserving Maps (Memoirs of the AMS no 272) (Providence, RI: American Mathematical Society)
47. Aguiar M A M D, Malta C P, Baranger M and Davies K T R 1987 Bifurcations of periodic trajectories in non-integrable Hamiltonian systems with two degrees of freedom: numerical and analytical results Ann. Phys. 180 167-205
48. Riddell R J Jr 1979 Boundary-distribution solution of the Helmholtz equation for a region with corners J. Comput. Phys. 31 21-41
49. Riddell R J Jr 1979 Numerical solution of the Helmholtz equation for two-dimensional polygonal regions J. Comput. Phys. 31 42-59
50. Berry M V and Wilkinson M 1984 Diabolical points in the spectra of triangles Proc. R. Soc. A 392 15-43
51. Primack H, Schanz H, Smilansky U and Ussishkin I 1996 Penumbra diffraction in the quantization of dispersing billiards Phys. Rev. Lett. 76 1615-18
52. Sieber M, Pavloff N and Schmit C 1997 Uniform approximation for diffractive contributions to the trace formula in billiard systems Phys. Rev. E 55 2279-99
53. Schomerus H and Sieber M 1997 Bifurcations of periodic orbits and uniform approximations J. Phys. A: Math. Gen. 30 4537-62