A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition

Communications in Mathematical Physics

Volumn 202, Issue 2, 1999, Pages 463-480

  Combescure, Monique ^a   Ralston, James ^b   Robert, Didier ^c  

^a UNIVERSITÉ PARIS SUD   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c Universite de Nantes   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033238569     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050591     Document Type: Article

References (39)

1. 1. Arnaud, J. A.: Hamiltonian theory of beam mode propagation. Progress in Optics XI, ed. E. Wolf, Amsterdam: North Holland, 1973, 249-304
2. 2. Babich, V.M.: Eigenfunctions concentrated in a neighborhood of a closed geodesic. In: Math. Problems in Wave Propagation Theory, V.M. Babich, ed., Sem. Math., V.A. Steklov Math. Inst. 9, 1968, Leningrad. Translated by Consultants Bureau, New York, 1970
3. 3. Babich, V.M., Buldyrev, V.S.: Asymptotic Methods in Short Wave Diffraction Problems. Vol. 1 (Russian), Moscow: Nauka, 1972
4. 4. Balian, R. and Bloch, C.: Distribution of eigenfrequencies for the wave equation in a finite domain. Ann. Phys. 69, vol. 1, 76-160 (1972)
5. 5. Balian, R. and Bloch, C.: Solution of the Schrödinger equation in terms of classical paths. Ann. Phys. 85, 514-545 (1974)
6. 6. Charbonnel, A-M., Popov, G.: Semiclassical asymptotics for several commuting operators. Comm. in PDE 24, 283-323 (1998)
7. 7. Chazarain, J.: Formule de Poisson pour les variétés Riemanniennes. Inv. Math. 24, 65-82 (1974)
8. 8. Colin de Verdière, Y.: Spectre du Laplacien et longueurs des géodésiques périodiques I. Compos. Math. 27, 83-106 (1973)
9. 9. Combescure, M. and Robert, D.: Semiclassical sum rules and generalized coherent states. J. Math. Phys. 36, 6596-6610 (1995)
10. 10. Combescure, M. and Robert, D.: Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. Asymptotic Anal. 14, 377-404 (1997)
11. 11. Combescure, M. and Robert, D.: Propagation d'états cohérents par l'équation de Schrödinger et approximation semi-classique. C. R. Acad. Sci. Paris, t. 323 Série I, 871-876 (1996)
12. 12. Dimassi, M., Sjöstrand, J.: Trace asymptotics via almost analytic extensions. PNLDE 21, Basel-Boston: Birkhäuser, pp. 126-142
13. 13. Dozias, S: Opérateurs h-pseudodifférentiels à flot périodique. Thèse, Paris 13, 1994
14. 14. Duistermaat, J. J. and Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29, 39-79 (1975)
15. 15. Folland, G. B.: Harmonic Analysis in Phase Space. Ann. of Math. Studies, 122, Princeton: Princeton University Press, 1989
16. 16. Guillemin, V. and Uribe, A.: Circular symmetry and the trace formula. Invent. Math. 96, 385-423 (1989)
17. 17. Gutzwiller, M.: Periodic orbits and classical quantization conditions. J. Math. Phys. 12, 343-358 (1971) and book Chaos in classical and quantum mechanics Berlin-Heidelberg-New York: Springer-Verlag, 1990
18. 17. Gutzwiller, M.: Periodic orbits and classical quantization conditions. J. Math. Phys. 12, 343-358 (1971) and book Chaos in classical and quantum mechanics Berlin-Heidelberg-New York: Springer-Verlag, 1990
19. 18. Hagedorn, G.: Semiclassical Quantum Mechanics. (I), Commun. Math. Phys. 71, 77-93 (1980); (II) Ann. Inst. H. Poincaré 42, 363-374 (1985)
20. 18. Hagedorn, G.: Semiclassical Quantum Mechanics. (I), Commun. Math. Phys. 71, 77-93 (1980); (II) Ann. Inst. H. Poincaré 42, 363-374 (1985)
21. 19. Hall, K.R, Meyer, G.R.: Introduction to Hamiltonian Dynamical Systems and the N-body problem. Applied Mathematical Sciences 90, Berlin-Heidelberg-New York: Springer-Verlag, 1991
22. 20. Heller, E. J.: Time dependent approach to semiclassical dynamics. J. Chem. Phys. 62, 1544-1555 (1975); Quantum localization and the rate of exploration of phase space. Phys. Rev. A35, 1360-1370 (1987)
23. 20. Heller, E. J.: Time dependent approach to semiclassical dynamics. J. Chem. Phys. 62, 1544-1555 (1975); Quantum localization and the rate of exploration of phase space. Phys. Rev. A35, 1360-1370 (1987)
24. 21. Helffer, B., Robert, D.: Calcul fonctionnel par la transformée de Mellin. J. Funct. Anal. V. 153, 246-268 (1983)
25. 22. Helffer, B and Sjöstrand, J.: Multiple wells in the semi-classical limit I. Com. in PDE, 9 (4), 337-408 (1984)
26. 23. Hörmander, L.: The analysis of partial differential operators. 1-4, Berlin: Springer, 1983
27. 24. Keller, J.B.: J. Opt. Soc. Am. 61, 40 (1971)
28. 25. Littlejohn, R.: The semiclassical evolution of wave packets. Physics Rep. 138, 193-291 (1986)
29. 26. Meinrenken, E.: Semiclassical principal symbols and Gutzwiller's trace formula. Reports on Math. Phys. 31, 279-295 (1992)
30. 27. Paul, T. and Uribe, A.: Sur la formule semi-classique des traces. C. R. Acad. Sci. Paris 313 I, 217-222 (1991)
31. 28. Paul, T. and Uribe, A.: A construction of quasimodes using coherent states. Ann. Inst. H. Poincaré 59, 357-381 (1993)
32. 29. Paul, T. and Uribe, A.: The semi-classical trace formula and propagation of wave packets. J. Funct. Anal. 132, 192-249 (1995)
33. 30. Paul, T. and Uribe, A.: On the pointwise behaviour of semiclassical measures. Commun. Math. Phys. 175, 229-258 (1996)
34. 31. Petkov, V, Popov, G.: Semiclassical trace formula and clustering of eigenvalues for Schrödinger operators. Ann. Inst. Henri Poincaré, sect. Phys. Th. 68, 17-83 (1998)
35. 32. Popov, G.: On the contribution of degenerate periodic trajectories to the wave-trace. Commun. Math. Phys. 196, 363-383 (1998)
36. 33. Ralston, J.: On the construction of quasimodes associated with stable periodic orbits. Commun. Math. Phys. 51, 219-242 (1976); Erratum, 67, 91
37. 34. Ralston, J.: Gaussian beams and the propagation of singularities. Studies in PDE, Stud. Math. 23, 207-248 (1982)
38. 35. Schrödinger, E.: Naturwissenschaften 14, 664 (1926)
39. 36. Wilkinson, M.: A semiclassical sum rule for matrix elements of classically chaotic systems. J. Phys. A: Math. Gen. 20, 2415-2423 (1987)