The semi-classical trace formula and propagation of wave packets

Journal of Functional Analysis

Volumn 132, Issue 1, 1995, Pages 192-249

  Paul, T ^a   Uribe, A ^b  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001493580     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1006/jfan.1995.1105     Document Type: Article

References (39)

1. L. Boutet de Monvel and V. Guillemin, “The Spectral Theory of Toeplitz Operators,” Annals of Mathematics Studies, Vol. 99, Princeton Univ. Press, Princeton, NJ, 1981.
2. R. Brummelhuis and A. Uribe. A trace formula for Schrodinger operators. Comm. Math. Phys. 136 (1991), 567-584.
3. R. Brummelhuis, T. Paul, and A. Uribe. Spectral estimates around a critical level, Duke Math. J., to appear.
4. J. Chazarain. Formule de Poisson pour les varietes Riemanniennes, Invent. Math. 24 (1974), 65-82.
5. Y. Colin de Verdure. Spectre du Laplacien et longueurs des geodesiques periodiques. Compos. Math. 27 (1973), 83-106.
6. Y. Colin de Verdière. Quasi-modes sur les varietes Riemanniennes. Invent. Math. 43 (1977), 15-42.
7. Y. Colin de Verdière, Spectre conjoint d’operateurs pseudodifferentiels qui commutent 1. Le cas non integrable. Duke Math. J. 46 (1979), 169-182.
8. Y. Colin de Virdière. Ergodicite et fonctions propres du Laplacien. Comm. Math. Phys. 102 (1985), 497-502.
9. A. Cordoba and C. Fefferman. Wave packets and Fourier integral operators. Comm. P.D.E. 3 (1978), 979-1005.
10. P.A.M. Dirac, “Principles of Quantum Mechanics,” Oxford Univ. Press, London, 1986.
11. S. Dozias, Memoire de Magistere, Eeole Normale Superieure, Paris, 1993.
12. S. Dozias, Operateurs A-pseudodifTerentiels a (lot periodique: Proprietes spectrales et asymptotic semi-classique, preprint 1994.
13. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974), 207-281.
14. J. Duistermaat, “Fourier Integral Operators." Courant Institute Publications, 1973.
15. J. J. Duistermaat and V. Guillemin, The spectrum of positive operators and periodic bicharacteristics. Inv. Math. 29 (1975), 39-79.
16. V. Guillemin, Symplectic spinors and partial differential equations, Coll. Inst. CNRS, Vol. 237, Geometrie Symplectique et Physique Mathematique, pp. 217-252.
17. V. Guillemin and S. Sternberg, “Geometric Asymptotics,” Math. Surveys, Vol. 14, Amer. Math. Soc., Providence, RI, 1977.
18. V. Guillemin and S. Sternberg. On the spectra of commuting pseudodifferential operators: Recent work of Kac-Spencer. Weinstein and others. In “Partal Differential Equations and Geometry,” proceedings of the Park City conference (C. I. Byrnes, Ed.), Dekker, New York, 1979.
19. V. Guillemin and A. Uribe. Circular symmetry and the trace formula. Invent. Math. 96 (1989), 385-423.
20. V. Guillemin and G. Uhlmann. Oscillatory integrals with singular symbols, Duke Math. J. 48 (1981), 251-267.
21. M. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. 12, (1971), 343-358.
22. G. A. Hagedorn, Semiclassical quantum mechanics I, Comm. Math. Phys. 71 (1980), 77-93.
23. B. Helffer, A. Martinez and D. Robert. Ergodicite et limit semi-classique. Comm. Math. Phys. 109 (1987), 313-326.
24. B. Helffer and D. Robert, Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques, Ann. Inst. Fourier XXXI, Fasc. 3 (1981), 169-223.
25. L. Hormander, “The Analysis of Linear Partial Differential Operators I,” Springer-Verlag, Berlin/New York, 1985.
26. I. Ivrn, Semi-classical microlocal analysis and precise spectral asymptotics, to appear.
27. M. Karasev and Y. Vorobjev, Integral representations over isotropic submanifolds and equations of zero curvature, preprint, 1993.
28. S. Kobayashi and K. Nomizu, “Foundations of Differential Geometry,” Vol. I, Interscience, New York, 1963.
29. G. Lion and M. Vergne, “The Weyl Representation, Maslov Index and Theta Series,” Birkhauser, Boston 1980.
30. E. Meinrenken, Semiclassical principal symbols and Gutzwiller’s trace formula, “Reports on Mathematical Physics,” Vol. 31 pp. 279-295.
31. R. B. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem. Comm. Pure Appl. Math. 32 (1979), 483-519.
32. T. Paul and A. Uribe, Sur la formule semi-classique des traces. C.R. Acad. Sci. Paris 313, No. 1 (1991), 217-222.
33. T. Paul and A. Uribe, On the pointwise behavior of the semi-classical measure, Comm. Math. Phys., to appear.
34. J. V. Ralston, On the construction of quasimodes associated with stable periodic orbits. Comm. Math. Phys. 51 (1976), 219-242.
35. D. Robert, Autour de l’approximation semi-classique, Progr. Math. 68, Birkhauser, Boston, 1987.
36. M. A. Schubin, “PseudodifTerential Operators and Spectral Theory,” Springer Verlag, New York, 1987.
37. M. Taylor and A. Uribe, Semiclassical spectra of gauge fields, J. Funct. Anal. 110 (1992), 1-46.
38. A. Uribe and S. Zelditch, Spectral statistics on Zoll surfaces, Comm. Math. Phys. 154, (1993), 313-346.
39. A. Voros, The return of the quartic oscillator: The complex WKB method, Ann. Inst. Henri Poincare Sect. A 39 (1983), 211-338.