Semi-classical trace formula and clustering of eigenvalues for Schrödinger operators

Vesselin Petkov; Georgi Popov

Annales de l'I.H.P. Physique théorique (1998)

  • Volume: 68, Issue: 1, page 17-83
  • ISSN: 0246-0211

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Petkov, Vesselin, and Popov, Georgi. "Semi-classical trace formula and clustering of eigenvalues for Schrödinger operators." Annales de l'I.H.P. Physique théorique 68.1 (1998): 17-83. <http://eudml.org/doc/76778>.

@article{Petkov1998,
author = {Petkov, Vesselin, Popov, Georgi},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {semi-classical asymptotics; periodic trajectories; counting function; quantization},
language = {eng},
number = {1},
pages = {17-83},
publisher = {Gauthier-Villars},
title = {Semi-classical trace formula and clustering of eigenvalues for Schrödinger operators},
url = {http://eudml.org/doc/76778},
volume = {68},
year = {1998},
}

TY - JOUR
AU - Petkov, Vesselin
AU - Popov, Georgi
TI - Semi-classical trace formula and clustering of eigenvalues for Schrödinger operators
JO - Annales de l'I.H.P. Physique théorique
PY - 1998
PB - Gauthier-Villars
VL - 68
IS - 1
SP - 17
EP - 83
LA - eng
KW - semi-classical asymptotics; periodic trajectories; counting function; quantization
UR - http://eudml.org/doc/76778
ER -

References

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  1. [1] V. Benci, , H. Hofer and P.H. Rabinowitz, A remark on a priori bound and existence of periodic solutions of hamiltonian systems, Periodic solutions of hamiltonian systems and related topics, NATO ASI Series, Series C, 209, 1987, pp. 85-88. Zbl0656.34033MR920609
  2. [2] R. Brcmmelhuis and A. Uribe, A semi-classical trace formula for Schrödinger operators, Commun. Math. Phys., Vol. 136, 1991, pp. 567-584. Zbl0729.35093MR1099696
  3. [3] F. Cardoso and G. Popov, Rayleigh quasimodes in linear elasticity, Comm. Partial Diff. Equations, Vol. 17, 1992, pp. 1327-1367. Zbl0795.35067MR1179289
  4. [4] A.-M. Charbonnel and G. Popov, A semi-classical trace formula for several commuting operators, Preprint, Université de Nantes, 1996. 
  5. [5] J. Chazarain, Spectre d'un hamiltonien quantique et mécanique classique, Comm. Partial Diff. Equations, Vol. 5, 1980, pp. 595-644. Zbl0437.70014MR578047
  6. [6] Y. Colin de Verdière, Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helv., Vol. 54, 1979, pp. 508-522. Zbl0459.58014MR543346
  7. [7] S. Dozias, Opérateurs h-pseudo-différentiels à flot périodique et asymptotique semi–classique, Thèse de Doctorat, Université Paris XIII, 1994. 
  8. [8] J. Duistermaat, Oscillatory integrals, Lagrange immersions and infolding of singularities, Comm. in Pure and Appl. Math., Vol. 27, 1974, pp. 207-281. Zbl0285.35010MR405513
  9. [9] J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Vol. 29, 1975, 39-79. Zbl0307.35071MR405514
  10. [10] J.P. Françoise and V. Guillemin, On the period spectrum of a symplectic map, J. Funct. Anal., Vol. 100, 1991, pp. 317-358. Zbl0739.58020MR1125229
  11. [11] V. Guillemin and A. Uribe, Circular symmetry and the trace formula, Invent. Math., Vol. 96, 1989, pp. 385-423. Zbl0686.58040MR989702
  12. [12] T. Guriev and Yu. Safarov, Sharp asymptotics of the spectrum of the Laplace operator on a manifold with periodic geodesics, Trudy Matem. Inst. Steklov, Vol. 179, 1988 (in Russian), English transl. in Proc. Steklov Institute of Mathematics, Vol. 179, 1989, pp. 35-53. Zbl0701.58058MR964912
  13. [13] M. Gutzwiller, Periodic orbits and classical quantization condition, J. Math. Phys., Vol. 12, 1971, pp. 345-358. 
  14. [14] B. Helffer, A. Martinez and D. Robert, Ergodicité et limite semi-classique, Commun. Math. Phys., Vol. 109, 1987, pp. 313-326. Zbl0624.58039MR880418
  15. [15] B. Helffer and D. Robert, Propriétes asymptotiques du spectre d'opérateurs pseudo–différentieles sur Rn, Comm. Partial Diff. Equations, Vol. 7, 1982, pp. 795-882. Zbl0501.35081MR662451
  16. [16] B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin, J. Funct. Anal., Vol. 53, 1983, pp. 246-268. Zbl0524.35103MR724029
  17. [17] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäser, Basel, 1994. Zbl0837.58013MR1306732
  18. [ 18] L. Hörmander, Fourier integral operators I, Acta Math. Vol. 127, 1971, pp. 79-183. Zbl0212.46601MR388463
  19. [19] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, Berlin - Heidelberg - New York, 1985. Zbl0601.35001MR781536
  20. [20] L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Springer, Berlin - Heidelberg - New York, 1985. Zbl0612.35001MR404822
  21. [21] V. Ivrii, Semi-classical microlocal analysis and precise spectral asymptotics, Ecole Polytechnique, Centre de Mathématiques, Preprints 1, 2, 3, 1991. Zbl0906.35003
  22. [22] E. Meinrenken, Semiclassical principal symbols and Gutzwiller's trace formula, Reports on Mathematical Physics, Vol. 31, 1992, pp. 279-295. Zbl0794.58046MR1232640
  23. [23] T. Paul et A. Uribe, Sur la formule semi-classique des traces, C. R, Acad. Sci. Paris, t. 313, Série I, 1991, pp. 217-222. Zbl0738.58046MR1126383
  24. [24] V. Petkov and G. Popov, On the Lebesgue measure of the periodic points of a contact manifold, Math. Z., Vol. 218, 1995, pp. 91-102. Zbl0816.58008MR1312579
  25. [25] V. Petkov et G. Popov, Une formule de trace semi-classique et asymptotiques de valeurs propres de l'opérateur de Schrödinger, C. R. Acad. Sci. Paris, t. 323, Série I, 1996, pp. 163-168. Zbl0858.35095MR1402536
  26. [26] V. Petkov and D. Robert, Asymptotiques semi-clasiques du spectre d'hamiltoniens quantiques et trajectoires classiques périodiques, Comm Part. Diff. Equations, Vol. 10, 1985, pp. 365-390. Zbl0574.35067MR784682
  27. [27] G. Popov, Length spectrum invariants of Riemannian manifolds, Math. Z., Vol. 213, 1993, pp. 311-351. Zbl0804.53068MR1221719
  28. [28] Yu Safarov, Asymptotics of the spectrum of a pseudodifferential operator with periodic characteristics, Zap. Nauchn. Sem. Leningrad. Otdel Mat. Inst. Steklov (LOMI). vol. 152. 1986, pp. 94-104 (in Russian), English translation in J. Soviet Math., Vol. 40, 1988. pp. 645-652. Zbl0621.35071MR869246
  29. [29] Yu. Safarov, Exact asymptotics of the spectrum of a boundary value problem and periodic billiards. Izv. AN SSSR, Ser. Mat,. Vol. 52, 1988, pp. 1230-1251 (in Russian), English translation in Math. USSR Izvestiya, Vol. 33, 1989, pp. 553-573. Zbl0682.35082MR984217
  30. [30] Yu. Safarov and D. Vasil'ev, Branching Hamiltonian billiards, Doklady Akad. Nauk SSSR, Vol. 301, 1988, pp. 271-274, English translation, Soviet Math. Dokl., Vol. 38, 1989, pp. 64-68. Zbl0671.58012MR967818
  31. [31] A. Uribe and St. Zelditch, Spectral statistics on Zoll surfaces, Commun. Math. Pxhysics, Vol. 154, 1993, pp. 313-346. Zbl0791.58102MR1224082
  32. [32] A. Weinstein, On the hypotheses of Rabinowitz' periodic orbit theorems, J. Diff. Equations, Vol. 33, 1979, pp. 353-358. Zbl0388.58020MR543704
  33. [33] St. Zelditch, Kuznecov sum formulae and Szegö limit formulae on manifolds, Comm. Partial Diff. Equations, Vol. 17, 1992, pp. 221-260. Zbl0749.58062MR1151262

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