Singular Bohr-Sommerfeld rules for 2D integrable systems
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 36 (2003) no. 1, pp. 1-55.
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     author = {Colin de Verdi\`ere, Yves and V\~{u} Ngọc, San},
     title = {Singular {Bohr-Sommerfeld} rules for {2D} integrable systems},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1--55},
     publisher = {Elsevier},
     volume = {Ser. 4, 36},
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     year = {2003},
     doi = {10.1016/S0012-9593(03)00002-8},
     zbl = {1028.81026},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S0012-9593(03)00002-8/}
}
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Colin de Verdière, Yves; Vũ Ngọc, San. Singular Bohr-Sommerfeld rules for 2D integrable systems. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 36 (2003) no. 1, pp. 1-55. doi : 10.1016/S0012-9593(03)00002-8. http://www.numdam.org/articles/10.1016/S0012-9593(03)00002-8/

[1] Ahlfors L., Sario S., Riemann Surfaces, Princeton University Press, 1960. | MR | Zbl

[2] Audin M., Courbes algébriques et systèmes intégrables: géodésiques des quadriques, Expositiones Math. 12 (1994) 193-226. | MR | Zbl

[3] Bates L., Cushman R., Global Aspects of Classical Integrable Systems, Birkhäuser, 1998. | MR | Zbl

[4] Bates S., Weinstein A., Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, 8, AMS, 1997. | MR | Zbl

[5] Child M.S., Semiclassical Mechanics with Molecular Applications, Oxford University Press, 1991.

[6] Colin De Verdière Y., Sur le spectre des opérateurs à bicaractéristiques toutes périodiques, Comment. Math. Helv. 54 (1979) 508-522. | MR | Zbl

[7] Colin De Verdière Y., Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Math. Z. 171 (1980) 51-73. | MR

[8] Colin De Verdière Y., Parisse B., Équilibre instable en régime semi-classique I: Concentration microlocale, Comm. Partial Differential Equations 19 (9-10) (1994) 1535-1563. | MR | Zbl

[9] Colin De Verdière Y., Parisse B., Équilibre instable en régime semi-classique II: Conditions de Bohr-Sommerfeld, Ann. Inst. H. Poincaré. Phys. Théor. 61 (3) (1994) 347-367. | Numdam | MR | Zbl

[10] Colin De Verdière Y., Parisse B., Singular Bohr-Sommerfeld rules, Comm. Math. Phys. 205 (1999) 459-500. | MR | Zbl

[11] Colin De Verdière Y., Vey J., Le lemme de Morse isochore, Topology 18 (1979) 283-293. | MR | Zbl

[12] Darboux G., Théorie générale des surfaces, Chelsea, 1972.

[13] Duistermaat J., Oscillatory integrals, Lagrange immersions and unfoldings of singularities, Comm. Pure Appl. Math. 27 (1974) 207-281. | MR | Zbl

[14] Fomenko A., Topological Classification of Integrable Systems, Advances in Soviet Mathematics, 6, AMS, 1991. | MR | Zbl

[15] Guillemin V., Some spectral results for the Laplace operator with potential on the n-sphere, Adv. in Math. 27 (1978) 273-286. | MR | Zbl

[16] Guillemin V., Some spectral results on rank one symmetric spaces, Adv. in Math. 28 (1978) 129-137. | MR | Zbl

[17] Guillemin V., Band asymptotics in two dimensions, Adv. in Math. 42 (1981) 248-282. | MR | Zbl

[18] Guillemin V., Schaeffer D., On a certain class of Fuchsian partial differential equations, Duke Math. J. 44 (1) (1977) 157-199. | MR | Zbl

[19] Hirzebruch F., Topological Methods in Algebraic Geometry, Grundlehren der math. W., 131, Springer, New York, 1966. | MR | Zbl

[20] Klingenberg W., Riemannian Geometry, de Gruyter, 1982. | MR | Zbl

[21] Moser J., Geometry of quadrics and spectral theory, in: The Chern Symposium, Springer, 1980, pp. 147-188. | MR | Zbl

[22] Nguyên Tiên Z., Singularities of integrable geodesic flows on multidimensional torus and sphere, J. Geom. Phys. 18 (1996) 147-162. | MR | Zbl

[23] Nguyên Tiên Z., Symplectic topology of integrable hamiltonian systems, I: Arnold-Liouville with singularities, Compositio Math. 101 (1996) 179-215. | Numdam | MR | Zbl

[24] Nguyên Tiên Z., Polyakova L., Selianova E., Topological classification of integrable geodesic flows on orientable two-dimensional manifolds, Funct. Anal. Appl. 27 (1993) 186-196. | MR | Zbl

[25] Ngọ Vũ, Formes normales semi-classiques des systèmes complètement intégrables au voisinage d'un point critique de l'application moment, Asympt. Analys. 24 (3,4) (2000) 319-342. | Zbl

[26] Vũ Ngọc S., Sur le spectre des systèmes complètement intégrables semi-classiques avec singularités, Ph.D. thesis, Université Grenoble 1, 1998.

[27] Vũ Ngọ S., Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math. 53 (2) (2000) 143-217. | Zbl

[28] Weinstein A., Lectures on Symplectic Manifolds, Regional Conference Series in Mathematics, 29, AMS, 1976. | MR | Zbl

[29] Weinstein A., Asymptotics of eigenvalue clusters for the laplacian plus a potential, Duke Math. J. 44 (4) (1977) 883-892. | MR | Zbl

[30] Weyl H., The Theory of Groups and Quantum Mechanics, Dover, 1950, Translated from the (second) German edition. | Zbl

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