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Article Dans Une Revue Annales Scientifiques de l'École Normale Supérieure Année : 2003

Singular Bohr-Sommerfeld Rules for 2D Integrable Systems

Yves Colin de Verdière
San Vu Ngoc

Résumé

In this paper, we describe Bohr-Sommerfeld rules for semi-classical completely integrable systems with 2 degrees of freedom with non degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian is non singular. The more singular case of {\it focus-focus} singularities is studied in [Vu Ngoc San, CPAM 2000] and [Vu Ngoc San, PhD 1998] The case of 1 degree of freedom has been studied in [Colin de Verdiere-Parisse, CMP 1999] Our theory is applied to some famous examples: the geodesics of the ellipsoid, the $1:2$-resonance, and Schroedinger operators on the sphere $S^2$. A numerical test shows that the semiclassical Bohr-Sommerfeld rules match very accurately the ''purely quantum'' computations.

Dates et versions

hal-00020086 , version 1 (05-03-2006)

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Citer

Yves Colin de Verdière, San Vu Ngoc. Singular Bohr-Sommerfeld Rules for 2D Integrable Systems. Annales Scientifiques de l'École Normale Supérieure, 2003, 36, pp.1-55. ⟨hal-00020086⟩
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