A P ] 2 6 M ay 2 00 0 SINGULAR BOHR-SOMMERFELD RULES FOR 2 D INTEGRABLE SYSTEMS

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 focus-focus singularities is studied in [27] and [26]. The case of 1 degree of freedom has been studied in [10]. Our theory is applied to some famous examples: the geodesics of the ellipsoid, the 1 : 2-resonance, and Schrödinger operators on the sphere S. A numerical test shows that the semi-classical Bohr-Sommerfeld rules match very accurately the “purely quantum” computations.

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