Singular Bohr-Sommerfeld rules for 2D integrable systems

This article gives Bohr-Sommerfeld rules for semiclassical completely integrable systems with two 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 was treated in previous works by San V? Ngc. The case of one degree of freedom has been studied by Colin de Verdière and Parisse.The results are applied to some famous examples: the geodesics of the ellipsoid, the 1:2-resonance, and Schrödinger operators on the sphere S². A numerical test shows that the semiclassical Bohr-Sommerfeld rules match very accurately the “purely quantum” computations.