Singular Bohr-Sommerfeld Rules for 2D Integrable Systems

TL;DR

This paper develops Bohr-Sommerfeld quantization rules for 2D integrable systems with non-degenerate singularities, extending previous work to new singularity types and validating the theory with numerical tests on classical examples.

Contribution

It introduces Bohr-Sommerfeld rules for 2D integrable systems with Morse-Bott singularities, expanding the scope of semi-classical quantization methods.

Findings

Rules match quantum computations accurately

Applied to geodesics on ellipsoid, 1:2 resonance, and Schrödinger operators on S^2

Numerical tests confirm theoretical predictions

Abstract

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.