Singular Bohr-Sommerfeld Rules for 2D Integrable Systems
摘要
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 -resonance, and Schroedinger operators on the sphere . A numerical test shows that the semiclassical Bohr-Sommerfeld rules match very accurately the ``purely quantum'' computations.
引用
@article{arxiv.math/0005264,
title = {Singular Bohr-Sommerfeld Rules for 2D Integrable Systems},
author = {Yves Colin de Verdiere and San Vu Ngoc},
journal= {arXiv preprint arXiv:math/0005264},
year = {2007}
}
备注
postscript, 61 pages, figures best seen in color. Preprint Institut Fourier