Singular Bohr-Sommerfeld Rules for 2D Integrable Systems
de Verdiere, Yves Colin ; Ngoc, San Vu
arXiv, 0005264 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2000-05-26
Classification:  Mathematics - Analysis of PDEs,  Mathematical Physics,  Mathematics - Symplectic Geometry,  Mathematics - Spectral Theory,  34E20,  34L25,  81Q20,  58F07,  58C40,  58C27 
@article{0005264,
     author = {de Verdiere, Yves Colin and Ngoc, San Vu},
     title = {Singular Bohr-Sommerfeld Rules for 2D Integrable Systems},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0005264}
}

de Verdiere, Yves Colin; Ngoc, San Vu. Singular Bohr-Sommerfeld Rules for 2D Integrable Systems. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0005264/