The purposes of this note are: 1) to propose a direct and ``elementary'' proof of the main result proved by Guillemin-Paul-Uribe [GPU], namely that the semi-classical spectrum near a global minimum of the classical Hamiltonian determines the whole semi-classical Birkhoff normal form (denoted the BNF) in the non-resonant case. 2) to present in the completely resonant case a similar problem.
Publié le : 2009-02-13
Classification:
semi-classics,
normal forms,
35P20, 35S30,
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph],
[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
@article{hal-00361271,
author = {Colin De Verdi\`ere, Yves},
title = {The semi-classical spectrum and the Birkhoff normal form},
journal = {HAL},
volume = {2009},
number = {0},
year = {2009},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00361271}
}
Colin De Verdière, Yves. The semi-classical spectrum and the Birkhoff normal form. HAL, Tome 2009 (2009) no. 0, . http://gdmltest.u-ga.fr/item/hal-00361271/