The semi-classical spectrum and the Birkhoff normal form

Yves Colin de Verdière (1)

1 IF - Institut Fourier

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.

Mots clés: semi-classics; normal forms

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Yves Colin de Verdière. The semi-classical spectrum and the Birkhoff normal form. 2009. ⟨hal-00361271⟩