The semi-classical spectrum and the Birkhoff normal form

TL;DR

This paper provides an elementary proof that the semi-classical spectrum near a global minimum uniquely determines the Birkhoff normal form in the non-resonant case and explores similar issues in the resonant case.

Contribution

It offers a new, elementary proof of the spectral determination of the Birkhoff normal form and extends the analysis to the resonant case.

Findings

Spectral data near a minimum determines the BNF in non-resonant cases.

Extension of spectral determination results to resonant cases.

Simplified proof approach for spectral and normal form relationship.

Abstract

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.