2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219965The 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.Mathematical PhysicsSpectral Theory35P20, 35S30text