2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115553Let $\hat H$ be an h-admissible pseudodifferential operator whose principal symbol, $H$, has a unique non-degenerate global minimum. We give a simple proof that the semi-classical asymptotics of the eigenvalues of $\hat H$ corresponding to the "bottom of the well" determine the Birkhoff normal form of $H$ at the minimum. We treat both the resonant and the non-resonant cases.9 pagesSpectral TheoryMathematical Physics58J50 (primary), 81Q20 (secondary)text