“Bottom of the well” semi-classical trace invariants

Abstract

Let $\hat H$ 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.