Mathematical Research Letters

Volume 14 (2007)

Number 4

“Bottom of the well” semi-classical trace invariants

Pages: 711 – 719



V. Guillemin (Massachusetts Institute of Technology)

T. Paul (École Normale Supérieure)

A. Uribe (University of Michigan)


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.

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