Mathematical Research Letters

Volume 14, Issue 4, July 2007 pp. 711-719.

``Bottom of the well" semi-classical trace invariants

Authors: V. Guillemin (1), T. Paul (2), and A. Uribe (3)
Author institution: Massachusetts Institute of Technology (1), École Normale Sup$#233;rieure (2), and University of Michigan (3)

Summary: 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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