, Volume 207, Issue 1, pp 173-195

Normal Forms and Quantization Formulae

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We consider the Schrödinger operator , where as , is Gevrey of order and has a unique non-degenerate minimum. A quantization formula up to an error of order is obtained for all eigenvalues of Q lying in any interval , with a>1 and 0<b<1 explicitly determined and c>0. For eigenvalues in , 0<δ<1, the error is of order . The proof is based upon uniform Nekhoroshev estimates on the quantum normal form constructed quantizing the Lie transformation.

Received: 9 January 1998 / Accepted: 21 April 1999