Normal Forms and Quantization Formulae

Abstract:

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.