Normal Forms and Quantization Formulae
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.
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