98-3 Dario Bambusi, Universita' di Milano, Italy. (bambusi@mat.unimi.it), Sandro Graffi, Universita' di Bologna, Italy. (graffi@dm.unibo.it), Thierry Paul, Universite' de Paris-IX, (France), (paulth@ceremade.dauphine.fr)

NORMAL FORMS AND QUANTIZATION FORMULAE (74K, Latex) Jan 2, 98

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Abstract. We consider the Schr\"odinger operator $ Q= -\hbar^2\Delta +V$ in $\R^n$ where $V(x)$ belongs to a Gevrey class, tends to $+\infty$ as $|x|\to +\infty$ and has a unique non-degenerate minimum. A quantization formula is obtained for all eigenvalues of $Q$ belonging to any interval $[0,\varphi(\hbar)]$ up to an error of order $\hbar^{\infty}$. Here $\varphi(x)$ is any positive, increasing function on $]0,1[$ such that $\varphi^b(x)\ln{x}\to 0$ as $x\to 0$ and $b$ an explicitly determined constant. The proof is based upon uniform Nekhoroshev estimates on the quantum normal form constructed quantizing the Lie transformation.

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