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)
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.