Normal forms and quantization formulae

We consider the Schrodinger operator Q = -h(2)Delta+V in R-n, where V(x) -- > +infinity as \ x \ --> +infinity, is Gevrey of order l and has a unique n on-degenerate minimum. A quantization formula up to an error of order e(-c) \ lnh \(-a) is obtained for-all eigenvalues of Q lying in any interval [0, \ lnh \(-b)], with a > 1 and 0 < b < 1 explicitly determined and c > 0. For eigenvalues in [0, h(delta)], 0 < delta < 1, the error is of order e(-c/h1 /l). The proof is based upon uniform Nekhoroshev estimates on the quantum n ormal form constructed quantizing the Lie transformation.