Normal forms and quantization formulae

We consider the Schrödinger operator Q = -ℏ2 Δ+V in ℝn, where V (x) → +∞ as \x | → +∞, is Gevrey of order ℓ and has a unique non-degenerate minimum. A quantization formula up to an error of order e-c|lnℏ|-a is obtained for all eigenvalues of Q lying in any interval [0, | \lnℏ-b], with a > 1 and 0 < b < 1 explicitly determined and c > 0. For eigenvalues in [O, ℏδ], 0 < δ < 1, the error is of order e-cℏl|ℓ. The proof is based upon uniform Nekhoroshev estimates on the quantum normal form constructed quantizing the Lie transformation.