Communications in Mathematical Physics

, Volume 207, Issue 1, pp 173–195

Normal Forms and Quantization Formulae


  • Dario Bambusi
    • Dipartimento di Matematica, Università di Milano, 20133 Milano, Italy. E-mail:
  • Sandro Graffi
    • Dipartimento di Matematica, Università di Bologna, 40127 Bologna, Italy. E-mail:
  • Thierry Paul
    • Ceremade, Université de Paris-IX, 75776 Paris, France. E-mail:

DOI: 10.1007/s002200050723

Cite this article as:
Bambusi, D., Graffi, S. & Paul, T. Comm Math Phys (1999) 207: 173. doi:10.1007/s002200050723


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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© Springer-Verlag Berlin Heidelberg 1999