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.
Normal forms and quantization formulae / D. Bambusi, S. Graffi, T. Paul. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 207:1(1999 Nov), pp. 173-195.
Normal forms and quantization formulae
D. BambusiPrimo
;
1999
Abstract
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.Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.