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Annales scientifiques de l'ENS - Parutions - série 4, 49 (2016)

Parutions < série 4, 49

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 49, fascicule 3 (2016)

Rémi Carles
On semi-classical limit of nonlinear quantum scattering
Annales scientifiques de l'ENS 49, fascicule 3 (2016), 711-756

Télécharger cet article : Fichier PDF

Résumé :
Limite semi-classique pour le scattering quantique non linéaire
Nous considérons l'équation de Schrödinger non linéaire en présence d'un potentiel à courte portée, en régime semi-classique. Lorsque la constante de Planck est fixée, une théorie du scattering permet d'établir qu'à la fois le potentiel et la non-linéarité sont négligeables en temps grand. Par ailleurs, pour des données sous la forme d'états cohérents, nous établissons une théorie du scattering pour l'équation d'enveloppe, elle-même non linéaire. Dans la limite semi-classique, les deux opérateurs de scattering peuvent être comparés, en faisant intervenir en outre la théorie du scattering classique, grâce à une estimation d'erreur uniforme en temps. Enfin, nous déduisons un phénomène de découplage en temps grand dans le cas d'un nombre fini d'états cohérents.

Mots-clefs : Équation de Schrödinger non linéaire, scattering, analyse semi-classique, états cohérents, oscillateur harmonique dépendant du temps.

Abstract:
We consider the nonlinear Schrödinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a complete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering theory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states.

Keywords: Nonlinear Schrödinger equation, scattering, semi-classical analysis, coherent states, time dependent harmonic oscillator.

Class. math. : 35Q55; 35B40, 35P25, 81Q20.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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