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Comptes Rendus Mathématique
Volume 346, n° 15-16
pages 849-852 (août 2008)
Doi : 10.1016/j.crma.2008.07.010
Received : 18 June 2008 ;  accepted : 17 July 2008
Analytic singularities for long range Schrödinger equations
Singularités analytiques pour des équations de Schrödinger à longue portée
 

André Martinez a, 1 , Shu Nakamura b , Vania Sordoni a
a Università di Bologna, Dipartimento di Matematica, Piazza di Porta San Donato 5, 40127 Bologna, Italy 
b Graduate School of Mathematical Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan 153-8914 

Abstract

We consider the Schrödinger equation associated to long range perturbations of the flat Euclidean metric (in particular, potentials growing subquadratically at infinity are allowed). We construct a modified quantum free evolution   acting on Sjöstrandʼs spaces, and we characterize the analytic wave front set of the solution   of the Schrödinger equation, in terms of the semiclassical exponential decay of  , where T stands for the Bargmann-transform. The result is valid for   near the forward non-trapping points, and for   near the backward non-trapping points. To cite this article: A. Martinez et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.
Résumé

On considère lʼéquation de Schrödinger associée à des perturbations à longue portée de la métrique euclidienne plate (en particulier, on autorise des potentiels qui croissent de manière sub-quadratique à lʼinfini). On construit une évolution quantique modifiée   agissant sur des espaces de Sjöstrand, et on caractérise le front dʼonde analytique de la solution   de lʼéquation de Schrödinger en termes de décroissance exponentielle semiclassique de  , où T désigne la tranformation de Bargmann. Le résultat est valable pour   près des points non captifs dans lʼavenir, et pour   près des points non captifs dans le passé. Pour citer cet article : A. Martinez et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.
1  Partly supported by Università di Bologna, Funds for Selected Research Topics and Founds for Agreements with Foreign Universities.


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