Analytic singularities for long range Schrödinger equations

Martinez, André; Nakamura, Shu; Sordoni, Vania

doi:10.1016/j.crma.2008.07.010

Partial Differential Equations

Analytic singularities for long range Schrödinger equations
[Singularités analytiques pour des équations de Schrödinger à longue portée]

Présenté par : Jean-Michel Bony

Martinez, André ¹ ; Nakamura, Shu ² ; Sordoni, Vania ¹

¹ Università di Bologna, Dipartimento di Matematica, Piazza di Porta San Donato 5, 40127 Bologna, Italy
² Graduate School of Mathematical Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan 153-8914

Comptes Rendus. Mathématique, Tome 346 (2008) no. 15-16, pp. 849-852.

Résumé
Abstract

On considère l'équation 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 l'infini). On construit une évolution quantique modifiée $G_{0} (s)$ agissant sur des espaces de Sjöstrand, et on caractérise le front d'onde analytique de la solution $e^{- i t H} u_{0}$ de l'équation de Schrödinger en termes de décroissance exponentielle semiclassique de $G_{0} (- t h^{−1}) T u_{0}$ , où T désigne la tranformation de Bargmann. Le résultat est valable pour $t < 0$ près des points non captifs dans l'avenir, et pour $t > 0$ près des points non captifs dans le passé.

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 $G_{0} (s)$ acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution $e^{- i t H} u_{0}$ of the Schrödinger equation, in terms of the semiclassical exponential decay of $G_{0} (- t h^{−1}) T u_{0}$ , where T stands for the Bargmann-transform. The result is valid for $t < 0$ near the forward non-trapping points, and for $t > 0$ near the backward non-trapping points.

Reçu le : 2008-06-18
Accepté le : 2008-07-17
Publié le : 2008-08-01

DOI : 10.1016/j.crma.2008.07.010

Affiliations des auteurs :

Martinez, André ¹ ; Nakamura, Shu ² ; Sordoni, Vania ¹

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     author = {Martinez, Andr\'e and Nakamura, Shu and Sordoni, Vania},
     title = {Analytic singularities for long range {Schr\"odinger} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {849--852},
     publisher = {Elsevier},
     volume = {346},
     number = {15-16},
     year = {2008},
     doi = {10.1016/j.crma.2008.07.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.07.010/}
}

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Martinez, André; Nakamura, Shu; Sordoni, Vania. Analytic singularities for long range Schrödinger equations. Comptes Rendus. Mathématique, Tome 346 (2008) no. 15-16, pp. 849-852. doi : 10.1016/j.crma.2008.07.010. http://www.numdam.org/articles/10.1016/j.crma.2008.07.010/

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