Partial Differential Equations
Analytic singularities for long range Schrödinger equations
Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 849-852.

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 G0(s) acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution eitHu0 of the Schrödinger equation, in terms of the semiclassical exponential decay of G0(th−1)Tu0, 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.

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 G0(s) agissant sur des espaces de Sjöstrand, et on caractérise le front d'onde analytique de la solution eitHu0 de l'équation de Schrödinger en termes de décroissance exponentielle semiclassique de G0(th−1)Tu0, 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.07.010
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
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     title = {Analytic singularities for long range {Schr\"odinger} equations},
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Martinez, André; Nakamura, Shu; Sordoni, Vania. Analytic singularities for long range Schrödinger equations. Comptes Rendus. Mathématique, Volume 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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