Analytic singularities for long range Schrödinger equations

André Martinez; Shu Nakamura; Vania Sordoni

doi:10.1016/j.crma.2008.07.010

Partial Differential Equations

Analytic singularities for long range Schrödinger equations

Jean-Michel Bony

André Martinez ¹; Shu Nakamura ²; Vania Sordoni ¹

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

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

Abstract (Original language)
Résumé

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.

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.

Article information
Cite this article

Received: 2008-06-18
Accepted: 2008-07-17
Published online: 2008-08-01

DOI: 10.1016/j.crma.2008.07.010

Author's affiliations:

André Martinez ¹ ; Shu Nakamura ² ; Vania Sordoni 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

André Martinez; Shu Nakamura; Vania Sordoni. 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

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     title = {Analytic singularities for long range {Schr\"odinger} equations},
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     year = {2008},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2008.07.010},
     language = {en},
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References
Cited by

[1] A. Hassel; J. Wunsch The Schrödinger propagator for scattering metrics, Ann. of Math. (2), Volume 162 (2005) no. 1, pp. 487-523

[2] A. Martinez An Introduction to Semiclassical and Microlocal Analysis, UTX Series, Springer-Verlag, New York, 2002

[3] A. Martinez; S. Nakamura; V. Sordoni Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math., Volume 59 (2006), pp. 1330-1351

[4] A. Martinez, S. Nakamura, V. Sordoni, Analytic wave front set for solutions to Schrödinger equations, Preprint, 2007

[5] A. Martinez, S., Nakamura, V. Sordoni, Analytic wave front set for solutions to Schrödinger equations II – long range perturbations, Preprint, 2008

[6] S. Nakamura Propagation of the homogeneous wave front set for Schrödinger equations, Duke Math. J., Volume 126 (2005), pp. 349-367

[7] S. Nakamura, Wave front set for solutions to Schrödinger equations, Preprint, 2004, J. Funct. Anal., in press

[8] S. Nakamura, Semiclassical singularities propagation properties for the Schrödinger equations, Preprint, 2006, J. Math. Soc. Japan, in press

[9] L. Robbiano; C. Zuily Microlocal analytic smoothing effect for Schrödinger equation, Duke Math. J., Volume 100 (1999), pp. 93-129

[10] L. Robbiano; C. Zuily Effet régularisant microlocal analytique pour l'équation de Schrödinger: le cas des données oscillantes, Comm. Partial Differential Equations, Volume 100 (2000), pp. 1891-1906

[11] L. Robbiano; C. Zuily Analytic theory for the quadratic scattering wave front set and application to the Schrödinger equation, Astérisque, Volume 283 (2002), pp. 1-128

[12] J. Sjöstrand Singularités analytiques microlocales, Astérisque, Volume 95 (1982), pp. 1-166

[13] J. Wunsch Propagation of singularities and growth for Schrödinger operators, Duke Math. J., Volume 98 (1999), pp. 137-186

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