Spectral asymptotics for magnetic Schrödinger operators with rapidly decreasing electric potentials

Georgi D. Raikov; Simone Warzel

doi:10.1016/S1631-073X(02)02554-2

Spectral asymptotics for magnetic Schrödinger operators with rapidly decreasing electric potentials
[Asymptotiques spectrales pour des opérateurs magnétiques de Schrödinger avec des potentiels électriques qui décroissent rapidement à l'infini]

Georgi D. Raikov ¹ ; Simone Warzel ²

¹ Departamento de Matemáticas, Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile
² Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany

Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 683-688.

Résumé
Abstract

On considère l'opérateur de Schrödinger H(V) agissant dans $L^{2} (ℝ^{2})$ ou $L^{2} (ℝ^{3})$ avec un champ magnétique constant et un potentiel électrique V qui génériquement décroı̂t à l'infini exponentiellement vite ou est à un support compact. On étudie le comportement asymptotique du spectre discret de H(V) en voisinage des points de la frontière de son spectre essentiel. Si la décroissance de V est gaussienne ou plus rapide ce comportement ne se décrit pas par les formules semi-classiques connues dans le cas où V décroı̂t comme une puissance.

We consider the Schrödinger operator H(V) on $L^{2} (ℝ^{2})$ or $L^{2} (ℝ^{3})$ with constant magnetic field, and a class of electric potentials V which typically decay at infinity exponentially fast or have a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H(V) near the boundary points of its essential spectrum. If V decays like a Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasi-classical formulas known for the case where V admits a power-like decay.

Reçu le : 2002-09-04
Accepté le : 2002-09-13
Publié le : 2002-10-01

DOI : 10.1016/S1631-073X(02)02554-2

Affiliations des auteurs :

Georgi D. Raikov ¹ ; Simone Warzel ²

¹ Departamento de Matemáticas, Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile
² Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany

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     author = {Georgi D. Raikov and Simone Warzel},
     title = {Spectral asymptotics for magnetic {Schr\"odinger} operators with rapidly decreasing electric potentials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {683--688},
     publisher = {Elsevier},
     volume = {335},
     number = {8},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02554-2},
     language = {en},
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Georgi D. Raikov; Simone Warzel. Spectral asymptotics for magnetic Schrödinger operators with rapidly decreasing electric potentials. Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 683-688. doi : 10.1016/S1631-073X(02)02554-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02554-2/

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