On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners

Virginie Bonnaillie

doi:10.1016/S1631-073X(03)00008-6

Partial Differential Equations

On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners
[Etat fondamental de l'opérateur de Schrödinger avec champ magnétique dans un domaine à coin]

Présenté par : Haı̈m Brézis

Virginie Bonnaillie ¹

¹ Département de mathématique, UMR CNRS 8625, bât. 425, Université Paris-Sud, 91405 Orsay cedex, France

Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 135-140.

Résumé
Abstract

Nous démontrons que la réalisation de Neumann de l'opérateur de Schrödinger avec un champ magnétique constant sur un secteur $Ω_{α} \subset ℝ^{2}$ d'angle α∈]0,π] admet au moins une valeur propre, en dessous du spectre essentiel, quand l'angle est suffisamment petit. Nous établissons un développement limité de la plus petite valeur propre pour α proche de 0. Cette étude permet de donner des estimations du bas du spectre dans le cas semi-classique pour des domaines à coin.

We show that the Neumann realization for the Schrödinger operator with a constant magnetic field in a sector has at least one eigenvalue below the essential spectrum, when the angle is sufficiently small. We establish the complete asymptotics of the lowest eigenvalue as the angle tends to 0. This study is applied to the analysis of the bottom of the spectrum in the semi-classical case for domains with edges.

Reçu le : 2002-11-26
Accepté le : 2002-11-26
Publié le : 2003-01-15

DOI : 10.1016/S1631-073X(03)00008-6

Affiliations des auteurs :

Virginie Bonnaillie ¹

¹ Département de mathématique, UMR CNRS 8625, bât. 425, Université Paris-Sud, 91405 Orsay cedex, France

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Virginie Bonnaillie. On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners. Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 135-140. doi : 10.1016/S1631-073X(03)00008-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00008-6/

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Virginie Bonnaillie No�l A posteriori error estimator for the eigenvalue problem associated to the Schr�dinger operator with magnetic field, Numerische Mathematik, Volume 99 (2004) no. 2, p. 325 | DOI:10.1007/s00211-004-0556-3

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