Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian

[Estimations asymptotiques précises pour le laplacien magnétique de Neumann]

Fournais, Soeren ; Helffer, Bernard

Annales de l'Institut Fourier, Tome 56 (2006), p. 1-67 / Harvested from Numdam

Résumé

Motivés par la théorie de la supraconductivité et plus précisément par le problème de l’apparition de la supraconductivité à la surface, de nombreux articles ont été consacrés récemment à l’analyse semi-classique de la plus petite valeur propre de l’opérateur de Schrödinger avec champ magnétique (Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg, Helffer-Morame et aussi Bauman-Phillips-Tang pour le cas du disque). Dans cet article, nous proposons des asymptotiques complètes pour les premières valeurs propres dans le cas d’un domaine de $\mathbb{R}^2$ dont la courbure du bord n’a qu’un unique maximum non-dégénéré.

Abstract

Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, many papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrödinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg and Helffer-Morame and also Bauman-Phillips-Tang for the case of a disc. In the present paper we settle one important part of this question completely by proving an asymptotic expansion to all orders for low-lying eigenvalues for generic domains. The word ‘generic’ means in this context that the curvature of the boundary of the domain has a unique non-degenerate maximum.

Détail

Publié le : 2006-01-01

MR 2228679 | Zbl 1097.47020 | 4 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2171
Classification:  47A75,  58C40,  35Q40,  81Q20
Mots clés: analyse semi-classique, supraconductivité, laplacien de Neumann, laplacien magnétique

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