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.
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 dont la courbure du bord n’a qu’un unique maximum non-dégénéré.
Keywords: semi-classical analysis, supraconductivity, Neumann Laplacian, magnetic Laplacian
Mot clés : analyse semi-classique, supraconductivité, laplacien de Neumann, laplacien magnétique
@article{AIF_2006__56_1_1_0, author = {Fournais, Soeren and Helffer, Bernard}, title = {Accurate eigenvalue asymptotics for the magnetic {Neumann} {Laplacian}}, journal = {Annales de l'Institut Fourier}, pages = {1--67}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {1}, year = {2006}, doi = {10.5802/aif.2171}, zbl = {1097.47020}, mrnumber = {2228679}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2171/} }
TY - JOUR AU - Fournais, Soeren AU - Helffer, Bernard TI - Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian JO - Annales de l'Institut Fourier PY - 2006 SP - 1 EP - 67 VL - 56 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2171/ DO - 10.5802/aif.2171 LA - en ID - AIF_2006__56_1_1_0 ER -
%0 Journal Article %A Fournais, Soeren %A Helffer, Bernard %T Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian %J Annales de l'Institut Fourier %D 2006 %P 1-67 %V 56 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2171/ %R 10.5802/aif.2171 %G en %F AIF_2006__56_1_1_0
Fournais, Soeren; Helffer, Bernard. Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian. Annales de l'Institut Fourier, Volume 56 (2006) no. 1, pp. 1-67. doi : 10.5802/aif.2171. http://www.numdam.org/articles/10.5802/aif.2171/
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