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

Nous démontrons que la réalisation de Neumann de l'opérateur de Schrödinger avec un champ magnétique constant sur un secteur Ω α 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.

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Accepté le :
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DOI : 10.1016/S1631-073X(03)00008-6
Bonnaillie, Virginie 1

1 Département de mathématique, UMR CNRS 8625, bât. 425, Université Paris-Sud, 91405 Orsay cedex, France
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Bonnaillie, Virginie. On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners. Comptes Rendus. Mathématique, Tome 336 (2003) no. 2, pp. 135-140. doi : 10.1016/S1631-073X(03)00008-6. http://www.numdam.org/articles/10.1016/S1631-073X(03)00008-6/

[1] Agmon, S. Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations, Math. Notes, Princeton University Press, 1982

[2] Bernoff, A.; Sternberg, P. Onset of superconductivity in decreasing fields for general domains, J. Math. Phys., Volume 39 (1998) no. 3, pp. 1272-1284

[3] Bolley, P.; Camus, J. Sur une classe d'opérateurs elliptiques et dégénérés à une variable, J. Math. Pures Appl., Volume 51 (1972), pp. 429-463

[4] V. Bonnaillie, On the fundamental state for a Schrödinger operator with magnetic field in domains with corners, Preprint, Mittag–Leffler Institute, January 2003

[5] Brosens, F.; Devreese, J.T.; Fomin, V.M.; Moshchalkov, V.V. Superconductivity in a wedge: analytical variational results, Solid State Comm., Volume 111 (1999) no. 12, pp. 565-569

[6] Helffer, B.; Morame, A. Magnetic bottles in connection with superconductivity, J. Funct. Anal., Volume 185 (2001), pp. 604-680

[7] H.-T. Jadallah, The Onset of superconductivity in a domain with a corner, Ph.D. thesis, Indiana University, 2001

[8] Pan, X.-B. Upper Critical Field for superconductors with edges and corners, Calc. Var. Partial Differential Equations, Volume 14 (2002), pp. 447-482

[9] Persson, A. Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand., Volume 8 (1960), pp. 143-153

[10] del Pino, M.; Felmer, P.L.; Sternberg, P. Boundary concentration for eigenvalue problems related to the onset of superconductivity, Comm. Math. Phys., Volume 210 (2000), pp. 413-446

[11] Schweigert, V.A.; Peeters, F.M. Influence of the confinement geometry on surface superconductivity, Phys. Rev. B, Volume 60 (1999) no. 5, pp. 3084-3087

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