00-452 Bernard Helffer and Abderemane Morame

Magnetic bottles in connection with superconductivity (171K, latex) Nov 13, 00

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Abstract. Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schr\"odinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan and Del Pino-Felmer-Sternberg. This recovers partially questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significatively the lower bounds and as a byproduct we solve a conjecture proposed by Bernoff-Sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.

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