03-510 Virginie Bonnaillie

On the fundamental state energy for a Schr\"odinger operator with magnetic field in domains with corners (523K, Postscript) Nov 25, 03

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Abstract. The superconducting properties of a sample submitted to an external magnetic field are mathematically described by the minimizers of the Ginzburg-Landau's functional. The analysis of the Hessian of the functional leads to estimate the fundamental state for the Schr\"odinger operator with intense magnetic field for which the superconductivity appears. So we are interested in the asymptotic behavior of the energy for the Schr\"odinger operator with a magnetic field. A lot of papers have been devoted to this problem, we can quote the works of Bernoff-Sternberg, Lu-Pan, Helffer-Mohamed. These papers deal with estimates of the energy in a regular domain and our goal is to establish similar results in a domain with corners. Although this problem is often mentioned in the physical literature, there are very few mathematical papers. We only know the contributions by Pan and Jadallah which deal with very particular domains like a square or a quarter plane. The Physicists Brosens, Devreese, Fomin, Moshchalkov, Schweigert and Peeters propose a non optimal upper bound for the energy. Here, we present a more rigourous analysis and give an asymptotics of the smallest eigenvalue of the operator in a sector $\Omega_\alpha$ of angle $\alpha$ when $\alpha$ is closed to 0, an estimate for the eigenfunctions and we use these results to study the fundamental state in the semi-classical case. A first version of this work was published by The Royal Swedish Academy of Sciences in 2003; some points are clarified and improved here

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