 03510 Virginie Bonnaillie
 On the fundamental state energy for a Schr\"odinger operator with magnetic field in domains with corners
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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 GinzburgLandau'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 BernoffSternberg, LuPan, HelfferMohamed. 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 semiclassical 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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