On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners

Article type: Research Article

Authors: Bonnaillie, Virginie

Affiliations: Département de Mathématiques, UMR CNRS 8628, Bât. 425, Université Paris‐Sud, 91405 Orsay cedex, France E‐mail: [email protected]‐psud.fr

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 Schrödinger operator with intense magnetic field for which the superconductivity appears. So we are interested in the asymptotic behavior of the energy for the Schrödinger 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 rigorous analysis and give an asymptotics of the smallest eigenvalue of the operator in a sector Ωα of angle α when α is closed to 0, an estimate for the eigenfunctions and we use these results to study the fundamental state in the semi‐classical case.

Journal: Asymptotic Analysis, vol. 41, no. 3-4, pp. 215-258, 2005