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

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