Lieb-Thirring inequalities for radial magnetic bottles in the disk
Résumé
We consider a Schrodinger operator H with a non vanishing radial magnetic field B and Dirichlet boundary conditions on the unit disk. We assume growth conditions on B near the boundary which guarantee in particular the compactness of the resolvent of this operator. Under some assumptions on an additional radial potential V the operator H + V has a discrete negative spectrum and we prove a Lieb-Thirring inequality on these negative eigenvalues. As a consequence we get an explicit upperbound of the number of eigenvalues of H less than any positive value, which depends on the minimum of B and on the integral of the square of any gauge associated to B.
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