Eigenvalue bounds for radial magnetic bottles on the disk

Abstract

We consider a Schrödinger operator H_A^D with a non-vanishing radial magnetic field B=dA 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_A^D−V has a discrete negative spectrum and we obtain an upper bound of the number of negative eigenvalues. As a consequence we get an upper bound of the number of eigenvalues of H_A^D smaller than any positive value λ, which involves the minimum of B and the square of the L²-norm of A(r)/r, where A(r) is the specific magnetic potential defined as the flux of the magnetic field through the disk of radius r centered in the origin.