Eigenvalue bounds for radial magnetic bottles on the disk

Article type: Research Article

Authors: Truc, Françoise

Affiliations: Institut Fourier, Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402 – Saint Martin d'Hères Cedex, France. E-mail: francoise.truc@ujf-grenoble.fr

Abstract: We consider a Schrödinger operator HAD 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 HAD−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 HAD 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.

Keywords: eigenvalue estimate, magnetic bottles, radial potential

DOI: 10.3233/ASY-2011-1077

Journal: Asymptotic Analysis, vol. 76, no. 3-4, pp. 233-248, 2012