Eigenvalue bounds for radial magnetic bottles on the disk

TL;DR

This paper derives bounds on the negative eigenvalues of a Schrödinger operator with a radial magnetic field on the disk, linking spectral properties to magnetic flux and potential growth conditions.

Contribution

It provides new upper bounds on the number of negative eigenvalues for Schrödinger operators with radial magnetic fields on the disk, under specific growth and boundary conditions.

Findings

Upper bounds on negative eigenvalues depend on magnetic flux and potential norms.

Conditions ensure the compactness of the resolvent of the operator.

Results connect magnetic field behavior near boundary to spectral properties.

Abstract

We consider a Schr\"odinger operator H 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 + V has a discrete negative spectrum and we obtain an upper bound on the number of negative eigenvalues. As a consequence we get an upperbound of the number of eigenvalues of H smaller than any positive value, which involves the minimum of B and the square of the L^2 -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 centerde in the origin.