Eigenvalue upper bounds for the magnetic Schroedinger operator

TL;DR

This paper derives bounds for the eigenvalues of the magnetic Schrödinger operator on compact Riemannian manifolds, linking spectral properties to geometric features and potentials, including magnetic and scalar fields.

Contribution

It introduces new eigenvalue bounds that incorporate geometric quantities and potential-related terms, extending previous spectral estimates to magnetic Schrödinger operators on manifolds.

Findings

Eigenvalue bounds depend on manifold dimension and volume.

The first eigenvalue of the Hodge-de Rham Laplacian influences spectral estimates.

Bounds incorporate mean scalar potential, magnetic field norm, and harmonic component distance.

Abstract

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain several bounds for the spectrum. Besides the dimension and the volume of the manifold, the geometric quantity which plays an important role in these estimates is the first eigenvalue of the Hodge-de Rham Laplacian acting on co-exact 1-forms. In the 2-dimensional case, this is nothing but the first positive eigenvalue of the Laplacian acting on functions. As for the dependence of the bounds on the potentials, it brings into play the mean value of the scalar potential q, the L^2-norm of the magnetic field B=dA, and the distance, taken in L^2, between the harmonic component of A and the subspace of all closed 1-forms whose cohomology class is integral (that is, having integral flux around any loop). In particular, this distance is zero when the first cohomology group is trivial.