Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area

TL;DR

This paper investigates the spectral properties of a magnetic Laplacian on non-compact hyperbolic surfaces with finite area, establishing conditions for discrete spectrum and confirming Weyl's law for eigenvalue counting.

Contribution

It proves the discreteness of the spectrum under certain conditions and confirms Weyl's law for the eigenvalues of the magnetic Laplacian on these surfaces.

Findings

Spectrum is discrete when harmonic component of A satisfies specific conditions.

Weyl's law holds for the eigenvalue counting function even when the magnetic field is zero.

Results extend classical spectral theory to magnetic Laplacians on non-compact hyperbolic surfaces.

Abstract

We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface $\mM $ with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0. $