Magnetic bottles on geometrically finite hyperbolic surfaces

TL;DR

This paper investigates the spectral properties of a magnetic Laplacian on geometrically finite hyperbolic surfaces with infinite boundary magnetic fields, revealing unique eigenvalue asymptotics for infinite-area surfaces.

Contribution

It provides new asymptotic results for eigenvalue counting functions in the setting of hyperbolic surfaces with infinite boundary magnetic fields, extending spectral theory.

Findings

Eigenvalue counting function exhibits unique asymptotic behavior for infinite-area surfaces.

Magnetic Laplacian's spectral properties are significantly influenced by boundary magnetic fields.

Results contribute to understanding quantum systems on hyperbolic geometries.

Abstract

We consider a magnetic Laplacian on a geometrically finite hyperbolic surface, when the corresponding magnetic field is infinite at the boundary at infinity. We prove that the counting function of the eigenvalues has a particular asymptotic behaviour when the surface has an infinite area.