Trace formula for the magnetic Laplacian on a compact hyperbolic surface

TL;DR

This paper derives a trace formula for the magnetic Laplacian on a compact hyperbolic surface with constant magnetic field, analyzing its asymptotic behavior near the Mane critical energy level.

Contribution

It provides the first explicit trace formula for the magnetic Laplacian in this geometric setting, including asymptotic analysis near critical energy levels.

Findings

Derived the trace formula for the magnetic Laplacian on hyperbolic surfaces.

Analyzed asymptotic behavior of coefficients near the Mane critical level.

Provides insights into spectral properties of magnetic Laplacians in hyperbolic geometry.

Abstract

We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level.