The trace formula for a point scatterer on a compact hyperbolic surface

TL;DR

This paper derives an exact trace formula for a Laplacian perturbed by a point scatterer on a compact hyperbolic surface, extending the Selberg trace formula to include diffractive orbits and their contributions.

Contribution

It introduces a novel trace formula for the Laplacian with a Dirac delta potential on hyperbolic surfaces, linking spectral data with diffractive orbits.

Findings

Derived an explicit trace formula for the perturbed Laplacian

Expressed the difference in traces as an identity plus diffractive orbit sum

Extended Selberg trace formula to include point scatterers

Abstract

An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed and unperturbed trace is expressed as an identity term plus a sum over combinations of diffractive orbits which visit the position of the potential.