The trace formula for a point scatterer on a hyperbolic surface with one cusp

TL;DR

This paper derives an exact trace formula for a point scatterer on a hyperbolic surface with one cusp, extending the Selberg trace formula to include delta potentials and diffractive orbits.

Contribution

It introduces a novel trace formula for Laplacians with delta potentials on non-compact hyperbolic surfaces, incorporating diffractive orbits.

Findings

Derived an explicit trace formula for point scatterers on hyperbolic surfaces.

Expressed the difference between perturbed and unperturbed traces using diffractive orbits.

Extended the Selberg trace formula to include delta potentials.

Abstract

We prove an exact trace formula for the Laplacian with a delta potential - also known as a point scatterer - on a non-compact hyperbolic surface of finite volume with one cusp. Our formula is an analogue of the Selberg trace formula. We express the difference of perturbed and unperturbed trace as a smooth term plus a sum over combinations of diffractive orbits.