The trace formula for singular perturbations of the Laplacian on hyperbolic surfaces

TL;DR

This paper develops a trace formula for hyperbolic surfaces with a delta potential, extending classical spectral theory results to include singular perturbations and deriving properties of associated zeta functions.

Contribution

It introduces a trace formula for Laplacian perturbations on hyperbolic surfaces and studies the spectral and zeta function implications of these singular perturbations.

Findings

Derived a trace formula for delta potentials on hyperbolic surfaces.

Established meromorphic continuation and functional equations for the perturbed zeta function.

Extended methods to surfaces with multiple cusps.

Abstract

We prove an analogue of Selberg's trace formula for a delta potential on a hyperbolic surface of finite volume. For simplicity we restrict ourselves to surfaces with at most one cusp, but our methods can easily be extended to any number of cusps. In the case of a noncompact surface we derive perturbative analogues of Maass cusp forms, residual Maass forms and nonholomorphic Eisenstein series. The latter satisfy a functional equation as in the classical case. We also introduce a perturbative analogue of Selberg's zeta function and apply the trace formula to prove its meromorphic continuation to the complex plane as well as a functional equation.