Counting Resonances on Hyperbolic Surfaces with Unitary Twists

TL;DR

This paper extends the analysis of the Laplace operator on hyperbolic surfaces with unitary twists to infinite-area cases, demonstrating meromorphic continuation and providing bounds on resonance counting functions.

Contribution

It introduces a new framework for the Laplace operator on infinite-area hyperbolic surfaces with unitary representations, extending previous finite-area results.

Findings

Meromorphic continuation of the resolvent to all of a2.

Optimal upper bounds for the resonance counting function.

Construction of a parametrix for the Laplacian on infinite-area surfaces.

Abstract

We present the Laplace operator associated to a hyperbolic surface $\Gamma\setminus\mathbb{H}$ and a unitary representation of the fundamental group $\Gamma$, extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of $\mathbb{C}$ by constructing a parametrix for the Laplacian, following the approach by Guillop\'e and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.