Scattering Theory with Unitary Twists

TL;DR

This paper investigates the spectral properties of the Laplace operator on hyperbolic surfaces with a focus on twisted scattering determinants, providing a factorization formula and analyzing the scattering matrix near a critical point.

Contribution

It introduces a new factorization formula for the twisted scattering determinant and characterizes the scattering matrix behavior near 1/2 for hyperbolic surfaces with unitary twists.

Findings

Derived a factorization formula for the twisted scattering determinant

Described the scattering matrix behavior near 1/2

Extended previous approaches to include unitary twists

Abstract

We study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillop\'e and Zworski, we establish a factorization formula for the twisted scattering determinant and describe the behavior of the scattering matrix in a neighborhood of $1/2$.