Zero is a resonance of every Schottky surface

TL;DR

This paper demonstrates that zero is a resonance for all Schottky surfaces by analyzing eigenfunctions of transfer operators and relating them to zeros of the Selberg zeta function.

Contribution

It provides a novel connection between transfer operator eigenfunctions and the spectral resonance structure of Schottky surfaces.

Findings

Zero is a resonance for every Schottky surface.

Explicit eigenfunctions of transfer operators are constructed for certain spectral parameters.

The dimension of eigenspaces matches the multiplicity of topological zeros of the Selberg zeta function.

Abstract

For certain spectral parameters we find explicit eigenfunctions of transfer operators for Schottky surfaces. Comparing the dimension of the eigenspace for the spectral parameter zero with the multiplicity of topological zeros of the Selberg zeta function, we deduce that zero is a resonance of every Schottky surface.