Distribution of resonances for hyperbolic surfaces

TL;DR

This paper numerically investigates the distribution of resonances for hyperbolic surfaces, focusing on the fractal Weyl law, spectral gap, and decay rate concentration, using Selberg zeta function computations.

Contribution

It provides the first numerical analysis of resonance distribution for geometrically finite hyperbolic surfaces, emphasizing key spectral properties.

Findings

Confirmation of the fractal Weyl law behavior

Identification of spectral gaps in resonance distribution

Observation of decay rate concentration phenomena

Abstract

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.