Density and localization of resonances for convex co-compact hyperbolic surfaces

TL;DR

This paper establishes improved bounds on the density of resonances for convex co-compact hyperbolic surfaces, showing fewer resonances in certain strips, which supports numerical findings in quantum chaos.

Contribution

It provides a new upper bound on the density of resonances in strips for convex co-compact hyperbolic surfaces, refining previous fractal Weyl bounds.

Findings

Resonance density is less than O(T^{1+δ−ε(σ)}) for σ > δ/2

Supports numerical results in quantum chaotic scattering

Improves upon previous bounds by Zworski

Abstract

Let $X$ be a convex co-compact hyperbolic surface and let $\delta$ be the Hausdorff dimension of the limit set of the underlying discrete group. We show that the density of the resonances of the Laplacian in strips ${\sigma\leq \re(s) \leq \delta}$ with $|\im(s)| \leq T$ is less than $O(T^{1+\delta-\epsilon(\sigma)})$ with $\epsilon>0$ as long as $\sigma>\delta/2$. This improves the fractal Weyl upper bounds of Zworski and supports numerical results obtained for various models of quantum chaotic scattering.