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

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 ${σ\leq \re(s) \leq δ}$ 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.