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Let $X$ be a convex co-compact hyperbolic surface and let $\delta$ be theHausdorff dimension of the limit set of the underlying discrete group. We showthat 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 andsupports numerical results obtained for various models of quantum chaoticscattering.

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