Density and localization of resonances for convex co-compact hyperbolic surfaces
Résumé
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 $\vert \im(s) \vert \leq T$ 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.
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