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

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

2012-03-20

arxiv.org/abs/1203.4378v1

Mathematics

Spectral Theory

Scientific paper

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.

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