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

Frédéric Naud (1)

1 LANLG - Laboratoire d'Analyse non linéaire et Géométrie

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.

Mots clés: Hyperbolic Laplacian on surfaces; resonances; Selberg's zeta function.

Mots clés: Selberg's zeta function

Domaines: Théorie spectrale [math.SP]

Frédéric Naud. Density and localization of resonances for convex co-compact hyperbolic surfaces. 2012. ⟨hal-00680801⟩