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

摘要

Let X = Γ H~2 be a convex co-compact hyperbolic surface and let δ be the Hausdorff dimension of the limit set. Let Δ_X be the hyperbolic Laplacian. We show that the density of resonances of the Laplacian Δ_X in rectangles{σ ≤ Re(s) ≤ δ, |Im(s)| ≤ T }is less than O(T ~(1+τ(σ))) in the limit T →∞, where τ(σ)<δ as long as σ >δ/2. This improves the previous fractal Weyl upper bound of Zworski (Invent. Math. 136(2):353-409, 1999) and goes in the direction of a conjecture stated in Jakobson and Naud (Geom. Funct. Anal. 22(2):352-368, 2012).