DENSITY AND LOCALIZATION OF RESONANCES FOR CONVEX CO-COMPACT HYPERBOLIC SURFACES

Abstract. Let X = Γ\H2 be a convex co-compact hyperbolic surface. We show that the density of resonances of the Laplacian ∆X in strips {σ ≤ Re(s) ≤ δ} with |Im(s) | ≤ T is less than O(T 1+δ−ε(σ)) with ε(σ)> 0 as long as σ> δ2. This improves the previous fractal Weyl upper bound of Zworski [28] and is in agreement with the conjecture of [13] on the essential spectral gap. 1. Introduction an