The Selberg Zeta Function for Convex Co-Compact Schottky Groups

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍⁿ⁺¹ : in strips parallel to the imaginary axis the zeta function is bounded by exp(C|s|^δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(C|s|ⁿ⁺¹), and it gives new bounds on the number of resonances (scattering poles) of Γ\Hⁿ⁺¹. The proof of this result is based on the application of holomorphic L²techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider γ\Hⁿ⁺¹ as the simplest model of quantum chaotic scattering.