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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics