The Selberg zeta function for convex co-compact Schottky groups

Details The Selberg zeta function for convex co-compact Schottky groups The Selberg zeta function for convex co-compact Schottky groups

2002-11-04

arxiv.org/abs/math/0211041v1

Communications in Mathematical Physics 245 (2004) 149-176

Mathematics

Differential Geometry

Scientific paper

10.1007/s00220-003-1007-1

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $ \exp (C |s|^\delta) $ where $ \delta $ 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 $ \Gamma \backslash {\mathbb H}^{n+1} $. The proof of this result is based on the application of holomorphic $ L^2$-techniques 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 $ \Gamma \backslash {\mathbb H}^{n+1} $ as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets.

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