The Selberg zeta function for convex co-compact Schottky groups
Résumé
We give a new upper bound on the Selberg zeta function for a convexco-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in stripsparallel to the imaginary axis the zeta function is bounded by $ \exp ( C|s|^\delta ) $ where $ \delta $ is the dimension of the limit set of thegroup. 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 proofof this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transferoperators and on the quasi-self-similarity of limit sets. We also studythis problem numerically and provide evidence that the bound may beoptimal. Our motivation comes from molecular dynamics and we consider $\Gamma \backslash {\mathbb H}^{n+1} $ as the simplest model of quantumchaotic scattering. The proof of this result is based on the applicationof holomorphic $L^2$-techniques to the study of the determinants of theRuelle transfer operators and on the quasi-self-similarity of limitsets.
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