The Selberg zeta function for convex co-compact Schottky groups

Abstract

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.

References in corpus (1)

• Fractal Weyl laws for chaotic open systems

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