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

Abstract : 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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Article dans une revue
Communications in Mathematical Physics, Springer Verlag, 2004, 245, pp.149-176. 〈10.1007/s00220-003-1007-1〉
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https://hal.archives-ouvertes.fr/hal-00000036
Contributeur : Laurent Guillopé <>
Soumis le : lundi 4 novembre 2002 - 10:28:04
Dernière modification le : jeudi 11 janvier 2018 - 06:12:18
Document(s) archivé(s) le : lundi 29 mars 2010 - 12:11:12

### Citation

Laurent Guillopé, Kevin Lin, Maciej Zworski. The Selberg zeta function for convex co-compact Schottky groups. Communications in Mathematical Physics, Springer Verlag, 2004, 245, pp.149-176. 〈10.1007/s00220-003-1007-1〉. 〈hal-00000036〉

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