### Abstract

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍ^{n+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\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 γ\H^{n+1} as the simplest model of quantum chaotic scattering.

Original language | English (US) |
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Pages (from-to) | 149-176 |

Number of pages | 28 |

Journal | Communications in Mathematical Physics |

Volume | 245 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2004 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications in Mathematical Physics*,

*245*(1), 149-176. https://doi.org/10.1007/s00220-003-1007-1