The Selberg Zeta Function for Convex Co-Compact Schottky Groups

Laurent Guillopé, Kevin K. Lin, Maciej Zworski

Research output: Contribution to journalArticle

60 Scopus citations

Abstract

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍn+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\Hn+1. The proof of this result is based on the application of holomorphic L2techniques 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\Hn+1 as the simplest model of quantum chaotic scattering.

Original languageEnglish (US)
Pages (from-to)149-176
Number of pages28
JournalCommunications in Mathematical Physics
Volume245
Issue number1
DOIs
StatePublished - Feb 2004
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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