HAL : hal-00000036, version 1

arXiv : math.DG/0211041

DOI : 10.1007/s00220-003-1007-1

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Communications in Mathematical Physics 245 (2004) 149-176

The Selberg zeta function for convex co-compact Schottky groups

Laurent Guillopé 1, Kevin Lin 2, Maciej Zworski 2

(2004)

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.

1 :	Laboratoire de Mathématiques Jean Leray (LMJL)
	CNRS : UMR6629 – Université de Nantes – École Centrale de Nantes
2 :	Mathematics Department
	University of California, Berkeley

Domaine

:

Mathématiques/Géométrie différentielle Mathématiques/Théorie spectrale Mathématiques/Physique mathématique Physique/Physique mathématique

Mots Clés : Selberg zeta function – Schottky group – limit set – Hausdorff dimension – Ruelle operator – Fredholm determinant – quasi-similarity – Markov partition – resonance

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Contributeur : Laurent Guillopé <>

Soumis le : Lundi 4 Novembre 2002, 10:28:04

Dernière modification le : Dimanche 21 Juin 2009, 16:52:34