Vol. 3, No. 4, 2010

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Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Vesselin Petkov and Luchezar Stoyanov

Vol. 3 (2010), No. 4, 427–489
Abstract

Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki N, i = 1,,κ0, κ0 3, and let

Rχ(z) = χ(ΔD z2)1χ,χ C 0(N),

be the cutoff resolvent of the Dirichlet Laplacian ΔD in the closure of N i=1κ0Ki. We prove that there exists σ1 < s0 such that the cutoff resolvent Rχ(z) has an analytic continuation for

Imz < σ1,|Rez| J1 > 0.

Keywords
open billiard, periodic rays, zeta function
Mathematical Subject Classification 2000
Primary: 35P20, 35P25
Secondary: 37D50
Milestones
Received: 30 March 2009
Revised: 20 February 2010
Accepted: 10 March 2010
Published: 8 September 2010
Authors
Vesselin Petkov
Université Bordeaux I
Institut de Mathématiques de Bordeaux
351, Cours de la Libération
33405 Talence
France
http://www.math.u-bordeaux1.fr/~petkov
Luchezar Stoyanov
School of Mathematics and Statistics
University of Western Australia
Perth, WA 6009
Australia
http://school.maths.uwa.edu.au/~stoyanov