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

Petkov, Vesselin; Stoyanov, Luchezar

doi:10.2140/apde.2010.3.427

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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

DOI: 10.2140/apde.2010.3.427

Abstract

Let $s_{0} < 0$ be the abscissa of absolute convergence of the dynamical zeta function $Z (s)$ for several disjoint strictly convex compact obstacles $K_{i} \subset ℝ^{N}$ , $i = 1, \dots, κ_{0}$ , $κ_{0} \geq 3$ , and let

R_{χ} (z) = χ {(- ‘ Δ_{D} - z^{2})}^{- 1} χ, χ \in C_{0}^{\infty} (ℝ^{N}),

be the cutoff resolvent of the Dirichlet Laplacian $- ‘ Δ_{D}$ in the closure of $ℝ^{N} ∖ ⋃_{i = 1}^{κ_{0}} K_{i}$ . We prove that there exists $σ_{1} < s_{0}$ such that the cutoff resolvent $R_{χ} (z)$ has an analytic continuation for

Im z < - σ_{1}, | Re z | \geq J_{1} > 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

Vol. 3, No. 4, 2010