#### Vol. 3, No. 4, 2010

 Recent Issues
 The Journal Cover About the Cover Editorial Board Editors’ Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print)
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 ${s}_{0}<0$ be the abscissa of absolute convergence of the dynamical zeta function $Z\left(s\right)$ for several disjoint strictly convex compact obstacles ${K}_{i}\subset {ℝ}^{N}$, $i=1,\dots ,{\kappa }_{0}$, ${\kappa }_{0}\ge 3$, and let

${R}_{\chi }\left(z\right)=\chi {\left(-‘{\Delta }_{D}-{z}^{2}\right)}^{-1}\chi ,\phantom{\rule{1em}{0ex}}\chi \in {C}_{0}^{\infty }\left({ℝ}^{N}\right),$

be the cutoff resolvent of the Dirichlet Laplacian $-‘{\Delta }_{D}$ in the closure of ${ℝ}^{N}\setminus {\bigcup }_{i=1}^{{\kappa }_{0}}{K}_{i}$. We prove that there exists ${\sigma }_{1}<{s}_{0}$ such that the cutoff resolvent ${R}_{\chi }\left(z\right)$ has an analytic continuation for

$Im\phantom{\rule{0.3em}{0ex}}z<-{\sigma }_{1},\phantom{\rule{1em}{0ex}}|Re\phantom{\rule{0.3em}{0ex}}z|\ge {J}_{1}>0.$

##### Keywords
open billiard, periodic rays, zeta function
##### Mathematical Subject Classification 2000
Primary: 35P20, 35P25
Secondary: 37D50