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 ${s}_{0}<0$ 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