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

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

$$R_{\chi }\left(z\right)=\chi {\left(-‘{\Delta }_D-z^2\right)}^{-1}\chi ,\chi \in C_0^{\infty }\left({ℝ}^N\right),$$

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

$$Imz<-{\sigma }_1,|Rez|\ge J_1>0.$$