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 $Z(s)$ for several disjoint strictly convex compact obstacles $K_i \subset \R^N, i = 1,..., κ_0,\: \ka_0 \geq 3,$ and let $R_χ(z) = χ(-Δ_D - z^2)^{-1}χ,\: χ\in C_0^{\infty}(\R^N),$ be the cut-off resolvent of the Dirichlet Laplacian $-Δ_D$ in $Ω= \bar{\R^N \setminus \cup_{i = 1}^{k_0} K_i}$. We prove that there exists $σ_1 < s_0$ such that $Z(s)$ is analytic for $\Re (s) \geq σ_1$ and the cut-off resolvent $R_χ(z)$ has an analytic continuation for $\Im (z) < - σ_1,\: |\Re (z)| \geq C > 0.$