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 $R_{\chi }(z)=\chi (-{\Delta }_D-z^2{)}^{-1}\chi ,\chi \in C_0^{\infty }({\mathbb{R}}^N),$ be the cut-off resolvent of the Dirichlet Laplacian $-{\Delta }_D$ in $\Omega =\overline{{\mathbb{R}}^N\setminus {\cup }_{i=1}^{k_0}{K}_i}$. We prove that there exists ${\sigma }_1<s_0$ such that $Z(s)$ is analytic for $\Re (s)\ge {\sigma }_1$ and the cut-off resolvent $R_{\chi }(z)$ has an analytic continuation for $\Im (z)<-{\sigma }_1,|\Re (z)|\ge C>0.$

References in corpus (3)

• Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems
• Numerical Study of Quantum Resonances in Chaotic Scattering
• Wave decay on convex co-compact hyperbolic manifolds