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

TL;DR

This paper proves the analytic continuation of the dynamical zeta function and the cut-off resolvent of the Laplacian for multiple convex obstacles, extending their domains beyond initial convergence regions.

Contribution

It establishes the existence of a broader domain where the dynamical zeta function and the resolvent are analytic, linking spectral properties with dynamical zeta functions in obstacle scattering.

Findings

Z(s) is analytic for Re(s) ≥ σ₁ < s₀

R_χ(z) has an analytic continuation for Im(z) < -σ₁

Extension of analyticity domains for resolvent and 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 $K_i \subset \R^N, i = 1,..., \kappa_0,\: \ka_0 \geq 3,$ and let $R_{\chi}(z) = \chi (-\Delta_D - z^2)^{-1}\chi,\: \chi \in C_0^{\infty}(\R^N),$ be the cut-off resolvent of the Dirichlet Laplacian $-\Delta_D$ in $\Omega = \bar{\R^N \setminus \cup_{i = 1}^{k_0} K_i}$. We prove that there exists $\sigma_1 < s_0$ such that $Z(s)$ is analytic for $\Re (s) \geq \sigma_1$ and the cut-off resolvent $R_{\chi}(z)$ has an analytic continuation for $\Im (z) < - \sigma_1,\: |\Re (z)| \geq C > 0.$