Dynamical zeta functions for billiards

TL;DR

This paper studies the distribution of resonances for billiard problems with convex obstacles, introducing dynamical zeta functions that extend meromorphically and reveal infinite resonances in certain strips.

Contribution

It establishes the meromorphic continuation of dynamical zeta functions for billiards with convex obstacles and analyzes the implications for resonance distribution.

Findings

Meromorphic continuation of zeta functions to the entire complex plane.

Existence of an infinite number of resonances in a specific strip.

Lower bounds for resonances in the case of real analytic boundaries.

Abstract

Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $\mu_j \in {\mathbb C},\: {\rm Im}\: \mu_j > 0,$ be the resonances of the Laplacian in the exterior of $D$ with Neumann or Dirichlet boundary condition on $\partial D$. For $d$ odd, $u(t) = \sum_j e^{i |t| \mu_j}$ is a distribution in $ \mathcal{D}'({\mathbb R} \setminus \{0\})$ and the Laplace transforms of the leading singularities of $u(t)$ yield the dynamical zeta functions $\eta_{\mathrm N},\: \eta_{\mathrm D}$ for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. Under the non-eclipse condition (1.1), for $d \geqslant 2$ we show that $\eta_{\mathrm N}$ and $\eta_\mathrm D$ admit a meromorphic continuation to the whole complex plane. In the particular case when the boundary $\partial D$ is real analytic, by using a result of Fried (1995), we prove that the function $\eta_\mathrm{D}$ cannot be entire. Following the result of Ikawa (1988), this implies the existence of a strip $\{z \in {\mathbb C}: \: 0 < {\rm Im}\: z \leq\alpha\}$ containing an infinite number of resonances $\mu_j$ for the Dirichlet problem. Moreover, for $\alpha \gg 1$ we obtain a lower bound for the resonances lying in this strip.