Resonances and weighted zeta functions for obstacle scattering via smooth models

TL;DR

This paper develops a smooth model for obstacle scattering in Riemannian manifolds, enabling the use of hyperbolic dynamics techniques to analyze resonances and zeta functions, with applications to Euclidean convex obstacles.

Contribution

It introduces a smooth model for geodesic billiards with obstacles, allowing the application of open hyperbolic systems techniques to study resonances and zeta functions.

Findings

Meromorphic resolvent for the billiard flow generator

Meromorphic continuation of weighted zeta functions

Explicit residue formulas for scattering by convex obstacles

Abstract

We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.