Resonances and weighted zeta functions for obstacle scattering via smooth models

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 i...

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Bibliographic Details
Published inarXiv.org
Main Authors Delarue, Benjamin, Schütte, Philipp, Weich, Tobias
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 08.06.2022
Subjects
Barriers
Euclidean geometry
Grazing
Hyperbolic systems
Manifolds (mathematics)
Resonance scattering
Riemann manifold
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Summary: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.
ISSN:2331-8422