Scattering phase asymptotics with fractal remainders

TL;DR

This paper establishes Weyl type asymptotics for the scattering phase on certain Riemannian manifolds, with remainders influenced by classical dynamical properties, and extends results to hyperbolic quotients using zeta functions.

Contribution

It introduces new asymptotic formulas for the scattering phase with remainders linked to classical escape rates and expansion rates, improving previous results for Axiom A flows.

Findings

Weyl type asymptotics with fractal remainders are proven.

Remainders are bounded by resonance counts near the real axis.

Asymptotics for hyperbolic quotients are derived using the Selberg zeta function.

Abstract

For a Riemannian manifold $(M,g)$ which is isometric to the Euclidean space outside of a compact set, and whose trapped set has Liouville measure zero, we prove Weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate. For Axiom A geodesic flows, this gives a polynomial improvement over the known remainders. We also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis, and provide similar asymptotics for hyperbolic quotients using the Selberg zeta function.