Distorted plane waves in chaotic scattering

TL;DR

This paper investigates the behavior of distorted plane waves, also known as Eisenstein functions, in the semiclassical limit on manifolds with hyperbolic classical dynamics near the trapped set, under certain topological conditions.

Contribution

It provides a general analysis of distorted plane waves in a setting including Euclidean manifolds at infinity with hyperbolic dynamics and negative topological pressure.

Findings

Analysis of distorted plane waves in hyperbolic scattering settings

Conditions under which Eisenstein functions exhibit specific asymptotic behaviors

Extension of semiclassical techniques to non-compact manifolds

Abstract

Distorted plane waves, sometimes called Eisenstein functions, are a family of eigenfunctions of a Schr\"odinger operator that are not square integrable. More precisely, they can be written as the sum of a plane wave and an outgoing wave. We shall study distorted plane waves in the semiclassical limit, in a general setting which includes manifolds that are Euclidean near infinity, under the hypothesis that the classical dynamics is hyperbolic close to the trapped set, and that some topological pressure is negative.