Stationary scattering theory on manifolds, II

TL;DR

This paper develops a comprehensive stationary scattering theory for Schrödinger operators on manifolds with ends, including Euclidean and hyperbolic types, deriving asymptotics and solving a conjecture on cross-ends transmission.

Contribution

It extends previous work by fully developing the scattering theory on manifolds with escape functions, including asymptotics and applications to conjectures.

Findings

Derived WKB asymptotics for generalized eigenfunctions

Developed scattering theory for manifolds with Euclidean/hyperbolic ends

Solved a conjecture on cross-ends transmissions

Abstract

Based on our previous study [IS2] we develop fully the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends, possibly with unbounded and non-smooth obstacles. We develop the theory largely along the classical lines [Sa, Co] and derive in particular WKB- asymptotics of appropriate generalized eigenfunctions. As an application we solve a conjecture of [HPW] on cross-ends transmissions in its natural and strong form within the framework of our theory.