Scattering for Schr\"{o}dinger operators with conical decay

TL;DR

This paper extends classical short-range scattering theory for Schr"odinger operators to potentials decaying along specific rays, providing a microlocal and spatial characterization of scattering states and their long-time behavior.

Contribution

It introduces a framework for analyzing Schr"odinger operators with potentials decaying along rays, generalizing classical results and offering new microlocal and spatial descriptions of scattering states.

Findings

States decompose into free and interacting parts over time.

Microlocal characterization of scattering states based on dynamics.

Spatial descriptions of scattering states in certain cases.

Abstract

We study the scattering properties of Schr\"{o}dinger operators with potentials that have short-range decay along a collection of rays in $\bbR^d$. This generalizes the classical setting of short-range scattering in which the potential is assumed to decay along \emph{all} rays. For these operators, we show that any state decomposes into an asymptotically free piece and a piece which may interact with the potential for long times. We give a microlocal characterization of the scattering states in terms of the dynamics and a corresponding description of their complement. We also show that in certain cases these characterizations can be purely spatial.