Propagation property and its application to inverse scattering for fractional powers of the negative Laplacian

TL;DR

This paper extends the Enss--Weder propagation estimate to fractional powers of the negative Laplacian and demonstrates its application in uniquely determining short-range interactions in inverse scattering.

Contribution

It introduces a new propagation estimate for fractional Laplacians and applies it to inverse scattering, generalizing previous methods for the standard Laplacian.

Findings

High-velocity limit of scattering operator determines short-range interactions

Extension of propagation estimates to fractional Laplacians

Application to inverse scattering problems

Abstract

Enss (1983) proved a propagation estimate for the usual free Schroedinger operator that turned out later to be very useful for inverse scattering in the work of Enss--Weder (1995). Since then, this method has been called the Enss--Weder time-dependent method. We study the same type of propagation estimate for the fractional powers of the negative Laplacian and, as with the Enss--Weder method, we apply our estimate to inverse scattering. We find that the high-velocity limit of the scattering operator uniquely determines the short-range interactions.