Inverse scattering on the line with incomplete scattering data

TL;DR

This paper investigates the inverse scattering problem for the Schrödinger equation on the line with compactly supported potentials, analyzing nonuniqueness and providing conditions for unique potential recovery from incomplete scattering data.

Contribution

It characterizes the nonuniqueness in potential recovery from a specific ratio of scattering data and shows how additional norm estimates can ensure unique identification of the potential.

Findings

Finite set of potentials corresponds to given data

L^2-norms of potentials are related and can be used for identification

Explicit examples illustrate the potential recovery process

Abstract

The Schroedinger equation is considered on the line when the potential is real valued, compactly supported, and square integrable. The nonuniqueness is analyzed in the recovery of such a potential from the data consisting of the ratio of a corresponding reflection coefficient to the transmission coefficient. It is shown that there are a discrete number of potentials corresponding to the data and that their L^2-norms are related to each other in a simple manner. All those potentials are identified, and it is shown how an additional estimate on the L^2-norm in the data can uniquely identify the corresponding potential. The recovery is illustrated with some explicit examples.