Inverse scattering problems where the potential is not absolutely continuous on the known interior subinterval

TL;DR

This paper proves that for inverse scattering problems with certain potentials, the potential can be uniquely reconstructed from reflection data if it is known on a finite interval and is not absolutely continuous there.

Contribution

It establishes uniqueness of potential reconstruction from reflection coefficients under the condition that the potential is not absolutely continuous on a known interval.

Findings

Potential is uniquely determined by reflection coefficient alone.

Uniqueness holds even when the potential is not absolutely continuous.

Results apply to Schrödinger operators with integrable potentials with finite first moment.

Abstract

The inverse scattering problem for the Schr$\mathrm{\ddot{o}}$dinger operators on the line is considered when the potential is real valued and integrable and has a finite first moment. It is shown that the potential on the line is uniquely determined by the left (or right) reflection coefficient alone provided that the potential is known on a finite interval and it is not absolutely continuous on this known interval.