Inverse scattering on the half-line revisited

TL;DR

This paper revisits the inverse scattering problem on the half-line, providing new proofs and conditions for the scattering data to correspond to a suitable potential, without relying on negative half-line equations.

Contribution

It introduces a novel proof and a new set of necessary and sufficient conditions for the inverse scattering problem on the half-line.

Findings

New proof of invertibility of the inversion steps

Necessary and sufficient conditions for scattering data

Avoids use of equations on the negative half-line

Abstract

The inverse scattering problem on the half-line has been studied in the literature in detail. V. Marchenko presented the solution to this problem. In this paper, the invertibility of the steps of the inversion procedure is discussed and a new set of necessary and sufficient conditions on the scattering data is given for the scattering data to be generated by a potential suitable potential. Our proof is new and in contrast with known proof it does not use equations on the negative half-line.