Inverse Scattering by the Stability Index Method

TL;DR

The paper introduces the stability index method for inverse scattering problems, providing a quantitative measure of the stability and uniqueness of potential reconstructions from fixed-energy data.

Contribution

It proposes a new stability index concept for inverse scattering, enabling assessment of the stability and uniqueness of solutions based on data.

Findings

The stability index effectively indicates the stability of the inversion.

The method successfully reconstructs potentials from phase shift data.

Practical efficiency demonstrated on model potentials.

Abstract

A novel numerical method for solving inverse scattering problem with fixed-energy data is proposed. The method contains a new important concept: the stability index of the inversion problem. This is a number, computed from the data, which shows how stable the inversion is. If this index is small, then the inversion provides a set of potentials which differ so little, that practically one can represent this set by one potential. If this index is larger than some threshold, then practically one concludes that with the given data the inversion is unstable and the potential cannot be identified uniquely from the data. Inversion of the fixed-energy phase shifts for several model potentials is considered. The results show practical efficiency of the proposed method. The method is of general nature and is applicable to a very wide variety of the inverse problems.