Stable identification of piecewise-constant potentials from fixed-energy phase shifts

TL;DR

This paper investigates the stability of identifying spherically symmetric potentials from fixed-energy phase shifts, introduces a quantitative instability measure, and proposes a numerical method for potential recovery, analyzing its performance with noisy data.

Contribution

It introduces a quantitative measure of instability for potential identification from phase shifts and develops a combined global-local search method for numerical recovery.

Findings

The instability measure D(k) quantifies the ill-posedness of the inverse problem.

The IRRS method effectively reconstructs potentials from noiseless and noisy data.

Stability varies with energy level k, influencing reconstruction accuracy.

Abstract

An identification of a spherically symmetric potential by its phase shifts is an important physical problem. Recent theoretical results assure that such a potential is uniquely defined by a sufficiently large subset of its phase shifts at any one fixed energy level. However, two different potentials can produce almost identical phase shifts. That is, the inverse problem of the identification of a potential from its phase shifts at one energy level $k^2$ is ill-posed, and the reconstruction is unstable. In this paper we introduce a quantitative measure $D(k)$ of this instability. The diameters of minimizing sets $D(k)$ are used to study the change in the stability with the change of $k$, and the influence of noise on the identification. They are also used in the stopping criterion for the nonlinear minimization method IRRS (Iterative Random Reduced Search). IRRS combines probabilistic global and deterministic local search methods and it is used for the numerical recovery of the potential by the set of its phase shifts. The results of the identification for noiseless as well as noise corrupted data are presented.