Quantum mechanical inverse scattering problem at fixed energy: a constructive method

TL;DR

This paper revisits and extends the Cox-Thompson constructive method for solving the inverse scattering problem of the 3D Schrödinger equation at fixed energy, focusing on potential recovery from limited phase shift data.

Contribution

It provides new insights into the asymptotics of potentials, statistical properties of colliding particles, and offers a systematic treatment of the Cox-Thompson method with numerical examples.

Findings

Unique potential recovery from a single phase shift with the Cox-Thompson scheme.

Numerical validation of potential reconstruction from two phase shifts.

Conditions for physically meaningful potentials in the class L_1,1.

Abstract

The inverse scattering problem of the three-dimensional Schroedinger equation is considered at fixed scattering energy with spherically symmetric potentials. The phase shifts determine the potential therefore a constructive scheme for recovering the scattering potential from a finite set of phase shifts at a fixed energy is of interest. Such a scheme is suggested by Cox and Thompson [3] and their method is revisited here. Also some new results are added arising from investigation of asymptotics of potentials and concerning statistics of colliding particles. A condition is given [2] for the construction of potentials belonging to the class L_1,1 which are the physically meaningful ones. An uniqueness theorem is obtained [2] in the special case of one given phase shift by applying the previous condition. It is shown that if only one phase shift is specified for the inversion procedure the unique potential obtained by the Cox-Thompson scheme yields the one specified phase shift while the others are small in a certain sense. The case of two given phase shifts is also discussed by numerical treatment and synthetic examples are given to illustrate the results. Besides the new results this contribution provides a systematic treatment of the CT method.