Inverted approach to the inverse scattering problem: complete solution of the Marchenko equation for a model system

TL;DR

This paper presents a complete analytical solution to the inverse scattering problem for a Morse potential model, demonstrating the effectiveness of the Marchenko method and unveiling new properties of the potential.

Contribution

It provides a full analytical solution to the inverse scattering problem using the Marchenko equation for a Morse potential, including detailed procedures and new analytic properties.

Findings

Successful implementation of the Marchenko method for the Morse potential

Derivation of an analytic algorithm for phase shift calculation

Discussion of combining Marchenko integral and differential equations

Abstract

An example of full solution of the inverse scattering problem on the half line is presented. For this purpose, a simple analytically solvable model system (Morse potential) is used, which is expected to be a reasonable approximation to a real potential. First one calculates all spectral characteristics for the fixed model system. This way one gets all the necessary input data (otherwise unobtainable) to implement powerful methods of the inverse scattering theory. In this paper, the multi-step procedure to solve the Marchenko integral equation is described in full details. Excellent performance of the method is demonstrated and its combination with the Marchenko differential equation is discussed. In addition to the main results, several important analytic properties of the Morse potential are unveiled. For example, a simple analytic algorithm to calculate the phase shift is derived.