An improved algorithm for reconstructing reflectionless potentials

TL;DR

This paper introduces a new algebraic method using determinants to efficiently reconstruct one-dimensional reflectionless potentials, simplifying the process especially for symmetric cases, and demonstrates its effectiveness through examples.

Contribution

The paper presents a novel algebraic formula for reconstructing reflectionless potentials, leveraging properties of determinants and simplifying the process for symmetric cases.

Findings

The method provides a simple, algebraic formula for potential reconstruction.

The approach is effective for symmetric reflectionless potentials.

Examples demonstrate the efficiency and applicability of the method.

Abstract

A fully algebraic approach to reconstructing one-dimensional reflectionless potentials is described. A simple and easily applicable general formula is derived, using the methods of the theory of determinants. In particular, useful properties of special determinants - the alternants - have been exploited. The main formula takes an especially simple form if one aims to reconstruct a symmetric reflectionless potential. Several examples are presented to illustrate the efficiency of the method.