An algebraic method for solving the inverse problem of quantum scattering theory

TL;DR

This paper introduces an algebraic approach leveraging Marchenko theory and a triangular wave basis to efficiently solve the inverse quantum scattering problem by reducing it to linear equations.

Contribution

It presents a novel algebraic method using a triangular wave basis for the Marchenko equation, simplifying the inverse scattering problem into a linear system.

Findings

Reduction of the Marchenko equation to linear equations

Explicit linear expression for kernel coefficients at zero angular momentum

Application of Fourier series coefficients in the solution

Abstract

We present a new algebraic method for solving the inverse problem of quantum scattering theory based on the Marchenko theory. We applied a triangular wave set for the Marchenko equation kernel expansion in a separable form. The separable form allows a reduction of the Marchenko equation to a system of linear equations. For the zero orbital angular momentum, a linear expression of the kernel expansion coefficients is obtained in terms of the Fourier series coefficients of a function depending on the momentum q and determined by the scattering data on the finite range of q.