An approximation of the $S$ matrix for solving the Marchenko equation

TL;DR

This paper introduces a novel approximation method for the S-matrix as a sum of rational and Sinc series components, enabling efficient solution of the Marchenko equation with high accuracy for various scattering data.

Contribution

The paper presents a new approximation of the S-matrix that simplifies the Marchenko equation into a linear system, improving computational efficiency and accuracy in inverse scattering problems.

Findings

Good convergence demonstrated numerically

Applicable to both unitary and non-unitary S matrices

Validated against an exactly solvable model

Abstract

I present a new approximation of the $S$-matrix dependence on momentum $q$, formulated as a sum of a rational function and a truncated Sinc series. This approach enables pointwise determination of the $S$ matrix with specified resolution, capturing essential features such as resonance behavior with high accuracy. The resulting approximation provides a separable kernel for the Marchenko equation (fixed-$l$ inversion), reducing it to a system of linear equations for the expansion coefficients of the output kernel. Numerical results demonstrate good convergence of this method, applicable to both unitary and non-unitary $S$ matrices. Convergence is further validated through comparisons with an exactly solvable square-well potential model. The method is applied to analyze $S_{31}$ $\pi N$ scattering data.