An algebraic form of the Marchenko inversion. Partial waves with orbital momentum $l\ge 0$

TL;DR

This paper generalizes the algebraic Marchenko inversion method for fixed angular momentum to any $l$, enabling potential reconstruction from scattering data with high accuracy using a separable kernel expansion.

Contribution

It introduces a new algebraic approach for Marchenko inversion applicable to all orbital angular momenta, simplifying the solution process.

Findings

Kernel expansion coefficients are expressed linearly in terms of scattering data.

The method achieves $h$-step accuracy in potential reconstruction.

Applicable to any orbital angular momentum $l$.

Abstract

We present a generalization of the algebraic method for solving the Marchenko equation (fixed-$l$ inversion) for any values of the orbital angular momentum $l$. We expand the Marchenko equation kernel in a separable form using a triangular wave set. The separable kernel allows a reduction of the equation to a system of linear equations. We obtained a linear expression of the kernel expansion coefficients in terms of the Fourier series coefficients of $q(1-S(q))$ function ($S(q)$ is the scattering matrix) depending on the momentum $q$. The linear expression is valid for any orbital angular momentum $l$. The kernel expansion coefficients are determined by the scattering data in the finite range $0\leq q\leq\pi/h$. In turn, the thus defined Marchenko kernel of the equation allows one to find the potential function of the radial Schr\"odinger equation with $h$-step accuracy.