Energy-independent optical $^{1}S_{0}NN$ potential from Marchenko equation

TL;DR

This paper introduces a new algebraic method based on Marchenko theory to reconstruct energy-independent optical potentials from scattering data, demonstrated on nucleon-nucleon interactions up to 3 GeV.

Contribution

A novel algebraic approach using a triangular wave set for solving the inverse scattering problem with Marchenko theory, enabling potential reconstruction from scattering data.

Findings

Successfully reconstructed energy-independent optical potential for $^{1}S_{0}NN$ data.

Demonstrated the method's accuracy with a finite momentum range Fourier series.

Applied the method to nucleon-nucleon scattering data up to 3 GeV.

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 in the finite range of $q$. It is shown that a Fourier series on a finite momentum range ($0<q<\pi/h$) of a $q(1-S)$ function ($S$ is the scattering matrix) defines the potential function of the corresponding radial Schr\"odinger equation with $h$-step accuracy. A numerical algorithm is developed for the reconstruction of the optical potential from scattering data. The developed procedure is applied to analyze the $^{1}S_{0}NN$ data up to 3 GeV. It is shown that these data are described by optical energy-independent partial potential.