Approximating the $S$ matrix for solving the Marchenko equation: the case of channels with different thresholds

TL;DR

This paper develops a method to approximate the multi-channel S-matrix in inverse scattering problems, accounting for different channel thresholds and relativistic effects, and demonstrates its effectiveness on theoretical and experimental data.

Contribution

It introduces a rational plus sinc series approximation for the S-matrix in multi-channel inverse scattering, including channels with different thresholds and relativistic kinematics.

Findings

The method accurately reconstructs potentials from simulated data.

It effectively handles channels with different thresholds.

Application to experimental data demonstrates practical utility.

Abstract

This work extends previous results on the inverse scattering problem within the framework of Marchenko theory (fixed-$l$ inversion). In particular, I approximate an $n$-channel $S$-matrix as a function of the first-channel momentum $q$ by a sum of a rational term and a truncated sinc series for each matrix element. Relativistic kinematics are taken into account through the correct momentum-energy relation, and the necessary minor generalization of Marchenko theory is given. For energies where only a subset of scattering channels is open, the analytic structure of the $S$-matrix is analyzed. I demonstrate that the submatrix corresponding to closed channels, particularly near their thresholds, can be reconstructed from the experimentally accessible submatrix of open channels.The convergence of the proposed method is verified by applying it to data generated from a direct solution of the scattering problem for a known potential, and comparing the reconstructed potential with the original one. Finally, the method is applied to the analysis of $S_{31}$ $\pi N$ scattering data.