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

TL;DR

The paper addresses the inverse scattering problem in a multichannel setting with different thresholds by extending fixed-$l$ Marchenko inversion to an $n$-channel $S$-matrix $S(q)$ and approximating each element as a rational term plus a truncated sinc correction to avoid unphysical poles. It incorporates relativistic kinematics through a mass-operator formulation, derives a Schrödinger-like multichannel framework, and analyzes the analytic structure across thresholds to relate open- and closed-channel data. A practical, separable-input-kernel approach is developed by combining a threshold-free rational $S^{(0)}(q)$ with a sinc-based correction, enabling stable reconstruction of the potential from $S$-matrix data for coupled channels. The method is validated on a two-channel test problem and applied to $ ext{πN}$ scattering data (S31), demonstrating convergent potentials and the potential for extrapolation/interpolation across thresholds, while highlighting areas for extension to more channels and three-body dynamics.

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 $S_{31}$ $πN$ scattering data.

Figures (7)

• Figure 1: The real parts of the $S$-matrix elements used to reconstruct the test potential $V_{1}(r)=V_{2}(r)=V_{12}(r)=V_{0}e^{-ar^{2}}$, with $V_{0}=-1.5\ \text{fm}^{-2}$ and $a=2\ \text{fm}^{-2}$, are shown. The solid thin curves were obtained from the direct solution of Eq. (\ref{['system2by2']}), while the thick dashed curves represent the approximation $S(q)=S^{(0)}(q)+\Delta S(q)$, and the dotted curves correspond to $S^{(0)}(q)$ alone.
• Figure 2: Continuation of Fig. \ref{['fig:testdataRe']} for imaginary parts of the $S$-matrix elements.
• Figure 3: $S$ matrix parameters $\gamma$ (bellow $W_2$) and $\sin(2\epsilon)$ (above $W_2$). Solid curves were calculated from direct solution of Eq. (\ref{['system2by2']}), while dashed curves represent approximation Eq. (\ref{['final_Smatrix_ap']}).
• Figure 4: Parameter $b$ defined from Eq. (\ref{['gamma_at_thres_left']}) bellow $W_2$ and from Eq. (\ref{['gamma_at_thres_right']}) above $W_2$. Solid curves were calculated from direct solution of Eq. \ref{['system2by2']}, while dashed curves represents approximation Eq. (\ref{['final_Smatrix_ap']}).
• Figure 5: Results of the inversion for the $S$-matrix fit presented in Figs. \ref{['fig:testdataRe']}-\ref{['fig:exp_gamma_koef']}. The dashed line corresponds to the input potential, and solid curves are results of the inversion.