Analyzing the contribution of individual resonance poles of the S-matrix to the two-channel scattering

TL;DR

This paper presents a method based on differential equations to analyze individual resonance poles of the S-matrix in two-channel scattering, enabling precise calculation of resonance parameters and their contributions to scattering cross sections.

Contribution

It introduces a differential equation approach for directly calculating the Jost matrix and resonance parameters, allowing detailed analysis of individual resonance contributions in two-channel scattering.

Findings

Accurate calculation of total and partial resonance widths.

Effective analysis of individual resonance pole contributions.

Poles far from the real axis can significantly influence scattering.

Abstract

A two-channel problem is considered within a method based on first order differential equations that are equivalent to the corresponding Schr\"odinger equation but are more convenient for dealing with resonant phenomena. Using these equations, it is possible to directly calculate the Jost matrix for practically any complex value of the energy. The spectral points (bound and resonant states) can therefore be located in a rigorous way, namely, as zeros of the Jost matrix determinant. When calculating the Jost matrix, the differential equations are solved and thus, at the same time, the wave function is obtained with the correct asymptotic behavior that is embedded in the solution analytically. The method offers very accurate way of calculating not only total widths of resonances but their partial widths as well. For each pole of the S-matrix, its residue can be calculated rather accurately, which makes it possible to obtain the Mittag-Leffler type expansion of the S-matrix as a sum of the singular terms (representing the resonances) and the background term (contour integral). As an example, the two-channel model by Noro and Taylor is considered. It is demonstrated how the contributions of individual resonance poles to the scattering cross section can be analyzed using the Mittag-Leffler expansion and the Argand plot technique. This example shows that even poles situated far away from the physical real axis may give significant contributions to the cross section.