Multi-channel analog of the effective-range expansion

TL;DR

This paper develops a generalized power-series expansion method for the multi-channel Jost-matrix, allowing analysis near any point on the Riemann surface, which aids in locating spectral points like bound and resonant states.

Contribution

It introduces a systematic, semi-analytic approach to expand the multi-channel Jost-matrix near arbitrary points on the Riemann surface, extending the effective-range expansion concept.

Findings

Enables expansion of the Jost-matrix near arbitrary complex energies.

Provides a systematic procedure for calculating expansion coefficients.

Facilitates locating spectral points such as bound and resonant states.

Abstract

Similarly to the standard effective range expansion that is done near the threshold energy, we obtain a generalized power-series expansion of the multi-channel Jost-matrix that can be done near an arbitrary point on the Riemann surface of the energy within the domain of its analyticity. In order to do this, we analytically factorize its momentum dependencies at all the branching points on the Riemann surface. The remaining single-valued matrix functions of the energy are then expanded in the power-series near an arbitrary point in the domain of the complex energy plane where it is analytic. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This means that near an arbitrary point in the domain of physically interesting complex energies it is possible to obtain a semi-analytic expression for the Jost-matrix (and therefore for the S-matrix) and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles.