Coupling between scattering channels with SUSY transformations for equal thresholds

TL;DR

This paper investigates how supersymmetric transformations can couple scattering channels in multi-channel quantum systems with equal thresholds, providing methods to invert scattering data and analyze bound states.

Contribution

It introduces a SUSY transformation framework that couples scattering channels starting from uncoupled potentials, with detailed analysis and examples for the 2x2 case.

Findings

SUSY transformations can induce channel coupling in scattering potentials.

Bound and virtual states with degeneracy are introduced at the factorization energy.

The method allows inversion of coupled-channel scattering data using SUSY transformations.

Abstract

Supersymmetric (SUSY) transformations of the multi-channel Schr\"odinger equation with equal thresholds and arbitrary partial waves in all channels are studied. The structures of the transformation function and the superpotential are analyzed. Relations between Jost and scattering matrices of superpartner potentials are obtained. In particular, we show that a special type of SUSY transformation allows us to introduce a coupling between scattering channels starting from a potential with an uncoupled scattering matrix. The possibility for this coupling to be trivial is discussed. We show that the transformation introduces bound and virtual states with a definite degeneracy at the factorization energy. A detailed study of the potential and scattering matrices is given for the $2\times 2$ case. The possibility of inverting coupled-channel scattering data by such a SUSY transformation is demonstrated by several examples ($s-s$, $s-p$ and $s-d$ partial waves).