Spectral properties of non-conservative multichannel SUSY partners of the zero potential

TL;DR

This paper investigates the spectral characteristics of non-conservative multichannel SUSY partner potentials derived from zero potential systems, revealing the structure of their spectral zeros, bound states, and resonances, with a focus on the inverse spectral problem.

Contribution

It provides a detailed analysis of the spectral zeros, bound states, and resonances of multichannel SUSY partner potentials, including a perturbation approach and inverse spectral problem solutions for the 2x2 case.

Findings

The Jost-matrix determinant has N2^{N-1} zeros, potentially all virtual states.

The number of bound states ranges from 0 to N.

Maximum resonances count is (N-1)2^{N-2}.

Abstract

Spectral properties of a coupled $N \times N$ potential model obtained with the help of a single non-conservative supersymmetric (SUSY) transformation starting from a system of $N$ radial Schr\"odinger equations with the zero potential and finite threshold differences between the channels are studied. The structure of the system of polynomial equations which determine the zeros of the Jost-matrix determinant is analyzed. In particular, we show that the Jost-matrix determinant has $N2^{N-1}$ zeros which may all correspond to virtual states. The number of bound states satisfies $0\leq n_b\leq N$. The maximal number of resonances is $n_r=(N-1)2^{N-2}$. A perturbation technique for a small coupling approximation is developed. A detailed study of the inverse spectral problem is given for the $2\times 2$ case.