Exactly-solvable coupled-channel potential models of atom-atom magnetic Feshbach resonances from supersymmetric quantum mechanics

TL;DR

This paper develops exactly-solvable coupled-channel potential models for atom-atom Feshbach resonances using supersymmetric quantum mechanics, providing analytical forms and inverse problem solutions relevant for cold atom physics.

Contribution

It introduces a new class of exactly-solvable coupled-channel potentials derived via supersymmetric transformations, with explicit parameterization and applications to Feshbach resonances.

Findings

Potential models depend on a finite set of parameters including bound states and resonances.

The models can reproduce large background scattering lengths near Feshbach resonances.

Explicit inverse solutions relate potential parameters to physical scattering properties.

Abstract

Starting from a system of $N$ radial Schr\"odinger equations with a vanishing potential and finite threshold differences between the channels, a coupled $N \times N$ exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on $N (N+1)/2$ unconstrained parameters and on one upper-bounded parameter, the factorization energy. A detailed study of the model is done for the $2\times 2$ case: a geometrical analysis of the zeros of the Jost-matrix determinant shows that the model has 0, 1 or 2 bound states, and 0 or 1 resonance; the potential parameters are explicitly expressed in terms of its bound-state energies, of its resonance energy and width, or of the open-channel scattering length, which solves schematic inverse problems. As a first physical application, exactly-solvable $2\times 2$ atom-atom interaction potentials are constructed, for cases where a magnetic Feshbach resonance interplays with a bound or virtual state close to threshold, which results in a large background scattering length.