Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems

TL;DR

This paper explores how scalings and translations affect the supersymmetric quantum structure of soliton systems, revealing a complex algebraic framework that simplifies under certain parameter coincidences and relates to integrable hierarchies.

Contribution

It uncovers the detailed supersymmetry structure of reflectionless quantum systems with multiple parameters and describes how this structure reduces or transforms under specific parameter conditions.

Findings

Superalgebra generated by differential supercharges of orders 2n and 2n+1.

Reduction of superalgebra order when energy levels coincide.

Identification of a special isospectral case linked to finite-gap Bogoliubov-de Gennes systems.

Abstract

We investigate a peculiar supersymmetry of the pairs of reflectionless quantum mechanical systems described by n-soliton potentials of a general form that depends on n scaling and n translation parameters. We show that if all the discrete energy levels of the subsystems are different, the superalgebra, being insensitive to translation parameters, is generated by two supercharges of differential order 2n, two supercharges of order 2n+1, and two bosonic integrals of order 2n+1 composed from Lax integrals of the partners. The exotic supersymmetry undergoes a reduction when r discrete energy levels of one subsystem coincide with any r discrete levels of the partner, the total order of the two independent intertwining generators reduces then to 4n-2r+1, and the nonlinear superalgebraic structure acquires a dependence on r relative translations. For a complete pairwise coincidence of the scaling parameters which control the energies of the bound states and the transmission scattering amplitudes, the emerging isospectrality is detected by a transmutation of one of the Lax integrals into a bosonic central charge. Within the isospectral class, we reveal a special case giving a new family of finite-gap first order Bogoliubov-de Gennes systems related to the AKNS integrable hierarchy.