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

Tunneling dynamics in exactly solvable models with triple-well potentials

“…Since such a non-monotonic dependence was obtained in the framework of supersymmetric quantum mechanics, it is potentially interesting to figure out this phenomenon in other supersymmetric systems, e.g., in [55].…”

“…In this paper, we have developed the previously proposed approach [36], based on exactly solvable quantum-mechanical models with multi-well potentials and the corresponding exact Specifying the initial Hamiltonian to that of the harmonic oscillator model, we applied 4 It is noteworthy to mention other recent applications of SQM to condensed matter physics problems [51], [52], [53], [54], [55]. 1 .…”

Soliton defects in one-gap periodic system and exotic supersymmetry

“…The same transformations provide an effective dressing method for construction of Lax-Novikov integrals for these systems. The condition of conservation of them generates the higher order nonlinear stationary equations for the KdV and mKdV hierarchies [28,29,30,31]. This picture also applies for a more general case of Zakharov-Shabat -Ablowitz-Kaup-Newell-Segur hierarchy [32].…”

“…At last, one can construct a reflectionless system completely isospectral to the system (2.7) but with the displaced potential ('soliton center'). Having at hands the building blocks corresponding to the described three possibilities, by the appropriate generalisation of the procedure, we can construct reflectionless system with arbitrary number of bound states and arbitrary positions of the corresponding soliton centres [30,29].…”

“…It is generated by two pairs of the supercharges, which are the 2 × 2 matrix differential operators of the odd and even orders. In addition, the exotic supersymmetric structure includes two bosonic generators composed from the Lax-Novikov integrals of subsystems, which are differential operators of higher odd-order [29,30].…”