Position-dependent mass, finite-gap systems, and supersymmetry

TL;DR

This paper explores how position-dependent mass systems can exhibit supersymmetry, including nonlinear types, by using similarity transformations and reduction procedures, leading to finite-gap models related to classical elliptic functions.

Contribution

It introduces a method to generate supersymmetry in PDM systems through similarity transformations and applies it to construct broad classes of finite-gap models via reduction techniques.

Findings

Finite-gap systems with PDM are constructed using reduction procedures.

Supersymmetry can be generated from the kinetic term alone in PDM systems.

Connections to classical elliptic functions and models like Higgs and Mathews-Lakshmanan are established.

Abstract

The ordering problem in quantum systems with position-dependent mass (PDM) is treated by inclusion of the classically fictitious similarity transformation into the kinetic term. This provides a generation of supersymmetry with the first order supercharges from the kinetic term alone, while inclusion of the potential term allows also to generate nonlinear supersymmetry with higher order supercharges. A broad class of finite-gap systems with PDM is obtained by different reduction procedures, and general results on supersymmetry generation are applied to them. We show that elliptic finite-gap systems of Lame and Darboux-Treibich-Verdier types can be obtained by reduction to Seiffert's spherical spiral and Bernoulli lemniscate in the presence of Calogero-like or harmonic oscillator potentials, or by angular momentum reduction of a free motion on some AdS_2-related surfaces in the presence of Aharonov-Bohm flux. The limiting cases include the Higgs and Mathews-Lakshmanan oscillator models as well as a reflectionless model with PDM exploited recently in the discussion of cosmological inflationary scenarios.