Dirac systems with magnetic field and position dependent mass: Darboux transformations and equivalence with generalized Dirac oscillators

TL;DR

This paper develops Darboux transformations for two-dimensional Dirac systems with position-dependent mass and potential, establishing equivalences with generalized Dirac oscillators and providing exactly solvable models at zero energy.

Contribution

It introduces a Darboux transformation framework for Dirac systems with variable mass and potential, linking them to generalized Dirac oscillators and enabling exact solutions.

Findings

Constructed Darboux transformations for zero-energy Dirac systems.

Derived exactly solvable Dirac equations with known zero-energy solutions.

Established equivalence between Dirac systems with magnetic field and generalized Dirac oscillators.

Abstract

We construct a Darboux transformation for a class of two-dimensional Dirac systems at zero energy. Our starting equation features a position-dependent mass, a matrix potential, and an additional degree of freedom that can be interpreted either as a magnetic field perpendicular to the plane or a generalized Dirac oscillator interaction. We obtain a number of Darbouxtransformed Dirac equations for which the zero energy solutions are exactly known.