Supersymmetric Dirac-Hamiltonians in $(1+1)$ dimensions revisited

TL;DR

This paper explores supersymmetric structures in (1+1)-dimensional Dirac Hamiltonians, revealing scalar or pseudo-scalar potentials and connecting their spectral properties to the non-relativistic Witten model, with applications to the Dirac oscillator.

Contribution

It provides a comprehensive analysis of supersymmetric Dirac Hamiltonians in (1+1) dimensions, including spectral properties, resolvents, and explicit solutions for the Dirac oscillator.

Findings

Supersymmetry permits scalar or pseudo-scalar potentials in Dirac Hamiltonians.

Spectral properties align with those of the non-relativistic Witten model.

Closed-form solutions for the Dirac oscillator's resolvent are derived.

Abstract

The most general Dirac Hamiltonians in $(1+1)$ dimensions are revisited under the requirement to exhibit a supersymmetric structure. It is found that supersymmetry allows either for a scalar or a pseudo-scalar potential. Their spectral properties are shown to be represented by those of the associated non-relativistic Witten model. The general discussion is extended to include the corresponding relativistic and non-relativistic resolvents. As example the well-studied relativistic Dirac oscillator is considered and the associated resolved kernel is found in a closed form expression by utilising the energy-dependent Green's function of the non-relativistic harmonic oscillator. The supersymmetric quasi-classical approximation for the Witten model is extended to the associated relativistic model.