Dirac-isotonic oscillators in (1 + 1) and (2 + 1) dimensions

TL;DR

This paper explores the Dirac oscillator in one and two spatial dimensions, generalizing it with supersymmetric quantum mechanics to connect it to isotonic oscillators and anti-Jaynes-Cummings models, providing exact solutions and non-relativistic limits.

Contribution

It introduces a supersymmetric quantum mechanics-based generalization of the Dirac oscillator in (1+1) and (2+1) dimensions, linking it to isotonic oscillators and anti-Jaynes-Cummings models.

Findings

Exact solutions for generalized Dirac oscillators.

Connection to isotonic oscillators in the non-relativistic limit.

Mapping to anti-Jaynes-Cummings-like Hamiltonian in 2D.

Abstract

We discuss the Dirac oscillator in $(1+1)$ and $(2+1)$ dimensions and generalize it in the spirit of the isotonic oscillator using supersymmetric quantum mechanics. In $(1+1)$ dimensions, the Dirac oscillator returns to the quantum harmonic oscillator in the non-relativistic limit, while its generalization maps to the isotonic oscillator. We describe exact solutions of these generalized systems and also present their non-relativistic limits. Finally, based on supersymmetric quantum mechanics, we show that a generalized Dirac oscillator in $(2+1)$ dimensions can be mapped to an anti-Jaynes-Cummings-like Hamiltonian in which the spin operators couple with the supercharges.