$\mathcal{PT}$ Symmetric Hamiltonian Model and Dirac Equation in 1+1 dimensions

TL;DR

This paper introduces a $ ext{PT}$ symmetric non-Hermitian Hamiltonian model linked to the Dirac equation in 1+1 dimensions, deriving real spectra and complex potentials through analytical and numerical methods.

Contribution

It presents a novel $ ext{PT}$ symmetric Hamiltonian model and connects it to the Dirac equation with position-dependent mass, deriving real spectra for complex potentials.

Findings

Real spectra obtained for complex vector potentials

Hamiltonian mapped to Dirac equation with position-dependent mass

Numerical methods confirmed real energy values for specific parameters

Abstract

In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=\omega (\hat{b}^{\dag}\hat{b}+1/2)+ \alpha (\hat{b}^{2}-(\hat{b}^{\dag})^{2})$ where $\omega$ and $\alpha$ are real constants, $\hat{b}$ and $\hat{b^{\dag}}$ are first order differential operators. The Hermitian form of the Hamiltonian $\mathcal{\hat{H}}$ is obtained by suitable mappings and it is interrelated to the time independent one dimensional Dirac equation in the presence of position dependent mass. Then, Dirac equation is reduced to a Schr\"{o}dinger-like equation and two new complex non-$\mathcal{PT}$ symmetric vector potentials are generated. We have obtained real spectrum for these new complex vector potentials using shape invariance method. We have searched the real energy values using numerical methods for the specific values of the parameters.