Inverted Dirac oscillator

The Dirac oscillator is obtained from the Dirac Hamiltonian $H^{\mathrm{D}}=(c\vec{\alpha }\cdot \vec{p}+m{c}^2\beta )$ by modifying the momentum through a non-Hermitian substitution $\overrightarrow{p}\to \overrightarrow{p}\pm i\omega \beta \overrightarrow{q}$. Despite the non-Hermitian nature of this momentum operator, the full Hamiltonian remains Hermitian due to the presence of the Dirac matrix $\vec{\alpha }$. However, if one instead introduces a Hermitian modification of the form $\vec{p}\to \vec{p}\pm \omega \beta \overrightarrow{q}$, the resulting Hamiltonian is no longer Hermitian. In this case, the system corresponds to an inverted Dirac oscillator $H^{\mathrm{r}}$, where the potential becomes unbounded from below, the energy spectrum becomes continuous, and the eigenfunctions fail to be square-integrable, leading to normalization difficulties. We show that the Hamiltonian $H^{\mathrm{r}}$ is a pseudo-$\mathcal{P}\mathcal{T}$-symmetric operator, and we introduce an unbounded, non-unitary transformation that establishes a connection between $H^{\mathrm{r}}$ and $H^{\mathrm{D}}$. The purpose of this work is to analyze this relativistic quantum system – known as the Dirac inverted oscillator – which, despite its various applications, admits an exact analytical solution