Spherical-separablility of non-Hermitian Dirac Hamiltonians and pseudo-PT-symmetry

TL;DR

This paper explores the spherical separability of non-Hermitian Dirac Hamiltonians with pseudo-PT symmetry, showing how certain symmetries affect energy eigenvalues and wavefunctions, and proposing methods to preserve quantum numbers.

Contribution

It introduces a framework for analyzing non-Hermitian Dirac Hamiltonians with spherical symmetry and pseudo-PT symmetry, highlighting the roles of descendant Hamiltonians and methods to maintain quantum numbers.

Findings

Conventional relativistic energy eigenvalues are recoverable.

Non-Hermitian Hamiltonians can preserve the magnetic quantum number m.

Changes in wavefunctions occur due to non-Hermitian symmetrization.

Abstract

A non-Hermitian P$_{\phi}$T$_{\phi}$-symmetrized spherically-separable Dirac Hamiltonian is considered. It is observed that the descendant Hamiltonians H$_{r}$, H$_{\theta}$, and H$_{\phi}$ play essential roles and offer some user-feriendly options as to which one (or ones) of them is (or are) non-Hermitian. Considering a P$_{\phi}$T$_{\phi}$-symmetrized H$_{\phi}$, we have shown that the conventional relativistic energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction $V(\theta)$=0 in the descendant Hamiltonian H$_{\theta}$ would manifest a change in the angular $\theta$-dependent part of the general solution too. Whilst some P$_{\phi}$T$_{\phi}$-symmetrized H$_{\phi}$ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the PT-symmetric ones (here the non-Hermitian