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

TL;DR

This paper explores non-Hermitian, P(phi)T(phi)-symmetrized spherically-separable Hamiltonians, revealing how their descendant Hamiltonians influence eigenvalues and wavefunctions, and proposing methods to maintain quantum numbers in pseudo-PT-symmetric systems.

Contribution

It introduces a framework for analyzing P(phi)T(phi)-symmetrized Hamiltonians, showing how to recover traditional energy spectra and maintain quantum numbers in non-Hermitian spherical systems.

Findings

Eigenvalues are recoverable in P(phi)T(phi)-symmetrized Hamiltonians.

Wavefunction components change with angular interactions.

A method to preserve the magnetic quantum number m is proposed.

Abstract

Non-Hermitian but P(phi)T(phi)-symmetrized spherically-separable Dirac and Schrodinger Hamiltonians are 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 Dirac (relativistic) and Schrodinger (non-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) 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 P(phi)T(phi)-symmetric Hamiltonians) are nicknamed as pseudo-PT-symmetric.