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

Non-Hermitian but PφTφ-symmetrized spherically-separable Dirac and Schrödinger Hamiltonians are considered. It is observed that the descendant Hamiltonians Hr, Hθ, and Hφ play essential roles and offer some ”user-feriendly” options as to which one (or ones) of them is (or are) nonHermitian. Considering a PφTφ-symmetrized Hφ, we have shown that the conventional Dirac (relativistic) and Schrödinger (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 (θ) 6= 0 in the descendant Hamiltonian Hθ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some PφTφ-symmetrized Hφ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamil-