Spherical-separablility of non-Hermitian Hamiltonians and pseudo- PT -symmetry Omar Mustafa 1 and S.Habib Mazharimousavi 2 Department of Physics, Eastern Mediterranean University, G Magusa, North Cyprus, Mersin 10,Turkey $^{1}$E-mail: omar.mustafa@emu.edu.tr $^{2}$E-mail: habib.mazhari@emu.edu.tr January 5, 2019 Abstract arXiv:0801.3572v2 [quant-ph] 17 Mar 2008 Non-Hermitian but P$_{ϕ}$T$_{ϕ}$ -symmetrized spherically-separable Dirac and Schr¨ odinger Hamiltonians are considered. It is observed that the descen- dant Hamiltonians H$_{r}$ , H$_{θ}$, and H$_{ϕ}$ 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$_{ϕ}$T$_{ϕ}$ -symmetrized H$_{ϕ}$ , we have shown that the conventional Dirac (relativistic) and Schr¨ odinger (non-relativistic) en- ergy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, set- ting a possible interaction V ( θ ) = 0 in the descendant Hamiltonian H$_{θ}$ would manifest a change in the angular θ -dependent part of the general solution too. Whilst some P$_{ϕ}$T$_{ϕ}$ -symmetrized H$_{ϕ}$ Hamiltonians are con- sidered, a recipe to keep the regular magnetic quantum number m , as defined in the regular traditional Hermitian settings, is suggested. Hamil- 1