probabilistic interpretations. This is due to the replacement of the regular spherical harmonics Y$_{ℓm}$ ( θ, ϕ ) part (for the radially symmetric 3D-Hamitonians) by the new P ϕ T $_{ϕ}$-symmetric part P m ℓ (cos θ )Φ$_{m}$ ( ϕ ) (defined above for our P ϕ T $_{ϕ}$- symmetric non-Hermitian Hamiltonian model). To see the effect of such a P ϕ T $_{ϕ}$-symmetrization on the probability density, we consider a radial Coulombic effective interaction V$_{eff}$ ( r ) = − 1 /r accom- panied by an azimuthal effective interaction V$_{eff}$ ( ϕ ) = − a $^{2}$e $^{iϕ}$(an illustrative example of fundamental nature). In figures 1 and 2 we plot the correspond- ing probability densities at different values of the coupling parameter a for the principle quantum numbers n = 1 and n = 2 for ℓ = 0 = m . It is clearly observed that whilst the probability density for small a imitates the Hermi- tian ϕ -independent probability density trends, it shifts and intensifies about | ϕ | = 0 as a increases (indicating that the corresponding state is more localized, therefore). In this case, of course, the rotational symmetry of a purely ”just- radially-symmetric” Coulombic interaction breaks down as a result of V$_{eff}$ ( ϕ ) . Taking V ( θ ) = 1 / 2 in (23) would imply where ˜ m = √$_{E}$ + M + m 2 for Dirac and ˜ m = √ 1 / 2 + m 2 for Schr¨ odinger settings. Similar equation was reported by Dutra and Hott [47]. The regular solution of which can (taking α = β = 0 and γ = 1 in equation (12) of ref. [47] to match with our settings) very well be copied and pasted to read (for Dirac 11 3.2 Consequences of V ( θ ) = 1 / 2 or V ( θ ) = 1 / (2 cos 2 θ ) in (23) [ 1 sin θ d dθ ( sin θ d dθ ) − ˜ m 2 sin 2 θ + Λ ] Θ ( θ ) = 0 , (37)