content_image/313681.jpg content_image/313679.jpg content_image/313680.jpg
and H$_{ϕ}$ from (19), (20), and (21), respectively. Moreover, if we substitute U ( r ) = R ( r ) /r in (19) then U (0) = 0 = U ( ∞ ). Yet, whilst Θ (0) and Θ ( π ) should be finite, Φ ( ϕ ) should satisfy the single-valuedness condition Φ ( ϕ ) = Φ ( ϕ + 2 π ). At this point, we argue that the reality of the spectrum of Dirac eigenvalue equation (6) is ensured not only by requiring m, Λ , λ ∈ R but also by requiring R ∋ λ + M 2 = E 2 > 0. With this understanding, we may now seek some PT -symmetrization recipe (be it L´ evai’s [37] regular PT -symmetrization or P ϕ T $_{ϕ}$-symmetrization of Mazharimousavi [36]) for each (at a time) of the descendant Hamiltonians in (26). 

## 3 Consequences of complexified P ϕ T $_{ϕ}$-symmetrized azimuthal Hamiltonians 

The eigenvalue equation in (21) with a ∈ R as a coupling parameter in a complexified-azimuthal effective interaction of the form 

would read 

Hence, a change of variable of the form z = e iϕ/ 2 would result in 

Obviously, equation (29) is the modified Bessel equation with imaginary argu- ment and has two independent solutions. The linear combination of which reads 

<img src='content_image/313681.jpg'>

<img src='content_image/313679.jpg'>

<img src='content_image/313680.jpg'>

