with M ( ρ, ϕ, z ) = M ( ρ ) = ρ − 2 and subjected to an interaction potential of the form would admit exact energy eigenvalues given by Obviously, the condition ( √$_{D/ϵ}$ − ˜ n$_{z}$ − 1 2 ) > 0 is manifested here and ought to be enforced, otherwise complex pairs of energy eigenvalues are obtained in the process. Moreover, a quantum particle with M ( ρ, ϕ, z ) = M ( ρ ) = ρ − 2 subjected to an interaction potential would indulge the exact energy eigenvalues We may now consider a PT -symmetrized ˜ V ( z ) Scarf II in (18) so that where the corresponding Hamiltonian is known to be a non-Hermitian PT - symmetric Hamiltonian that admits exact eigenvalues (cf., e.g., Mustafa and 11 3.3 P T -symmetrized ˜ V ( z ) spectral signatures ˜ V ( z ) = − 3 + A 2 4 cosh $^{2}$z − i A sinh z cosh $^{2}$z , (37) E$_{n}$$_{ρ}$$_{,m,}$ ˜ n$_{z}$ = ( m 2 + 3 2 ) − ( ζ − β ) − 1 2 ( 1 a [ √$_{D}$ ϵ − ˜ n$_{z}$ − 1 2 ] + 2 n$_{ρ}$ + 1 ) 2 . (36) V ( ρ, ϕ, z ) = a $^{2}$ρ$^{4}$/ 4 + Dρ 2 ( e − 2 ϵz − 2 e − $^{ϵz}$) ; D > 0 , (35) E$_{n}$$_{ρ}$$_{,m,}$ ˜ n$_{z}$ = ( m 2 + 3 2 ) − ( ζ − β ) − 1 2   1 √ √$_{D}$ ϵ − ˜ n$_{z}$ − 1 2 − n$_{ρ}$ − 1  $^{2}$ . (34) V ( ρ, ϕ, z ) = − 2 ρ + Dρ 2 ( e − 2 ϵz − 2 e − $^{ϵz}$) ; D > 0 , (33)