Under such settings, equation (3) would read Hereby, it should mentioned that the case where υ = − 3 / 2, b = 2 (i.e., g ( ρ ) = 1 /ρ $^{2}$) and k ( z ) = 1 = f ( ϕ ) is the case we have considered in [31]. It is now just a special case of the current, though more general, recipe g ( ρ ) = bρ 2 υ $^{+1}$/ 2; υ, b ∈ R . The results and examples reported therein are recoverable and hold true for the current work, therefore. Yet, an obvious manifestation of the energy term bρ 2 υ $^{+1}$f ( ϕ ) k ( z ) E towards separability of (11) is that, in addition to the three feasible separable cases f ( ϕ ) = 1 = k ( z ), k ( z ) = 1 = g ( ρ ), and f ( ϕ ) = 1 = g ( ρ ) (reported in [31]), one finds two more feasibly separable cases. They are, for υ = − 3 / 2, k ( z ) = 1, f ( ϕ ) = 1 (which, in turn, would break azimuthal symmetry) and for υ = − 3 / 2, k ( z ) = 1, f ( ϕ ) = 1. Therefore, the separability of (11) may be facilitated by the forms of the position-dependent mass and of the interaction potential V ( ρ, ϕ, z ). To secure azimuthal symmetrization of the problem at hand we substitute ˜ V ( ϕ ) = 0 and f ( ϕ ) = 1. Moreover, we choose k ( z ) = 1 to imply that ˜ Φ $^{′′}$( ϕ ) ˜ Φ ( ϕ ) = k 2 ϕ ; k 2 ϕ = − m 2 ; | m | = 0 , 1 , 2 , · · · , (12) [ − ∂ 2 z + ˜ V ( z ) ] ˜ Z ( z ) = k 2 z ˜ Z ( z ) , (13) 7 0 = bρ 2 υ $^{+1}$f ( ϕ ) k ( z ) E + [ U $^{′′}$( ρ ) U ( ρ ) + (2 υ + 1) $^{2}$[ ζ − β − 1] − 2 υ ( υ + 1) 2 ρ 2 − ˜ V ( ρ ) ] + [ ˜ Z $^{′′}$( z ) ˜ Z ( z ) + (2 ζ − 3) 4 ( k $^{′}$( z ) k ( z ) ) 2 − β 2 k ′′ ( z ) k ( z ) − ˜ V ( z ) ] + 1 ρ 2 [ ˜ Φ $^{′′}$( ϕ ) ˜ Φ ( ϕ ) + (2 ζ − 3) 4 ( f $^{′}$( ϕ ) f ( ϕ ) ) 2 − β 2 f ′′ ( ϕ ) f ( ϕ ) − ˜ V ( ϕ ) ] . (11)