content_image/302013.jpg content_image/302010.jpg content_image/302011.jpg
Under such settings, equation (3) would read 

<img src='content_image/302013.jpg'>

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 

<img src='content_image/302010.jpg'>

<img src='content_image/302011.jpg'>

