content_image/301663.jpg content_image/301662.jpg content_image/301661.jpg
to ease coordinates separability of (8), we obtain 

<img src='content_image/301663.jpg'>

It is obvious that separability is granted through a variety of choices. The simplest of which may be sought in an obviously ”manifested-by-equation (12)” general identity of the form 

<img src='content_image/301662.jpg'>

In this case, we may avoid any specifications on the forms of g ( ρ ), f ( ϕ ), and k ( z ) rather than being mathematically and quantum mechanically ”very well” defined. However, the energy term, 2 g ( ρ ) f ( ϕ ) k ( z ) E , in (12) suggests three feasible separabilities for f ( ϕ ) = 1 = k ( z ), k ( z ) = 1 = g ( ρ ), and f ( ϕ ) = 1 = g ( ρ ). We focus on one of these cases in the sequel. 

Let us consider the position-dependent-mass function to be only an explicit function of ρ . Namely, we choose f ( ϕ ) = 1 = k ( z ) and g ( ρ ) = ρ − 2 so that M ( ρ, ϕ, z ) = M ( ρ ) = ρ − $^{2}$. Under these settings, equation(12) collapses into a simple separable form 

<img src='content_image/301661.jpg'>

