2 Cylindrical coordinates and separability Let us consider the kinetic energy operator of the PDM Hamiltonian in (1) and a PDM function of the form m ( ⃗ r ) = m ◦ M ( ρ, ϕ, z ) = M ( ρ, ϕ, z ) (where ℏ = m ◦ = 1 units are to be used hereinafter). Moreover, we consider the following substitutions to imply Using the above identities, one (with M ( ρ, ϕ, z ) ≡ M for simplicity of notations) may rewrite and 4 M γ ⃗ ∇ M $^{β}$· ⃗ ∇ M α = M γ ( ⃗ B · ⃗ A ) + M α + γ ( ⃗ B · ⃗ ∇ ) + M γ + β [ 2 ⃗ A · ⃗ ∇ + ⃗ ∇ · ⃗ A ] + M − 1 ⃗ ∇ $^{2}$. (6) M α ⃗ ∇ M $^{β}$· ⃗ ∇ M γ = M α ( ⃗ B · ⃗ C ) + M α + γ ( ⃗ B · ⃗ ∇ ) + M α + β [ 2 ⃗ C · ⃗ ∇ + ⃗ ∇ · ⃗ C ] + M − 1 ⃗ ∇ $^{2}$, (5) ⃗ ∇ M ( ρ, ϕ, z ) α = ⃗ A + M ( ρ, ϕ, z ) α ⃗ ∇ ⃗ ∇ M ( ρ, ϕ, z ) β = ⃗ B + M ( ρ, ϕ, z ) β ⃗ ∇ ⃗ ∇ M ( ρ, ϕ, z ) γ = ⃗ C + M ( ρ, ϕ, z ) γ ⃗ ∇ . (4) ⃗ A = αM ( ρ, ϕ, z ) $^{α}$− 1 [ ˆ ρ ∂$_{ρ}$ + ˆ ϕ ρ ∂$_{ϕ}$ + ˆ z ∂$_{z}$ ] M ( ρ, ϕ, z ) , ⃗ B = βM ( ρ, ϕ, z ) $^{β}$− 1 ˆ ρ ∂$_{ρ}$ + ˆ ϕ $_{ρ}$∂$_{ϕ}$ + ˆ z ∂$_{z}$ M ( ρ, ϕ, z ) , ⃗ C = γM ( ρ, ϕ, z ) $^{γ}$− 1 [ ˆ ρ ∂$_{ρ}$ + ˆ ϕ $_{ρ}$∂$_{ϕ}$ + ˆ z ∂$_{z}$ ] M ( ρ, ϕ, z ) , (3)