tures of different ˜ V ( z ) potentials on the overall spectrum. The choice of υ = 1 / 2 along with ˜ V ( ρ ) = a $^{2}$ρ$^{2}$/ 4 would imply that equation (14) reads 3.1 The radial harmonic oscillator g ( ρ ) = bρ $^{2}$/ 2 [ − ∂ 2 ρ + ˜ ℓ 2 1 / 2 − 1 / 4 ρ 2 + ( a 2 − 4 bE ) 4 ρ 2 ] U ( ρ ) = − k 2 $_{z}$U ( ρ ) , (17) ∣ ∣$^{˜}$ ℓ$_{1}$$_{/}$$_{2}$ ∣ = √ ( m 2 + 3) − 2 ( ζ − β ) . (18) k 2 z = − √ ( a 2 − 4 bE ) [ 2 n$_{ρ}$ + ∣ ∣$^{˜}$ ℓ$_{1}$$_{/}$$_{2}$ ∣ + 1 ] 2 , (19) E = a 2 4 b − 1 4 b [ k 2 z 2 n$_{ρ}$ + √ ( m 2 + 3) − 2 ( ζ − β ) + 1 ] 2 . (20) and (15), in turn, yields Obviously, Eq.(17) has exact eigenvalues in the form and implies that We observe that an auxiliary constraint ( ζ − β ) = α ( α − 1) + γ ( γ − 1) − β ( β + 2) ≤ ( m 2 + 3 ) / 2 (21) on the ambiguity parameters is manifested here by the requirement that E ∈ R . 9