Equation (14) with azimuthal symmetry (i.e., ˜ V ( ϕ ) = 0) would immediately imply that and In due course, the solution of (15) reads Φ ( ϕ ) = exp ( imϕ ) where m = 0 , ± 1 , ± 2 , · · · is the magnetic quantum number and Φ ( ϕ ) satisfies the single valued condition Φ ( ϕ ) = Φ ( ϕ + 2 π ). Moreover, we obtain Consequently, one may cast and In the following section, we consider ˜ V ( ρ ) to represent a ”Coulombic” and a ”harmonic oscillator” and find the spectral signatures of different ˜ V ( z ) poten- tials of (18) on the over all spectra. [ Φ ′′ ( ϕ ) Φ ( ϕ ) + 2 E + 2 ( ζ − β − 1) ] = K 2 $_{ϕ}$, (15) [ R ′′ ( ρ ) R ( ρ ) + 3 ρ R ′ ( ρ ) R ( ρ ) + K 2 ϕ ρ 2 − ˜ V ( ρ ) ] + [ Z ′′ ( z ) Z ( z ) − ˜ V ( z ) ] = 0 . (16) K 2 ϕ = 2 E + 2 ( ζ − β − 1) − m $^{2}$. (17) Z ′′ ( z ) Z ( z ) − ˜ V ( z ) = − K 2 z (18) R ′′ ( ρ ) R ( ρ ) + 3 ρ R ′ ( ρ ) R ( ρ ) + K 2 ϕ ρ 2 − ˜ V ( ρ ) = K 2 z (19) 7