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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. 

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