3 Two examples; the radial cylindrical Coulom- bic and the harmonic-oscillator A priori, we remove the first-order derivative in the radial cylindrical part of (19) and redefine to obtain In fact, this 1D radial cylindrical Schr¨ odinger equation provides an effective tool to study the effect of different ˜ V ( z ) settings of (18) on the spectra of two interesting models of fundamental nature. The Coulombic and the harmonic oscillator [30]. Of course, such effects could be tested for other models. Let us take a Coulombic radial cylindrical model ˜ V ( ρ ) = − 2 /ρ . In this case, equation (21) would read where ℓ = ( 1 − K 2 ϕ ) 1 / $^{2}$, K$_{z}$ = ( n$_{ρ}$ + ℓ + 1) − $^{1}$, and n$_{ρ}$ = 0 , 1 , 2 , · · · is the radial quantum number. Hence, K$_{z}$ = 1 / ( n$_{ρ}$ + √ 1 − K 2 ϕ + 1 ) and where K$_{z}$ is to be determined through the solution of (18) under different ˜ V ( z ) settings. Next, we consider the radial cylindrical harmonic oscillator model ˜ V ( ρ ) = − U ′′ ( ρ ) + [ ℓ 2 − 1 / 4 ρ 2 − 2 ρ ] U ( ρ ) = − K 2 $_{z}$U ( ρ ) , (22) E = ( m 2 + 3 2 ) − ( ζ − β ) − 1 2 ( 1 K$_{z}$ − n$_{ρ}$ − 1 ) 2 , (23) 8 R ( ρ ) = ρ − 3 / $^{2}$U ( ρ ) , (20) − U ′′ ( ρ ) + [ 3 / 4 − K 2 ϕ ρ 2 + ˜ V ( ρ ) ] U ( ρ ) = − K 2 $_{z}$U ( ρ ) . (21)