Ring potential generated from the central hyperbolic Manning-Rosen potential using the transformation method

Systematic search of exactly solvable ring-shaped potential using the transformation method

“…Again the comparison shows that λ = 0, which means l = 0 and from equations ( 14) and ( 41) we have got a constraint relation as m = ± n 2 + α 2 θ 4n 2 . The wave functions of the regenerated Makarov potential are obtained using equations ( 13), (42) and η = 1 in (17) as…”

“…where α = s − n + σ s−n and β = s − n − σ s−n . Choosing η = 1 for mathematical simplification, and using equations ( 13) and (50) in ( 16) and comparing both sides, the regenerated ring-shaped potential is found to be [42] V…”

“…In the similar spirit of the SUSY approach, the extended transformation (ET) method [41,42] has been developed to generate (regenerate) exactly solvable potentials from already known exactly solvable central potentials (ESCPs) for physical QSs, where we need neither to solve the Schrödinger equation nor to be concerned about the idea of supersymmetry and shape-invariance in deriving the solutions of the Schrödinger equation. The SUSY approach contains a coordinate transformation (CT) [32,33], while the ET method comprises a functional transformation (FT) in addition to the CT and a plausible ansatz, providing us with a straightforward way of obtaining the angular wave functions for the generated exactly solvable ring-shaped potentials (ESRPs) from the radial wave functions of the original ESCPs.…”