Shape invariant Natanzon potential hierarchy and its twin: A novel approach

Shape invariant supersymmetric partner potentials corresponding to the Natanzon superpotential Atanhγr−Bcothγr and its twin i.e. −(A+B)tanhγr−Bsechγrcschγr are obtained as 2-fold hierarchies whose solutions are a common set of the quantum states. It is shown that they are derived by using simultaneous laddering relations with respect to two different parameters n and m of some associated Gegenbauer functions via two different methods including (a) simultaneous increasing or decreasing of the parameters, and (b) increasing one of the parameters and decreasing the other one. The shape invariance condition in the concept of Gendenshtein on the 2-fold hierarchies of supersymmetric partner potentials with respect to the parameters n and m is inherited from those laddering relations by two different ways.