Supersymmetry, operator transformations and exactly solvable potentials

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Abstract
A large class of potentials can be solved algebraically by using supersymmetry and shape invariance. The authors apply operator transformations (f transformations) to these algebraically solvable problems to obtain a larger class of solvable potentials-the Natanzon class of potentials which are not shape invariant. The important condition (which is independent of supersymmetry) for finding new potentials from old ones using operator transformations is that the resulting Schrodinger equation has a potential which does not depend on the state. As a special case of the f transformation they rederive the previously known connection between the 3D harmonic oscillator, the hydrogen atom and the Morse potential. They also discuss the lack of commutivity of SUSY and the f transformations.