Close connections between the methods of Laplace transform, quantum canonical transform, and supersymmetry shape-invariant potentials in solving schrödinger equations

For all commonly known solvable models of the Schrödinger equation, three different methods, Laplace transform, quantum canonical transform, and supersymetry shape-invariant potential, can be employed to obtain solutions. In contrast to the method of power expansion, these methods systematically reduce the Schrödinger equation to a first order differential equation, followed by integration to yield a closed form analytic solution. We analyze the correspondence between these methods and show: (1) All the commonly known solvable models can be divided into two classes. One corresponds to the hypergeometric equation and the other the con uent hypergeometric equation. For each class the sequential steps leading to the solutions are systematic and universal. (2) In both classes there is a precise correspondence between the steps of each method. Such a close connection offers insight into the long standing problem of explaining why solvable models are not abundant and why all these three analytical methods share a common set of solvable models.