Strictly Isospectral Susy Potentials and the Riccati Superposition Principle

Generating quasi-isospectral potentials by the methods of supersymmetric quantum mechanics (SUSYQM) is a simple and well-known mathematical technique [1]. Its relationships with intertwining, factorization, Darboux covariance, and inverse scattering procedures have been recently reviewed by the author [2]. Also, the author, either in collaboration or alone [3] dealt extensively with a SUSYQM procedure based on the general Riccati solution [4], that may be called the double Darboux general Riccati method (henceforth DDGR). For details the reader is directed to my previous works on DDGR and the references therein. The purpose of this note is to display the connection of the DDGR method with the Riccati nonlinear superposition principle. The DDGR procedure has been first used in physics by Mielnik for the quantum harmonic oscillator case [5]. For simplicity reasons, I shall deal with the case of unbroken SUSY, i.e., when one has at hand a true zero mode u0 of the initial Schroedinger problem. DDGR leads to families of strictly isospectral potentials to the initial (bosonic) one. Asking for the most general superpotential (i.e., the general Riccati solution) such that the fermionic potential fulfills V+(x) = w 2 g + dwg dx , it is obvious that one particular solution to this equation is wp, i.e., Witten’s superpotential. Thus, one is led to consider the following Riccati equation w g + dwg dx = w 2 p+ dwp dx , whose general solution can be written down as wg(x) = wp(x)+ 1 v(x) , where v(x) is an unknown function. Using this ansatz, one obtains for the function v(x) the following Bernoulli equation