Nonlinear Supersymmetric (Darboux) Covariance of the Ermakov-Milne-Pinney Equation

Abstract

It is shown that the nonlinear Ermakov-Milne-Pinney equation $\rho^{\prime\prime}+v(x)\rho=a/\rho^3$ obeys the property of covariance under a class of transformations of its coefficient function. This property is derived by using supersymmetric, or Darboux, transformations. The general solution of the transformed equation is expressed in terms of the solution of the original one. Both iterations of these transformations and irreducible transformations of second order in derivatives are considered to obtain the chain of mutually related Ermakov-Milne-Pinney equations. The behaviour of the Lewis invariant and the quantum number function for bound states is investigated. This construction is illustrated by the simple example of an infinite square well.