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

TL;DR

This paper demonstrates that the nonlinear Ermakov-Milne-Pinney equation exhibits covariance under specific transformations using supersymmetric (Darboux) methods, linking solutions and invariants across transformed equations.

Contribution

It introduces a supersymmetric transformation framework for the Ermakov-Milne-Pinney equation, showing covariance and deriving related solutions and invariants.

Findings

Covariance of the equation under Darboux transformations

Explicit relation between original and transformed solutions

Analysis of invariants and quantum states in examples

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.