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.

8 pages

References in corpus (5)

• N-fold Supersymmetry in Quantum Mechanics - General Formalism -
• General Forms of a N-fold Supersymmetric Family
• New Methods for Two-Dimensional Schrödinger Equation: SUSY-separation of Variables and Shape Invariance
• Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
• sl(2) Construction of Type A N-fold supersymmetry

Cited by in corpus (11)

• Cosmological aspects of the Eisenhart-Duval lift
• New supersymmetry-generated complex potentials with real spectra
• Interplay between Riccati, Ermakov and Schroedinger equations to produce complex-valued potentials with real energy spectrum
• Metric versus observable operator representation, higher spin models
• On Lie systems and Kummer-Schwarz equations
• An amplitude-phase (Ermakov-Lewis) approach for the Jackiw-Pi model of bilayer graphene
• Milne quantization for non-Hermitian systems
• A geometric Hamilton--Jacobi theory for a Nambu--Poisson structure
• Remarks on the method of comparison equations (generalized WKB method) and the generalized Ermakov-Pinney equation
• Lie Point Symmetries for Reduced Ermakov Systems
• On the Time Dependent Oscillator and the Nonlinear Realizations of the Virasoro Group