Lie symmetries, quantisation and c-isochronous nonlinear oscillators

Abstract

In a series of papers Calogero and Graffi [F. Calogero, S. Graffi, On the quantisation of a nonlinear Hamiltonian oscillator, Phys. Lett. A 313 (2003) 356–362] and Calogero [F. Calogero, On the quantisation of two other nonlinear harmonic oscillators, Phys. Lett. A 319 (2003) 240–245; F. Calogero, On the quantisation of yet another two nonlinear harmonic oscillators, J. Nonlinear Math. Phys. 11 (2004) 1–6] treated the quantisation of several one-degree-of-freedom Hamiltonians containing a parameter, c. Two of these systems possess the Lie algebra sl ( 2 , R ) characteristic of the Ermakov–Pinney problem and are related to the Hamiltonian of that problem by an autonomous canonical transformation. Calogero found that the ground-state energy eigenvalues of the corresponding three Schrödinger equations differed when the standard quantisation procedures were used. We examine three simpler c-isochronous oscillators to determine if the method of quantisation is responsible for this unexpected result. We propose a quantisation scheme based on the preservation of the algebraic properties of the Lie point symmetries of the kinetic energy. We find that this criterion removes the dependence of the ground-state eigenvalue on the parameter c and that in fact the eigenvalues are the same for the three systems. Similarly for the Ermakov–Pinney problem and the two derivate models of Calogero we find consistency of ground-state eigenvalues.