Lie symmetries, quantisation and c-isochronous nonlinear oscillators

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.