Some group theoretical aspects of nonlinear quantal oscillators

We investigate the algebraic properties of the time-dependent Schrödinger equations of certain nonlinear oscillators introduced by Calogero and Graffi (Calogero F & Graffi S, On the quantisation of a nonlinear Hamiltonian oscillator Physics Letters A 313 (2003) 356-362; Calogero F, On the quantisation of two other nonlinear harmonic oscillators Physics Letters A 319 (2003) 240-245; Calogero F, On the quantisation of yet another two nonlinear harmonic oscillators Journal of Nonlinear Mathematical Physics 11 (2004) 1-6). Although all of the corresponding classical Hamiltonians are characterised by the Lie algebra sl(2, R), we find that the algebras in the quantal case are not unique and depend upon the choice of parameters made in the quantisation process