Quantum solvability of quadratic Liénard type nonlinear oscillators possessing maximal Lie point symmetries: An implication of arbitrariness of ordering parameters

Abstract

In this paper, we investigate the quantum dynamics of underlying two one-dimensional quadratic Liénard type nonlinear oscillators which are classified under the category of maximal (eight parameter) Lie point symmetry group [Tiwari A K, Pandey S N, Senthilvelan M and Lakshmanan M 2013 J. Math. Phys. 54, 053 506]. Classically, both the systems were also shown to be linearizable as well as isochronic. In this work, we study the quantum dynamics of the nonlinear oscillators by considering a general ordered position dependent mass Hamiltonian. The ordering parameters of the mass term are treated to be arbitrary to start with. We observe that the quantum version of these nonlinear oscillators are exactly solvable provided that the ordering parameters of the mass term are subjected to certain constraints imposed on the arbitrariness of the ordering parameters. We obtain the eigenvalues and eigenfunctions associated with both the systems. We also consider briefly the quantum versions of other examples of quadratic Liénard oscillators which are classically linearizable.