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

TL;DR

This paper explores the quantum behavior of quadratic Li'enard nonlinear oscillators with maximal Lie symmetries, showing they are exactly solvable under specific ordering parameter constraints and deriving their eigenvalues and eigenfunctions.

Contribution

It introduces a general ordered Hamiltonian framework for these oscillators and identifies conditions for exact solvability based on ordering parameters.

Findings

Quantum versions are exactly solvable with specific parameter constraints

Eigenvalues and eigenfunctions are explicitly derived

Classically linearizable oscillators are also examined in quantum context

Abstract

In this paper, we investigate the quantum dynamics of underlying two one-dimensional quadratic Li'enard type nonlinear oscillators which are classified under the category of maximal (eight parameter) Lie point symmetry group (J. Math. Phys.54 , 053506 (2013)). 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 Li'enard oscillators which are classically linearizable.