Classical and quantum systems with position-dependent mass: An application to a Mathews-Lakshmanan-type oscillator

Position-dependent mass (PDM) systems have diverse applications in physics, attracting significant scientific interest in the last decades. Classical PDM systems pose analytical challenges with their nonlinear equations of motion, while the quantum case is complicated because the kinetic operator is non-Hermitian. To address these challenges, this study explores Hamiltonian factorization and canonical transformations to investigate classical and quantum PDM systems, applying it to a Mathews-Lakshmanan-type oscillator with $m(x)=1/[1+{(\lambda x)}^2]$. Classically, the phase space is examined, revealing increasingly pronounced deformities in the trajectories as energy and $\lambda$ values increase. In the quantum realm, the solution to the ambiguous ordering problem for the PDM oscillator is presented, accompanied by an analysis of wave functions and probability densities. Further, the tunneling probabilities are analyzed. As $\lambda$ increases, findings indicate that the tunneling probabilities of the PDM system decrease fast for higher excited states, offering novel insights into its behavior.