Quantum solvability of a general ordered position dependent mass system: Mathews-Lakshmanan oscillator

TL;DR

This paper solves the quantum Mathews-Lakshmanan oscillator with a general ordered position-dependent mass Hamiltonian, revealing eigenfunctions in terms of non-integral degree associated Legendre polynomials and connecting them to other orthogonal polynomials.

Contribution

It provides a comprehensive solution for all orderings of the PDM Hamiltonian of the Mathews-Lakshmanan oscillator, including non-Hermitian cases, and explores the eigenfunctions in terms of non-integral associated Legendre polynomials.

Findings

Eigenfunctions are expressed in terms of non-integral degree associated Legendre polynomials.

The Schrödinger equation reduces to the associated Legendre differential equation for all orderings.

Connections established between non-integral Legendre polynomials and Jacobi, Gegenbauer polynomials.

Abstract

In position dependent mass (PDM) problems, the quantum dynamics of the associated systems have been understood well in the literature for particular orderings. However, no efforts seem to have been made to solve such PDM problems for general orderings to obtain a global picture. In this connection, we here consider the general ordered quantum Hamiltonian of an interesting position dependent mass problem, namely the Mathews-Lakshmanan oscillator, and try to solve the quantum problem for all possible orderings including Hermitian and non-Hermitian ones. The other interesting point in our study is that for all possible orderings, although the Schr\"odinger equation of this Mathews-Lakshmanan oscillator is uniquely reduced to the associated Legendre differential equation, their eigenfunctions cannot be represented in terms of the associated Legendre polynomials with integral degree and order. Rather the eigenfunctions are represented in terms of associated Legendre polynomials with non-integral degree and order. We here explore such polynomials and represent the discrete and continuum states of the system. We also exploit the connection between associated Legendre polynomials with non-integral degree with other orthogonal polynomials such as Jacobi and Gegenbauer polynomials.