An exactly-solvable three-dimensional nonlinear quantum oscillator

TL;DR

This paper provides exact analytical solutions for a three-dimensional nonlinear quantum oscillator with position-dependent mass, extending previous one-dimensional models and offering explicit wave functions and energy spectra.

Contribution

It introduces a novel exact solution method for a 3D nonlinear quantum oscillator with position-dependent mass, including complete wave functions and energy spectra.

Findings

Exact solutions in terms of special functions are derived.

Wave functions include spherical harmonics and associated Legendre functions.

Complete energy spectrum for bound states is obtained.

Abstract

Exact analytical, closed-form solutions, expressed in terms of special functions, are presented for the case of a three-dimensional nonlinear quantum oscillator with a position dependent mass. This system is the generalization of the corresponding one-dimensional system, which has been the focus of recent attention. In contrast to other approaches, we are able to obtain solutions in terms of special functions, without a reliance upon a Rodrigues-type of formula. The wave functions of the quantum oscillator have the familiar spherical harmonic solutions for the angular part. For the s-states of the system, the radial equation accepts solutions that have been recently found for the one-dimensional nonlinear quantum oscillator, given in terms of associated Legendre functions, along with a constant shift in the energy eigenvalues. Radial solutions are obtained for all angular momentum states, along with the complete energy spectrum of the bound states.