A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

TL;DR

This paper explores classical and quantum super-integrable systems as deformations of known oscillators in curved spaces, demonstrating their exact solvability and super-separability in quantum form.

Contribution

It introduces a new class of super-integrable, super-separable systems in curved spaces and provides their exact quantum solutions, extending the understanding of oscillator models.

Findings

Classical systems are deformations of harmonic oscillator and Smorodinsky-Winternitz systems.

Quantum analogs are exactly solvable with Schrödinger equations reducible to single-variable equations.

The systems are super-integrable and super-separable in multiple coordinate systems.

Abstract

Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schr\"odinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom.