Superintegrable quantum oscillator and Kepler-Coulomb systems on curved spaces

TL;DR

This paper explores superintegrable quantum systems on curved spaces, demonstrating two equivalent quantization methods that preserve superintegrability for oscillator and Kepler-Coulomb systems.

Contribution

It introduces two gauge-equivalent quantization prescriptions for superintegrable systems on curved spaces, ensuring superintegrability is maintained after quantization.

Findings

Two quantization methods are gauge equivalent.

Superintegrability is preserved under both quantizations.

Classical systems can be viewed as oscillator or Kepler-Coulomb types.

Abstract

An overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces is presented. It turns out that each of these systems can be considered either as an oscillator or as a Kepler-Coulomb Hamiltonian. We show that two possible quantization prescriptions for all these curved systems arise if we impose that superintegrability is preserved after quantization, and we prove that both possibilities are gauge equivalent.