Superintegrable quantum oscillator and Kepler-Coulomb systems on curved spaces

We present an overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces (arXiv:1304.4544). 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.

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