A formulation of quantum mechanics on spaces of constant curvature is studied by quantizing the Noether momenta and using these to form the quantum Hamiltonian. This approach gives the opportunity of studying a superintegrable quantum system. It is shown there are three different ways of obtaining a Hilbert space of common eigenstates. Three different orthogonal coordinate systems are determined, one for each case. It is shown how the Schrödinger equation can be rendered separable in each of the cases.
"Schrödinger Equation for a Particle on a Curved Space and Superintegrability." J. Geom. Symmetry Phys. 38 25 - 37, 2015. https://doi.org/10.7546/jgsp-38-2015-25-37