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Schrödinger equation for a particle on a curved space and superintegrability. (English) Zbl 1326.81083

Summary: 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.

MSC:

81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81S05 Commutation relations and statistics as related to quantum mechanics (general)
51P05 Classical or axiomatic geometry and physics
35P05 General topics in linear spectral theory for PDEs
81Q80 Special quantum systems, such as solvable systems
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