Classical and Quantum Superintegrable Hamiltonian Systems on curved spaces.

In 1992 a theoretical astrophysicist, V.Perlick posed the following question: "What are the most general systems, autonomous and with radial symmetry, that satisfy Bertrand Theorem" He gave the answer as well, exhibiting two multi-parametric families of systems of such type that do the job. Of course, he had to relax the hypothesis of   living in a flat space, and identified both the conformal
factor characterizing the metric and the corresponding potential in the Hamiltonian. In the present talk, after a terse review of Perlick's results, I will consider in detail two prototype systems, both at the classical and at the quantum level, the Darboux III and the Taub-Nut system, that in fact provide simple exactly solvable examples of a generalisation of the Harmonic   Oscillator and the Kepler Coulomb systems to spaces of nonconstant curvature. At the end I will outline some possibly interesting open problems.
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