Adding potentials to superintegrable systems with symmetry

In previous work, we have considered Hamiltonians associated with three-dimensional conformally flat spaces, possessing two-, three- and four-dimensional isometry algebras. Previously, our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the three-dimensional space reduce to 3 or 4 parameter potentials for Darboux–Koenigs Hamiltonians. Other three-dimensional coordinate systems reveal connections between Darboux–Koenigs and other well-known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.

Application of the exact quantization rule for some noncentral separable potentials

The energy spectrum of some noncentral separable potentials are obtained using the exact quantization rule in r and θ dimensions. The results are consistent with those obtained by other methods. PACS Nos.: 03.65.Ca, 03.65.Ge