(Quasi)-exact-solvability on the sphere Sn

An Exactly Solvable (ES) potential on the sphere Sn is reviewed and a related Quasi-Exactly Solvable (QES) potential is found and studied. After mapping the sphere to a simplex, it is found that the metric (of constant curvature) is in polynomial form, and both the ES and the QES potentials are rational functions. Their hidden algebra is gln, realized in a finite-dimensional representation by first order differential operators acting on RPn. It is shown that variables in the Schrödinger eigenvalue equation can be separated in polyspherical coordinates and there is always complete integrability. The QES system is completely integrable for n = 2 and non-maximally superintegrable for n ≥ 3. There is no separable coordinate system in which it is exactly solvable. We point out that by taking contractions of superintegrable systems, such as induced by Inönü-Wigner Lie algebra contractions, we can find other QES superintegrable systems, and we illustrate this by contracting our Sn system to a QES non-maximal superintegrable system on Euclidean space En, an extension of the Smorodinsky-Winternitz potential.