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(Quasi)-exact-solvability on the sphere $S^n$

An Exactly-Solvable (ES) potential on the sphere $S^n$ is reviewed and the related Quasi-Exactly-Solvable (QES) potential is found and studied. 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 $gl_n$ in a finite-dimensional representation realized by first order differential operators acting on $RP^n$. It is shown that variables in the Schrödinger eigenvalue equation can be separated in spherical coordinates and a number of the integrals of the second order exists assuring the complete integrability. The QES system is completely-integrable for $n=2$ and non-maximally superintegrable for $n\ge 3$. 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 Wigner-Inönü Lie algebra contractions, we can find other QES superintegrable systems, and we illustrate this by contracting our $S^n$ system to a QES non-maximal superintegrable system on Euclidean space $E^n$, an extension of the Smorodinsky-Winternitz potential.