Quantum superintegrability and exact solvability in N dimensions

A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in N-dimensional Euclidean space. Two different sets of N commuting second order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and N further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.