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We show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators on L2false(S2false) have a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog of the Laplace–Runge–Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect, we construct a classical integrable system that is the semiclassical limit of the quantum integrable system of commuting operators. We show that this is a generalized semitoric system with a nondegenerate focus–focus point, such that there is monodromy in the classical and the quantum systems.

This paper investigates the symmetry reduction of the regularised n-body problem. The three body problem, regularised through quaternions, is examined in detail. We show that for a suitably chosen symmetry group action the space of quadratic invariants is closed and the Hamiltonian can be written in terms of the quadratic invariants. The corresponding Lie-Poisson structure is isomorphic to the Lie algebra u(3, 3). Finally, we generalise this result to the n-body problem for n > 3.

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