We study the free quantum motion on the three-dimensional sphere in ellipso-cylindrical coordinates, where we distinguish between prolate elliptic and oblate elliptic coordinates. The oblate and prolate elliptic wavefunctions are constructed in terms of the hyper-spherical and cylindrical wavefunctions. A perturbation theory for the elliptic wavefunctions in powers of the elliptic parameter is established. Finally, the interbasis expansions are used to set up path integral identities for the two ellipso-cylindrical coordinate systems on .
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Grosche et al. (1997) studied this question.
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