We classify the Hamiltonians H = p 2 x + p 2 y + V (x, y) of all classical superintegrable systems in two dimensional complex Euclidean space with second-order constants of the motion. We similarly classify the superintegrable Hamiltonians H = J 2 1 + J 2 2 + J 2 3 + V (x, y, z) on the complex 2-sphere where x 2 + y 2 + z 2 = 1. This is achieved in all generality using properties of the complex Euclidean group and the complex orthogonal group.
In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two-dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup Cartesian and polar coordinate cases, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the n-dimensional isotropic quantum oscillator.
Path integral formulations for the Smorodinsky‐Winternitz potentials in two‐ and three‐dimensional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky‐Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave‐functions.
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