Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space

Abstract

Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimen\-sional 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.

LaTeX 60 pages, DESY 94-018

References in corpus (3)

• Deformed oscillator algebras for two dimensional quantum superintegrable systems
• Periodicity and Quasi-Periodicity for Super-Integrable Hamiltonian Systems
• Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

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