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

TL;DR

This paper develops path integral formulations for Smorodinsky-Winternitz potentials in 2D and 3D Euclidean spaces, detailing coordinate systems, explicit evaluations, and spectral expansions for solvable cases.

Contribution

It provides a comprehensive analysis of path integral formulations for Smorodinsky-Winternitz potentials across various coordinate systems, including explicit solutions where possible.

Findings

Explicit path integral formulations for certain coordinate systems.

Spectral expansions into wave-functions for solvable cases.

Identification of coordinate systems where explicit path integrals are not possible.

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.