On the Path Integral Treatment for an Aharonov-Bohm Field on the Hyperbolic Plane

In this paper I discuss by means of path integrals the quantum dynamics of a charged particle on the hyperbolic plane under the influence of an Aharonov-Bohm gauge field. The path integral can be solved in terms of an expansion of the homotopy classes of paths. I discuss the interference pattern of scattering by an Aharonov-Bohm gauge field in the flat space limit, yielding a characteristic oscillating behavior in terms of the field strength. In addition, the cases of the isotropic Higgs-oscillator and the Kepler-Coulomb potential on the hyperbolic plane are shortly sketched.Comment: LaTeX 12 pp., one figur

Path Integration on Darboux Spaces

In this paper the Feynman path integral technique is applied to two-dimensional spaces of non-constant curvature: these spaces are called Darboux spaces \DI--\DIV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases, the exceptions being quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified P\"oschl--Teller potential, and for spheroidal wave-functions, respectively. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green's functions and the expansions into the wave-functions, respectively. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.Comment: 48 pages, 3 Tables In revised version typos correcte

Path Integrals with Kinetic Coupling Potentials

Path integral solutions with kinetic coupling potentials $\propto p_1p_2$ are evaluated. As examples I give a Morse oscillator, i.e., a model in molecular physics, and the double pendulum in the harmonic approximation. The former is solved by some well-known path integral techniques, whereas the latter by an affine transformation.Comment: 8 pages., LateX, 1 figure (postscript

Classification of Solvable Feynman Path Integrals

A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last ten years or so, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given. Explicit formul\ae\ for the so-called basic path integrals are presented on which our general scheme to classify and calculate path integrals in quantum mechanics is based.Comment: 13 pages, amstex, preprint DESY 92--189, and SISSA/1/93/F

Alternative Solution of the Path Integral for the Radial Coulomb Problem

In this Letter I present an alternative solution of the path integral for the radial Coulomb problem which is based on a two-dimensional singular version of the Levi-Civita transformation.Comment: 7 pages, Late

Conditionally solvable path integral problems

Abstract. New classes of exactly solvable potentials are discussed within the path integral formalism. They are constructed from the hypergeometric and confluent Natanzon potentials, respectively. It is found that they allow incorporation of four free parameters, which give rise to fractional power behaviour, long-range and strongly anharmonic terms. We find six different classes of such potentials. 1