Path integral approach for spaces of nonconstant curvature in three dimensions

Abstract

In this contribution a path integral approach for the quantum motion on three-dimensional spaces according to Koenigs, for short"Koenigs-Spaces", is discussed. Their construction is simple: One takes a Hamiltonian from three-dimensional flat space and divides it by a three-dimensional superintegrable potential. Such superintegrable potentials will be the isotropic singular oscillator, the Holt-potential, the Coulomb potential, or two centrifugal potentials, respectively. In all cases a non-trivial space of non-constant curvature is generated. In order to obtain a proper quantum theory a curvature term has to be incorporated into the quantum Hamiltonian. For possible bound-state solutions we find equations up to twelfth order in the energy E.

Path Integral Representations on the Complex Sphere

“…There are several three-dimensional other spaces where a path integral treatment for various coordinate systems is possible: these are the single-sheeted hyperboloid, the O(2,2)-hyperboloid [12], Darboux spaces in three dimensions [14], and Koenigs spaces in two and three dimensions [15]. The latter two open the possibility to discuss quantum motion on spaces of non-constant curvature, whereas the former two have the property that in addition to the continuous spectrum also an infinite discrete spectrum exists.…”

Path integral approach for superintegrable potentials on spaces of non-constant curvature: II. Darboux spaces D III and D IV

“…The cases for two-dimensional flat space with two-dimensional superintegrable potentials is discussed in[16]. It turns out that the quantization conditions for the bound energy states is always determined by an equation of eighth order in E.…”