Some new exact path integral treatments are presented. These are the path integrals for the symmetric top, the 'smooth step', the Rosen-Morse- and the Manning-Rosen-potential. By appropriate spacetime transformations these path integral problems can be reduced to the path integral problem of the Poschl-Teller potential.
I discuss in this paper the behaviour of the solutions of the so-called q-hyperbolic potentials, i.e. Pöschl–Teller-like and conditionally solvable potentials, in terms of the path integral formalism. The differences in comparison to the usual Pöschl–Teller-like potentials are investigated, including the discrete energy spectra and the bound state wave-functions. 1 Introduction. In this paper I want to discuss some specific generalisations of Pöschl–Teller related potentials. They are based on a q-deformation of the usual hyperbolic potentials, and are denoted by (we assume without loss of generality q> 0) Consequently we define
In this paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half-plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half-plane to the existing one. All the calculations are mainly based on Fourier-expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Pöschl-Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave-functions and the energy-spectrum are derived. The energy-spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p -momentum). The “zero-point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.
Towards the Classification of Exactly Solvable Feynman Path Integrals: δ-Function Perturbations and Boundary-Problems as Miscellaneous Solvable Models
Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) Pöschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals.