607 research outputs found
Path Integral Approach for Spaces of Non-constant Curvature in Three Dimensions
In this contribution I show that it is possible to construct
three-dimensional spaces of non-constant curvature, i.e. three-dimensional
Darboux-spaces. Two-dimensional Darboux spaces have been introduced by Kalnins
et al., with a path integral approach by the present author. In comparison to
two dimensions, in three dimensions it is necessary to add a curvature term in
the Lagrangian in order that the quantum motion can be properly defined. Once
this is done, it turns out that in the two three-dimensional Darboux spaces,
which are discussed in this paper, the quantum motion is similar to the
two-dimensional case. In \threedDI we find seven coordinate systems which
separate the Schr\"odinger equation. For the second space, \threedDII, all
coordinate systems of flat three-dimensional Euclidean space which separate the
Schr\"odinger equation also separate the Schr\"odinger equation in
\threedDII. I solve the path integral on \threedDI in the -system,
and on \threedDII in the -system and in spherical coordinates
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
On the Path Integral in Imaginary Lobachevsky Space
The path integral on the single-sheeted hyperboloid, i.e.\ in -dimensional
imaginary Lobachevsky space, is evaluated. A potential problem which we call
``Kepler-problem'', and the case of a constant magnetic field are also
discussed.Comment: 16 pages, LATEX, DESY 93-14
Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV
This is the second paper on the path integral approach of superintegrable
systems on Darboux spaces, spaces of non-constant curvature. We analyze in the
spaces \DIII and \DIV five respectively four superintegrable potentials,
which were first given by Kalnins et al. We are able to evaluate the path
integral in most of the separating coordinate systems, leading to expressions
for the Green functions, the discrete and continuous wave-functions, and the
discrete energy-spectra. In some cases, however, the discrete spectrum cannot
be stated explicitly, because it is determined by a higher order polynomial
equation.
We show that also the free motion in Darboux space of type III can contain
bound states, provided the boundary conditions are appropriate. We state the
energy spectrum and the wave-functions, respectively
On the Green function of linear evolution equations for a region with a boundary
We derive a closed-form expression for the Green function of linear evolution
equations with the Dirichlet boundary condition for an arbitrary region, based
on the singular perturbation approach to boundary problems.Comment: 9 page
Proper incorporation of self-adjoint extension method to Green's function formalism : one-dimensional -function potential case
One-dimensional -function potential is discussed in the framework
of Green's function formalism without invoking perturbation expansion. It is
shown that the energy-dependent Green's function for this case is crucially
dependent on the boundary conditions which are provided by self-adjoint
extension method. The most general Green's function which contains four real
self-adjoint extension parameters is constructed. Also the relation between the
bare coupling constant and self-adjoint extension parameter is derived.Comment: LATEX, 13 page
The Coulomb-Oscillator Relation on n-Dimensional Spheres and Hyperboloids
In this paper we establish a relation between Coulomb and oscillator systems
on -dimensional spheres and hyperboloids for . We show that, as in
Euclidean space, the quasiradial equation for the dimensional Coulomb
problem coincides with the -dimensional quasiradial oscillator equation on
spheres and hyperboloids. Using the solution of the Schr\"odinger equation for
the oscillator system, we construct the energy spectrum and wave functions for
the Coulomb problem.Comment: 15 pages, LaTe
Compromise of Localized Graviton with a Small Cosmological Constant in Randall-Sundrum Scenario
A new mechanism which leads to a linearized massless graviton localized on
the brane is found in the /CFT setting, {\it i.e.} in a single copy of
spacetime with a singular brane on the boundary, within the
Randall-Sundrum brane-world scenario. With an help of a recent development in
path-integral techniques, a one-parameter family of propagators for linearized
gravity is obtained analytically, in which a parameter reflects various
kinds of boundary conditions that arise as a result of the half-line
constraint. In the case of a Dirichlet boundary condition () the
graviton localized on the brane can be massless {\it via} coupling constant
renormalization. Our result supports a conjecture that the usual
Randall-Sundrum scenario is a regularized version of a certain underlying
theory.Comment: 6 pages, no figure, V2 12 pages, one more author added, will appear
in PL
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