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

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 $(u,v,w)$-system, and on \threedDII in the $(u,v,w)$-system and in spherical coordinates

On the Path Integral in Imaginary Lobachevsky Space

The path integral on the single-sheeted hyperboloid, i.e.\ in $D$-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

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

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

Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds

There are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on a Liouville manifold and the Schr\"odinger equation can be solved by separation of variables in one coordinate system. The Lie integrable systems are defined on a Lie manifold and are not generally separable ones but the can be solved. Therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems. The quantum analogues of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems are classified as special cases of these six general classes. The coefficients of the associative algebra of the general cases are calculated. These coefficients are the same as the coefficients of the classical case multiplied by $-\hbar^2$ plus quantum corrections of order $\hbar^4$ and $\hbar^6$.Comment: LaTeX file, 25 page

Magnetic Transition in the Kondo Lattice System CeRhSn2

Our resistivity, magnetoresistance, magnetization and specific heat data provide unambiguous evidence that CeRhSn2 is a Kondo lattice system which undergoes magnetic transition below 4 K.Comment: 3 pages text and 5 figure

Multi-Channel Electron Transfer Reactions: An Analytically Solvable Model

We propose an analytical method for understanding the problem of multi-channel electron transfer reaction in solution, modeled by a particle undergoing diffusive motion under the influence of one donor and several acceptor potentials. The coupling between the donor potential and acceptor potentials are assumed to be represented by Dirac Delta functions. The diffusive motion in this paper is represented by the Smoluchowski equation. Our solution requires the knowledge of the Laplace transform of the Green's function for the motion in all the uncoupled potentials.Comment: arXiv admin note: substantial text overlap with arXiv:0903.306

Proper incorporation of self-adjoint extension method to Green's function formalism : one-dimensional δ′\delta^{'}-function potential case

One-dimensional $\delta^{'}$-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 $n$-dimensional spheres and hyperboloids for $n\geq 2$. We show that, as in Euclidean space, the quasiradial equation for the $n+1$ dimensional Coulomb problem coincides with the $2n$-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