Path integral approach for spaces of nonconstant curvature in three dimensions

In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant 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 D _3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D _3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D _3d-II. I solve the path integral on D _3d-I in the (u, v, w) system and on D _3d-II in the (u, v, w) system and in spherical coordinates.     

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimensional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions.     

Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

In this paper path integration in two- and three-dimensional spaces of constant curvature is discussed: i.e. the flat spaces R² and R³, the two- and three-dimensional sphere and the two- and three-dimensional pseudosphere. We are going to discuss all coordinates systems where the Laplace operator admits separation of variables. In all of them the path integral formulation will be stated, however in most of them an explicit solution in terms of the spectral expansion can be given only on a formal level. What can be stated in all cases, are the propagator and the corresponding Green function, respectively, depending on the invariant distance which is a coordinate independent quantity. This property gives rise to numerous identities connecting the corresponding path integral representations and propagators in various coordinate systems with each other.     

Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces <Emphasis Type="Italic">D</Emphasis><Subscript>I</Subscript> and <Emphasis Type="Italic">D</Emphasis><Subscript>II</Subscript>

In this paper, the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces D _I and D _II. On D _I, there are three, and on D _II four such potentials. 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 either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D _I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge. The text was submitted by the authors in English.     

On the interbasis expansion for the Kaluza-Klein monopole system

We study the interbasis expansion of the wave-functions of the Kaluza-Klein monopole system in the parabolic coordinate system with respect to the spherical coordinate system, and vice versa. We show that the coefficients of the expansion are proportional to Clebsch-Gordan coefficients. We analyse the discrete and continuous spectrum as well, briefly discuss the feature that the (reduced) Kaluza-Klein monopole system is separable in three coordinate systems, and the fact that there are five functionally independent integrals of motion, respectively observables, a property which characterizes this system as super-integrable.     

Non-Fermi liquid states in the pressurized CeCu2(Si1-xGex)2 system: two critical points

In the archetypal strongly correlated electron superconductor CeCu2Si2 and its Ge-substituted alloys CeCu2(Si1-xGex)2 two quantum phase transitions--one magnetic and one of so far unknown origin-can be crossed as a function of pressure. We examine the associated anomalous normal state by detailed measurements of the low temperature resistivity (rho) power-law exponent alpha. At the lower critical point (at pcl, 1相似文献   

Path integral for Koenigs spaces

I discuss a path-integral approach for the quantum motion on two-dimensional spaces according to Koenigs, for short “Koenigs spaces”. Their construction is simple: one takes a Hamiltonian from a two-dimensional flat space and divides it by a two-dimensional superintegrable potential. These superintegrable potentials are the isotropic singular oscillator, the Holt potential, and the Coulomb potential. In all cases, a nontrivial space of nonconstant curvature is generated. We can study free motion and the motion with an additional superintegrable potential. For possible bound-state solutions, we find in all three cases an equation of the eighth order in the energy E. The special cases of the Darboux spaces are easily recovered by choosing the parameters accordingly. The text was submitted by the authors in English.