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Christian Grosche

Publications and source records attributed to Christian Grosche.

At least 19 recordsLinked to original sources
arXiv2007

Path Integral Representations on the Complex Sphere

In this paper we discuss the path integral representations for the coordinate systems on the complex sphere S3C. The Schroedinger equation, respectively the path integral, separates in exactly 21 orthogonal coordinate systems. We enumerate these coordinate systems and we are able to present the path integral representations explicitly in the majority of the cases. In each solution the expansion into the wave-functions is stated. Also, the kernel and the corresponding Green function can be stated in closed form in terms of the invariant distance on the sphere, respectively on the hyperboloid.

quant-ph↗
arXiv2007

Path Integral Approach for for Quantum Motion on Spaces of Non-constant Curvature According to Koenigs: Three Dimensions

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.

quant-ph↗
arXiv2006

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.

quant-ph↗
arXiv2006

Path Integral Approach for Quantum Motion on Spaces of Non-constant Curvature According to Koenigs

In this contribution 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 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 non-trivial space of non-constant 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 eighth order in the energy E. The special cases of the Darboux spaces are easily recovered by choosing the parameters accordingly.

quant-ph↗
arXiv2006

Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII

In this paper the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of non-constant curvature: these spaces are Darboux spaces D_I and D_II, respectively. On D_I there are three and on D_II four such potentials, respectively. 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.

quant-ph↗
arXiv2005

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 Schrödinger equation. For the second space, $\threedDII$, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger 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.

quant-ph↗
arXiv2005

Path Integration on Hermitian Hyperbolic Space

In this paper the path integral technique is applied to the quantum motion on the Hermitian hyperbolic space HH(2). The Schrödinger equation on this space separates in 12 coordinate systems which are closely related to the coordinate systems on the two-dimensional hyperboloid. For six coordinate systems out of the twelve it is possible to find a path integral solution.

nlin.SI↗
arXiv2005

Path Integration on Darboux Spaces

In this paper the Feynman path integral technique is applied to two-dimensional spaces of non-constant curvature: these spaces are called Darboux spaces $\DI$--$\DIV$. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases, the exceptions being quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl--Teller potential, and for spheroidal wave-functions, respectively. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green's functions and the expansions into the wave-functions, respectively. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.

quant-ph↗
arXiv1998

On the Path Integral Treatment for an Aharonov-Bohm Field on the Hyperbolic Plane

In this paper I discuss by means of path integrals the quantum dynamics of a charged particle on the hyperbolic plane under the influence of an Aharonov-Bohm gauge field. The path integral can be solved in terms of an expansion of the homotopy classes of paths. I discuss the interference pattern of scattering by an Aharonov-Bohm gauge field in the flat space limit, yielding a characteristic oscillating behavior in terms of the field strength. In addition, the cases of the isotropic Higgs-oscillator and the Kepler-Coulomb potential on the hyperbolic plane are shortly sketched.

quant-ph↗
arXiv1998

Path Integrals with Kinetic Coupling Potentials

Path integral solutions with kinetic coupling potentials $\propto p_1p_2$ are evaluated. As examples I give a Morse oscillator, i.e., a model in molecular physics, and the double pendulum in the harmonic approximation. The former is solved by some well-known path integral techniques, whereas the latter by an affine transformation.

quant-ph↗
arXiv1994

$δ'$-Function Perturbations and Neumann Boundary-Conditions by Path Integration

$δ'$-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual $δ$-function or a $δ'$-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral.

hep-th↗
arXiv1993

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 $\bbbr^2$ and $\bbbr^3$, the two- and three-dimensional sphere and the two- and three dimensional pseudosphere. The Laplace operator in these spaces admits separation of variables in various coordinate systems. In all these coordinate systems 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.

hep-th↗
arXiv1993

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.

hep-th↗
arXiv1993

Path Integral Solution of a Class of Explicitly Time-Dependent Potentials

A specific class of explicitly time-dependent potentials is studied by means of path integrals. For this purpose a general formalism to treat explicitly time-dependent space-time transformations in path integrals is sketched. An explicit time-dependent model under consideration is of the form $V(q,t)=V[q/ζ(t)]/ζ^2(t)$, where $V$ is a usual potential, and $ζ(t)=(at^2+2bt+c)^{1/2}$. A recent result of Dodonov et al.\ for calculating corresponding propagators is incorporated into the path integral formalism by performing a space-time transformation. Some examples illustrate the formalism.

hep-th↗
arXiv1993

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.

hep-th↗
arXiv1993

Path Integral Discussion of Two and- Three-Dimensional $δ$-Function Perturbations

The incorporation of two- and three-dimensional $δ$-function perturbations into the path-integral formalism is discussed. In contrast to the one-dimensional case, a regularization procedure is needed due to the divergence of the Green-function $G^{(V)}(\vec x,\vec y;E)$, ($\vec x,\vec y\in\bbbr^2,\bbbr^3$) for $\vec x=\vec y$, corresponding to a potential problem $V(\vec x)$. The known procedure to define proper self-adjoint extensions for Hamiltonians with deficiency indices can be used to regularize the path integral, giving a perturbative approach for $δ$-function perturbations in two and three dimensions in the context of path integrals. Several examples illustrate the formalism.

hep-th↗
arXiv1993

Path Integration Via Summation of Perturbation Expansions and Applications to Totally Reflecting Boundaries, and Potential Steps

The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of $δ$-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the $δ$-function perturbation infinite repulsive, produces a totally reflecting boundary, hence giving a path integral solution in half-spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step-potential and the finite potential-well in the half-space, respectively.

hep-th↗