Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems

We propose the integrable (pseudo)spherical generalization of the four-dimensional anisotropic oscillator with additional nonlinear potential. Performing its Kustaanheimo-Stiefel transformation we then obtain the pseudospherical generalization of the MICZ-Kepler system with linear and $\cos\theta$ potential terms. We also present the generalization of the parabolic coordinates, in which this system admits the separation of variables. Finally, we get the spherical analog of the presented MICZ-Kepler-like system.Comment: 7 page

Two-body quantum mechanical problem on spheres

The quantum mechanical two-body problem with a central interaction on the sphere ${\bf S}^{n}$ is considered. Using recent results in representation theory an ordinary differential equation for some energy levels is found. For several interactive potentials these energy levels are calculated in explicit form.Comment: 41 pages, no figures, typos corrected; appendix D was adde

Absorption and quasinormal modes of classical fields propagating on 3D and 4D de Sitter spacetime

We extensively study the exact solutions of the massless Dirac equation in 3D de Sitter spacetime that we published recently. Using the Newman-Penrose formalism, we find exact solutions of the equations of motion for the massless classical fields of spin s=1/2,1,2 and to the massive Dirac equation in 4D de Sitter metric. Employing these solutions, we analyze the absorption by the cosmological horizon and de Sitter quasinormal modes. We also comment on the results given by other authors.Comment: 31 page

On exact solutions for quantum particles with spin S= 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space-time model are examined in detail. Firstly, for a scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient $R_{\epsilon j}$ on the background of the de Sitter space-time model is analyzed. It is shown that the determination of R_{\epsilon j} requires an additional constrain on quantum numbers \epsilon \rho / \hbar c >> j, where \rho is a curvature radius. When taken into account of this condition, the R_{\epsilon j} vanishes identically. It is claimed that the calculation of the reflection coefficient R_{\epsilon j} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space-time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.Comment: 30 pages. This paper is an updated and more comprehensive version of the old paper V.M. Red'kov. On Particle penetrating through de Sitter horizon. Minsk (1991) 22 pages Deposited in VINITI 30.09.91, 3842 - B9

Theory of Quantum Mechanical Scattering in Hyperbolic Space

The theory of quantum mechanical scattering in hyperbolic space is developed. General formulas based on usage of asymptotic form of the solution of the Shrödinger equation in hyperbolic space are derived. The concept of scattering length in hyperbolic space, a convenient measurable in describing low-energy nuclear interactions is introduced. It is shown that, in the limit of the flat space, i.e., when ρ→∞, the obtained expressions for quantum mechanical scattering in hyperbolic space transform to corresponding formulas in three-dimensional Euclidean space