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 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 j on the background of the de Sitter space-time model is analyzed. It is shown that the determination of R j requires an additional constrain on quantum numbers ρ/hc j, where ρ is a curvature radius. When taken into account of this condition, the R j vanishes identically. It is claimed that the calculation of the reflection coefficient R 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.

Abstract. We study motion of a relativistic particle in the 3-dimensional Lobachevsky space in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in the Euclidean space. Three integrals of motion are found and equations of motion are solved exactly in the special cylindrical coordinates. Motion on surface of the cylinder of constant radius is considered in detail.

Complex formalism of Riemann -Silberstein -Majorana -Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space -time in accordance with the tetrad recipe of Tetrode -Weyl -Fock -Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations.Similar treatment is given for Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.

In this paper we present the distinguished (d-) Riemannian geometry (in the sense of nonlinear connection, Cartan canonical linear connection, together with its d-torsions and d-curvatures) for a possible Lagrangian inspired by optics in non-uniform media. The corresponding equations of motion are also exposed, and some particular solutions are given. For instance, we obtain as geodesic trajectories some circular helices (depending on an angular velocity ω), certain circles situated in some planes (ones are parallel with xOy, and other ones are orthogonal on xOy), or some straight lines which are parallel with the axis Oz. All these geometrical geodesics are very specific because they are completely determined by the non-constant index of refraction n(x). (2010): 53C60, 53C80, 83C10. Mathematics Subject Classification

The Riemann-Silberstein-Majorana-Oppenheimer complex approach to the Maxwell electrodynamics is investigated within the matrix formalism. Within the squaring procedure we construct four types of formal solutions of the Maxwell equations on the base of scalar D'Alembert solutions. General problem of separating physical electromagnetic solutions in the linear space λ 0 0 + λ 1 1 + λ 2 2 + λ 3 3 is investigated, the Maxwell equations reduce to a new form including parameters λ a . Several particular cases, plane waves and cylindrical waves, are considered in detail.Possible extension of the technique to a curved space-time models is discussed.