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In this paper the equations of motion associated with a Lagrangian inspired by relativistic optics in nonuniform moving media are considered. The model describes optical effects in the nonuniform dispersionless moving medium. When using the optical metric restricted to the Minkowski manifold, we have established the Euler-Lagrange equations for geodesics. We have specified the general model to the special case when the refractive index increases along the direction $Z$. The exact analytical solutions of the corresponding Euler-Lagrange equations have been constructed. Analysis of the solutions shows that the light beams are bending to the axes $Z$ along which the refractive index increases.

A particle with spin 1/2 is investigated both in expanding and oscillating cosmological de Sitter models. It is shown that these space-time geometries admit existence of the non-relativistic limit in the covariant Dirac equation. Procedure for transition to the Pauli approximation is conducted in the equations in the variables $(t, r)$, obtained after separating the angular dependence of $(\theta, \phi)$ from the wave function. The non-relativistic systems of equations in the variables $(t, r)$ is solved exactly in both models. The constructed wave functions do not represent stationary states with fixed energy, however the corresponding probability density does not depend on the time., Comment: 12 pages

In the context of the composite boson interpretation, we construct the exact general solution of the Dirac--K\"ahler equation for the case of the spherical Riemann space of constant positive curvature, for which due to the geometry itself one may expect to have a discrete energy spectrum. In the case of the minimal value of the total angular momentum, $j=0$, the radial equations are reduced to second-order ordinary differential equations, which are straightforwardly solved in terms of the hypergeometric functions. For non-zero values of the total angular momentum, however, the radial equations are reduced to a pair of complicated fourth-order differential equations. Employing the factorization approach, we derive the general solution of these equations involving four independent fundamental solutions written in terms of combinations of the hypergeometric functions. The corresponding discrete energy spectrum is then determined via termination of the involved hypergeometric series, resulting in quasi-polynomial wave-functions. The constructed solutions lead to notable observations when compared with those for the ordinary Dirac particle. The energy spectrum for the Dirac-K\"ahler particle in spherical space is much more complicated. Its structure substantially differs from that for the Dirac particle since it consists of two paralleled energy level series each of which is twofold degenerate. Besides, none of the two separate series coincides with the series for the Dirac particle. Thus, the Dirac--K\"ahler field cannot be interpreted as a system of four Dirac fermions. Additional arguments supporting this conclusion are discussed.

Relativistic theory of the Cox's scalar not point-like particle with intrinsic structure is developed on the background of arbitrary curved space-time. It is shown that in the most general form, the extended Proca-like tensor first order system of equations contains non minimal interaction terms through electromagnetic tensor F_{\alpha \beta} and Ricci tensor R_{\alpha \beta}. In relativistic Cox's theory, the limiting procedure to non-relativistic approximation is performed in a special class of curved space-time models. This theory is specified in simple geometrical backgrounds: Euclid's, Lobachevsky's, and Rie\-mann's. Wave equation for the Cox's particle is solved exactly in presence of external uniform magnetic and electric fields in the case of Minkowski space. Non-trivial additional structure of the particle modifies the frequency of a quantum oscillator arising effectively in presence if external magnetic field. Extension of these problems to the case of the hyperbolic Lobachevsky space is examined. In presence of the magnetic field, the quantum problem in radial variable has been solved exactly; the quantum motion in z-direction is described by 1-dimensional Schr\"{o}dinger-like equation in an effective potential which turns out to be too difficult for analytical treatment. In the presence of electric field, the situation is similar. The same analysis has been performed for spherical Riemann space model., Comment: 36 pages. Report to: The International conference in honor of Ya.B. Zeldovich 100-th Anniversary: Subatomic particles, Nucleons, Atoms, Universe: Processes and Structure, held March 10-14, 2014, Minsk, Belarus (a shortened version submitted to: NPCS)

The spin 1 particle is treated in the presence of the Dirac magnetic monopole in the Minkowski and Lobachevsky spaces. Separating the variables in the frame of the matrix 10-component Duffin-Kemer-Petiau approach (wave equation) and making a nonrelativistic approximation in the corresponding radial equations, a system of three coupled second order linear differential equations is derived for each type of geometry. For the Minkowski space, the nonrelativistic equations are disconnected using a linear transformation, which makes the mixing matrix diagonal. The resultant three unconnected equations involve three routs of a cubic algebraic equation as parameters. The approach allows extension to the case of additional external spherically symmetric fields. The Coulomb and oscillator potentials are considered and for each of these cases three series of energy spectra are derived. A special attention is given to the states with minimum value of the total angular momentum. In the case of the curved background of the Lobachevsky geometry, the mentioned linear transformation does not disconnect the nonrelativistic equations in the presence of the monopole. Nevertheless, we derive the solution of the problem in the case of minimum total angular momentum. In this case, we additionally involve a Coulomb or oscillator field. Finally, considering the case without the monopole field, we show that for both Coulomb and oscillator potentials the problem is reduced to a system of three differential equations involving a hypergeometric and two general Heun equations. Imposing on the parameters of the latter equations a specific requirement, reasonable from the physical standpoint, we derive the corresponding energy spectra., Comment: 25 pages. Report to: XVII International Conference Foundations & Advances in Nonlinear Science September 29 - October 3, Minsk, 2014. Submitted to: Nonlinear Phenomena in Complex Systems

The hydrogen atom theory is developed for the de Sitter and anti de Sitter spaces on the basis of the Klein-Gordon-Fock wave equation in static coordinates. In both models, after separation of the variables, the problem is reduced to the general Heun equation, a second order linear differential equation having four regular singular points. A qualitative examination shows that the energy spectrum for the hydrogen atom in the de Sitter space should be quasi-stationary, and the atom should be unstable. We derive an approximate expression for energy levels within the quasi-classical approach and estimate the probability of decay of the atom. A similar analysis shows that in the anti de Sitter model the hydrogen atom should be stable in the quantum-mechanical sense. Using the quasi-classical approach, we derive approximate formulas for energy levels for this case as well. Finally, we present the extension to the case of a spin 1/2 particle for both de Sitter models. This extension leads to complicated differential equations with 8 singular points., Comment: 17 pages, Report to International Conference and Workshop "Quanta and %Matter: Through Physics to Future Emerging Technologies. 22-26 September 2014. Yerevan-Tsaghkadzor, Armenia

Tetrad-based generalized complex formalism by Majorana--Oppenheimer is applied to treat electromagnetic field in extending de Sitter Universe in on-static spherically-symmetric coordinates. With the help of Wigner D-functions, we separate angular dependence in the complex vector field E_{j}(t,r)+i B_{j}(t,r) from (t,r)-dependence. The separation parameter arising here instead of frequency \omega in Minkowski space-time is quantized, non-static geometry of the de Sitter model leads to definite dependence of electromagnetic modes on the time variable. Relation of 3-vector complex approach to 10-dimensional Duffin-Kemmer-Petiau formalism is considered. On this base, the electromagnetic waves of magnetic and electric type have been constructed in both approaches. In Duffin-Kemmer-Petiau approach, there are constructed gradient-type solutions in Lorentz gauge., Comment: 35 pages

In the paper, the well-known quantum mechanical problem of a spin 1/2 particle in external Coulomb potential, reduced to a system of two first-order differential equations, is studied from the point of view of possible applications of the Heun function theory to treat this system. It is shown that in addition to the standard way to solve the problem in terms of the confluent hypergeometric functions (proposed in 1928 by G. Darvin and W. Gordon), there are possible several other possibilities which rely on applying the confluent Heun functions. Namely, in the paper there are elaborated two combined possibilities to construct solutions: the first applies when one equation of the pair of relevant functions is expressed trough hypergeometric functions, and another constructed in terms of confluent Heun functions. In this respect, certain relations between the two classes of functions are established. It is shown that both functions of the system may be expressed in terms of confluent Heun functions. All the ways to study this problem lead us to a single energy spectrum, which indicates their correctness., Comment: 22 pages, submitted to: Nonlinear Phenomena in Complex Systems

In the context of applying the Lorentz group theory to polarization optics in the frames of Stokes-Mueller formalism, some properties of the Lorentz group are investigated. We start with the factorized form of arbitrary Lorentz matrix as a product of two commuting and conjugate $4\times 4$-matrices, $L(q,q)= A(q_{a}) A^*(q_{a}); a= 0,1,2,3$. Mueller matrices of the Lorentzian type M=L are pointed out as a special sub-class i n the total set of $4\times 4$ matrices of the linear group GL(4,R). Any arbitrary Lorentz matrix is presented as a linear combination of 16 elements of the Dirac basis. On this ground, a method to construct parameters q_a by an explicitly given Lorentz matrix L is elaborated. It is shown that the factorized form of L=M matrices provides us with a number of simple transitivity equations relating couples of initial and final 4-vectors, which are defined in terms of parameters q_a of the Lorentz group. Some of these transitivity relations can be interpreted within polarization optics and can be applied to the group-theoretic analysis of the problem of measuring Mueller matrices in optical experiments., Comment: 15 pages. Report to The International Conference "Differential Geometry and Dynamical Systems - 2012". 29 August - 2 September 2012, Mangalia, Romania

Quantum-mechanical wave equation for a particle with spin 1 is investigated in presence of external magnetic field in spaces with non-Euclidean geometry with constant positive curvature. Separation of the variable is performed; differential equations in the variable r are solved in hypergeometric functions. T he study o f z-dependence of the wave function has been reduced to a system of three linked ordinary differential 2-nd order equations; till now the system in z variable is not solved., Comment: 27 pages; Nauchno-Technicheskie Vedomosti. Saint Petersburg State Polytechnical University

A complete system of solutions for a field with spin 1 in the space of constant negative curvature, Lobachevsky space H_{3}, has been constructed. The treatment is based on 10-dimensional Duffin-Kemmer formalism extended to curved model according to tetrad method by Tetrode-Weyl-Fock-Ivanenko, and specified in horospherical coordinates. The solving procedure substantially uses a generalized helicity operator. The Lobachevsky geometry acts along z axis as a medium with simple reflecting properties. Restriction to massless case is performed as well., Comment: 13 pages

Spin 1 particle in Pauli approximation is investigated on the background of the curved space of constant negative curvature, Lobachevsky space. Nonrelativistic approximation is performed in the system of 10 equations resulted from separating the variables in Duffin-Kemmer equation specified in quasi-cartesian coordinates. The problem is solved exactly in Bessel functions, the quantum states are determined by four quantum numbers. The treatment is substantially based on the use of a generalized helicity operator in Lobachevsky space model., Comment: 14 pages

Spin 1 particle is investigated in 3-dimensional curved space of constant negative curvature. An extended helicity operator is defined and the variables are separated in a tetrad-based 10-dimensional Duffin--Kemmer equation in quasi Cartesian coordinates. The problem is solved exactly in hypergeometric functions, the quantum states are determined by three quantum numbers. It is shown that Lobachevsky geometry acts effectively as a medium with simple reflecting properties. Transition to a massless case of electromagnetic field is performed., Comment: 13 pages

Spin 1 particle is investigated in 3-dimensional curved space of constant positive curvature. An extended helicity operator is defined and the variables are separated in a tetrad-based 10-dimensional Duffin-Kemmer equation in quasi cylindrical coordinates. The problem is solved exactly in hypergeometric functions, the energy spectrum determined by three discrete quantum numbers is obtained. Transition to a massless case of electromagnetic field is performed., Comment: 21 pages

In the paper complete systems of exact solutions for Dirac and Weyl equations in the Lobachevsky space are constructed on the base of the method of separation of the variables in quasi-cartesian coordinates. An extended helicity operator is introduced. It is shown that solution constructed when translating to the limit of vanishing curvature coincide with common plane wave solutions on Minkowski space going in opposite z-directions. It is shown the problem posed in Lobachevsky space simulates a situation in the flat space for a quantum-mechanical particle of spin 1/2 in a 2-dimensional potential barrier smoothly rising to infinity on the right., Comment: 33 pages

In polarization optics, an important role play Mueller matrices -- real four-dimensional matrices which describe the effect of action of optical elements on the polarization state of the light, described by 4-dimensional Stokes vectors. An important issue is to classify possible classes of the Mueller matrices. In particular, of special interest are degenerate Mueller matrices with vanishing determinants. Earlier, it was developed a special technique of parameterizing arbitrary 4-dimensional matrices with the use of four 4-dimensional vector (k, m, l, n). In the paper, a classification of degenerate 4-dimensional real matrices of rank 1, 2, 3. is elaborated. To separate possible classes of degenerate matrices of ranks 1 and 2, we impose linear restrictions on (k, m, l, n), which are compatible with the group multiplication law. All the subsets of matrices obtained by this method, are either sub-groups or semigroups. To obtain singular matrices of rank 3, we specify 16 independent possibilities to get 4-dimensional matrices with zero determinant., Comment: 12 pages

With assumption that an optical element is described by a Mueller matrix of the Lorentzian type, a method to find a 3-dimensional complex vector parameter for a corresponding Mueller matrix from results of four specially chosen polarization measurements has been elaborated., Comment: 11 pages

Any one measurement with polarized light makes it possible to fix the Mueller matrices of the Lorentz type with up to four arbitrary numeric parameters (x, u; z, w). These parameters are subject to the quadratic condition. It is demonstrated that the quadratic form can be diagonalized; in the case of partially polarized light four diagonal coefficients turn out to benon-zero and positive; in the case of completely polarized light two diagonal coefficients equal to zero., Comment: 9 pges, 2 figures

Cylindrically symmetric analogue of the Volkov problem is examined. In presence of the field of a cylindrical electromagnetic wave, classical motion of a non-relativistic particle on a cylindrical surface is described exactly in terms of elliptic functions., Comment: 13 pages

There are constructed exact solutions of the quantum-mechanical equation for a spin S=1 particle in 2-dimensional Riemannian space of constant negative curvature, hyperbolic plane, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels describing quantization of the motion of the vector particle in magnetic field on the 2-dimensional space H_{2} has been found, nonrelativistic and relativistic equations have been solved., Comment: 20 pages

The Dirac monopole string is specified for anti de Sitter cosmological model. Dirac equation for spin 1/2 particle in presence of this monopole has been examined on the background of anti de Sitter space-time in static coordinates. Instead of spinor monopole harmonics, the technique of Wigner D-functions is used. After separation of the variables radial equations have been solved exactly in terms of hypergeometric functions. The complete set of spinor wave solutions has been constructed, the most attention is given to treating the states of minimal values for total moment quantum number j_{min}. At all values of j, the energy spectrum is discrete., Comment: 21 pages

Transition to a nonrelativistic Pauli equation in Riemann space of constant positive curvature for a Dirac particle in presence of the Coulomb field is performed in the system of radial equations, exact solutions are constructed in terms of Hein functions, the energy spectrum is derived. The same is done for the Kepler quantum problem in hyperbolic Lobachevsky space, solutions are constructed in terms of Hein functions, the energy spectrum is obtained., Comment: 17 pages

It is shown that there exist several ways to treat the quantum-mechanical Coulomb problem for a Dirac particle in flat Minkowski space with the help of the Heun differential equation, Fuchs's equation with four singular points. When extending the problem to curved spaces of constant curvature, Lobachevsky H_{3} and Rimann S_{3}, there arise 2-nd order differential equations of the Fuchs type with 6 singular points, a method to get relevant equations with five singular points has been elaborated., Comment: 22 pages

There are constructed exact solutions of the quantum-mechanical equation for a spin S=1 particle in 2-dimensional Riemannian space of constant positive curvature, spherical plane, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels describing quantization of the motion of the vector particle in magnetic field on the 2-dimensional space S_{2} has been found, nonrelativistic and relativistic equations have been solved., Comment: 11 pages, 34 references

The Dirac monopole string is specified for de Sitter cosmological model. Dirac equation for spin 1/2 particle in presence of this monopole has been examined on the background of de Sitter space-time in static coordinates. Instead of spinor monopole harmonics, the technique of Wigner D-functions is used. After separation of the variables, detailed analysis of the radial equations is performed; four types of solutions, singular, regular, in- and out- running waves, are constructed in terms of hypergeometric functions. The complete set of spinor wave solutions \Psi_{\epsilon, j,m, \lambda}(t,r, \theta, \phi) has been constructed, special attention is given to treating the states of minimal values of the total angular moment j_{\min}., Comment: 27 pages, 46 references; Report to 18 Conference NPCS-2011, May 17-20, 2011, Minsk, Belarus

Tensor 50-component form of the first order relativistic wave equation for a particle with spin 2 and anomalous magnetic moment is extended to the case of an arbitrary curved space-time geometry. An additional parameter considered in the presence of only electromagnetic field as related to anomalous magnetic moment, turns to determine additional interaction terms with external geometrical background through Ricci R_{kl} and Riemann R_{klmn} tensors., Comment: 11 pages, 49 references

Three possible techniques to deal with a vector particle in the anti de Sitter cosmological model are viewed: Duffin-Kemmer-Petiau matrix formalism based on the general tetrad recipe, group theory 5-dimensional approach based on the symmetry group SO(3.2), anda tetrad form of Maxwell equations in complex Riemann-Silberstein-Majorana-Oppenheimer representation. In the first part, a spin 1 massive field is considered in static coordinates of the anti de Sitter space-time in tetrad-based approach. The complete set of spherical wave solutions with quantum numbers (\epsilon, j,m,l) is constructed; angular dependence in wave functions is described with the help of Wigner functions. The energy quantization rule has been found. Transition to massless case of electromagnetic field is specified, and electromagnetic solutions in Lorentz gauge have been constructed. In the second part, the problem of a particle with spin 1 is considered on the base of 5-dimensional wave equation specified in the same static coordinates. In the third part, a rarely used approach, based on tetrad form of Maxwell equations in complex representation is examined in the anti de Sitter model., Comment: 32 pages

With the use of the general covariant matrix 10-dimensional Petiau-Duffin-Kemmer formalism in cylindrical coordinates exact solutions of the quantum-mechanical equation for a particle with spin 1 in the presence of an external homogeneous magnetic field are constructed. Three linearly independent types of solutions are separated; in each case the formula for the energy levels has been found.Within similar technique for the quantum-mechanical equation for a particle with spin 1 and additional intrinsic electromagnetic characteristics - polarizability, exact solutions are found in the presence of an external homogeneous magnetic field., Comment: 15 pages, 16 references

Lobachewsky geometry simulates a medium with special constitutive relations. The situation is specified in quasi-cartesian coordinates (x,y,z). Exact solutions of the Maxwell equations in complex 3-vector form, extended to curved space models within the tetrad formalism, have been found in Lobachevsky space. The problem reduces to a second order differential equation which can be associated with an 1-dimensional Schrodinger problem for a particle in external potential field U(z) = U_{0} e^{2z}. In quantum mechanics, curved geometry acts as an effective potential barrier with reflection coefficient R=1; in electrodynamic context results similar to quantum-mechanical ones arise: the Lobachevsky geometry simulates a medium that effectively acts as an ideal mirror. Penetration of the electromagnetic field into the effective medium, depends on the parameters of an electromagnetic wave, \omega, k_{1}^{2} + k_{2}^{2}, and the curvature radius \rho., Comment: 20 pages, 3 figures, 37 references

The Coulomb problem for Schr\"{o}dinger equation is examined, in spaces of constant curvature, Lobachevsky H_{3} and Riemann S_{3} models, on the base of generalized parabolic coordinates. In contrast to the hyperbolic case, in spherical space S_{3} such parabolic coordinates turn to be complex-valued, with additional constraint on them. The technique of the use of such real and complex coordinates in two space models within the method of separation of variables in Schr\"{o}dinger equation with Kepler potential is developed in detail; the energy spectra and corresponding wave functions for bound states have been constructed in explicit form for both spaces; connections with Runge-Lenz operators in both curved space models are described., Comment: 28 pages, 2 figures, 60 references

Quantum-mechanical system -- spin 1 particle in external Coulomb field is studied on the base of the matrix Duffin-Kemmer-Petiau formalism with the use of the tetrad technique. Separation of the variables is performed with the help of Wigner functions. With the help of parity operator, the radial 10-equation system is divided into two subsystem of 4 and 6 equations that correspond to parity P=(-1)^{j+1} and P= (-1)^{j} respectively. The system of 4 equation is reduced to a second order differential equation which coincides with that arising in the case of a scalar particle in Coulomb potential. It is shown that the 6-equation system reduces to two different second order differential equations for a "main" function. One main equation reduces to a confluent Heun equation and provides us with energy spectrum. Another main equation is a more complex one, and any solutions for it are not constructed., Comment: 19 pages

Quantum-mechanical WKB-method is elaborated for the known quantum oscillator problem in curved 3-spaces models Euclid, Riemann, and Lobachevsky E_{3}, H_{3}, S_{3} in the framework of the complex variable function theory. Generalized generally covariant Schr\"{o}dinger equation is considered. In all three space models, exact energy levels are found with the help of constructing special formal WKB-sieries., Comment: 12 pages, Contrivution to BGL-7: International Conference on Non-Euclidean Geometry and its applications. 5 -- 9 July 2010, Kluj-Napoca, ROMANIA

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the transversal motion of the particle in magnetic field has been obtained. The same problem is solved for spin 1/2 particle in the space of constant positive curvature, spherical Riemann space. A generalized formula for energy levels, describing quantization of the transversal and along the magnetic field motions of the particle on the background of the Riemann space geometry, is obtained., Comment: 34 pages, Submitted to SIGMA

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., Comment: 4 pages

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in the space of constant positive curvature, spherical Riemann space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Riemann space geometry, is obtained., Comment: 7 pages, International Seminar NPCS-2010, 18-21 May, 2010, Minsk, Belarus

With the use of the general covariant matrix 10-dimensional Petiau -- Duffin -- Kemmer formalism in cylindrical coordinates and tetrad there are constructed exact solutions of the quantum-mechanical equation for a particle with spin 1 in presence of an external homogeneous magnetic field. There are separated three linearly independent types of solutions; in each case the formula for energy levels has been found., Comment: 8 pages, International Seminar NPCS-2010, 18-21 May, 2010, Minsj, Belarus

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the motion of the particle in magnetic field on the background of the Lobachevsky space geometry, has been obtained., Comment: 8 pages

The aim of the article to clarify the status of Shapiro plane wave solutions of the Schr\"odinger's equation in the frames of the well-known general method of separation of variables. To solve this task, we use the well-known cylindrical coordinates in Riemann and Lobachevsky spaces, naturally related with Euler angle-parameters. Conclusion may be drawn: the general method of separation of variables embraces the all plane wave solutions; the plane waves in Lobachevsky and Riemann space consist of a small part of the whole set of basis wave functions of Schr\"odinger equation. In space of constant positive curvature $S_{3}$, a complex analog of horospherical coordinates of Lobachevsky space $H_{3}$ is introduced. To parameterize real space $S_{3}$, two complex coordinates $(r,z)$ must obey additional restriction in the form of the equation $r^{2} = e^{z-z^{*}} - e^{2z} $. The metrical tensor of space $S_{3}$ is expressed in terms of $(r,z)$ with additional constraint, or through pairs of conjugate variables $(r,r^{*})$ or $(z,z^{*})$; correspondingly exist three different representations for Schr\"{o}dinger Hamiltonian. Shapiro plane waves are determined and explored as solutions of Schr\"odinger equation in complex horosperical coordinates of $S_{3}$. In particular, two oppositely directed plane waves may be presented as exponentials in conjugated coordinates. $\Psi_{-}= e^{-\alpha z}$ and $\Psi_{+}= e^{-\alpha z^{*}}$. Solutions constructed are single-valued, finite, and continuous functions in spherical space and correspond to discrete energy levels., Comment: 19 pages

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 - B91

Motion of a classical particle in 3-dimensional Lobachevsky and Riemann spaces is studied in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in Euclidean space. In both cases three integrals of motions are constructed and equations of motion are solved exactly in the special cylindrical coordinates on the base of the method of separation of variables. In Lobachevsky space there exist trajectories of two types, finite and infinite in radial variable, in Riemann space all motions are finite and periodical. The invariance of the uniform magnetic field in tensor description and gauge invariance of corresponding 4-potential description is demonstrated explicitly. The role of the symmetry is clarified in classification of all possible solutions, based on the geometric symmetry group, SO(3,1) and SO(4) respectively.

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., Comment: 39 pages

The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell electrodynamics in vacuum is investigated within the matrix formalism. The matrix form of electrodynamics includes three real 4 \times 4 matrices. Within the squaring procedure we construct four formal solutions of the Maxwell equations on the base of scalar Klein -- Fock -- Gordon solutions. The problem of separating physical electromagnetic waves in the linear space \lambda_{0}\Psi^{0}+\lambda_{1}\Psi^{1}+\lambda_{2}\Psi^{2}+ lambda_{3}\Psi^{3} is investigated, several particular cases, plane waves and cylindrical waves, are considered in detail., Comment: 26 pages 16 International Seminar NCPC, May 19-22, 2009, Minsk, Belarus

The Duffin-Kemmer form of massless vector field (Maxwell field) 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 simplest static cosmological model, space of constant curvature of Riemann parameterized by spherical coordinates. Separation of variables is realized in the basis of Schr\"odinger- Pauli type, description of angular dependence in electromagnetic filed functions is given in terms of Wigner D-functions. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three discrete parameters is found. 4-potentials for spherical electro- magnetic waves of magnetic and electric type have been constructed., Comment: 10 pages, Report to XVI Annual Seminar NPCS - 2009, May 19-22, 2009, Minsk, Belarus

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 simplest static cosmological models, spaces of constant curvature of Riemann and Lobachevsky parameterized by spherical coordinates. Separation of variables is realized in the basis of Schr\"odinger -- Pauli type, description of angular dependence in electromagnetic complex 3-vectors is given in terms of Wigner D-functions. In the case of compact Riemann model a discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three discrete parameters is found. In the case of hyperbolic Lobachevsky model no discrete spectrum for frequencies of electromagnetic modes arises., Comment: 15 pages. Report to XVI Annual Seminar Nonlinear Phenomena in Complex Systems, Minsk, Belarus, May 19-22, 2009