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E. M. Ovsiyuk

Publications and source records attributed to E. M. Ovsiyuk.

At least 19 recordsLinked to original sources
arXiv2021

Optics in nonuniform media and Lagrange geometry

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.

physics.optics↗
arXiv2015

Spin 1 particle in the magnetic monopole potential, nonrelativistic approximation. Minkowski and Lobachevski spaces

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.

quant-ph↗
arXiv2015

On solutions of the Pauli equation in non-static de Sitter metrics

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 $(θ, ϕ)$ 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.

quant-ph↗
arXiv2014

Hydrogen atom in de Sitter spaces

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.

math-ph↗
arXiv2014

Dirac-Kähler particle in Riemann spherical space: boson interpretation

In the context of the composite boson interpretation, we construct the exact general solution of the Dirac--Kähler 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-Kähler 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--Kähler field cannot be interpreted as a system of four Dirac fermions. Additional arguments supporting this conclusion are discussed.

math-ph↗
arXiv2014

On geometry influence on the behavior of a quantum mechanical scalar particle with intrinsic structure in external magnetic and electric fields

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_{αβ} and Ricci tensor R_{αβ}. 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 Schrödinger-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.

quant-ph↗
arXiv2014

Confluent Heun functions and the Coulomb problem for spin 1/2 particle in Minkowski space

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.

math-ph↗
arXiv2014

Electromagnetic Field in de Sitter Expanding Universe: Majorana--Oppenheimer Formalism, Exact Solutions in non-Static Coordinates

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 ωin 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.

math-ph↗
arXiv2012

Parameters of Lorentz Matrices and Transitivity in Polarization Optics

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.

physics.optics↗
arXiv2012

Quantum mechanics for a vector particle in the magnetic field on 4-dimensional sphere

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.

math-ph↗
arXiv2012

Spin 1 field in the Lobachevsky space H_{3}: horospherical coordinates, exact solutions

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.

math-ph↗
arXiv2012

Nonrelativistic approximation for quasi-planes waves of a spin 1 particle in Lobachevsky space

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.

math-ph↗
arXiv2012

Quasi-plane waves for spin 1 field in Lobachevsky space and a generalized helicity operator

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.

math-ph↗
arXiv2012

Spin 1 particle on 4-dimensional sphere: extended helicity operator, separation of the variables, and exact solutions

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.

math-ph↗
arXiv2012

Quasi-plane waves for a particle with spin 1/2 on the background of Lobachevsky geometry: simulating of a special medium

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.

math-ph↗
arXiv2012

Classification of degenerate 4-dimensional matrices with semi-group structure and polarization optics

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.

math-ph↗
arXiv2012

Transitivity in polarization optics and the diagonalization of quadratic forms

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.

physics.optics↗