Classification of F algebras and noncommutative integration of the Klein-Gordon equation in Riemannian spaces

The method of noncommutative integration of linear differential equations [A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 4, 116; No. 5, 100 (1991)] is used to integrate the Klein-Gordon equation in Riemannian spaces. The situation is investigated where the set of noncommuting symmetry operators of the Klein-Gordon equation consists of first-order operators and one second-order operator and forms a so-called F algebra, which generalizes the concept of a Lie algebra. The F algebra is a quadratic algebra in the given situation. A classification of four- and five-dimensional F algebras is given. The integration of the Klein-Gordon equation in a Riemannian space, which does not admit separation of variables, is demonstrated in a nontrivial example.V. V. Kuibyshev State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 45–50, January, 1993.     

Use of noncommutative integration to construct the basis of wave-equation solutions

This study continues an earlier investigation of applications of the method of noncommutative integration of linear partial differential equations [A. V. Shapovalov and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 1995; No. 5, 33 (1991)], which was a generalization of the analogous method for Hamiltonian systems. The method of noncommutative integration uses nonabelian algebra to characterize the symmetry of the equation, which makes it possible to construct exact solutions going beyond the framework of the method of separation of variables. The condition of noncommutative integrability is used to select the algebras of waveequation symmetry needed for the given method in Minkowski space R_1,2. Nonequivalent noncommutative subalgebras of conformal algebra k_1,2 are used to construct the basis of solutions of the three-dimensional wave equation.Tomsk University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 54–60, May, 1995.     

Free Schrödinger equation analyzed in terms of the wave equation

The free time-dependent Schrödinger equation in three-dimensional space is analyzed as a special case of the wave equation in five-dimensional spacetime. This approach transforms the separation of variables in the parabolic Schrödinger equation into the separation of variables in a nonparabolic equation. Then one can solve the problem using the last theorem of V. N. Shapovalov (see Differents. Uravn., No. 10, 1864 (1980)) on the necessary and sufficient conditions for the complete separation of variables. Other advantages of this approach are also discussed.Translated from Izvestiya Vysshikh uchebnykh Zavedenii, Fizika, No. 7, pp. 59–64, July 1990.     

Commutative subalgebras of three first-order symmetry operators and separation of variables in the wave equation

The problem of complex separation of variables in the wave equation is considered in four-dimensional Minkowskii space-time. In contrast to the known series of researches by Kalnins and Miller (see Ref. Zh., Fiz., 2B9 (1978); 1B208 and 1B209 (1979), e.g.), underlying this research is a theorem on the necessary and sufficient conditions of total separation of variables in the non-parabolic V. N. Shapovalov equation (Differents. Uravn.,16, No. 10, 1864–1874 (1980)). Nonequivalent complete sets of three differential first-order symmetry operators are constructed, appropriate coordinate systems are found, and complete separation of variables is performed in the wave equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 79–84, May, 1990.     

Orientational dependence of microeddy losses due to rotation processes in four-axis ferromagnets

On the basis of the common solution of the wave equation and the equations of rotational moments for four-axis multidomain ferromagnets with a given magneticphase distribution we find the orientational dependence of the microeddy losses caused by the reverse rotations of the spontaneous magnetization vectors of the domains.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 68–72, August, 1991.     

Noncommutative integration of quantum Euler equations on the Lie algebraso (4)

The quantum analogue is considered of the classical Euler equations on the Lie algebraso (4) for the integrable case. With the help of the method of non-commutative integration (A. V. Shapovalov, I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, p. 95; No. 5, p. 33 (1991)) and the method of separation of variables, the quantum equations are reduced to the solution of a system of ordinary differential equations.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 45–50, November, 1992.     

Newton's equation as a consequence of a semiclassical method of solving the Schrödinger equation

Using the one-dimensional Schrbdinger equation as an example, it is shown that the classical equations of motion necessarily arise during the construction of semiclassical solutions of quantum mechanical equations with the aid of V. P. Maslov's complex-germ method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 77–80, July, 1991.     

On the Dirac equation

A connection is established between solutions of the Dirac equation (for an electron in a constant magnetic field) and those of the wave equation for a particle with zero mass.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 96–100, December, 1973.     

Enveloping algebra identities on solutions of conformally invariant wave equations

We study identities in the enveloping algebra of the conformal group, which is the symmetry group of many wave equations: d'Alambert, Weyl, Maxwell, etc. We find all second-order identities for these equations and, in addition, the dimension of the space of nontrivial symmetry operators of any order for the d'Alambert equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 14–18, September, 1991.     

Scalar particle with polarizability in the field of a plane electromagnetic wave and in an electric field

Exact solutions are obtained for the wave equations for a scalar particle that possesses polarizability in the field of a plane electromagnetic wave of arbitrary polarization and in a constant electric field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 40–43, February, 1991.     

Complete sets of symmetry operators containing a second-order operator and the problem of separation of variables in the wave equation

For the wave equation in Minkowski space, a space is defined of nontrivial local second-order differential symmetry operators. The algebraic conditions, which, in accordance with the general theorems on the separation of variables, must be satisfied by the commutative subalgebras, including two first-order operators and second-order operator, are formulated in coordinate-free form. On the basis of these subalgebras, there are obtained all the complete sets of symmetry operators of types (2.0), (2.1). Sets are presented which do not have analogues in papers of other authors.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 115–119, April, 1991.     

Classification of quadratic symmetry algebras of the Schrödinger equation

Noncommuntative quadratic symmetry algebras of a certain class for the Schrödinger equation are classified. For each such algebra, the permissible potential is found. The application of noncommuntative integration of partial differential equations by means of quadratic algebras is demonstrated for a nontrivial example. The solution obtained forms the basis for the representation of quadratic algebras.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 11–17, August, 1995.     

Noncommutative integration of the Dirac equation in Riemann spaces possessing a group of automorphisms

Applying the method of noncommutative integration for linear differential equations, we build exact solutions for the Dirac equation in 4-dimensional Riemann spaces, which have a 5-parameter group of automorphisms and where the Klein-Gordon and the Dirac equations are nonintegrable using the technique of complete separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 43–46, September, 1991.     

Relaxation of the electret state of A1-(Anodic oxide film)-A1 systems

On the basis of a solution of the Debye equation we have obtained an expression that describes the relaxation of the electret potential difference U_e of A1-(anodic oxide film)-A1 systems, which is probably due to the volume electron charge formed in the field. The derived equation, which describes the relaxation of U_e in the initial stage of storage of samples, accords well with experimental results.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 109–111, July, 1991.     

Barium titanate single crystal electrooptic modulator

In this work, the magnitudes of the critical voltages are determined for various mutual orientations of the optical axis of the crystal, the vector of the applied electrical field intensity, and the wave vector of the optical radiation being modulated both for the linear and quadratic electrooptic effects in BaTiO₃ crystals. It is shown that the critical voltages are considerably lower for the linear electrooptic effect than for the quadratic. The minimal critical voltage which ensures a modulation depth of 100% for the linear electrooptic effect is equal to 30 V.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 16, No. 7, pp. 35—38, July, 1973.     

Solution of the Dirac equation in quaternions

A technique is considered for solving in quaternions the Dirac equation for particles with an anomalous magnetic moment in the field of a plane electromagnetic wave.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 43–46, October, 1981.     

Integrals of the motion for an electron in a quantized plane electromagnetic wave

The structure of the integrals of motion of an electron moving in a plane quantized electromagnetic wave is discussed, using the general solution of the integrals of motion of a system whose Hamiltonian is a quadratic form of creation and annihilation operators with constant coefficients.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 116–121, February, 1977.     

Production of electron-positron pairs on nuclei in the field of a bichromatic plane electromagnetic wave

The exact solution of the Dirac equation for an electron in the field of a bichromatic plane electromagnetic wave — with components that are monochromatic waves propagating in one direction — is used to calculate the probability of e⁺e^– pair production on a nucleus in the field of such a wave. This influence of the nucleus is treated perturbatively as an external Coulomb field. The cases of small and large values of the intensity parameters of the wave components are considered and the passage to the limit of a crossed field is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 33–40, September, 1978.We are very grateful to V. R. Khalilov for a fruitful discussion of the results.     

Electron in the field of a plane quantized monochromatic electromagnetic wave

An exact solution of the Dirac equation is found for an electron moving in the field of a plane quantized monochromatic electromagnetic wave of arbitrary polarization.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55–58, August, 1973.     

Noncqmmutative integration of Klein-Gordon and Dirac equations in Riemannian spaces with a group of motions

By the method of non-coimmutative integration of linear differential equations proposed by the authors [Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 95 (1991)] the Klein-Gordon and Dirac equations are integrated in four-dimensional Riemannian spaces, not admitting separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 33–38, May, 1991.