Separation of variables in the Klein-gordon equations. III

This paper supplements [1] and [2]. All kinds of external electromagnetic fields are found which contain arbitrary functions admitting complete separation of variables in the Klein-Gordon equations by using one first and two second order differential symmetry operators. The curvilinear coordinates in which the variables separate are presented, and equations in the separated variables are written down.     

Separation of variables in the Klein-Gordon equations. II

Two kinds of external nonstationary electromagnetic fields are found containing arbitrary functions which admit of total separation of variables in the Klein-Gordon equations by using two differential symmetry operators and one second order operator. Curvilinear coordinates are presented in which the variables are divided, and equations are written down in the separated variables.Translated from Izvestiya VUZ, Fizika, No. 12, pp. 45–52, December, 1973.     

A new closure hypothesis for the BBGKY system of equations

The first part of this paper deals with the Frey-Salmon irreversibility hypothesis, the kinetic equation that is deduced therefrom, and the comparison with experiment for the viscosity and thermal conductivity coefficients. The relaxation time which has been introduced is the average value of the duration of a collision. The second part deals with a new irreversibility hypothesis which leads to a kinetic equation of the same form as the first but in which the relaxation time is deduced from the interparticle potential.     

A scaling method for deriving kinetic equations from the BBGKY hierarchy

On the basis of the scale covariance of correlation functions under a coarsegraining in space and time, the Boltzmann equation for neutral gases, the Balescu-Lenard-Boltzmann-Landau equation for dilute plasmas, and linear equations for the variances of fluctuations are derived from the BBGKY hierarchy equations with no short-range correlations at the initial time. This is done by using Mori's scaling method in an extended form. Thus it is shown that the scale invariance of macroscopic features affords a useful principle in nonequilibrium statistical mechanics. It is also shown that there existtwo kinds of correlation functions, one describing the interlevel correlations of the kinetic level with its sublevels and the other representing the fluctuations in the kinetic level.Partially financed by the Scientific Research Fund of the Ministry of Education.     

A scaling method for obtaining kinetic equations from the BBGKY hierarchy

Kinetic equations for the one-particle distribution function of dilute gases and plasmas and evolution equations for the variance of their fluctuations are obtained from the BBGKY hierarchy equations by using Mori's scaling method in an extended form.     

Separation of variables in 0(3)-invariant wave equations for quadrupole coordinates

Linear, 0(3) invariant equations for a multicomponent wave function, which depends on quadrupole coordinates, are examined. Vibrational, rotational, and spin parts in the Hamiltonian are separated, on which basis angular and internal variables are separated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 80–83, July, 1982.I thank S. A. Vladimirov for formulating the problem and for useful discussions.     

Separation of variables in the Dirac equation

In this paper we investigate the possibility of separating variables in the Dirac equation by using complete sets, containing second-order differential operators and integral operators having a definite structure.Translated from Izvestiya VUZ. Fizika, No. 6, pp. 94–100, June, 1973.     

The cauchy problem for BBGKY hierarchy of quantum kinetic equations with coulomb potential

The existence of a unique solution, in terms of initial data of the BBGKY hierarchy of quantum kinetic equations with coulomb potential, is proved. This is based on non-relativistic quantum mechanics and utilizing a semigroup method.     

Wick rotation of the time variables for two-point functions on analytic backgrounds

Letters in Mathematical Physics - We set up a general framework for Calderón projectors (and their generalization to non-compact manifolds), associated with complex Laplacians obtained by Wick...     

Separation of variables in the diffusion equation. II

The covariant and symmetry properties of the linear diffusion equation having a scalar matrix of variable diffusion coefficients are studied. By means of differential symmetry operators of order no higher than two, a complete separation of variables is effected for the stationary and nonstationary cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 40–45, December, 1976.     

Separation of variables in the diffusion equation. I

For both stationary and nonstationary cases, all the coefficients of the linear diffusion equation with a constant diffusion tensor are enumerated for which total separation of the variables is possible by means of differential symmetry operators of a definite structure.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 46–53, November, 1976.     

Separation of variables in the diffusion equation. III

The Separation of variables of type N = 2 is completed, and all complete sets of type N = 1 are constructed for the diffusion equation with a scalar matrix for the leading derivatives.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 48–50, July, 1979.In conclusion, we take this opportunity to thank V. N. Shapovalov for continued interest.     

Elimination of variables in simple laser equations

The usual way of obtaining rate equations (RE) and a single equation for the field amplitude (EFA) from the semiclassical laser equations (Lorenz-Haken model) is reexamined by undertaking a systematic elimination procedure developed in synergetics. The RE and EFA are justified in the case 1 (, ) and case 2 (, ), respectively. We show that, because the eliminated variable happens to contain a considerable contribution from an unstable mode, the usual elimination technique in the case 3 (, ) leads to an inconsistency. As important by-products we obtain the RE and EFA for arbitrary cavity relaxation constant (). Some remarks are given on the direct elimination technique in the non-diagonal representation in the study of instabilities.     

Separation of variables in the stationary Schrödinger equation

The problem of separation of variables in the stationary Schrödinger equation is considered for a charge moving in an external electromagnetic field. On the basis of the definition formulated, necessary and sufficient conditions are found for separation of variables in equations of elliptic type to which the stationary Schrödinger equation belongs. Application of general theorems made it possible to enumerate all types of electromagnetic fields and systems of coordinates in which separation of variables in the stationary Schrödinger equation is possible. Systems of ordinary differential equations which the wave function in the separated variables satisfies are written down to explicit form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 45–50, August, 1972.     

Separation of variables in the nonstationary Schrödinger equation

Separation of variables in the Schrödinger equation is performed by using complete sets of differential operators of symmetry with operators not higher than second order, and all types of electromagnetic field potentials for which separation of variables is possible are listed.     

Separation of variables in Kolmogorov's equation. I

Calculations are made of the coefficients of the nonstationary Kolmogorov equation for which a solution can be found by separation of variables using a complete set of differential symmetry operators of first order; the variables are separated in accordance with this procedure. A solution is found to the equation describing the motion of a dynamical system containing a random parameter.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 16–20, August, 1977.