Method of green's functions in the theory of quantum kinetic equations. II

The kinetic and antikinetic equations are obtained for the single-particle Wigner function in the context of the method of Green's time-temperature functions for an inhomogeneous system of weakly interacting particles situated in a time-dependent electric field. The kinetic equation is derived here from the equation of motion for Green's function, satisfying the causality condition.     

Localized magnetic excitations of coupled impurities in a Heisenberg ferromagnet: Optical defect modes

A Green's function method is used to obtain the spectrum of excitations of two neighboring impurities in a semi-infinite ferromagnet. The equations of motion for Green's functions are determined in the framework of the Heisenberg model. The energies of non-resonant localized modes are calculated as a function of the interaction parameters for the exchange coupling between impurity-spin pairs, host-spin pairs, and impurity-host neighbors, as well as the effective field parameter at the impurity sites. With two impurities the system is less symmetric and has more localized modes when compared with a single impurity case.     

Covariant interaction operators for a composite system ofN scalar particles

The explicit form of relativistic, energy-dependent interaction operators for a composite system of N scalar particles in the second order of perturbation theory is obtained within the framework of the simultaneous covariant approach of quantum field theory on the basis of the retarding part of two-time Green's functions. Explicitly covariant equations for the wave function of the bound state and the conditions for its normalization are obtained. Frantsiska Skorina State University, Gomel'. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 17–24, August, 1997.     

Localized magnetic excitations for a line of magnetic impurities in a transverse Ising thin film ferromagnet

A Green's function method is used to obtain the spectrum of spin excitations associated with a linear array of magnetic impurities implanted in a ferromagnetic thin film. The equations of motion for the Green's functions of the anisotropic film are written in the framework of the Ising model in a transverse field. The frequencies of localized modes are calculated as a function of the interaction parameters for the exchange coupling between impurity-spin pairs, host-spin pairs, and impurity-host neighbors, as well as the effective field parameter at the impurity sites.     

Relation between phase function and Green's function

A relation is established between the phase of the wave function and Green's function of onedimensional motion. On the basis of this relation, a series of accurate equations are obtained, including a formula for obtaining the Green's function at different one-dimensional coordinates from its form at coinciding one-dimensional coordinates.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 20–24, November, 1979.It remains to thank V. L. Bonch-Bruevich for useful discussions.     

带电细圆环与导体球壳系统的场分布

先依电象法,推导均匀带电圆环在金属导体球壳内的"象电荷";再在球坐标系下,根据电场强度的计算公式与Tay-lor展开式,计算出均匀带电细圆环在全空间的电场分布的级数形式解;进而结合唯一性定理和电场的叠加原理,获得带电细圆环与导体球壳系统的空间场分布.     

Dynamics of Acoustooptic Modulation in a Gyrotropic Medium

A system of equations describing the transformation of dynamic optical transfer functions in gyrotropic crystals with cubic syngony perturbed by a spatially-dependent dynamic field of a slowly varying acoustic signal is derived in the first approximation of the method of slowly varying amplitudes (SVA). The general solution of these equations is obtained in the form of a converging functional series. It can be used to calculate the dynamics of polarization parameters in the angular spectra of interacting light beams for arbitrary power and spatiotemporal structure of the acoustic signal field.     

On the sound field of a shallow spherical shell in an infinite baffle

A method is presented for calculating the far field sound radiation from a shallow spherical shell in an acoustic medium. The shell has a concentrated ring mass boundary condition at its perimeter representing a loudspeaker voice coil and is excited by a concentrated ring force exerted by the end of the voice coil. A Green's function is developed for a shallow spherical shell, which is based upon Reissner's solution to the shell wave equation [Q. Appl. Math. 13, 279-290 (1955)]. The shell is then coupled to the surrounding acoustic medium using an eigenfunction expansion, with unknown coefficients, for its deflection. The resulting surface pressure distribution is solved using the King integral together with the free space Green's function in cylindrical coordinates. In order to eliminate the need for numerical integration, the radiation (coupling) integrals are solved analytically to yield fast converging expansions. Hence, a set of simultaneous equations is obtained which is solved for the coefficients of the eigenfunction expansion. These coefficients are finally used in formulas for the far field sound radiation.     

Dependence on crystal parameters of the correlation time between signal and idler beams in parametric down conversion calculated in the Wigner representation

The theory of parametric down conversion within the framework of the Wigner representation has been treated recently in a series of papers using the standard model Hamiltonian. Here we take a more fundamental point of view studying the mechanism, inside the crystal, for the production of the signal and idler beams. We begin from the evolution equations for the quantum field operators, pass to the Wigner function and solve the resulting (Maxwell) equations with the use of the Green's function method. We derive the time dependence of the coincidence detection probability as a function of the parameters of the nonlinear crystal (in particular the length) the radius of the pumping beam, and the bandwidth of the filters in front of the detectors. Received 24 January 2000 and Received in final form 24 March 2000     

Discrete phase space II. Relativistic lattice wave equations and divergence-free green's functions

Therelativistic lattice Klein-Gordon equation, Dirac equation, electromagnetic equations, and gauge field equations are presented as partialdifference equations. Various lattice Green's functions are constructed (except for non-abelian gauge fields). It is proved that many of the lattice Green's functions are non-singular or divergence-free. Moreover, it is conjectured that all lattice Green's functions are non-singular.     

A reliable treatment of the homotopy analysis method for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].      

On self-consistent approximations for random anharmonic systems

A model with random 1-, 2-, and 3-point potentials is used to study the elementary excitations in substitutionally disordered crystals. The equations of motion for the 1- and 2-point Green's functionsG(1),G(12) are derived and averaged over the ensemble of random configurations. An extension of the coherent potential method is proposed, which leads to a self-consistent set of equations for the averaged 1- and 2-point Green's functions, including corresponding conditional averages. The theory takes into account that randomness effects the anharmonic interactions both via the explicit configuration dependence of the cubic vertices and via the implicit dependence through the Green's functions. The final equations take a similar form as in the usual CPA if the harmonic potential of the pure system and the harmonic single-site impurity potential are replaced by corresponding functionals of averaged and conditionally averaged 1- and 2-point functions, and the definition of the single-site mass-operator is appropriately generalized.Work supported by the Deutsche ForschungsgemeinschaftExtract from thesis, D 26     

On the formal expansion of the possibilities of the method of integral transformations in investigating linear distributed systems with constant parameters

The Green's functions for a series of boundary-value problems are formulated in transforms with respect to t by means of matrix functions that are introduced and allow the scalar dependence on the weighted differences of the coordinate argument to be isolated in them. It was possible in general form to carry out the procedure of the method of integral transformations in solving the integral equations for the function that generates periodic motion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 323–331, March, 1972.     

GREEN'S FUNCTION THEORY OF THE KONDO LATTICE(I) —MAGNETIC ORDERING AND KONDO EFFECT

The periodic s-d model is investigated by Green's function methods. A closed set of equations for Green's function of Kondo-lattice system and a formula of single electron self-energy are obtained in the presence of a broken rotational symmetry. The formula can-be used to discuss low temperature anomalous properties of Kondo-lattice system with spontaneous magnetization or in an external magnetic field. We have evaluated the influence of magnetic ordering on logarithmic sin-gularity of electron self-energy by perturbative method to Leading logarithmic order (LLO).The result obtained indiGates there exists a competition between Kondo effect and Zong range magnetic ordering. The static susceptibility in the paramagnetic case has been evaluated, too.     

The hyperspherical-expansion approach and shell models

By means of Talmi integrals the matrix elements of nuclear interaction in the set of radial differential equations of the hyperspherical-expansion approach are related to the elements of usual shell models. The introduced average field equated to the diagonal element of the core reproduces the form of the effective (two-body) cut-off oscillator. The corresponding eigenvalue problem with a flat (heavy nuclei) and rounded (light nuclei) alternative for the well is solved and compared with the phenomenological relations: the asymptotic saturation can be achieved for the flat well. Using these results the starting set is transformed into the set of integral equations by means of the Green's functions. Then it is, successively, reduced to the eigenvalue equations of usual shell models identical for both the well alternatives. Compared with the usual models some improvements apply and the oscillator length is given automatically.     

The dyadic Green's function for the cylindrical chirostrip antenna

Using the eigenfunction expansion of dyadic Green's function in unbounded chiral medium, and the method of scattering superposition, the expressions of dyadic Green's function for the one and two—layer cylindrical chirostrip antennas are derived, when the field sources are placed in any layer of cylindrical chirostrip antenna. Also, using the method of saddle point integration, the asymptotic expression of dyadic Green's function for the wraparound chirostrip antenna is obtained. The results can be directly used to analyse the radiation characteristics of cylindrical chirostrip dipolc antenna, et al.     

Method for breaking chains of equations for green's functions

This article presents a new method for closing infinite systems of equations for Green's functions. Its basis is a procedure for simultaneously splitting two higher-order Green's functions. There is a particular relationship between the three Green's functions that result. These Green's functions are self-consistent. Known Green's functions are used to reconstruct the corresponding correlation functions. An example is provided by the problem of magnetization for a spin system in the Heisenberg model with first-and secondorder approximations in the spin-spin interaction. The first-order approximation corresponds to the molecular-field approximation. The second-order computation corresponds to obtaining general relations determining the magnetization of a system. V. I. Lenin Moscow State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 42–47, August, 1996.     

Poincaré-invariant gravitational field and equations of motion of two pointlike objects: The postlinear approximation of general relativity

Using a fast-motion approximation method we obtain the second-order gravitational field and equations of motion for two pointlike objects in algebraically closed form. A regularization procedure is used which is shown to guarantee the consistency of the approximation scheme. The equations of motion are then transformed within the framework of relativistic predictive mechanics into a system of ordinary differential equations.     

Dispersion and nonlinear frequancy shift of natural waves in the Bose–Einstein condensate

Quantum mechanics equations for a system of the Bose particles are represented in the form of material field equations. A nonlinear equation for the macroscopic one-particle wave function is derived. Using the Krylov–Bogolyubov–Mitropol’skii method for equations in partial derivatives, nonlinear waves in the Bose–Einstein condensate are investigated. In the cubic approximation, dispersion relations for waves are derived and nonlinear frequency shift is calculated in the first- and third-order approximations for the interaction radius.     

Influence of anisotropy of the diffusion coefficient on the asymptotic behavior of a spatial atom distribution in structures with complex geometry

Obtained by the method of continual integration in the optimal trajectory approximation is the asymptotic behavior of the Green's function of the Fick equation with anisotropic diffusion coefficient in structures with complex geometry when the domain boundary is impermeable to the diffusing particles. Using the expression found for the Green's function, topological properties of the equal-concentration surfaces, particularly the topological singularities of a diffusion p-n-junction, are determined by means of the given initial distribution of the diffusing atoms.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 9–12, May, 1988.