The method of exact diagonalization preserving the total spin and taking the point symmetry of the two-dimensional isotropic Heisenberg magnet into account

We propose a double-pass method of exact diagonalization of a finite cluster on the base of functions that have a definite total spin and transform by a definite irreducible representation of the point symmetry group of the lattice. We also propose the method for approximating the energy spectrum in the thermodynamic limit using the spectrum of the surrounding states, which increases the calculation accuracy leaving the cluster size invariant. The algorithm details are extensively illustrated with an example of clusters of spin 1/2 on a simple square lattice. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 262–280, November, 2006.     

Parapositronium atom in the field of annihilation and optical photons as a nonlinear quantum system

A parapositronium atom in an optical laser field is described beyond the perturbation theory framework by a closed system of Heisenberg equations on operators of atoms and photons. Wwe consider the annihilation of the parapositronium atom, which starts from one or two quantum states; optical quantum transitions between these states are caused by one or two optical photons. Mean occupation numbers of these states are governed by a system of two nonlinear equations. We investigated particular stationary and nonstationary solutions of this system and find that annihilation photons substantially affect the annihilation process. We show that definite optical laser radiation may stabilize the parapositronium atom and make its lifetime hundreds of times longer than the lifetime of the free parapositronium atom in the 1s state. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 148–168, July, 2000.     

The heat equation on a matrix space

On the space of real matrices, an analog of the heat equation is studied. On the basis of properties of a heat source, the solution of the Cauchy problem with an initial radiation defined at the vertex of the cone of positive definite symmetric matrices is constructed. A relation between the Riesz potentials associated with composite power functions on a matrix space and the solutions of the heat equation is established. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 268–285.     

Existence of Lyapunov functions of variable sign for linear extensions of dynamical systems on a torus

The problem of the existence of quadratic forms that have a positive definite derivative along the solutions of linear extensions of dynamical systems on a torus is considered. Assuming the existence of quadratic forms whose derivative is positive definite only with respect to part of the variables, conditions ensuring the existence of a quadratic form whose derivative is already positive definite with respect to all variables are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1713–1717, December, 1990.     

Radiation beyond four space-time dimensions

We present a set of formulas describing classical radiation of a rank-s tensor field from an accelerated pointlike source in a flat space-time of an arbitrary even dimension d. These formulas allow straightforwardly and algorithmically evaluating the total intensity and radiated momentum for any s and d by hand or using a computer. The practical application of the obtained results is limited for s > 1 because the energy-momentum tensor for the pointlike source is not conserved. This usually means that contributions to the radiation from tensions of the forces causing the acceleration of the radiation source cannot be neglected. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 282–291, August, 2008.     

Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model

The work gives a consistent and uniform exposition of all known results related to Heisenberg model. The classification of excitations is presented and their scattering is described both in ferromagnetic and the antiferromagnetic cases. It is shown that in the antiferromagnetic case there exists only one excitation with spin 1/2 which is a kink in the following sense: in physical states there is only an even number of kinks-spin waves, therefore they always have an integer spin. Thus, it is shown that the conventional picture of excitations Is wrong in the antiferromagnetic case and the spin wave has spin 1/2, matrix is calculated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 134–178, 1981.     

Describing spinors using probability distribution functions

Irreducible representations of the rotation group are realized using a family of positive probability distributions of the spin projections for an arbitrary value of the spin. The family is parametrized by the points on the sphere. An invertible mapping of the spinors onto the probability distribution functions is constructed. Examples of probability distributions for the well-known states with the spins 1/2 and 1 are presented. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 185–198. May, 1998.     

On extensions of positive definite integral Fredholm operators

The main result of this paper states that a positive definite Fredholm integral operator acting on L²([0,1]) can be modified on a Lebesque measurable set D\mit\Delta in [0,1]² such that the resulting operator is positive definite and its resolvent kernel is zero on D\mit\Delta . This answers a question raised in [3]. The proof is based on extension results for positive definite operator matrices and their connection to generalized determinants.     

The acoustic field of a high-frequency source moving in a waveguide

The acoustic field of a source moving at a subsonic velocity in a regular waveguide with perfectly reflecting boundaries is considered. The acceleration of the source is assumed to be small. In a moving coordinate system, the asymptotics of the wave field is obtained. This asymptotics is inapplicable near the critical cross sections, for which the Doppler frequency of the source coincides with the frequency of the waveguide mode under consideration. It is demonstrated that, in this case, the wave field can be represented locally by a special type of integral, which is analyzed by the saddle-point method.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 71–78.This work was supported in part by the Russian Foundation for Basic Research under grant 95-01-01285a.     

Equilibrium in quantum systems of particles with magnetic interaction

We compute density matrices determining equilibrium states in the thermodynamic limit for a class of quantum systems. Interaction in these systems is described as a collective electromagnetic field whose scalar potential is equal to zero.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1686–1691, December, 1995.     

Electron in the Aharonov-Bohm potential and in the Coulomb field in 2+1 dimensions

We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006. An erratum to this article is available at .     

On norm dependent positive definite functions

In the first part of the paper we prove a decomposition theorem for positive definite functions (Theorem 2.3) generalizing a result of de Leeuw and Glicksberg. Using this theorem, we then show (Theorem 3.1) that certain norm dependent positive definite functions are automatically continuous at every point different from zero.     

Classification of the string solutions of Bethe equations in an XXZ model of arbitrary spin

One investigates the structure of the string solutions of a Heisenberg XXZ model of arbitrary spin. It is discovered that there exist values of the spin for which there exist values of the spin for which there arise constraints on the disposition of the centers of certain string solutions. One finds a class of spins, commensurable in a definite manner with the anisotropy parameter, for which such constraints do not arise. One gives a hypothetical formulation of the classification of the string solutions over a ferromagnetic vacuum in the general case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 31–36, 1985.     

Orbital topological classification of analytic differential equations in a neighborhood of a degenerate elementary singular point on the two-dimensional complex plane

We consider the germ of an analytic differential equation in (C², 0) defined by a vector field with a singularity at zero. A singular point of the equation is called degenerate elementary if one eigenvalue of the linear part of the field is equal to zero and the other is different from zero. The orbital topological classification of differential equations in a neighborhood of a degenerate elementary singular point is obtained for all cases except the so-called Liouville case. This case defines a subset of equations which does not separate the corresponding function space.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 137–165, 1988.     

Estimate of the precision of determination of elastic parameters by the diffraction tomography method with the use of the finite-difference method (the SV-problem)

In the present paper, the finite-difference method is used for the solution of the direct problem, which enables us to take into account the diffraction phenomenon on local inhomogeneities of not weak contrast. Examples that allow us to estimate the precision of restoration of the parameters of such inhomogeneities, depending on the degree of their contrast, are given. The possibility of restoring the parameters of a local inhomogeneity in a medium containing interfaces with stepwise change of elastic properties for the case where source and receiver points are located on the free surface is demonstrated. An example of the numerical estimation of the precision of calculation of a scattered field in the Born approximation, approximating the wave field by the zero term of the ray method, is offered. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 136–152. Translated by Yu. V. Kiselev     

Tomography of Spin States, the Entanglement Criterion, and Bell's Inequalities

We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 172–185, January, 2006.     

Quantization of planar ferromagnets in the Chern-Simons representation

We formulate the two-dimensional planar classical continuous Heisenberg spin model as a constrained Chern-Simons gauged nonlinear Schrödinger system. Several physical consequences in the framework of the anyon field theory are discussed. We study the Hamiltonian structure of the model, which is quantized using the gauge invariant approach. A preliminary study of the quantum states is presented.Dipartimento di Fisica dell'Universita' and Sezione INFN, 73100 Lecce, Italy. Joint Institute for Nuclear Research, 141980 Dubna, Russia. Translated from Teoreticheskaya i Matematicheskaya, Fizika, Vol. 99. No. 3, pp. 450–461, June, 1994.     

The fundamental theorem on the relation between spin and statistics

The fundamental theorem on the connection of spin and statistics is deduced from the basic symmetry between emission and absorption processes in quantum field theory. The new theorem which prescribes the physical connection of Bose fields with integral spin and Fermi fields with half-integral spin, is valid for all field theories; and it contains Pauli’s theorem as a special case.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

Decomposition of variables and duality in non-Abelian models

We consider inhomogeneous current states in low-dimensional systems characterized by spatial separation of phase states with ordered spin and charge degrees of freedom. We show that near the self-duality point in the Ginzburg-Landau spinor model, the inhomogeneity degree of non-Abelian states is higher than that of states with an Abelian distribution of degrees of freedom. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 510–517, June, 2007.