Uniform convergence of Green’s functions

Given a sequence of regular planar domains converging in the sense of kernel, we prove that the corresponding Green’s functions converge uniformly on the complex sphere, provided the limit domain is also regular, and the connectivity is uniformly bounded.     

Temperature Green’s functions in Fermi systems: The superconducting phase transition

We consider the model of an equilibrium Fermi system of arbitrary-spin particles with the density-densitytype interaction. Based on the microscopic Hamiltonian in the formalism of temperature Green’s functions, we find critical modes and construct an effective action describing a neighborhood of the phase transition point. A renormalization group analysis of the obtained model leads to the standard critical behavior indices for spin-1/2 fermions but shows that in the system of higher-spin fermions, a first-order phase transition occurs whose temperature exceeds the standard estimates for the temperature of a second-order phase transition.     

Two-time Green’s functions in superconductivity theory

Based on the method of the equations of motion for two-time Green’s functions, we derive superconductivity equations for different types of interactions related to the scattering of electrons on phonons and spin fluctuations or caused by strong Coulomb correlations in the Hubbard model. We derive an exact Dyson equation for the matrix Green’s function with the self-energy operator in the form of the multiparticle Green’s function. Calculating the self-energy operator in the approximation of noncrossing diagrams leads to a closed system of equations corresponding to the Migdal-Eliashberg strong-coupling theory. We propose a theory of high-temperature superconductivity due to kinematic interaction in the Hubbard model. We show that two pairing channels occur in systems with a strong Coulomb correlation: one due to the antiferromagnetic exchange in interband hopping and the other due to the coupling to spin and charge fluctuations in hopping within one Hubbard band. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 129–146, January, 2008.     

Special values of canonical Green’s functions

We give a precise formula for the value of the canonical Green’s function at a pair of Weierstrass points on a hyperelliptic Riemann surface. Further we express the ‘energy’ of the Weierstrass points in terms of a spectral invariant recently introduced by N. Kawazumi and S. Zhang. It follows that the energy is strictly larger than log 2. Our results generalize known formulas for elliptic curves.     

Two-time temperature Green’s functions in kinetic theory and molecular hydrodynamics: II. Equations for pair-interaction systems

We consider two approaches to the calculation of correlation functions for a system of particles with direct pair interaction. The first is based on a chain of equations that determines a Boltzmann-type kinetic equation; the second is based on a chain of molecular hydrodynamic equations. We demonstrate that the two approaches are equivalent in the sense that they completely describe the system under consideration. We discuss the advantages of the approach based on the molecular hydrodynamic equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 142–166, April, 1999.     

Two-dimensional Coulomb scattering of a quantum particle: Wave functions and Green’s functions

We solve the problem of the propagation of a charged quantum particle in a two-dimensional plane embedded in the three-dimensional coordinate space. We consider scattering of this particle by a stable Coulomb center situated in the same plane. We study the wave function of this particle, its Green’s function, and all radial components of these functions. We derive uniform majorant bounds on absolute values of these functions and find the wave function representation in terms of regular radial Coulomb functions and the scattering amplitude representation via partial phases. We obtain integral representations of the Greens’s function and all its radial components.     

Contour-ordered Green’s functions in stochastic field theory

We briefly review the functional formulation of the perturbation theory for various Green’s functions in quantum field theory. In particular, we discuss the contour-ordered representation of Green’s functions at a finite temperature. We show that the perturbation expansion of time-dependent Green’s functions at a finite temperature can be constructed using the standard Wick rules in the functional form without introducing complex time and evolution backward in time. We discuss the factorization problem for the corresponding functional integral. We construct the Green’s functions of the solution of stochastic differential equations in the Schwinger-Keldysh form with a functional-integral representation with explicitly intertwined physical and auxiliary fields.     

Equations for two-temperature Green’s functions for a Heisenberg ferromagnet in the mode-coupling approximation

We use the mode-coupling approximation to derive a closed system of equations for the Green’s functions for the transverse and longitudinal spin components taking the sum rules related to the finiteness of the spin into account exactly. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 197–207, January, 2008.     

3D Green’s functions for a transversely isotropic thermoelastic cone

We use the compact harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional Green’s functions of a steady point heat source on the apex of a transversely isotropic thermoelastic cone by three newly introduced harmonic functions. All components of thermoelastic field are expressed in terms of elementary functions and are convenient to use. When the apex angle 2α equals to π, the solution reduce to the important solution of semi-infinite body with a surface point heat source. Numerical results are given graphically by contours.     

Green’s functions for discrete mTH-order problems*

We investigate an mth-order discrete problem with additional conditions, described by m linearly independent linear functionals. We find the solution to this problem and present a formula and the existence condition of Green??s function if the general solution of a homogeneous equation is known. We obtain a relation between Green??s functions of two nonhomogeneous problems. It allows us to find Green??s function for the same equation, but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.     

The method of two-time Green’s functions in the magnetism theory of metals with strong correlations

We apply the Bogoliubov-Tyablikov method of two-time Green’s functions to a multielectron theory of metals in the framework of the models with strong correlations, which are the Hubbard model and the Vonsovskii s-d(f) exchange model. We discuss the atomic representation in the problem of describing electron states, the metal-insulator transition, the problem of noncoherent states, and the Kondo effect. We analyze the ferromagnetism criterion in systems of strongly correlated electrons, which differs drastically from the Stoner condition.     

Edge Green’s functions on a branched surface. Asymptotics of solutions of coordinate and spectral equations

The problems of diffraction by a slit or a strip having ideal boundary conditions, and some other problems, can be reduced to the problem of wave propagation on a multisheet surface by applying the method of reflections. Further simplifications of the problem can be achieved by applying an embedding formula. As a result, the solution of the problem with a plane wave incidence becomes expressed in terms of the edge Green’s functions, i.e., in terms of the fields generated by dipole sources localized at branchpoints of the surface. The present paper is devoted to finding the edge Green’s functions. For this problem, two sets of differential equations, namely, the coordinate and spectral equations, are used. The properties of solutions of these equations are studied. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 233–256.     

Baxter’s difference systems and orthogonal rational functions

Rational functions orthogonal on the unit circle are considered beginning with their recurrence formulas. Various summability conditions are imposed on the recurrence coefficients and the asymptotics of the solutions are studied and the orthogonality measure is recovered. The techniques developed by Baxter and Benzaid and Lutz are used.     

Two-dimensional Green’s functions for semi-infinite isotropic thermoelastic plane

Based on the two-dimensional steady-state governing equations of isotropic thermoelastic material and the compact general solution expressed in three harmonic functions, the corresponding three harmonic functions contain nine undetermined constants are constructed for a line heat source applied in the interior of a semi-infinite thermoelastic plane. All components of thermoelastic field in the semi-infinite plane can be derived by substituting the harmonic functions into the general solution. And the undetermined constants can be obtained by the compatibility conditions, equilibrium conditions and the different boundary conditions for extended Mindlin problem and extended Lorentz problem. Thus, the Green’s functions in above two cases are obtained, and the numerical results are given graphically by contours.     

Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.