Self-consistent cluster theory of the electron spectra of alloys

A method of self-consistent cluster expansion for the mass operator of the Green's function with allowance for short-and longrange order in alloys is proposed. A self-consistent system of equations for the coherent potential and the mass operator is obtained in the approximation in which the contributions of processes involving electron scattering by clusters of three or more atoms are ignored. The contributions of the scattering processes to the Green's function of the alloy decrease with increasing number of particles in a cluster and can be estimated by means of a certain small parameter. Analytic and numerical investigations of the energy dependence of the density of single-electron states of a binary alloy are made for different values of the parameter of the short-range order.Institute of Metal Physics, Ukrainian Academy of Sciences; Kiev University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 2, pp. 304–319, November, 1993.     

Preconditioning the Poincaré-Steklov operator by using Green's function

This paper is concerned with the Poincaré-Steklov operator that is widely used in domain decomposition methods. It is proved that the inverse of the Poincaré-Steklov operator can be expressed explicitly by an integral operator with a kernel being the Green's function restricted to the interface. As an application, for the discrete Poincaré-Steklov operator with respect to either a line (edge) or a star-shaped web associated with a single vertex point, a preconditioner can be constructed by first imbedding the line as the diameter of a disk, or the web as a union of radii of a disk, and then using the Green's function on the disk. The proposed technique can be effectively used in conjunction with various existing domain decomposition techniques, especially with the methods based on vertex spaces (from multi-subdomain decomposition). Some numerical results are reported.

     

Linear self-adjoint multipoint boundary value problems and related approximation schemes

Summary Linear self-adjoint multipoint boundary value problems are investigated. The case of the homogeneous equation is shown to lead to spline solutions, which are then utilized to construct a Green's function for the case of homogeneous boundary conditions. An approximation scheme is described in terms of the eigen-functions of the inverse of the Green's operator and is shown to be optimal in the sense of then-widths of Kolmogorov. Convergence rates are given and generalizations to more general boundary value problems are discussed.     

Solution to a Class of Coupled Linear Partial Differential Equations

The solution to a coupled system of partial differential equationsinvolving a general linear time-independent operator L is presented.Examples of these equations include coupled diffusion equationsor coupled convection–dispersion equations. The solutionconsists of a convolution of the Green's function appropriatefor the operator L and a function independent of the operatorL. The method enables one to write software to calculate thesolution to a wide range of problems. The change of solutionupon changing the problem often only involves a substitutionof the Green's function. A specific example of physical significanceis given.     

Diffusion in layered media at large times

The large-time asymptotic behavior of the Green's function for the one-dimensional diffusion equation is found in two cases: 1) the potential is a function with compact support; 2) the potential is a periodic function of the coordinates. In the first case, the asymptotic behavior of the Green's function can be expressed in terms of the elements of theS matrix of the corresponding Schrödinger operator for negative values of the energy on the spectral plane. In the second case, the asymptotic behavior can be expressed in terms of Floquet-Bloch functions of the corresponding Hille operator at negative values of the energy on the spectral plane. The results are used to study diffusion in layered media at large times. The case of external force is also considered. In the periodic case, the Seeley coefficients are found.Institute of Problems of Mechanical Engineering, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 106–148, January, 1994.     

Green's functions for polynomials in the Laplacian

A Green's function is constructed for an arbitrary polynomial in theN-dimensional Laplacian operator, subject only to the condition that no root of the polynomial may be real and negative.     

Killed Random Processes and Heat Kernels

Let V(x) ≥ 0 be a given function tending to a constant at infinity. It is well known that the density of the Brownian motion B_t killed at the infinitesimal rate V is a Green's function for the heat operator with such a potential. With an appropriate generalization, its Laplace transform also gives the density of ∫ ₀ ^t V(B_s)ds. We construct such a Green's function via spectral analysis of the classical one-dimensional stationary Schrodinger operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 423–432, August, 2005. An erratum to this article is available at .     

Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments

In this paper we obtain the expression of the Green's function related with a first order periodic differential equation with piecewise constant argument. We derive comparison results for the treated linear operator by studying the sign of the obtained Green's function.     

On the operator representation of the Green's functions of the dynamic theory of elasticity

Starting from the operator notation for the dyadic Green's function for elastic displacement of the nonstationary theory of elasticity, we propose a method of factoring the mutually orthogonal components of the Green's function in the form of tensor products of operators acting respectively on the coordinates of the points of observation and the source. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 78–80.     

Linear Second Order Ordinary Differential Equations with Nonsmooth Coefficients

This paper deals with a Dirichlet boundary value problem for a linear second order ordinary differential operator, whose coefficients belong to certainL^p-spaces. Its solution is to be understood in the sense of Sobolev, so that the Fredholm alternative holds. The main purpose of this paper is, in case of unique solvability, to introduce a Green's function by means of which the solution can be given explicitly by integrals. We give the precise definition of the Green's function via Riesz' Representation Theorem and establish some of its basic properties. As a preliminary tool the Cauchy initial value problem is considered.     

Local Conservation Laws and the Hamiltonian Formalism for the Ablowitz–Ladik Hierarchy

We derive a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik (AL) hierarchy. Our methods rely on a recursive approach to the AL hierarchy using Laurent polynomials and on asymptotic expansions of the Green's function of the AL Lax operator, a five-diagonal finite difference operator.     

Molecular hydrodynamics of degenerate weakly nonideal bose gas. I. Generalized equations of two-fluid hydrodynamics

A degenerate weakly nonideal Bose gas is investigated at temperatures near zero. Relations connecting the irreducible Green's functions are used to obtain exact expressions for the two-time temperature-dependent Green's functions. In the case of weak nonideality an expression possessing interpolation properties with respect to the frequency and momentum is obtained for the density—density Green's function. At low frequencies, the results of two-fluid hydrodynamics are reproduced. At wavelengths less than the mean free path the energy of the elementary excitations and the damping obtained in the paper agree with the results of perturbation theory with respect to the coupling constant. An expression for the operator of the superfluid velocity is obtained.In memory of Nikolai Nikolaevich BogolyubovV. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 412–465, December, 1992.     

Averaging the resolvent with a space-correlated random potential

The problem of the quasi-particle spectrum in a binary disordered alloy with a space-correlated random potential is considered. The extended space formalism is used to represent the average resolvent. To calculate the mass operator, some self-consistent approximation procedures are suggested that coincide with the well-known self-consistent approximations for α=0 (where α is the short-range order parameter). The elaborated theory ensures the correct passage to the Green's function of a perfect crystal in the limits α→1 and α→−1 for any concentration and 50% concentration, respectively. The approximations possess the correct analytic properties for all values of the parameter α. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 2, pp. 296–313, February, 1998.     

A generalized Green's formula for elliptic problems in domains with edges

The usual Green's formula connected with the operator of a boundary-value problem fails when both of the solutions u and v that occur in it have singularities that are too strong at a conic point or at an edge on the boundary of the domain. We deduce a generalized Green's formula that acquires an additional bilinear form in u and v and is determined by the coefficients in the expansion of solutions near singularities of the boundary. We obtain improved asymptotic representations of solutions in a neighborhood of an edge of positive dimension, which together with the generalized Green's formula makes it possible, for example, to describe the infinite-dimensional kernel of the operator of an elliptic problem in a domain with edge. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 106–147.     

Linear differential systems with infinitely many boundary points

Summary The article extends results previously known for boundary value problems involving only a finite number of boundary points to those which involve an infinite number of (possibly dense) boundary points. Specifically, the system , is discussed in the Hilbert space L²(0, 1). Suitable conditions for inverting the operator L are found, and the Green's function is exhibited. It is shown to have the standard properties as well as some which are new, when considered as a function of its second variable It is further shown to be the limit a.e. of Green's function for problems involving only a finite number of boundary points, as those points increase in number. Finally it is shown that L⁻¹ is compact. By using the Green's function the domain of L is shown to be dense in L²(0, 1). and the adjoint L* and its domain are found. L is also shown to be closed. Lastly, by using some theorems concerning entire functions, the eigenvalues of L are shown to lie in a vertical strip with infinity as their only limit point. This in turn implies that if L⁻¹ fails to exist, a slight perturbation in P will result in an invertible L, and the assumption made earlier concerning the existence of the Green's function is reasonable. Entrata in Redazione il 25 febbraio 1971.     

On variations in the solutions of integro-differential equations of mixed problems of elasticity theory for domain variations

The behaviour of the solution of the boundary value problem for a pseudodifferential equation (PDE), Green's function of this problem, and also some of their local and global characteristics, during variation of the domain is investigated. Formulas are proposed that enable the solution of a broad class of PDE in a domain to be expressed in terms of the solution in the near domain. Local characteristics of the solution are expressed in terms of the local characteristics of the solution in the near domain. A double asymptotic form of Green's function for both arguments tending to the domain boundary occurs in the variation formula. The variation of this double asymptotic form as the domain varies is expressed in terms of this same asymptotic form. The system of variation formulas obtained is closed. It enables the PDE solution in the domain to be reduced to the solution of an ordinary differential equation in functional space. The local characteristics of the solution can also be found by this method without calculating the solution itself. If there is sufficient symmetry in the initial operator, then conservation laws in the Noether sense are obtained for its Green's function and its asymptotic form. The behaviour of the quantities under investigation is studied under inversion.

The investigation of variations of the solutions of problems for the variation of the domain occurs in the paper by Hadamard /1/, who studied the variation in conformal mapping and obtained a formula similar to (1.4). The formula for the variation of the solution of the boundary value problem for an elliptic differential equation is obtained in /2/. Variation formulas for the case of the operator of the problem about a crack and a circular domain are obtained in /3, 4/. The Irwin formula /5/ is obtained from formulas (1.4) and (1.21) by substitution.     

Identities for eigenfunctions of ordinary differential operators

The paper contains formulae for regularized k-sums of residues of the Green's function for an ordinary differential operator with regular boundary conditions.Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 107–117, 1984.The author would like to express his gratitude to Victor Antonovich Sadovnichii for his interest in this work.     

Positive solution for m-point boundary value problems

In this paper, we investigate an m-point boundary value problem with sign changing nonlinearity. The existence of an interval of parameters which ensures the problem has at least one positive solution is determined by constructing available operator and combining the method of lower solution with the method of topology degree. Moreover, the associated Green's function for the above problem is also given.     

A generalized Green's function

Summary We present a method in which the given boundary conditions are replaced by integral boundary conditions, so chosen that the Green's function can be constructed without changing the principal spectral properties of the operator.Translated from Litovskii Matematicheskii Sbornik, Vol. 13, No. 1, pp. 109–114, January–March, 1973.