Functions Of Differential Operators Research Articles

A functional-differential operator with involution

A functional-differential operator with involution

Description of spectral functions of differential operators with arbitrary deficiency indices

Description of spectral functions of differential operators with arbitrary deficiency indices

О базисах Рисса из собственных и присоединенных функций функционально-дифференциального оператора переменной структуры

О базисах Рисса из собственных и присоединенных функций функционально-дифференциального оператора переменной структуры

О сходимости средних Рисса разложений по собственным функциям функционально-дифференциального оператора на графе-цикле

О сходимости средних Рисса разложений по собственным функциям функционально-дифференциального оператора на графе-цикле

Differential inequalities for functional perturbations of first-order ordinary differential equations

The theory of differential inequalities plays a central role in the qualitative and quantitative study of differential equations. In this paper, we present several comparison results for a class of functional differential equations of first order with periodic boundary value conditions. The inequalities obtained are, generally speaking, of the following type: Pv ≤ 0 implies that v ≤ 0, where P is a functional differential operator subject to some boundary conditions, and v is an element of a prescribed space of functions. We first obtain several new results for the linear problem. Then, we consider a nonlinear differential equation as a functional perturbation of the original differential equation and give different comparison results. Our results improve and generalize previous estimates described in the literature.

A MeanValueFormula for Root Functions of Differential Operators with Locally Integrable Coefficients: I

A MeanValueFormula for Root Functions of Differential Operators with Locally Integrable Coefficients: I

Solvability and Regularity of Solutions for Some Classes of Elliptic Functional-Differential Equations

1. This review is devoted to the theory of boundary-value problems for some classes of elliptic functionaldifferential equations. Boundary-value problems for elliptic functional-differential equations have some astonishing properties. For example, unlike elliptic differential equations, the smoothness of generalized solutions can be broken in a bounded domain and is preserved only in some subdomains. The symbol of a self-adjoint semibounded functional-differential operator can change its sign. Boundary-value problems for elliptic functional-differential equations are closely related to nonlocal elliptic problems. On the other hand, elliptic functional-differential equations have important applications to nonlinear optics, elasticity theory, control theory, and diffusion processes. Elliptic functional-differential equations containing transformations of arguments were studied by A. B. Antonevich [2], D. Przeworska-Rolewicz [36], and V. S. Rabinovich [37]. The authors assumed that the transformations of arguments map a domain onto itself. Therefore, their results were similar to the well-known results for elliptic differential equations. The situation changes if the equation has these shifts in the highest derivatives, and the shifts map the points of the boundary into the domain. For an introduction to the theory of these equations, see A. L. Skubachevskii [67]. 2. In this paper, we mainly study boundary-value problems for elliptic differential-difference equations and for elliptic functional-differential equations with contractions and expansions. In order to illustrate the properties of elliptic differential-difference equations, we consider the following example:

Equiconvergence of biorthogonal expansions in root functions of differential operators: II

Equiconvergence of biorthogonal expansions in root functions of differential operators: II

On normality of some elliptic functional differential operators

On normality of some elliptic functional differential operators

Equivalence in Ln[0, 1] of the system ei2?kx (k=0, �1, ...) and the system of the eigenfunctions of an ordinary functional-differential operator

Equivalence in Ln[0, 1] of the system ei2?kx (k=0, �1, ...) and the system of the eigenfunctions of an ordinary functional-differential operator

Consistent and covariant gauge anomalies as solutions of the same extended algebraic problem

We show that the usual consistent and covariant gauge anomalies are different solutions of the same algebraic problem related to the complex noncompact extension of the vector and axial vector gauge group. They obtained as nontrivial integrals of the relevant Wess-Zumino consistency conditions as well as representations of the algebra of generators acting as functional differential operators upon a fermionic effective action extended to complex vector and axial vector potentials.

Functional analysis and two-point differential operators

Functional analysis and two-point differential operators

Necessary and sufficient conditions for invertibility of uniformly c-continuous functional-differential operators

Necessary and sufficient conditions for invertibility of uniformly c-continuous functional-differential operators

The Kac-moody and Virasoro commutation equations

The commutation equation of the Kac-Moody generators realized as functional differential operators is explicitly verified. The anomaly of the loop group generators is briefly discussed and the Virasoro commutation equation is then derived. No use is made of normal ordering in obtaining the central extensions of the above commutation equations.

Necessary and sufficient conditions for invertibility of nonautonomous functional-differential operators

Necessary and sufficient conditions for invertibility of nonautonomous functional-differential operators

An exact realization of the Kac-Moody algebra

The Kac-Moody generators are explicitly realized as functional differential operators expressed in terms of the coordinates of the (infinite-dimensional) group manifold. As an application of this realization, the two-dimensional Wess-Zumino chiral action is derived from the hamiltonian formulation.

On the Spectral Properties of a Class of Elliptic Functional Differential Operators Arising in Feedback Control Theory for Diffusion Processes

In this paper Dirichlet boundary value problems are considered for certain operators of the form $L + \Pi $, where L is a 2nd order, elliptic, formally self-adjoint $PDO$ and $\Pi $ is a feedback operator with finite-dimensional range.>The results concern mainly the resolvent $(L + \Pi - \lambda )^{ - 1} $, the analyticity of the semigroup generated by $L + \Pi $, the location of the spectrum $\sigma $$\sigma (L + \Pi )$ and the completeness of the eigenspaces and the eigenprojections associated to a $(L + \Pi )$.As a consequence of the completeness of the eigenprojections it is possible to derive quite a number of interesting formulas of the “resolution of the identity” type.

Abschätzungssätze für parabolische Funktional-Differential-Operatoren mit Funktional-Randbedingungen

For nonlinear parabolic functional-differential operators with nonlinear functional-boundary conditions of first and third kind, involving time-derivatives on the boundary, a uniform conception for one- and two-sided estimations on unbounded domains is introduced. In the infinity variable growth conditions are given. As a conclusion stability, uniqueness and a maximum principle follow.

Schrödinger representation and Casimir effect in renormalizable quantum field theory

We show that the Schrödinger representation exists in renormalizable quantum field theory to all orders in the perturbation expansion. In this sense, completeness of the Schrödinger states also holds. However, the field operator that is being diagonalized on a smooth three-dimensional hypersurface differs from the usual renormalized one by a factor that diverges logarithmically if the distance from the hypersurface goes to zero. This requires a limit procedure to be employed if expectation values of the renormalized field operator are to be computed in this representation. The Schrödinger functional differential operator involves point splitting Δ and has coefficients depending logarithmically on Δ, and also some by factors Δ −1, Δ −2, Δ −4. Details are given for the massless φ 4 4 theory, but the extension to other models, in particular with spin- 1 2 fermions, is outlined. The Casimir potential for disjoint surfaces is shown to be finite to all orders in the perturbation expansion, and computed for a pair of parallel plates to first order in massless φ ν 4.

Notes on generalized boundary value problems in Banach spaces. I. Adjoint and extension theory

Abstract : Multipoint boundary conditions arise in the theory of beams or plates with interior point loads, and also the mathematical theory of splines. Interface conditions arise in problems of diffusion through parallel 'slabs' with different properties (e.g., nuclear reactors or the study of shock waves). Adjoints of such differential operators also are encountered when one attempts to derive Euler-Lagrange equations for constrained minimization problems. Our method is very general and is designed to work for partial differential, integral and functional differential operators as well as differential operators. Part I presents some of the abstract machinery to solve the problem. Part II will apply this machinery to concrete and applied problems of the type mentioned above.