System Of Root Functions Research Articles

Criteria for the Uniqueness of a Solution to a Differential-Operator Equation with Non-Degenerate Conditions

In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and B. The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers, the criterion for the uniqueness of the boundary value problem formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required.

Inverse problems of determining an unknown depending on time coefficient for a parabolic equation with involution and periodicity conditions

The solution of the investigated problem with an unknown coefficient in the equation was constructed by using the method of separation of variables. The properties of the induced spectral problem for the second-order differential equation with involution are studied. The dependence on the equation involutive part of the spectrum and its multiplicity as well as the structure of the system of root functions and partial solutions of the problem were investigated. The conditions for the existence and uniqueness of the solution of the inverse problem have been established. To determine the required coefficient, Volterra's integral equation of the second kind was found and solved.

Uniform Basis Property of Root Vector Systems of Dirac Operators

On the segment [0, π], we study the one-dimensional Dirac operator ℒ with boundary conditions U regular in the Birkhoff sense and with a complex-valued summable potential P = (pij(x)), i, j = 1, 2.We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator ℒ, assuming that boundary conditions U and the number \( \int_0^{\pi}\left({p}_1(x)-{p}_4(x)\right) dx \) are fixed and the potential P takes values from the ball B(0, R) of radius R in the space Lϰ, ϰ > 1. We succeed to select the system of root functions such that it consists of eigenfunctions of the operator ℒ apart from a finite set of root vectors such that their number can be estimated uniformly with respect to the ball ‖P‖ϰ ≤ R.

On a Problem that Does not Have Basis Property of Root Vectors, Associated with a Perturbed Regular Operator of Multiple Differentiation

A spectral problem for a multiple differentiation operator with integral perturbation of boundary value conditions which are regular but not strongly regular is considered in the paper. The feature of the problem is the absence of the basis property of the system of root vectors. A characteristic determinant of the spectral problem is constructed. It is shown that absence of the basis property of the system of root functions of the problem is unstable with respect to the integral perturbation of the boundary value condition

On Multiple Completeness of Root Functions for Ordinary Differential Polynomial Pencils with Constant Coefficients

In the space of square integrable functions on a finite segment, we consider a class of polynomial pencils of nth-order ordinary differential operators with constant coefficients and two-point boundary conditions (at the edges of the segment). We suppose that all roots of the characteristic equations of pencils of the said class are simple and nonzero. We find sufficient conditions for the m-multiple completeness (1 ≤ m ≤ n) of the system of root functions of pencils from the specified class in the space of square integrable functions on the said segment.

On Bessel property of the system of root functions of the second order differential operator

In this paper studied the Bessel property of the system of root functions of a second order differential operator with summable coefficients. Established the criterion of Bessel property in L 2 (G), G = (0,1) of the system of root functions of the given operator.

Riesz Inequality for the System of Root Functions of Second Order Ordinary Differential Operator

An ordinary differential operator of second order with coefficients is considered. The Riesz property of the system of root functions of the given operator is studied. The criterion of Bessel property in 2 L , of root functions system is established and use it to obtain sufficient conditions for the Riesz property of a system of normalized root functions of this operator in p L .

Nonlocal Multipoint Problem with Multiple Spectrum for an Ordinary (2n)TH Order Differential Equation

We study the spectral properties of a nonself-adjoint problem for a (2n )th order differentiation operator with nonlocal conditions obtained as perturbations of regular but not strongly regular conditions and generalizing the periodicity conditions. The cases of problems with regular and irregular (in Birkhoff’s sense) perturbed boundary conditions are analyzed. The system of root functions of the multipoint problem is constructed. We also establish sufficient conditions under which this system is complete and forms a Riesz basis under certain additional assumptions.

Basis properties of root functions of a regular fourth order boundary value problem

In this paper, we consider the following boundary value problem\[ y^{(4)}+q(x) y=\lambda y,~\ \ \ 0<x<1, \]\[ y^{\prime\prime\prime}\left(1\right)-\left(-1\right)^{\sigma}y^{\prime\prime\prime}\left(0\right)+\alpha y\left(0\right) =0, \]\[ y^{(s)}(1) -( -1) ^{\sigma}y^{(s) }( 0) =0,\ \ \ s=\overline{0,2}, \]where $\lambda $ is a spectral parameter, $q( x)\in L_{1}(0,1)$ is complex-valued function and $\sigma =0,1$. The boundary conditions of this problem are regular but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. When $\alpha\ne 0$, we proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a Riesz basis in the space $L_{2}( 0,1)$. Furthermore, we show that the system of root functions forms a basis in the space $L_{p}( 0,1)$, $1<p<\infty$ $(p\neq 2)$, under the conditions $\alpha\ne 0$ and $q( x) \in W_{1}^{1}( 0,1)$.

Some results on eigenvalues of finite type, resolvents and Riesz projections

The present paper considers a separable Hilbert space H and a bounded linear operator A:H↦H that has an eigenvalue of finite type at λ0∈C. Using the image and the kernel of certain operators that are defined recursively starting from λ0−A, a construction of the local Smith form and extended canonical system of root functions of λ−A at λ0 is provided. This allows to express the partial multiplicities, the algebraic multiplicity, the Jordan structure and Jordan chains of A at λ0 in terms of the quantities defined by the recursion. Similarly, formulas for the rank, image and kernel of the operators in the principal part of the resolvent (λ−A)−1 at λ0, and thus also of the Riesz projection, are given in terms of the quantities defined by the recursion.

The System of Root Functions is a Basis in a Problem with a Multiple Eigenvalue

We consider a problem that describes the biorthogonal adjoint system for the classical system of root functions for a loaded string with one free end in the presence of a multiple eigenvalue. We show that the complete and minimal subsystems isolated from the system of root functions constitute a basis in the presence of one or two associated functions.

Convergence of Spectral Decompositions for a Singular Differential Operator with General Boundary Conditions

We investigate the general boundary-value problem for the operator lu = −u′′ + q(x)u , 0 < x < 1, If the complex-valued coefficients q(x) is summable on (0,1), the integral $$ {\int}_0^1x\left(1-x\right)\left|q(x)\right| dx $$ converges. The asymptotic solutions of the equation lu = μ2u derived in this article are used to construct the asymptotic spectrum of the problem, to classify the boundary conditions, and to prove theorems asserting that the system of root functions is complete and forms an unconditional basis in L2 (0,1).

ON THE MINIMALITY OF SYSTEMS OF ROOT FUNCTIONS OF THE LAPLACE OPERATOR IN THE PUNCTURED DOMAIN

In this paper, we consider the Laplace operator in a punctured domain, which generates a class of new correctly solvable boundary value problems. And for this class of problems the resolvent formula is obtained. Also described are meromorphic functions that generate the root functions of the class of problems studied. The main goal is to study the minimality of root function systems. The paper is a continuation of [8], where a description is given of correctly solvable boundary value problems for the Laplace operator in punctured domains The Laplace operator in the punctured domain, which generates the class of new correctly solvable boundary value problems, is considered, and the resolvent formula is obtained for the generated problems, and meromorphic functions are described that induce systems of functions. One of these systems is a system of eigenfunctions and associated functions. The last section is devoted to the study of the minimality of the system of root functions.

Uniform Basis Property of the System of Root Vectors of the Dirac Operator

We study one-dimensional Dirac operator L on the segment [0,π] with regular in the sense of Birkhoff boundary conditions U and complex-valued summable potential P=(pij(x)), i,j=1,2. We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator L assuming that boundary-value conditions U and the number ∫(p1(x)-p4(x))dx are fixed and the potential P takes values from the ball B(0,R) of radius R in the space Lϰ for ϰ&gt;1. Moreover, we can choose the system of root functions so that it consists of eigenfunctions of the operator L except for a finite number of root vectors that can be uniformly estimated over the ball ∥P∥ϰ≤R.

Spectral Properties of Nonself-Adjoint Nonlocal Boundary-Value Problems for the Operator of Differentiation of Even Order

We study spectral properties of an essentially nonself-adjoint problem generated by nonlocal multipoint conditions for the operator of differentiation of order 2n and analyze the cases of regular and irregular Birkhoff two-point boundary conditions. A system of root functions of the problem and elements of biorthogonal systems are constructed. We also establish sufficient conditions under which these systems are complete and conditions under which they form a Riesz basis (under certain additional assumptions).

The Basis Properties of the System of Root Functions of the Oblique Derivative Problem

The Basis Properties of the System of Root Functions of the Oblique Derivative Problem

The Basis Property of the System of Root Functions of the Oblique Derivative Problem

The oblique-derivative spectral problem for the Laplacian in a disk D is studied. The asymptotic properties of its eigenvalues are established, and the system of root functions of this problem is proved to be a basis with brackets in L2 (D).

On Multiple Completeness of the Root Functions of Ordinary Differential Polynomial Pencil with Constant Coefficients

In the space of square integrable functions on a finite segment we consider a class of polynomial pencils of nth-order ordinary differential operators with constant coefficients and two-point boundary-value conditions (at the edges of the segment). We suppose that roots of the characteristic equation of pencils of this class are simple and nonzero. We establish sufficient conditions for m-multiple completeness (1≤m≤n) of the system of root functions of pencils from this class in the space of square integrable functions on this segment.

Basis Properties in $${\varvec{L}}_{{\varvec{p}}}$$ L p of Root Functions of Sturm–Liouville Problem with Spectral Parameter-Dependent Boundary Conditions

In this paper, we consider the Sturm–Liouville problem with spectral parameter in the boundary conditions. We associate this problem with a self-adjoint operator in the Pontryagin space \(\Pi _{2}\). Using this operator-theoretic formulation and analytic methods, we study the basis properties in the space \(L_{p} (0,1),\,1<p < \infty \), of systems of root functions of this problem.

Spectral properties of the complex airy operator on the half-line

We prove a theorem on the completeness of the system of root functions of the Schrodinger operator L = −d 2/dx 2 + p(x) on the half-line R+ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L c = −d 2/dx 2 + cx, c = const, implies the completeness of the system of eigenfunctions of L c for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6.