System Of Root Vectors Research Articles

On the completeness property of root vector systems for 2 × 2 Dirac type operators with non-regular boundary conditions

The paper is concerned with the completeness property of a system of root vectors of a boundary value problem for the following 2×2 Dirac type equationLy=−iB−1y′+Q(x)y=λy,y=col(y1,y2),x∈[0,1],B=diag(b1,b2),b1<0<b2,andQ∈W1n[0,1]⊗C2×2, subject to general non-regular two-point boundary conditions Cy(0)+Dy(1)=0. If b2=−b1=1, this equation is equivalent to the one dimensional Dirac equation.We establish a new completeness result for the system of root vectors of such boundary value problem with non-regular and even degenerate boundary conditions. We also present several explicit completeness results in terms of values Q(j)(0) and Q(j)(1). In the case of degenerate boundary conditions and the analytic Q(⋅), the criterion of the completeness property is established. We demonstrate our results on the explicit example of a complete system of vector quasi-exponential polynomials.Applications to the spectral synthesis for Dirac type operators are discussed. Moreover, applications to the completeness property for the damped string equation are provided.

On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function px, which belongs to the space L20,1. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when px≡0) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions px, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L20,1, is dense within the space L20,1. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.

Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

AbstractThe paper is concerned with the Bari basis property of a boundary value problem associated in with the following 2 × 2 Dirac‐type equation for : with a potential matrix and subject to the strictly regular boundary conditions . If , this equation is equivalent to one‐dimensional Dirac equation. We show that the normalized system of root vectors of the operator is a Bari basis in if and only if the unperturbed operator is self‐adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let , , boundary conditions be strictly regular, and let be the sequence biorthogonal to the normalized system of root vectors of the operator . Then, These abstract results are applied to noncanonical initial‐boundary value problem for a damped string equation.

Dilation, model, scattering and spectral problems of second-order matrix difference operator

In the Hilbert space ?2 ?(Z; E) (Z := {0,? 1, ? 2, ...}, dim E = N &lt; ?), the maximal dissipative singular second-order matrix difference operators that the extensions of a minimal symmetric operator with maximal deficiency indices (2N, 2N) (in limit-circle cases at ? ?) are considered. The maximal dissipative operators with general boundary conditions are investigated. For the dissipative operator, a self-adjoint dilation and is its incoming and outgoing spectral representations are constructed. These constructions make it possible to determine the scattering matrix of the dilation. Also a functional model of the dissipative operator is constructed. Then its characteristic function in terms of the scattering matrix of the dilation is set. Finally, a theorem on the completeness of the system of root vectors of the dissipative operator is proved.

An abstract theorem on completeness of systems of root vectors of correct restrictions

An abstract theorem on completeness of systems of root vectors of correct restrictions

Spectral analysis of the multidimensional diffusion operator with random jumps from the boundary

We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.

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

Spectral Analysis of Singular Matrix-Valued Sturm–Liouville Operators

In the Hilbert space $$L_{W}^{2}([a,b);E)$$ ( $$-\infty<a<b\le +\infty ,$$ $$\dim E=N<+\infty ,$$ $$W>0$$ ) a space of boundary values of the symmetric singular matrix-valued Sturm–Liouville operator with maximal deficiency indices (2N, 2N) (in limit-circle case at singular end point b) is constructed. With the help of the boundary conditions at a and b, all maximal dissipative, maximal accumulative and self-adjoint extensions of such a symmetric operator are established. In particular, the maximal dissipative operators with separated boundary conditions, called ‘dissipative at a’ and ‘self-adjoint at b’ are investigated. A self-adjoint dilation of the dissipative operator is constructed and then its incoming and outgoing spectral representations are determined. This representation allows us to determine the scattering matrix of the dilation with the help of the Weyl matrix-valued function of a self-adjoint matrix-valued Sturm–Liouville operator. Further a functional model of the dissipative operator is determined and its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl function) is established. Finally, a theorem on completeness of the system of root vectors of the dissipative operator is proved.

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.

Extensions, dilations, and spectral problems of singular Hamiltonian systems

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit‐point case at a(b) and limit‐circle case at b(a)) acting in the Hilbert space In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self‐adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at a” and “dissipative at b.” For 2 cases, a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl‐Titchmarsh function of the corresponding self‐adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.

The one‐dimensional Schrödinger operator on bounded time scales

In this paper, we consider the one‐dimensional Schrödinger operator on bounded time scales. We construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self‐adjoint, and other extensions of the dissipative Schrödinger operators in terms of boundary conditions. In particular, using Lidskii's theorem, we prove a theorem on completeness of the system of root vectors of the dissipative Schrödinger operators on bounded time scales. Copyright © 2016 John Wiley &amp; Sons, Ltd.

On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications

The paper is concerned with the completeness property of root functions of general boundary value problems for n \times n first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper [37] we substantially relax the assumptions on boundary conditions guaranteeing the completeness of root vectors, allowing them to be non-weakly regular and even degenerate. Emphasize that in this case the completeness property substantially depends on the values of a potential matrix at the endpoints of the interval. It is also shown that the system of root vectors of the general n \times n Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. Finally, we apply our results to the dynamic generator of spatially non-homogenous damped Timoshenko beam model.

Completeness Theorem for Discontinuous Dirac Systems

In this paper, we investigate the completeness of the system of rootvectors for discontinuous Dirac systems in the Weyl’s limit circle case, using Krein’s theorem.

Completeness of the rootvectors of a dissipative Sturm–Liouville operators on time scales

In this article, we consider dissipative Sturm–Liouville operators in the limit-circle case on time scales. Then, using the Livšic’s Theorem, we prove the completeness of the system of root vectors for dissipative Sturm–Liouville operators.

On the completeness of systems of root vectors for first-order systems: Application to the Regge problem

On the completeness of systems of root vectors for first-order systems: Application to the Regge problem

SPECTRAL PROBLEMS OF NONSELFADJOINT 1D SINGULAR HAMILTONIAN SYSTEMS

In this paper, the maximal dissipativeone dimensional singular Hamiltonian operators (in limit-circle case atsingular end point $b$) are considered in the Hilbert space $\mathcal{L}_{W}^{2}(\left[ a,b\right) ;\mathbb{C}^{2}) (-\infty<a<b\leq \infty ).$ The maximal dissipative operators with general boundary conditions are investigated. A selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations are constructed. These representations allows us to determine the scattering matrix of the dilation. Further a functional model of the dissipative operator is constructed and its characteristic function in terms of the scattering matrix of dilation is considered. Finally, the theorem on completeness of the system of root vectors of the dissipative operators is proved.

Extensions and spectral problems of 1D discrete Hamiltonian systems

In this paper, we construct a space of boundary values of the minimal symmetric discrete Hamiltonian operator with defect index (2,2), which is known as Weyl's limit‐circle cases at ± ∞ , acting in the Hilbert space , where . With the help of the space of the boundary values, we describe all maximal dissipative (accretive), self‐adjoint, and other extensions of such a symmetric operator. In these descriptions, we investigate maximal dissipative operators with general boundary conditions. For maximal dissipative operator, a self‐adjoint dilation is constructed. Further, following the scattering theory, its incoming and outgoing spectral representations are set. These representations allow us to determine the scattering matrix of the dilation. Moreover, we construct a functional model of the maximal dissipative operator, and we define its characteristic function in terms of the scattering matrix of the dilation. Finally, we prove a completeness theorem about the system of root vectors of the maximal dissipative operator. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. II. Application

This paper is a continuation of the author’s paper in 2009, where the abstract theory of fold completeness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions. Moreover, equations and boundary conditions may contain abstract operators as well. So, we deal, generally, with integro-differential equations, functional-differential equations, nonlocal boundary conditions, multipoint boundary conditions, integro-differential boundary conditions. We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach L q -framework, in contrast to previously known results in the Hilbert L 2-framework. Some concrete mechanical problems are also presented.

Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

We analyze the perturbations T + B of a selfadjoint operator T in a Hilbert space H with discrete spectrum \({\{ t_k\}, T \phi_k = t_k \phi_k}\). In particular, if tk+1 − tk ≥ ckα - 1, α > 1/2 and \({\| B \phi_k \| = o(k^{\alpha - 1})}\) then the system of root vectors of T + B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H (Theorem 6). Under the assumptions \({t_{k+p} - t_k \geq d > 0, \forall k}\) (with d and p fixed) and \({\| B \phi_k \| \rightarrow 0}\) a Riesz system {Pk} of projections on invariant subspaces of T + B, Rank Pk ≤ p, is constructed (Theorem 3).

On the characteristic values of the real component of a dissipative boundary value transmission problem

In this paper, using Krein’s theorem, we investigate the completeness of the system of root vectors of a Bessel-type singular dissipative boundary value transmission problem in the Weyl’s limit circle case.