Vladimir Derkach

Vasyl Stus Donetsk National University

We propose a general approach for deriving transparent boundary conditions for the stationary Schroedinger equation with arbitrary potential. It is proven that the transparent boundary conditions can be written in terms of the Weyl-Titchmarsh coefficients. As examples for the application of the proposed approach, two special cases for the stationar...

An isometric operator V in a Pontryagin space $${{{\mathfrak {H}}}}$$ H is called standard, if its domain and the range are nondegenerate subspaces in $${{{\mathfrak {H}}}}$$ H . A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteri...

An isometric operator V in a Pontryagin space H is called standard, if its domain and the range are nondegenerate subspaces in H. A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized...

Full indefinite Stieltjes moment problem is studied via the step-by-step Schur algorithm. Naturally associated with indefinite Stieltjes moment problem are generalized Stieltjes continued fraction and a system of difference equations, which, in turn, lead to factorization of resolvent matrices of indefinite Stieltjes moment problem. A criterion for...

De Branges–Pontryagin spaces B(E) with negative index κ of entire p×1 vector valued functions based on an entire p×2p entire matrix valued function E(λ) (called the de Branges matrix) are studied. An explicit description of these spaces and an explicit formula for the indefinite inner product are presented. A characterization of those spaces B(E) t...

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class Nκ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel Nωz≔fz−fω¯z−ω¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\...

With a closed symmetric operator $A$ in a Hilbert space ${\mathfrak H}$ a triple $\Pi=\{{\mathcal H},\Gamma_0,\Gamma_1\}$ of a Hilbert space ${\mathcal H}$ and two abstract trace operators $\Gamma_0$ and $\Gamma_1$ from $A^*$ to ${\mathcal H}$ is called a generalized boundary triple for $A^*$ if an abstract analogue of the second Green's formula ho...

Indefinite Sturm-Liouville operators defined on the real line are often considered as a coupling of two semibounded symmetric operators defined on the positive and the negative half axis. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension. In this paper...

Let J be an m × m signature matrix, i.e., J = J* = J⁻¹. An m × m mvf (matrix valued function) W(λ) that is meromorphic in the unit disk D is called J-inner if W(λ)JW(λ)* ≤ J for every λ from hW⁺, the domain of holomorphy of W, in D, and W(μ)JW(μ)* = J for a.e. μ ∈ T = ∂D. A J-inner mvf W(λ) is called A-singular if it is outer and it is called right...

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension $T$ of a symmetric operator $S$ in a Hilbert space $\mathfrak H$, employing the technique of quasi boundary triples for $T$. The general results are illustrated with couplings of Schr\"{o}dinger operators on Lipschitz domains...

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension $T$ of a symmetric operator $S$ in a Hilbert space $\mathfrak H$, employing the technique of quasi boundary triples for $T$. The general results are illustrated with couplings of Schr\"{o}dinger operators on Lipschitz domains...

Nondegenerate truncated indefinite Stieltjes moment problem in the class $\mathbf{N}_{\kappa}^{k}$ of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step-by-step algorythm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recentl...

Nondegenerate truncated indefinite Stieltjes moment problem in the class $\mathbf{N}_{\kappa}^{k}$ of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step-by-step algorythm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recentl...

This chapter contains a short review of the theory of boundary triplets, and the corresponding Weyl functions, of symmetric operators in Hilbert and Kreĭn spaces. The theory of generalized resolvents of such operators is exposed from the point of view of boundary triplets approach. Applications to different continuation problems related to the exte...

Let $A$ be a densely defined symmetric operator with equal deficiency indices in a Hilbert space. We introduce the notion of a Weyl function $M(z)$ of $A$ corresponding to an ordinary boundary triplet of the operator $A^*$ and then investigate its basic properties. In particular, a connection with Krein-Langer Q-functions and Krein's type formula f...

A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real value $a$ in a single point $z_0\in {\mathbb C}_+$ should be identically equal to $a$. In the present note we p...

A generalized bitangential interpolation problem of Nevanlinna-Pick type in the class of generalized Schur matrix valued functions (mvf's) is considered both in the unit disc and in the right half plane. Linear fractional representations of the set of solutions to these problems are presented in the strictly completely indeterminate case and in the...

A self-adjoint operator $A$ in a Krein space $\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$ is called partially fundamentally reducible if there exist a fundamental decomposition ${\mathcal K} = {\mathcal K}_+ [\dot{+}] {\mathcal K}_-$ (which does not reduce $A$) and densely defined symmetric operators $S_+$ and $S_-$ in the Hilbert spaces $\bigl({...

Presented in this volume are a number of new results concerning the extension theory and spectral theory of unbounded operators using the recent notions of boundary triplets and boundary relations. This approach relies on linear single-valued and multi-valued maps, isometric in a Krein space sense, and offers a basic framework for recent developmen...

Truncated moment problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established, covering both the even and odd problems, including complete parametrizations of solutions. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new...

Boundary relations for a symmetric relation in a Pontryagin space are studied and the corresponding Weyl families are characterized. In partic-ular, it is shown that every generalized Nevanlinna family can be realized as the Weyl family of a boundary relation in a Pontryagin space.

Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix JC=UL is a monic generalized Jacobi matrix associated with the function FC(λ)=λF(λ)+1. It turns out that the Christoffel transformation JC of a bou...

Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_c=UL is a monic generalized Jacobi matrix associated with the function F_c(z)=zF(z)+1. It turns out that the Christoffel transformation J_c of a...

Convergence of diagonal Padé approximants is studied for a class of functions which admit the integral representation \( \mathfrak{F}(\lambda ) = r_1 (\lambda )\int_{ - 1}^1 {\frac{{td\sigma (t)}} {{t - \lambda }} + r_2 (\lambda )} \), where δ is a finite nonnegative measure on [−1, 1], r 1, r 2 are real rational functions bounded at ∞, and r 1 is...

A class \({\mathcal{U}}_{\kappa 1} (J)\) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of \(p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q}\) and some associated reproducing kernel Pontryagin spaces are studied. Th...

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these...

In this paper the realization problems for the Kreĭn–Langer class Nκ of matrix-valued functions are being considered. We found the criterion when a given matrix-valued function from the class Nκ can be realized as linear-fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii–Livsic rigged...

Convergence of diagonal Pad\'e approximants is studied for a class of functions which admit the integral representation $ {\mathfrak F}(\lambda)=r_1(\lambda)\int_{-1}^1\frac{td\sigma(t)}{t-\lambda}+r_2(\lambda), $ where $\sigma$ is a finite nonnegative measure on $[-1,1]$, $r_1$, $r_2$ are real rational functions bounded at $\infty$, and $r_1$ is n...

Asymptotic expansions of generalized Nevanlinna functions Q are investigated by means of a factorization model involving a part of the generalized zeros and poles of nonpositive type of the function Q. The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R∞(S F) of the so-called generaliz...

We study a stepwise algorithm for solving the indefinite truncated mo-ment problem and obtain the factorization of the matrix describing the solution of this problem into elementary factors. We consider the generalized Jacobi matrix corresponding to Magnus' continuous P -fraction that appears in this algorithm and the polynomials of the first and s...

The abstract interpolation problem (AIP) in the Schur class was posed V. Katznelson, A. Kheifets and P. Yuditskii in 1987. In the present paper an analog of the AIP for Nevanlinna classes is considered. The description of solutions of the AIP is reduced to the description of \( \mathcal{L} \)-resolvents of some model symmetric operator associated w...

The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the...

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in $\dC_+$) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to...

In this paper the realization problems for the Krein-Langer class $N_\kappa$ of matrix-valued functions are being considered. We found the criterion when a given matrix-valued function from the class $N_\kappa$ can be realized as linear-fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodsk...

A new class of generalized Jacobi matrices is introduced. Every proper real rational function is proved to be the m-function of a unique finite generalized Jacobi matrix. Moreover, every generalized Nevanlinna function m(·) which is a solution of a determinate indefinite moment problem turns out to be the m-function of a unique infinite generalized...

When the singular finite rank perturbations of an unbounded selfadjoint operator Ao in a Hilbert space c, formally defined by A(α) = A 0 + GαG*, are lifted to an exit Pontryagin space ℌ by means of an operator model, they become ordinary range perturbations of a self-adjoint operator H ∞ in ℌ ⊃ ℌ0:H T = H ∞-ΩT-1Ω*. Here G is a mapping from ℂd into...

A new model for generalized Nevanlinna functions Q∈ N_κ will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of Q. The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and the corresponding W...

A new model for generalized Nevanlinna functions Q ∈ N κ Q\in \mathbf {N}_\kappa will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of Q Q . The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and t...

The Schur–Nevanlinna–Pick interpolation problem is considered in the class of generalized Schur functions and reduced to the problem of the extension of a certain isometric operator V that acts in the Pontryagin space. The description of the solutions of this problem is based on the theory of the resolvent matrix developed by Krein.

Singular finite rank perturbations of an unbounded self-adjoint operator A 0 in a Hilbert space ℌ0 are defined formally as A (α)=A 0+GαG *, where G is an injective linear mapping from ℋ=ℂd to the scale space ℌ-k(A0)k ∈ ℕ, k∈N, of generalized elements associated with the self-adjoint operator A 0, and where α is a self-adjoint operator in ℋ. The cas...

Let N1 denote the class of generalized Nevanlinna functions with one negative square and let N1, 0 be the subclass of functions Q(z)∈N1 with the additional properties limy→∞ Q(iy)/y=0 and lim supy→∞ y |Im Q(iy)|<∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ)=A+τ[·, ω] ω in a Pontryagin space setti...

In the present paper a refined version of the approach by A. V. Shtraus [Izv. Akad. Nauk SSSR, Ser. Mat. 24, 43-74 (1960; Zbl 0151.19404)] to the theory of characteristic functions is proposed. A new definition of the boundary mapping ℬ for a pair of linear relations A, A * in the Krein space. The boundary mappings ℬ are proved to be in a one-to-on...

For an arbitrary symmetric operator in a Hilbert space with the block Weyl function exact formulas are proposed for determining the Weyl functions of symmetric extensions of the operator. These formulas combined with couplings of symmetric operators give rise to some new results concerning generalized resolvents of a symmetric operator: a geometric...

Let NK be the class of meromorphic functions Q(z) defined on ℂ\ℝ with \( Q\left( {\overline z } \right) = \overline {Q\left( z \right)}, \) and such that on its domain of holomorphy the kernel $$ {N_Q}\left( {z,w} \right) = \frac{{Q\left( z \right) - \overline {Q\left( w \right)} }}{{z - \overline w }}, z \ne \overline w {N_Q}\left( {z,\overline z...

Without Abstract

Without Abstract

A two-sided indefinite interpolation problem in the class of gen- eralized Nevanlinna pairs is considered. In the case where the Pick matrix is nondegenerate a solvability criterion for the problem is given. All solutions of the problem are described as a fractional linear transformations of a parame- ter from a subclass of the Nevanlinna class. Th...

The Laguerre-Sonin polynomialsL n () are orthogonal in linear spaces with indefinite inner product if<–1. We construct the completion () of this space and describe self-adjoint extensions of the Laguerre operatorl(y)=xy+(1+–x)y,<–1, in the space (). In particular, we write out the self-adjoint extension of the Laguerre operator whose eigenfunctio...

Let (sj)∞j=0 be a sequence of real numbers such that the Hankel matrices (si+j)∞0, (Si+j+i)∞0 have finite numbers of negative eigenvalues. The indefinite moment problem with the moments Sj (j = 0,1,2, …) and the corresponding Stieltjes string are investigated. We use the approach via the Kreîn — Langer extension theory of symmetric operators in spa...

Let {sj}0∞ be a sequence of real numbers such that Hankel matrices S = (si+j)0∞, S(1) = (si+j+1)0∞ have finite numbers κ, κ1 of negative eigenvalues. An indefinite moment problem with the moments sj (j = 0, 1, 2, . . .) and the corresponding Stieltjes string are investigated. We use the approach via the Krein-Langer extension theory of symmetric op...

A description of generalized resolvents for a densely defined Hermitian operatorA in a Krein space K\mathcal{K} is given under explicit consideration of the number of negative squares of the inner product on the extending space and of the forms [A,], [A,],A being a selfadjoint extension ofA which corresponds to the generalized resolvent. New classe...

Without Abstract

Various classes of extensions and generalized resolvents of Hermitian operators acting in Krein spaces are described in terms of abstract boundary conditions.

The characteristic operator-functions W() are studied of the almost solvable extensions of an Hermitian operator. The inverse problem is solved, a multiplication theorem is proved, and a formula is derived expressing W() in terms of the Weyl function and the boundary operator. Characteristic functions are computed of various differential and differ...

A Hermitian operator A with gaps (αj, βj) (1 ⩽ j ⩽ m ⩽ ∞) is studied. The self-adjoint extensions which put exactly kj < ∞ eigenvalues into each gap (αj, βj), in particular (for kj = 0, 1 ⩽ j ⩽ m) the extensions preserving the gaps, are described in terms of boundary conditions. The generalized resolvents of the extensions with the indicated proper...

Let H be a separable Hilbert space, let A be a positive Hermitian operator (A > 0) with defect indices n+(A) = n_(A) = n 0 y f e D(T). Further, T is m-sectorial (m-accretive) if, furthermore, it does not have sectorial (accretive) extensions. The collection of m-sectorial operators satisfying condition (i) will be denoted by SH(~), while that of th...

An operator model for the generalized Friedrichs extension in the Pontryagin space setting is presented. The model is based on a factorization of the associated Weyl function (or Q-function) and it carries the information on the asymptotic behavior of the Weyl function at z = ∞.

New classes of operator functions generalizing the well-known Krein-Stieltjes classes are introduced. Operator functions from these classes are characterized in terms of their poles and zeros on negative semiaxes. The results are applied to a description of some classes of generalized resolvents of a nondensely defined Hermitian operator with a gap...

Selfadjoint extensions of a closed symmetric operator in a Hilbert space with equal deficiency indices are described by means of ordinary boundary triplets. In certain problems the more general notion of a boundary triplet of bounded type is needed. It will be shown that such triplets correspond in a certain way with the, in general infinite dimens...

The authors continue their study [see Dokl. Akad. Nauk SSSR 293, No. 5, 1041-1046 (1987; Zbl 0655.47005)] of classes S ℌ ±x of operator-valued functions F(z) analytic in ℂ + which have a nonnegative imaginary part and are such that z ±1 F(z)∈N x . At x=0, classes S ℌ ±0 coincide with the familiar Krein-Stieltjes classes. In terms of zeros and poles...