arXiv subjects

Sergey Belyi

Publications and source records attributed to Sergey Belyi.

16 recordsLinked to original sources

arXiv2026

Dissipation and c-Entropy in Nevanlinna-Pick Interpolation

We study interpolation L-systems realizing finite Nevanlinna-Pick data sets and analyze their structural and quantitative characteristics. Explicit formulas are derived for the c-entropy and dissipation coefficient, two intrinsic invariants that describe the dissipative structure of interpolation L-systems. These quantities depend only on the geometric placement of interpolation nodes in $\mathbb{C}_+$, attaining maximal finite values for purely imaginary nodes. The interpolation model $\Theta_\Delta$ and its unitary equivalents reveal that these invariants form a direct link between analytic interpolation data and the dynamical properties of L-systems. Particular attention is given to symmetric configurations, where the impedance function admits an explicit rational representation and a natural physical interpretation in terms of equivalent $LC$-networks.

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arXiv2025

The c-Entropy of non-dissipative L-systems

In this paper, we extend the definition of c-entropy to canonical L-systems with non-dissipative state-space operators. We also introduce the concepts of dissipation and accumulation coefficients for such systems. In addition, we examine the coupling of these L-systems and derive closed form expressions for the corresponding c-entropy.

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arXiv2025

L-systems with Multiplication Operator and c-Entropy

In this note, we utilize the concepts of c-entropy and the dissipation coefficient in connection with canonical L-systems based on the multiplication (by a scalar) operator. Additionally, we examine the coupling of such L-systems and derive explicit formulas for the associated c-entropy and dissipation coefficient. In this context, we also introduce the concept of a skew-adjoint L-system and analyze its coupling with the original L-system.

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arXiv2024

The c-Entropy optimality of Donoghue classes

In this note we evaluate c-Entropy of perturbed L-systems introduced in [5]. Explicit formulas relating the c-Entropy of the L-systems and the perturbation parameter are established. We also show that c-Entropy attains its maximum value (finite or infinite) whenever the perturbation parameter vanishes so that the impedance function of such a L-system belongs to one of the generalized (or regular) Donoghue classes.

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arXiv2023

The L-system representation and c-entropy

Given a symmetric operator $\dot A$ with deficiency indices $(1,1)$ and its self-adjoint extension $A$ in a Hilbert space $\mathcal{H}$, we construct a (unique) L-system with the main operator in $\mathcal{H}$ such that its impedance mapping coincides with the Weyl-Titchmarsh function $M_{(\dot A, A)}(z)$ or its linear-fractional transformation $M_{(\dot A, A_\alpha)}(z)$. Similar L-system constructions are provided for the Weyl-Titchmarsh function $aM_{(\dot A, A)}(z)$ with $a>0$. We also evaluate c-entropy and the main operator dissipation coefficient for the obtained L-systems.

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arXiv2022

Realization of inverse Stieltjes functions $(-m_α(z))$ by Schrodinger L-systems

We study L-system realizations of the original Weyl-Titchmarsh functions $(-m_α(z))$. In the case when the minimal symmetric Schrödinger operator is non-negative, we describe the Schrödinger L-systems that realize inverse Stieltjes functions $(-m_α(z))$. This approach allows to derive a necessary and sufficient conditions for the functions $(-m_α(z))$ to be inverse Stieltjes. In particular, the criteria when $(-m_\infty(z))$ is an inverse Stieltjes function is provided. Moreover, the value $m_\infty(-0)$ and parameter $α$ allow us to describe the geometric structure of the realizing $(-m_α(z))$ L-system. Additionally, we present the conditions in terms of the parameter $α$ when the main and associated operators of a realizing $(-m_α(z))$ L-system have the same or different angle of sectoriality which sets connections with the Kato problem on sectorial extensions of sectorial forms.

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arXiv2021

On the c-entropy of L-systems with Schrodinger operator

We study L-systems whose main operators are extensions of one-dimensional half-line Schrödinger operators with deficiency indices $(1, 1)$, the Schrödinger L-systems. Introducing new concepts of an c-entropy and dissipation coefficient for an L-system we discuss the following dual problems: describe Schrödinger L-systems (1) with a given c-entropy and minimal dissipation coefficient, and (2) with a given dissipation coefficient and maximal c-entropy. Also, we analyze in detail the dual c-entropy problems for Schrödinger L-systems with sectorial and extremal main operators.

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arXiv2021

The original Weyl-Titchmarsh functions and sectorial Shrödinger L-systems

In this paper we study the L-system realizations generated by the original Weyl-Titchmarsh functions $m_α(z)$ in the case when the minimal symmetric Shrö\-dinger operator in $L_2[\ell,+\infty)$ is non-negative. We realize functions $(-m_α(z))$ as impe\-dance functions of Shrödinger L-systems and derive necessary and sufficient conditions for $(-m_α(z))$ to fall into sectorial classes $S^{β_1,β_2}$ of Stieltjes functions. Moreover, it is shown that the knowledge of the value $m_\infty(-0)$ and parameter $α$ allows us to describe the geometric structure of the L-system that realizes $(-m_α(z))$. Conditions when the main and state space operators of the L-system realizing $(-m_α(z))$ have the same or not angle of sectoriality are presented in terms of the parameter $α$. Example that illustrates the obtained results is presented in the end of the paper.

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arXiv2020

On realization of the original Weyl-Titchmarsh functions by Shrödinger L-systems

We study realizations generated by the original Weyl-Titchmarsh functions $m_\infty(z)$ and $m_α(z)$. It is shown that the Herglotz-Nevanlinna functions $(-m_\infty(z))$ and $(1/m_\infty(z))$ can be realized as the impedance functions of the corresponding Shrödinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz-Nevanlinna functions $(-m_α(z))$ and $(1/m_α(z))$. We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrödinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrödinger L-systems with equal boun\-dary parameters. A condition for two Shrödinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.

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arXiv2018

Perturbations of Donoghue classes and inverse problems for L-systems

We study linear perturbations of Donoghue classes of scalar Herglotz-Nevanlinna functions by a real parameter $Q$ and their representations as impedance of conservative L-systems. Perturbation classes $\mathfrak M^Q$, $\mathfrak M^Q_κ$, $\mathfrak M^{-1,Q}_κ$ are introduced and for each class the realization theorem is stated and proved. We use a new approach that leads to explicit new formulas describing the von Neumann parameter of the main operator of a realizing L-system and the unimodular one corresponding to a self-adjoint extension of the symmetric part of the main operator. The dynamics of the presented formulas as functions of $Q$ is obtained. As a result, we substantially enhance the existing realization theorem for scalar Herglotz-Nevanlinna functions. In addition, we solve the inverse problem (with uniqueness condition) of recovering the perturbed L-system knowing the perturbation parameter $Q$ and the corresponding non-perturbed L-system. Resolvent formulas describing the resolvents of main operators of perturbed L-systems are presented. A concept of a unimodular transformation as well as conditions of transformability of one perturbed L-system into another one are discussed. Examples that illustrate the obtained results are presented.

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arXiv2017

On Sectorial L-systems with Shrödinger operator

We study L-systems with sectorial main operator and connections of their impedance functions with sectorial Stieltjes and inverse Stieltjes functions. Conditions when the main and state space operators (the main and associated state space operators) of a given L-system have the same or not angle of sectoriality are presented in terms of their impedance functions with discussion provided. Detailed analysis of L-systems with one-dimensional sectorial Shroödinger operator on half-line is given as well as connections with the Kato problem on sectorial extensions of sectorial forms. Examples that illustrate the obtained results are presented.

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arXiv2016

On unimodular transformations of conservative L-systems

We study unimodular transformations of conservative $L$-systems. Classes $\sM^Q$, $\sM^Q_κ$, $\sM^{-1,Q}_κ$ that are impedance functions of the corresponding $L$-systems are introduced. A unique unimodular transformation of a given $L$-system with impedance function from the mentioned above classes is found such that the impedance function of a new $L$-system belongs to $\sM^{(-Q)}$, $\sM^{(-Q)}_κ$, $\sM^{-1,(-Q)}_κ$, respectively. As a result we get that considered classes (that are perturbations of the Donoghue classes of Herglotz-Nevanlinna functions with an arbitrary real constant $Q$) are invariant under the corresponding unimodular transformations of $L$-systems. We define a coupling of an $L$-system and a so called $F$-system and on its basis obtain a multiplication theorem for their transfer functions. In particular, it is shown that any unimodular transformation of a given $L$-system is equivalent to a coupling of this system and the corresponding controller, an $F$-system with a constant unimodular transfer function. In addition, we derive an explicit form of a controller responsible for a corresponding unimodular transformation of an $L$-system. Examples that illustrate the developed approach are presented.

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arXiv2015

A system coupling and Donoghue classes of Herglotz-Nevanlinna functions

We study the impedance functions of conservative L-systems with the unbounded main operators. In addition to the generalized Donoghue class $\sM_κ$ of Herglotz-Nevanlinna functions considered by the authors earlier, we introduce "inverse" generalized Donoghue classes $\sM_κ^{-1}$ of functions satisfying a different normalization condition on the generating measure, with a criterion for the impedance function $V_Θ(z)$ of an L-system $Θ$ to belong to the class $\sM_κ^{-1}$ presented. In addition, we establish a connection between "geometrical" properties of two L-systems whose impedance functions belong to the classes $\sM_κ$ and $\sM_κ^{-1}$, respectively. In the second part of the paper we introduce a coupling of two L-system and show that if the impedance functions of two L-systems belong to the generalized Donoghue classes $\sM_{κ_1}$($\sM_{κ_1}^{-1}$) and $\sM_{κ_2}$($\sM_{κ_2}^{-1}$), then the impedance function of the coupling falls into the class $\sM_{κ_1κ_2}$. Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class $\sM=\sM_0$ is coupled with any other L-system, the impedance function of the coupling belongs to $\sM$ (the absorbtion property). Observing the result of coupling of $n$ L-systems as $n$ goes to infinity, we put forward the concept of a limit coupling which leads to the notion of the system attractor, two models of which (in the position and momentum representations) are presented. All major results are illustrated by various examples.

math.FA↗

arXiv2007

Stieltjes like functions and inverse problems for systems with Schrödinger operator

A class of scalar Stieltjes like functions is realized as linear-fractional transformations of transfer functions of conservative systems based on a Schrödinger operator T_h in $L_2[a,+\infty)$ with a non-selfadjoint boundary condition. In particular it is shown that any Stieltjes function of this class can be realized in the unique way so that the main operator $\bA$ of a system is an accretive (*)-extension of a Schrödinger operator T_h. We derive formulas that restore the system uniquely and allow to find the exact value of a non-real parameter h in the definition of T_h as well as a real parameter $μ$ that appears in the construction of the elements of the realizing system. An elaborate investigation of these formulas shows the dynamics of the restored parameters h and $μ$ in terms of the changing free term $γ$ from the integral representation of the realizable function. It turns our that the parametric equations for the restored parameter h represent different circles whose centers and radii are determined by the realizable function. Similarly, the behavior of the restored parameter $μ$ are described by hyperbolas.

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arXiv2005

A general realization theorem for matrix-valued Herglotz-Nevanlinna functions

New special types of stationary conservative impedance and scattering systems, the so-called non-canonical systems, involving triplets of Hilbert spaces and projection operators, are considered. It is established that every matrix-valued Herglotz-Nevanlinna function of the form V(z)=Q+Lz+\int_{\dR}(\frac{1}{t-z}-\frac{t}{1+t^2})dΣ(t) can be realized as a transfer function of such a new type of conservative impedance system. In this case it is shown that the realization can be chosen such that the main and the projection operators of the realizing system satisfy a certain commutativity condition if and only if L=0. It is also shown that $V(z)$ with an additional condition (namely, $L$ is invertible or L=0), can be realized as a linear fractional transformation of the transfer function of a non-canonical scattering $F_+$-system. In particular, this means that every scalar Herglotz-Nevanlinna function can be realized in the above sense. Moreover, the classical Livsic systems (Brodskii-Livsic operator colligations) can be derived from $F_+$-systems as a special case when $F_+=I$ and the spectral measure $dΣ(t)$ is compactly supported. The realization theorems proved in this paper are strongly connected with, and complement the recent results by Ball and Staffans.

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arXiv2005

On realization of the Krein-Langer class Nk of matrix-valued functions in Hilbert spaces with indefinite metric

In this paper the realization problems for the Krein-Langer class $N_κ$ of matrix-valued functions are being considered. We found the criterion when a given matrix-valued function from the class $N_κ$ can be realized as linear-fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii-Livsic rigged operator colligation) with the main operator acting on a rigged Pontryagin space $\Pk$ with indefinite metric. We specify three subclasses of the class $N_κ(R)$ of all realizable matrix-valued functions that correspond to different properties of a realizing system, in particular, when the domains of the main operator of a system and its conjugate coincide, when the domain of the hermitian part of a main operator is dense in $Πκ$. Alternatively we show that the class $N_κ(R)$ can be realized as transfer matrix-functions of some canonical impedance systems with self-adjoint main operators in rigged spaces $\Pk$. The case of scalar functions of the class $N_κ(R)$ is considered in details and some examples are presented.

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