arXiv subjects

Konstantin Makarov

Publications and source records attributed to Konstantin Makarov.

6 recordsLinked to original sources

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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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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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.

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arXiv2013

On the Weyl-Titchmarsh and Livšic functions

We establish a mutual relationship between main analytic objects for the dissipative extension theory of a symmetric operator $\dot A$ with deficiency indices $(1,1)$. In particular, we introduce the Weyl-Titchmarsh function $\cM$ of a maximal dissipative extension $\hat A$ of the symmetric operator $\dot A$. Given a reference self-adjoint extension $A$ of $\dot A$, we introduce a von Neumann parameter $κ$, $|κ|<1$, characterizing the domain of the dissipative extension $\hat A$ against $\Dom (A)$ and show that the pair $(κ, \cM)$ is a complete unitary invariant of the triple $(\dot A, A, \hat A)$, unless $κ=0$. As a by-product of our considerations we obtain a relevant functional model for a dissipative operator and get an analog of the formula of Krein for its resolvent.

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