arXiv subjects

Sergey Buterin

Publications and source records attributed to Sergey Buterin.

16 recordsLinked to original sources

arXiv2025

Operator-differential expressions: regularization and completeness of the root functions

We consider an operator-differential expression of the form $$ \ell y=\frac{d^m}{dx^m}\Big(By^{(n)}+Cy\Big), \quad 0<x<1, $$ where $B$ is a linear bounded invertible operator, while $C$ is some finite-dimensional linear operator relatively bounded to the operator of $n$-fold differentiation. To such a form, we can reduce, in particular, various singular differential expressions with the coefficients in negative Sobolev spaces, which creates an alternative to their regularization. In the case when $B$ is an integral Volterra operator of the second kind with a continuous kernel vanishing at the diagonal, we establish completeness of the root functions of an operator generated by the expression $\ell y$ and irregular semi-separated boundary conditions.

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arXiv2024

On a control system on an infinite temporal tree

We discuss the stochastic interpretation of a control system determined by a system of differential equations on a tree. For example, such a system on a finite tree arises after replacing the coefficients of the equation on an interval with stochastic processes in discrete time and with finitely many states. The countable number of states will correspond to a more complicated and, at the same time, more general case of an infinite tree, which is under consideration.

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arXiv2023

On damping a control system of arbitrary order with global aftereffect on a tree

We study a problem of damping a control system described by functional-differential equations of natural order $n$ and neutral type with non-smooth complex coefficients on an arbitrary tree with global delay. The latter means that the delay propagates through internal vertices of the tree. Minimization of the energy functional of the system leads to a variational problem. We establish its equivalence to a certain self-adjoint boundary value problem on the tree for equations of order $2n$ with nonlocal quasi-derivatives and multidirectional shifts of the argument, as well as Kirchhoff-type conditions emerging at the internal vertices. The unique solvability of both problems is proved.

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arXiv2023

On damping a control system with global aftereffect on quantum graphs. Stochastic interpretation

Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as {\it temporal} networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff-type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first-order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self-adjoint boundary value problem on the tree for some second-order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree.

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arXiv2023

On recovering Sturm--Liouville-type operators with global delay on graphs from two spectra

We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, being measured in the direction to a specific boundary vertex, called the root, propagates through the internal vertex to other edges. Now, we intend to recover the potentials given the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except the root. We prove the uniqueness theorem and obtain a constructive procedure for solving this inverse problem assuming that the common boundary conditions are of the Robin type and they are pairwise linearly independent. Although we focus on graphs with equal edges when the delay parameter also coincides with their length, the proposed formulation is expected to be relevant for more general situations including non-star trees with non-equal edges and with a wide range for the global delay parameter.

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arXiv2023

An inverse Sturm--Liouville-type problem with constant delay and non-zero initial function

We suggest a new statement of the inverse spectral problem for Sturm--Liouville-type operators with constant delay. This inverse problem consists in recovering the coefficient (often referred to as potential) of the delayed term in the corresponding equation from the spectra of two boundary value problems with one common boundary condition. However, all studies in this direction focus on the case of the zero initial function, i.e. they exploit the assumption that the potential vanishes on the corresponding subinterval. In the present paper, we waive that assumption in favor of a continuously matching initial function, which leads to appearing an additional term with frozen argument in the equation. For the resulting new inverse problem, we pay a special attention to the situation when one of the spectra is given only partially. Sufficient conditions and necessary conditions on the corresponding subspectrum for the unique determination of the potential are obtained, and a constructive procedure for solving the inverse problem is given. In parallel, we obtain the characterization of the spectra for the zero initial function and the Neumann common boundary condition, which is found to include an additional restriction as compared with the case of the Dirichlet common condition.

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arXiv2022

Functional-differential operators on geometrical graphs with global delay and inverse spectral problems

We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves {\it global} delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a {\it locally} nonlocal version. The latter means that each edge of the graph has its own delay parameter, which does not affect any other edge. In this paper, we introduce {\it globally} nonlocal operators that are expected to be more natural for modelling nonlocal processes on graphs. We also extend this idea to arbitrary trees, which opens a wide area of further research. Another goal of the paper is to study inverse spectral problems for operators with global delay in one illustrative case by addressing a wide range of questions including uniqueness, characterization of the spectral data as well as the uniform stability.

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arXiv2022

Inverse problems for Dirac operators with constant delay: uniqueness, characterization, uniform stability

We initiate studying inverse spectral problems for Dirac-type functional-differential operators with constant delay. For simplicity, we restrict ourselves to the case when the delay parameter is not less than one half of the interval. For the considered case, however, we give answers to the full range of questions usually raised in the inverse spectral theory. Specifically, reconstruction of two complex $L_2$-potentials is studied from either complete spectra or subspectra of two boundary value problems with one common boundary condition. We give conditions on the subspectra that are necessary and sufficient for the unique determination of the potentials. Moreover, necessary and sufficient conditions for the solvability of both inverse problems are obtained. For the inverse problem involving the complete spectra, we establish also uniform stability in each ball. For this purpose, we use recent results on uniform stability of sine-type functions with asymptotically separated zeros.

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arXiv2021

On the uniform stability of recovering sine-type functions with asymptotically separated zeros

We obtain a uniform stability of recovering entire functions of a special form from their zeros. To this form, one can reduce the characteristic determinants of strongly regular differential operators and pencils of the first and the second orders, including differential systems with asymptotically separated eigenvalues whose characteristic numbers lie on a line containing the origin, and their non-local perturbations. We establish that the dependence of such functions on the sequences of their zeros possesses the Lipschitz property with respect to natural metrics on each ball of a finite radius. Results of this type can be used for studying the uniform stability of inverse spectral problems. In addition, general theorems on the asymptotics of zeros of functions of this class and on their equivalent representation via an infinite product are obtained, which give the corresponding results for many specific operators.

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arXiv2021

Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

The paper deals with Sturm-Liouville-type operators with frozen argument of the form $\ell y:=-y''(x)+q(x)y(a),$ $y^{(α)}(0)=y^{(β)}(1)=0,$ where $α,β\in\{0,1\}$ and $a\in[0,1]$ is an arbitrary fixed rational number. Such nonlocal operators belong to the so-called loaded differential operators, which often appear in mathematical physics. We focus on the inverse problem of recovering the potential $q(x)$ from the spectrum of the operator $\ell.$ Our goal is two-fold. Firstly, we establish a deep connection between the so-called main equation of this inverse problem and Chebyshev polynomials of the first and the second kinds. This connection gives a new perspective method for solving the inverse problem. In particular, it allows one to completely describe all non-degenerate and degenerate cases, i.e. when the solution of the inverse problem is unique or not, respectively. Secondly, we give a complete and convenient description of iso-spectral potentials in the space of complex-valued integrable functions.

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arXiv2021

Inverse spectral problems for Hill-type operators with frozen argument

The paper deals with nonlocal differential operators possessing a term with frozen (fixed) argument appearing, in particular, in modelling various physical systems with feedback. The presence of a feedback means that the external affect on the system depends on its current state. If this state is taken into account only at some fixed physical point, then mathematically this corresponds to an operator with frozen argument. In the present paper, we consider the operator $Ly\equiv-y''(x)+q(x)y(a),$ $y^{(ν)}(0)=γy^{(ν)}(1),$ $ν=0,1,$ where $γ\in{\mathbb C}\setminus\{0\}.$ The operator $L$ is a nonlocal analog of the classical Hill operator describing various processes in cyclic or periodic media. We study two inverse problems of recovering the complex-valued square-integrable potential $q(x)$ from some spectral information about $L.$ The first problem involves only single spectrum as the input data. We obtain complete characterization of the spectrum and prove that its specification determines $q(x)$ uniquely if and only if $γ\ne\pm1.$ For the rest (periodic and antiperiodic) cases, we describe classes of iso-spectral potentials and provide restrictions under which the uniqueness holds. The second inverse problem deals with recovering $q(x)$ from the two spectra related to $γ=\pm1.$ We obtain necessary and sufficient conditions for its solvability and establish that uniqueness holds if and only if $a=0,1.$ For $a\in(0,1),$ we describe classes of iso-bispectral potentials and give restrictions under which the uniqueness resumes. Algorithms for solving both inverse problems are provided. In the appendix, we prove Riesz-basisness of an auxiliary two-sided sequence of sines.

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arXiv2021

Iso-bispectral potentials for Sturm-Liouville-type operators with small delay

In recent years, there appeared a considerable interest in the inverse spectral theory for functional-differential operators with constant delay. In particular, it is well known that, for each fixed $ν\in\{0,1\},$ the spectra of two operators generated by one and the expression $-y''(x)+q(x)y(x-a)$ and the boundary conditions $y^{(ν)}(0)=y^{(j)}(π)=0,$ $j=0,1,$ uniquely determine the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[π/2,π).$ For $a<π/2,$ the main equation of the corresponding inverse problem is nonlinear, and it actually became the basic question of the inverse spectral theory for Sturm-Liouville operators with constant delay whether the uniqueness holds also in this nonlinear case. A few years ago, a positive answer was obtained for $a\in[2π/5,π/2).$ Recently, the authors gave, however, a negative answer for $a\in[π/3,2π/5)$ by constructing infinite families of iso-bispectral potentials. Meanwhile, the question remained open for the most difficult nonlinear case $a\in(0,π/3),$ allowing the parameter $a$ to approach the classical situation $a=0,$ in which the uniqueness is well known. In the present paper, we address this gap and give a negative answer in this remarkable case by constructing appropriate iso-bispectral potentials.

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arXiv2021

On non-uniqueness of recovering Sturm-Liouville operators with delay and the Neumann boundary condition at zero

As is known, for each fixed $ν\in\{0,1\},$ the spectra of two operators generated by $-y''(x)+q(x)y(x-a)$ and the boundary conditions $y^{(ν)}(0)=y^{(j)}(π)=0,$ $j=0,1,$ uniquely determine the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[2π/5,π).$ Meanwhile, it actually became the main question of the inverse spectral theory for Sturm-Liouville operators with constant delay whether the uniqueness holds also for smaller values of $a.$ Recently, a negative answer was given by the authors [Appl. Math. Lett. 113 (2021) 106862] for $a\in[π/3,2π/5)$ in the case $ν=0$ by constructing an infinite family of iso-bispectral potentials. Moreover, an essential and dramatic reason was established why this strategy, generally speaking, fails in the remarkable case when $ν=1.$ Here we construct a counterexample giving a negative answer for $ν=1,$ which is an important subcase of the Robin boundary condition at zero. We also refine the former counterexample for $ν=0$ to $W_2^1$-potentials.

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arXiv2020

On an open question in recovering Sturm-Liouville-type operators with delay

In recent years, there appeared a considerable interest in the inverse spectral theory for functional-differential operators with constant delay. In particular, it is well known that specification of the spectra of two operators $\ell_j,$ $j=0,1,$ generated by one and the same functional-differential expression $-y''(x)+q(x)y(x-a)$ under the boundary conditions $y(0)=y^{(j)}(π)=0$ uniquely determines the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[π/2,π).$ For many years, it has been a challenging {\it open question} whether this uniqueness result would remain true also when $a\in(0,π/2).$ Recently, a positive answer was obtained for the case $a\in[2π/5,π/2).$ In this paper, we give, however, a {\it negative} answer to this question for $a\in[π/3,2π/5)$ by constructing an infinite family of iso-bispectral potentials. Some discussion on a possibility of constructing a similar counterexample for other types of boundary conditions is provided, and new open questions are outlined.

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arXiv2020

Uniform full stability of recovering convolutional perturbation of the Sturm-Liouville operator from the spectrum

The perturbation of the Sturm-Liouville operator on a finite interval with Dirichlet boundary conditions by a convolution operator is considered. Local stability and global unique solvability of the inverse problem of recovering the convolution kernel from the spectrum, provided that the potential is given a priori, is known. In the present work, we establish uniform full stability of this inverse problem involving a uniform estimate of deviations of the convolution kernel via deviations of the spectrum and the potential within balls of any fixed radii.

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arXiv2020

Uniform stability of the inverse spectral problem for a convolution integro-differential operator

The operator of double differentiation, perturbed by the composition of the differentiation operator and a convolution one, on a finite interval with Dirichlet boundary conditions is considered. We obtain uniform stability of recovering the convolution kernel from the spectrum in a weighted $L_2$-norm and in a weighted uniform norm. For this purpose, we successively prove uniform stability of each step of the algorithm for solving this inverse problem in both the norms. Besides justifying the numerical computations, the obtained results reveal some essential difference from the classical inverse Sturm-Liouville problem.

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