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Natalia P. Bondarenko

Publications and source records attributed to Natalia P. Bondarenko.

At least 37 records · Page 2Linked to original sources
arXiv2023

Local solvability and stability for the inverse Sturm-Liouville problem with polynomials in the boundary conditions

In this paper, we for the first time prove local solvability and stability of the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof method is constructive. It is based on the reduction of the inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation of the inverse problem remains uniquely solvable. Furthermore, we derive new reconstruction formulas for obtaining the problem coefficients from the solution of the main equation and get stability estimates for the recovered coefficients.

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arXiv2023

Regularization and inverse spectral problems for differential operators with distribution coefficients

In this paper, we consider a class of matrix functions, which contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order $n \ge 2$. We show that every matrix function of this class is associated with some differential expression. Moreover, we construct the family of associated matrices for a fixed differential expression. Furthermore, our regularization results are applied to inverse spectral theory. We study a new type of inverse spectral problems, which consist in the recovery of distribution coefficients from the spectral data independently of the associated matrix. The uniqueness theorems are proved for the inverse problems by the Weyl-Yurko matrix and by the discrete spectral data. As an example, we consider the case $n = 2$ in detail.

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arXiv2023

Solving Barcilon's inverse problems by the method of spectral mappings

In this paper, we consider Barcilon's inverse problem, which consists of the recovery of the fourth-order differential operator from three spectra. We obtain the relationship of Barcilon's three spectra with the Weyl-Yurko matrix. Moreover, we prove the uniqueness theorem for the inverse problem solution by developing the ideas of the method of spectral mappings. Our approach allows us to obtain the result for the general case of complex-valued distributional coefficients. In the future, the methods and the results of this paper can be generalized to differential operators of orders greater than 4 and used for further development of the inverse problem theory for higher-order differential operators.

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arXiv2023

Inverse spectral problem for the third-order differential equation

This paper is concerned with the inverse spectral problem for the third-order differential equation with distribution coefficient. The inverse problem consists in the recovery of the differential expression coefficients from the spectral data of two boundary value problems with separated boundary conditions. For this inverse problem, we solve the most fundamental question of the inverse spectral theory about the necessary and sufficient conditions of solvability. In addition, we prove the local solvability and stability of the inverse problem. Furthermore, we obtain very simple sufficient conditions of solvability in the self-adjoint case. The main results are proved by a constructive method that reduces the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. In the future, our results can be generalized to various classes of higher-order differential operators with integrable or distribution coefficients.

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arXiv2022

Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition

In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a star-shaped graph with nonlocal integral matching condition. This operator is adjoint to the functional-differential operator with frozen argument at the central vertex of the graph. We study the inverse problem that consists in the recovery of the integral condition coefficients from the eigenvalues. We obtain the spectrum characterization, reconstruction algorithms, and prove the uniqueness of the inverse problem solution.

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arXiv2022

Reconstruction of higher-order differential operators by their spectral data

This paper is concerned with inverse spectral problems for higher-order ($n > 2$) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our approach is based on the reduction of an inverse problem to a linear equation in the Banach space of bounded infinite sequences. This equation is derived in a general form that can be applied to various classes of differential operators. The unique solvability of the linear main equation is also proved. By using the solution of the main equation, we derive reconstruction formulas for the differential expression coefficients in the form of series and prove the convergence of these series for several classes of operators. The results of this paper can be used for constructive solution of inverse spectral problems and for investigation of their solvability and stability.

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arXiv2022

Spectral data asymptotics for the higher-order differential operators with distribution coefficients

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author's knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance.

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arXiv2022

Linear differential operators with distribution coefficients of various singularity orders

In this paper, the linear differential expression of order $n \ge 2$ with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we present the new statements of inverse spectral problems that consist in the recovery of differential operators with distribution coefficients from the Weyl matrix on the half-line and on a finite interval. The uniqueness theorems for these inverse problems are proved by developing the method of spectral mappings.

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arXiv2021

A new approach to the inverse discrete transmission eigenvalue problem

A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.

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arXiv2021

Finite-difference approximation of the inverse Sturm-Liouville problem with frozen argument

This paper deals with the discrete system being the finite-difference approximation of the Sturm-Liouville problem with frozen argument. The inverse problem theory is developed for this discrete system. We describe the two principal cases: degenerate and non-degenerate. For these two cases, appropriate inverse problems statements are provided, uniqueness theorems are proved, and reconstruction algorithms are obtained. Moreover, the relationship between the eigenvalues of the continuous problem and its finite-difference approximation is investigated. We obtain the "correction terms" for approximation of the discrete problem eigenvalues by using the eigenvalues of the continuous problem. Relying on these results, we develop a numerical algorithm for recovering the potential of the Sturm-Liouville operator with frozen argument from a finite set of eigenvalues. The effectiveness of this algorithm is illustrated by numerical examples.

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arXiv2021

Inverse spectral problems for functional-differential operators with involution

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems.

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arXiv2021

Direct and inverse problems for the matrix Sturm-Liouville operator with the general self-adjoint boundary conditions

The matrix Sturm-Liouville operator on a finite interval with the boundary conditions in the general self-adjoint form and with the singular potential from the class $W_2^{-1}$ is studied. This operator generalizes Sturm-Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.

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arXiv2021

Solvability and stability of the inverse problem for the quadratic differential pencil

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

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arXiv2020

Spectral data characterization for the Sturm-Liouville operator on the star-shaped graph

The inverse spectral problems are studied for the Sturm-Liouville operator on the star-shaped graph and for the matrix Sturm-Liouville operator with the boundary condition in the general self-adjoint form. We obtain necessary and sufficient conditions of solvability for these two inverse problems, and also prove their local solvability and stability.

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arXiv2020

Inverse problem solution and spectral data characterization for the matrix Sturm-Liouville operator with singular potential

The matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for the studied operator.

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arXiv2020

Partial inverse problems for quadratic differential pencils on a graph with a loop

In this paper, partial inverse problems for the quadratic pencil of Sturm-Liouville operators on a graph with a loop were studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for the partial inverse problems.

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arXiv2020

Local solvability and stability of the inverse problem for the non-self-adjoint Sturm-Liouville operator

We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local solvability and stability of this inverse problem, relying on the method of spectral mappings. Possible splitting of multiple eigenvalues is taken into account.

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