Spectral Data Asymptotics for the Higher-Order Differential Operators With Distribution Coefficients

Bondarenko, Natalia P.

doi:10.1007/s10958-022-06118-x

Cited by 6 publications

(16 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, n − 1}, the spectrum of the problem L k is a countable set of eigenvalues {λ l,k } l≥1 . They are supposed to be numbered counting with multiplicities according to the asymptotics obtained in [49]:…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

Bondarenko

2022

Mathematics

Self Cite

View full text Add to dashboard Cite

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 the constructive solution of inverse spectral problems and for the investigation of their solvability and stability.

show abstract

Section: Resultsmentioning

confidence: 99%

“…The spectral data {λ l,k , β l,k } l≥1, k=1,n−1 uniquely specify the coefficients p = (p k ) n−2 k=0 ∈ W . By Theorem 6.2 of [49], the weight numbers have the asymptotics…”

Section: Spectral Datamentioning

confidence: 96%

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

Bondarenko

2022

Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The estimate of Proposition 1 is valid for Φ⋆ n−k+1 (x, λ). Taking the asymptotics (10) for λ l,k,ε and the definition (43) into account, we obtain…”

Section: Main Equation Solvabilitymentioning

confidence: 99%

“…For n = 2, 3, our conditions coincide with the previously known results (see [29,42]). For n = 4, we obtain a novel theorem (Theorem 3), which completely characterizes the corresponding spectral data in terms of the asymptotics recently derived in [43] and structural properties.…”

Section: Introductionmentioning

confidence: 95%

Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators

Bondarenko

2023

Mathematics

View full text Add to dashboard Cite

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties.

show abstract

“…The methods of [24][25][26] rely on the regularization of differential operators with distribution coefficients (see [27][28][29]) and on the ideas of the method of spectral mappings [20]. In the reconstruction technique of [26], an important role is played by the spectral data asymptotics which have been deduced in [30] by using the Birkhoff-type solutions recently obtained in [31]. However, necessary and sufficient conditions (NSC) of inverse problem solvability for higherorder differential operators with distribution coefficients, to the best of the author's knowledge, have not been investigated yet.…”

Section: Introductionmentioning

confidence: 99%

Inverse spectral problem for the third-order differential equation

Bondarenko¹

2023

Preprint

View full text Add to dashboard Cite

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.

show abstract

Spectral Data Asymptotics for the Higher-Order Differential Operators With Distribution Coefficients

Cited by 6 publications

References 19 publications

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators

Inverse spectral problem for the third-order differential equation

Contact Info

Product

Resources

About