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Reconstruction of Higher-Order Differential Operators by Their Spectral Data

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  • Natalia P. Bondarenko

    (Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia
    Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia)

Abstract

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.

Suggested Citation

  • Natalia P. Bondarenko, 2022. "Reconstruction of Higher-Order Differential Operators by Their Spectral Data," Mathematics, MDPI, vol. 10(20), pages 1-32, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:20:p:3882-:d:946866
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    References listed on IDEAS

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    1. Upeksha Perera & Christine Böckmann, 2020. "Solutions of Sturm-Liouville Problems," Mathematics, MDPI, vol. 8(11), pages 1-14, November.
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    Cited by:

    1. Natalia P. Bondarenko, 2023. "Local Solvability and Stability of an Inverse Spectral Problem for Higher-Order Differential Operators," Mathematics, MDPI, vol. 11(18), pages 1-22, September.
    2. Natalia P. Bondarenko, 2023. "Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators," Mathematics, MDPI, vol. 12(1), pages 1-27, December.
    3. Natalia P. Bondarenko, 2023. "Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients," Mathematics, MDPI, vol. 11(16), pages 1-23, August.

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    1. Elena Corina Cipu & Cosmin Dănuţ Barbu, 2022. "Variational Estimation Methods for Sturm–Liouville Problems," Mathematics, MDPI, vol. 10(20), pages 1-18, October.
    2. Natalia P. Bondarenko, 2023. "Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators," Mathematics, MDPI, vol. 12(1), pages 1-27, December.

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