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Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients

Author

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  • Natalia P. Bondarenko

    (Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia
    Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia
    S.M. Nikolskii Mathematical Institute, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow 117198, Russia)

Abstract

In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order n ≥ 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 of 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 examples, we consider the cases n = 2 and n = 4 in more detail.

Suggested Citation

  • Natalia P. Bondarenko, 2023. "Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients," Mathematics, MDPI, vol. 11(16), pages 1-23, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3455-:d:1213778
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    References listed on IDEAS

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    1. Natalia P. Bondarenko, 2022. "Reconstruction of Higher-Order Differential Operators by Their Spectral Data," Mathematics, MDPI, vol. 10(20), pages 1-32, October.
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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.

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    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.

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