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

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

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.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:61-:d:1306670
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    References listed on IDEAS

    as
    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, 2022. "Reconstruction of Higher-Order Differential Operators by Their Spectral Data," Mathematics, MDPI, vol. 10(20), pages 1-32, October.
    3. Upeksha Perera & Christine Böckmann, 2020. "Solutions of Sturm-Liouville Problems," Mathematics, MDPI, vol. 8(11), pages 1-14, November.
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