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Inverse Spectral Problem for the Third-Order Differential Equation

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Abstract

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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Funding

This work was supported by Grant 21-71-10001 of the Russian Science Foundation, https://rscf.ru/en/project/21-71-10001/.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov, Russia, 410012

    Natalia P. Bondarenko

  2. S.M. Nikolskii Mathematical Institute, Peoples’ Friendship University of Russia (RUDN University), Miklukho-Maklaya Street 6, Moscow, Russia, 117198

    Natalia P. Bondarenko

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Correspondence to Natalia P. Bondarenko.

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Bondarenko, N.P. Inverse Spectral Problem for the Third-Order Differential Equation. Results Math 78, 179 (2023). https://doi.org/10.1007/s00025-023-01955-x

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