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Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions

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Abstract

In this paper, we for the first time get constructive solution for the inverse Sturm–Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl function is proved. An algorithm of solving the inverse problem is obtained and justified. More concretely, we reduce the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences and then derive reconstruction formulas for the problem coefficients, which are new even for the case of regular potential. Note that the spectral problem in this paper is investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. In the future, our results can be applied to investigation of the inverse problem solvability and stability as well as to development of numerical methods for the reconstruction.

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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. Institute of IT and Cybernetics, Samara National Research University, Moskovskoye Shosse 34, Samara, Russia, 443086

    Egor E. Chitorkin

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

    Egor E. Chitorkin & Natalia P. Bondarenko

  3. Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara, Russia, 443086

    Natalia P. Bondarenko

  4. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, Russia, 117198

    Natalia P. Bondarenko

Authors
  1. Egor E. Chitorkin
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  2. Natalia P. Bondarenko
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Contributions

Conceptualization, NPB; Methodology, NPB; Validation, NPB; Investigation, EEC; Writing—original draft, EEC; Writing—review and editing, NPB; Supervision, NPB. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Egor E. Chitorkin.

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Chitorkin, E.E., Bondarenko, N.P. Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions. Anal.Math.Phys. 13, 79 (2023). https://doi.org/10.1007/s13324-023-00845-3

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