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On the inverse problem for Sturm–Liouville-type operators with frozen argument: rational case

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Abstract

We study the inverse problem of recovering the potential q(x) from the spectrum of the operator \(-y''(x)+q(x)y(a),\) \(y^{(\alpha )}(0)=y^{(\beta )}(1)=0,\) where \(\alpha ,\beta \in \{0,1\}\) and \(a\in [0,1]\) is an arbitrary fixed rational number. We completely describe the cases when the solution of the inverse problem is unique and non-unique. In the last case, we describe sets of iso-spectral potentials and provide various restrictions on the potential under which the uniqueness holds. Moreover, we obtain an algorithm for solving the inverse problem along with necessary and sufficient conditions for its solvability in terms of characterization of the spectrum.

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Acknowledgements

This research was supported by the Ministry of Education and Science of Russian Federation (Grant 1.1660.2017/4.6).

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Authors and Affiliations

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

    Sergey Buterin & Maria Kuznetsova

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  1. Sergey Buterin
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  2. Maria Kuznetsova
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Correspondence to Sergey Buterin.

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Communicated by Luz de Teresa.

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Buterin, S., Kuznetsova, M. On the inverse problem for Sturm–Liouville-type operators with frozen argument: rational case. Comp. Appl. Math. 39, 5 (2020). https://doi.org/10.1007/s40314-019-0972-8

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  • DOI: https://doi.org/10.1007/s40314-019-0972-8

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