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

  • Sergey ButerinEmail author
  • Maria Kuznetsova


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.


Sturm–Liouville-type operator Functional-differential operator Frozen argument Inverse spectral problem 

Mathematics Subject Classification

34A55 34K29 



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


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© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of MathematicsSaratov State UniversitySaratovRussia

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