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This talk is devoted to recovering the Sturm--Liouville operator with frozen argument from its spectrum. Unique solvability of this inverse problem depends on the position of frozen argument and the boundary conditions. We compare different approaches to the inverse problem and the corresponding results in two cases of rational and irrational frozen argument. Further, we suggest a new unified approach to operators with frozen argument, which is effective in both cases.

Applying it, we obtain new-type asymptotic formulae completely characterizing the class of sequences that can be spectra of the considered operators.

This talk is based on the paper: Kuznetsova, M. Necessary and sufficient conditions for the spectra of the Sturm--Liouville operators with frozen argument, Applied Mathematics Letters 131 (2022), article 108035.

The paper is available via the link https://authors.elsevier.com/a/1enGg3BGwfEDzY

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