Geodesic
Differential operators on graphs with a cycle
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 3, pp. 343-351.

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An inverse problem of spectral analysis is studied for Sturm – Liouville differential operators on a graph with a cycle. We pay the main attention to the most important nonlinear inverse problem of recovering coefficients of differential equations provided that the structure of the graph is known a priori. We use the standard matching conditions in the interior vertices and Robin boundary conditions in the boundary vertices. For this class of operators properties of spectral characteristics are established, a constructive procedure is obtained for the solution of the inverse problem of recovering coefficients of differential operators from spectra, and the uniqueness of the solution is proved. For solving this inverse problem we use the method of spectral mappings, which allows one to construct the potential on each fixed edge. For transition to the next edge we use a special representation of the characteristic functions. 
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V. A. Yurko. Differential operators on graphs with a cycle. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 3, pp. 343-351. http://geodesic.mathdoc.fr/item/ISU_2021_21_3_a6/

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