We study the inverse problem for the Strum-Liouville equation on a graph that consists of two quasione-dimensional loops of the same length having a common vertex. As spectral data, we consider the set of eigenvalues of the entire system together with the sets of eigenvalues of two Dirichlet problems for the Sturm-Liouville equations with the condition of total reflection at the vertex of the graph. We obtain conditions for three sequences of real numbers that enable one to reconstruct a pair of real potentials from L 2 corresponding to each loop. We give an algorithm for the construction of the entire set of potentials corresponding to this triple of spectra.
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1168–1188, September, 2008.
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Gomilko, A.M., Pivovarchik, V.N. Inverse Sturm-Liouville problem on a figure-eight graph. Ukr Math J 60, 1360–1385 (2008). https://doi.org/10.1007/s11253-009-0145-9
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DOI: https://doi.org/10.1007/s11253-009-0145-9