We consider a Schrödinger operator on a model graph with small loops assuming the violation of the typical nonresonance condition which guarantees the holomorphy property of the resolvents of elliptic operators on graphs with small edges in the general case. We study the behavior of parts of the resolvent corresponding to finite edges and small loops in a certain sense. We show that for the parts of the resolvent the holomorphy property is preserved with respect to a small parameter characterizing the length of small loops. Unlike the nonresonance case, for the part of the resolvent corresponding to small loops an additional term appears in the leading term of the Taylor series, which leads to a certain localization of the resolvent on these loops.
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Translated from Problemy Matematicheskogo Analiza 125, 2023, pp. 47-57.
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Borisov, D.I. Resolvent of a Schrödinger Operator on a Model Graph with Small Loops. J Math Sci 276, 48–60 (2023). https://doi.org/10.1007/s10958-023-06724-3
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DOI: https://doi.org/10.1007/s10958-023-06724-3