We consider a perturbed graph consisting of two infinite edges, a loop, and a glued arbitrary finite graph γε with small edges, where γε is obtained by ε−1 times contraction of some fixed graph and ε is a small parameter. On the perturbed graph, we consider the Schrödinger operator whose potential on small edges can singularly depend on ε with the Kirchhoff condition at internal vertices and the Dirichlet or Neumann condition at the boundary vertices. For the perturbed eigenvalue and the corresponding eigenfunction we prove the holomorphy with respect to ε and propose a recurrent algorithm for finding all coefficients of their Taylor series.
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Translated from Problemy Matematicheskogo Analiza 111, 2021, pp. 3-17.
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Borisov, D.I., Konyrkulzhaeva, M.N. & Mukhametrakhimova, A.I. On Discrete Spectrum of a Model Graph with Loop and Small Edges. J Math Sci 257, 551–568 (2021). https://doi.org/10.1007/s10958-021-05503-2
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DOI: https://doi.org/10.1007/s10958-021-05503-2