On Discrete Spectrum of a Model Graph with Loop and Small Edges

Borisov D.I. Konyrkulzhaeva M.N. Mukhametrakhimova A.I.
September 2021 Springer

Journal of Mathematical Sciences (United States)
2021 #257 Issue 5 551 - 568 pp.

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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Institute of Mathematics, UFRC RAS, 112, Chernyshevskii St, Ufa, 450008, Russian Federation
Bashkir State University, 32, Zaki Validi St, Ufa, 450000, Russian Federation
University of Hradec Králové, 62, Rokitansk´eho, Hradec Králové, 50003, Czech Republic
Al-Farabi Kazakh National University, 71, al-Farabi Ave, Almaty, 050040, Kazakhstan
International Information Technology University, 8, Manas St, Almaty, 050000, Kazakhstan
Bashkir State Pedagogical University, 3a, October Revolution St, Ufa, 450000, Russian Federation

Institute of Mathematics
Bashkir State University
University of Hradec Králové
Al-Farabi Kazakh National University
International Information Technology University
Bashkir State Pedagogical University

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