Abstract:
We consider a general scalar self-adjoint elliptic second order operator with general boundary conditions on an arbitrary metric graph containing a subgraph with edges of lengths proportional to a small parameter. We show that the resolvent of such operator is holomorphic in the small parameter and provide its representations by Taylor series. The coefficients of the series are found rather explicitly.
Keywords:graph, small edge, resolvent, holomorphy in a small parameter.
Citation:
D. I. Borisov, “Quantum graphs with small edges: holomorphy of resolvents”, Dokl. RAN. Math. Inf. Proc. Upr., 498 (2021), 21–26; Dokl. Math., 103:3 (2021), 113–117
This publication is cited in the following 3 articles:
D. I. Borisov, “Resolvent of a Schrödinger operator on a model graph with small loops”, J. Math. Sci., 276:1 (2023), 48
D. I. Borisov, M. N. Konyrkulzhaeva, A. I. Mukhametrakhimova, “On Discrete Spectrum of a Model Graph with Loop and Small Edges”, J Math Sci, 257:5 (2021), 551
Denis I. Borisov, “Spectra of Elliptic Operators on Quantum Graphs with Small Edges”, Mathematics, 9:16 (2021), 1874