Borisov, D. I.; Mukhametrakhimova, A. I. On a model graph with a loop and small edges. Holomorphy property of resolvent. (English. Russian original) Zbl 1458.34059 J. Math. Sci., New York 251, No. 5, 573-601 (2020); translation from Probl. Mat. Anal. 106, 17-41 (2020). The resolvent of the Sturm-Liouville operator on a geometrical graph with a small parameter is studied. It is shown that the resolvent is holomorphic with respect to the small parameter, and the asymptotics of the resolvent is obtained. Reviewer: Vjacheslav Yurko (Saratov) Cited in 5 Documents MSC: 34B45 Boundary value problems on graphs and networks for ordinary differential equations 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47E05 General theory of ordinary differential operators Keywords:Sturm-Liouville operators; geometrical graph; resolvent; asymptotics PDFBibTeX XMLCite \textit{D. I. Borisov} and \textit{A. I. Mukhametrakhimova}, J. Math. Sci., New York 251, No. 5, 573--601 (2020; Zbl 1458.34059); translation from Probl. Mat. Anal. 106, 17--41 (2020) Full Text: DOI References: [1] Yu. V. Pokornyi et al. Differential Equations on Geometric Graghs [in Russian], Fizmatlit, Moscow (2005). [2] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Am. Math. Soc., Providence, RI (2013). · Zbl 1318.81005 [3] Cheon, T.; Exner, P.; Turek, O., Approximation of a general singular vertex coupling in quantum graphs, Ann. Phys., 325, 3, 548-578 (2010) · Zbl 1192.81159 [4] Zhikov, VV, Homogenization of elasticity problems on singular structures, Izv. Math., 66, 2, 299-365 (2002) · Zbl 1043.35031 [5] Berkolaiko, G.; Latushkin, Y.; Sukhtaiev, S., Limits of quantum graph operators with shrinking edges, Adv. Math., 352, 632-669 (2019) · Zbl 1422.81101 [6] Borisov, DI; Konyrkulzhaeva, MN, Simplest graphs with small edges: Asymptotics for resolvents and holomorphic dependence of the spectrum, Ufa Math. J., 11, 2, 3-17 (2019) [7] Borisov, DI; Konyrkulzhayeva, MN, Perturbation of threshold of the essential spectrum of the Schrödinger operator on the simplest graph with a small edge, J. Math. Sci., New York, 239, 3, 248-267 (2019) · Zbl 1426.34039 [8] Maz’ya, VG; Nazarov, SA; Plamenevskii, BA, Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes, Math. USSR-Izv., 24, 2, 321-345 (1985) · Zbl 0566.35031 [9] A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Am. Math. Soc., Providence, RI (1992). [10] Borisov, DI; Mukhametrakhimova, AI, On norm resolvent convergence for elliptic operators in multi-dimensional domains with small holes, J. Math. Sci., New York, 232, 3, 283-298 (2018) · Zbl 1403.35023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.