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 Tr. Mosk. Mat. Obs., 2019, Volume 80, Issue 2, Pages 147–156 (Mi mmo626)

The finiteness of the spectrum of boundary value problems defined on a geometric graph

V. A. Sadovnichiia, Ya. T. Sultanaevb, A. M. Akhtyamovcd

a Lomonosov Moscow State University, Moscow, Russia 119234
b Bashkir State Pedagogical University n. a. M. Akmulla, Ufa, Russia
c Bashkir State University, Ufa, Russia
d Mavlyutov Institute of Mechanics, Ufa Investigation Center R.A.S., Ufa, Russia

Abstract: We consider boundary value problems on a geometric graph with a polynomial occurrence of spectral parameter in the differential equation. It has previously been shown (see A. M. Akhtyamov [Differ. Equ.55 (2019), no. 1, pp. 142-144]) that a boundary value problem for one differential equation whose characteristic equation has simple roots cannot have a finite spectrum, and a boundary value problem for one differential equation can have any given finite spectrum when the characteristic polynomial has multiple roots. In this paper, we obtain a similar result for differential equations defined on a geometric graph. We show that a boundary value problem on a geometric graph cannot have a finite spectrum if all its characteristic equations have simple roots, and a boundary value problem has a finite spectrum if at least one characteristic equation has multiple roots. We also give results showing that a boundary value problem can have any given finite spectrum.

Key words and phrases: Boundary value problem on a geometric graph, characteristic equation, finite spectrum.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-51-06002_Aç_a18-01-00250_a17-41-020230_ð_a17-41-020195_ð_à This work was supported by the Russian Foundation for Basic Research, grants. no. 18-51-06002-Az_a, 18-01-00250-a, 17-41-020230-p_a, and 17-41-020195-p_a.

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English version:
Transactions of the Moscow Mathematical Society, 2019, 80, 123–131

UDC: 517.984
MSC: 34B45, 47E05

Citation: V. A. Sadovnichii, Ya. T. Sultanaev, A. M. Akhtyamov, “The finiteness of the spectrum of boundary value problems defined on a geometric graph”, Tr. Mosk. Mat. Obs., 80, no. 2, MCCME, M., 2019, 147–156; Trans. Moscow Math. Soc., 80 (2019), 123–131

Citation in format AMSBIB
\Bibitem{SadSulAkh19} \by V.~A.~Sadovnichii, Ya.~T.~Sultanaev, A.~M.~Akhtyamov \paper The finiteness of the spectrum of boundary value problems defined on a geometric graph \serial Tr. Mosk. Mat. Obs. \yr 2019 \vol 80 \issue 2 \pages 147--156 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo626} \transl \jour Trans. Moscow Math. Soc. \yr 2019 \vol 80 \pages 123--131 \crossref{https://doi.org/10.1090/mosc/293} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85083770043}