Geodesic
The finiteness of the spectrum of boundary value problems defined on a geometric graph
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 80 (2019) no. 2, pp. 147-156.

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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. 
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V. A. Sadovnichii; Ya. T. Sultanaev; A. M. Akhtyamov. The finiteness of the spectrum of boundary value problems defined on a geometric graph. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 80 (2019) no. 2, pp. 147-156. http://geodesic.mathdoc.fr/item/MMO_2019_80_2_a1/

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