The finiteness of the spectrum of boundary value problems defined on a geometric graphстатья
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Аннотация: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 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.