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

Sadovnichii, V.; Sultanaev, Ya.; Akhtyamov, A.

doi:10.1090/mosc/293

The finiteness of the spectrum of boundary value problems defined on a geometric graph
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by V. A. Sadovnichii, Ya. T. Sultanaev and A. M. Akhtyamov
Translated by: Taras Panov

Trans. Moscow Math. Soc. 2019, 123-131

DOI: https://doi.org/10.1090/mosc/293

Published electronically: April 1, 2020

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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.

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