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 Tr. Mosk. Mat. Obs., 2014, Volume 75, Issue 2, Pages 107–123 (Mi mmo559)

Distribution of the eigenvalues of singular differential operators in a space of vector-functions

N. F. Valeeva, È. A. Nazirovab, Ya. T. Sultanaevc

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa
b Bashkir State University, Ufa
c Bashkir State Pedagogical University, Ufa

Abstract: A significant part of B. M. Levitan's scientific activity dealt with questions on the distribution of the eigenvalues of differential operators [1]. To study the spectral density, he mainly used Carleman's method, which he perfected. As a rule, he considered scalar differential operators. The purpose of this paper is to study the spectral density of differential operators in a space of vector-functions. The paper consists of two sections. In the first we study the asymptotics of a fourth-order differential operator
$$y^{(4)}+Q(x)y=\lambda y,$$
both taking account of the rotational velocity of the eigenvectors of the matrix $Q(x)$ and without taking the rotational velocity of these vectors into account. In Section 2 we study the asymptotics of the spectrum of a non-semi-bounded Sturm–Liouville operator in a space of vector-functions of any finite dimension.

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English version:
Transactions of the Moscow Mathematical Society, 2014, 75, 89–102

UDC: 517.926, 517.928, 517.984.5
MSC: 47B39, 34L05, 34L02, 34B25
Revised: 16.06.2014

Citation: N. F. Valeev, È. A. Nazirova, Ya. T. Sultanaev, “Distribution of the eigenvalues of singular differential operators in a space of vector-functions”, Tr. Mosk. Mat. Obs., 75, no. 2, MCCME, M., 2014, 107–123; Trans. Moscow Math. Soc., 75 (2014), 89–102

Citation in format AMSBIB
\Bibitem{ValNazSul14} \by N.~F.~Valeev, \E.~A.~Nazirova, Ya.~T.~Sultanaev \paper Distribution of the eigenvalues of singular differential operators in a space of vector-functions \serial Tr. Mosk. Mat. Obs. \yr 2014 \vol 75 \issue 2 \pages 107--123 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo559} \elib{http://elibrary.ru/item.asp?id=23780158} \transl \jour Trans. Moscow Math. Soc. \yr 2014 \vol 75 \pages 89--102 \crossref{https://doi.org/10.1090/S0077-1554-2014-00238-7} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84959053257} `

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This publication is cited in the following articles:
1. N. F. Valeev, E. A. Nazirova, Ya. T. Sultanaev, “On a new approach for studying asymptotic behavior of solutions to singular differential equations”, Ufa Math. J., 7:3 (2015), 9–14
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