This article is cited in 2 papers

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

N. F. Valeev^a, È. A. Nazirova^b, Ya. T. Sultanaev<sup>c

a</sup> 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.

UDC: 517.926, 517.928, 517.984.5

MSC: 47B39, 34L05, 34L02, 34B25

Received: 24.12.2013
Revised: 16.06.2014

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

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