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

Valeev, N.; Nazirova, È.; Sultanaev, Ya.

doi:10.1090/S0077-1554-2014-00238-7

Distribution of the eigenvalues of singular differential operators in a space of vector-functions
HTML articles powered by AMS MathViewer

by N. F. Valeev, È. A. Nazirova and Ya. T. Sultanaev
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2014, 89-102

DOI: https://doi.org/10.1090/S0077-1554-2014-00238-7

Published electronically: November 6, 2014

PDF

Abstract:

A significant part of B. M. Levitan’s scientific activity dealt with questions on the distribution of the eigenvalues of differential operators \cite{lev}. 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.

References

B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I., 1975. Translated from the Russian by Amiel Feinstein. MR 0369797
M. A. Naĭmark, Lineĭ nye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1969 (Russian). Second edition, revised and augmented; With an appendix by V. È. Ljance. MR 0353061
A. G. Kostjučenko and B. M. Levitan, Asymptotic behavior of eigenvalues of the operator Sturm-Liouville problem, Funkcional. Anal. i Priložen. 1 (1967), 86–96 (Russian). MR 0208419
M. Markus and Kh. Mink, Survey of the theory of matrices and matrix inequalities, Nauka, Moscow, 1972. (Russian)
F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
A. G. Kostjučenko and I. S. Sargsjan, Raspredelenie sobstvennykh znacheniĭ, “Nauka”, Moscow, 1979 (Russian). Samosopryazhennye obyknovennye differentsial′nye operatory. [Selfadjoint ordinary differential operators]. MR 560900
Ja. T. Sultanaev, The asymptotic behavior of the spectrum of a differential operator in a space of vector-valued functions, Differencial′nye Uravnenija 10 (1974), 1673–1683, 1733 (Russian). MR 0372682
R. S. Ismagilov, The asymptotic behavior of the spectrum of a differential operator in a space of vector valued fnctions, Mat. Zametki 9 (1971), 667–676 (Russian). MR 285757
R. S. Ismagilov and A. G. Kostyuchenko, On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator, Funktsional. Anal. i Prilozhen. 42 (2008), no. 2, 11–22, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 42 (2008), no. 2, 89–97. MR 2438014, DOI 10.1007/s10688-008-0014-6
R. S. Ismagilov and A. G. Kostyuchenko, On the spectrum of a vector Schrödinger operator, Funktsional. Anal. i Prilozhen. 41 (2007), no. 1, 39–51, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 41 (2007), no. 1, 31–41. MR 2333981, DOI 10.1007/s10688-007-0003-1