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


Authors: N. F. Valeev, È. A. Nazirova and Ya. T. Sultanaev
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2014, 89-102
MSC (2010): Primary 47B39; Secondary 34L05
DOI: https://doi.org/10.1090/S0077-1554-2014-00238-7
Published electronically: November 6, 2014
MathSciNet review: 3308603
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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

$\displaystyle 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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  • 1. B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators, American Mathematical Society, Providence, R.I., 1975. Translated from the Russian by Amiel Feinstein; Translations of Mathematical Monographs, Vol. 39. MR 0369797
  • 2. M. A. Naĭmark, \cyr Lineĭnye differentsial′nye operatory, Izdat. “Nauka”, Moscow, 1969 (Russian). Second edition, revised and augmented; With an appendix by V. È. Ljance. MR 0353061
    M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Frederick Ungar Publishing Co., New York, 1967. MR 0216050
    M. A. Naĭmark, Linear differential operators. Part II: Linear differential operators in Hilbert space, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968. MR 0262880
  • 3. 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
  • 4. M. Markus and Kh. Mink, Survey of the theory of matrices and matrix inequalities, Nauka, Moscow, 1972. (Russian)
  • 5. F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
  • 6. A. G. Kostjučenko and I. S. Sargsjan, \cyr Raspredelenie sobstvennykh znacheniĭ, “Nauka”, Moscow, 1979 (Russian). \cyr Samosopryazhennye obyknovennye differentsial′nye operatory. [Selfadjoint ordinary differential operators]. MR 560900
  • 7. 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
  • 8. 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 0285757
  • 9. 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, https://doi.org/10.1007/s10688-008-0014-6
  • 10. 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, https://doi.org/10.1007/s10688-007-0003-1

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Additional Information

N. F. Valeev
Affiliation: Institute of Mathematics with Computing Centre of the Ufa Science Centre of the Russian Academy of Sciences, Bashkir State University, Ufa
Email: valeevnf@mail.ru

È. A. Nazirova
Affiliation: Bashkir State University, Ufa
Email: ellkid@gmail.com

Ya. T. Sultanaev
Affiliation: Bashkir State Pedagogical University, Ufa
Email: sultanaevyt@gmail.com

DOI: https://doi.org/10.1090/S0077-1554-2014-00238-7
Keywords: Spectral theory of differential operators, distribution of eigenvalues, asymptotics of the spectrum of a differential operator in a space of vector-functions
Published electronically: November 6, 2014
Article copyright: © Copyright 2014 N. F. Valeev, È. A. Nazirova, Ya.T.Sultanaev