Geodesic
Distribution of the eigenvalues of singular differential operators in a space of vector-functions
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 107-123.

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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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N. F. Valeev; È. A. Nazirova; Ya. T. Sultanaev. Distribution of the eigenvalues of singular differential operators in a space of vector-functions. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 107-123. http://geodesic.mathdoc.fr/item/MMO_2014_75_2_a1/

[1] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu (samosopryazhennye obyknovennye differentsialnye operatory), Nauka, M., 1970

[2] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR | Zbl

[3] Kostyuchenko A. G., Levitan B. M., “Ob asimptoticheskom povedenii sobstvennykh znachenii operatornoi zadachi Shturma–Liuvillya”, Funkts. analiz i ego pril., 1:1 (1967), 86–96 | MR | Zbl

[4] Markus M., Mink Kh., Obzor po teorii matrits i matrichnykh neravenstv, Nauka, M., 1972 | MR

[5] Atkinson F., Diskretnye i nepreryvnye granichnye zadachi, Mir, M., 1968 | MR | Zbl

[6] Kostyuchenko A. G., Sargsyan I. S., Raspredelenie sobstvennykh znachenii (samosopryazhennye obyknovennye differentsialnye operatory), Nauka, M., 1979

[7] Sultanaev Ya. T., “Ob asimptotike spektra differentsialnogo operatora v prostranstve vektor-funktsii”, Diff. uravneniya, 10:9 (1974), 1673–1683 | MR | Zbl

[8] Ismagilov R. S., “Ob asimptotike spektra differentsialnogo operatora v prostranstve vektor-funktsii”, Matem. zametki, 9:6 (1971), 667–676 | Zbl

[9] Ismagilov R. S., Kostyuchenko A. G., “Ob asimptotike spektra nepoluogranichennogo vektornogo operatora Shturma–Liuvillya”, Funkts. analiz i ego pril., 42:2 (2008), 11–22 | DOI | MR

[10] Ismagilov R. S., Kostyuchenko A. G., “O spektre vektornogo operatora Shredingera”, Funkts. analiz i ego pril., 41:1 (2007), 39–51 | DOI | MR | Zbl