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 Ufimsk. Mat. Zh., 2017, Volume 9, Issue 1, Pages 18–28 (Mi ufa362)

On deficiency index for some second order vector differential operators

I. N. Braeutigama, K. A. Mirzoevb, T. A. Safonovaa

a Northern (Arctic) Federal University named after M. V. Lomonosov, Severnaya Dvina Emb. 17, 163002, Arkhangelsk, Russia
b Lomonosov Moscow State University, Leninskie Gory, 1, 119991, Moscow, Russia

Abstract: In this paper we consider the operators generated by the second order matrix linear symmetric quasi-differential expression
$$l[y]=-(P(y'-Ry))'-R^*P(y'-Ry)+Qy$$
on the set $[1,+\infty)$, where $P^{-1}(x)$, $Q(x)$ are Hermitian matrix functions and $R(x)$ is a complex matrix function of order $n$ with entries $p_{ij}(x),q_{ij}(x),r_{ij}(x)\in L^1_{loc}[1,+\infty)$ ($i,j=1,2,…,n$). We describe the minimal closed symmetric operator $L_0$ generated by this expression in the Hilbert space $L^2_n[1,+\infty)$. For this operator we prove an analogue of the Orlov's theorem on the deficiency index of linear scalar differential operators.

Keywords: quasi-derivative, quasi-differential expression, minimal closed symmetric differential operator, deficiency numbers, asymptotic of the fundamental system of solutions.

 Funding Agency Grant Number German Academic Exchange Service (DAAD) 1.728.2016/DAAD Russian Science Foundation 14-11-00754 Ministry of Education and Science of the Russian Federation ÌÊ-3941.2015.1 The first author is supported by the grant of the Ministery of Educations and Science of Russia and German Academic Exchange Service (DAAD) under the program “Mikhail Lomonosov” (no. 1.728.2016/DAAD). The second author is supported by the grant of RSF (no. 14-11-00754). The third author is supported by Ministery of Educations and Science of Russia (the grant of the President of Russia no. MK-3941.2015.1).

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English version:
Ufa Mathematical Journal, 2017, 9:1, 18–28 (PDF, 359 kB); https://doi.org/10.13108/2017-9-1-18

Bibliographic databases:

Document Type: Article
UDC: 517.984
MSC: 34A30, 34L05, 47E05

Citation: I. N. Braeutigam, K. A. Mirzoev, T. A. Safonova, “On deficiency index for some second order vector differential operators”, Ufimsk. Mat. Zh., 9:1 (2017), 18–28; Ufa Math. J., 9:1 (2017), 18–28

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Budyika V., Malamud M., Posilicano A., “Nonrelativistic Limit For 2P X 2P-Dirac Operators With Point Interactions on a Discrete Set”, Russ. J. Math. Phys., 24:4 (2017), 426–435
2. I. N. Braeutigam, K. A. Mirzoev, “Asymptotics of Solutions of Matrix Differential Equations with Nonsmooth Coefficients”, Math. Notes, 104:1 (2018), 150–155
3. N. N. Konechnaja, K. A. Mirzoev, A. A. Shkalikov, “On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients”, Math. Notes, 104:2 (2018), 244–252
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