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 Mat. Zametki, 2016, Volume 99, Issue 2, Pages 262–277 (Mi mz10854)

On the Deficiency Index of the Vector-Valued Sturm–Liouville Operator

K. A. Mirzoeva, T. A. Safonovab

a Lomonosov Moscow State University
b Nothern (Arctic) Federal University named after M. V. Lomonosov, Arkhangelsk

Abstract: Let $\mathbb R_+:=[0,+\infty)$, and let the matrix functions $P$, $Q$, and $R$ of order $n$, $n\in\mathbb N$, defined on the semiaxis $\mathbb R_+$ be such that $P(x)$ is a nondegenerate matrix, $P(x)$ and $Q(x)$ are Hermitian matrices for $x\in\mathbb R_+$ and the elements of the matrix functions $P^{-1}$, $Q$, and $R$ are measurable on $\mathbb R_+$ and summable on each of its closed finite subintervals. We study the operators generated in the space $\mathscr L^2_n(\mathbb R_+)$ by formal expressions of the form
$$l[f]=-(P(f'-Rf))'-R^*P(f'-Rf)+Qf$$
and, as a particular case, operators generated by expressions of the form
$$l[f]=-(P_0f')'+i((Q_0f)'+Q_0f')+P'_1f,$$
where everywhere the derivatives are understood in the sense of distributions and $P_0$, $Q_0$, and $P_1$ are Hermitian matrix functions of order $n$ with Lebesgue measurable elements such that $P^{-1}_0$ exists and $\|P_0\|,\|P^{-1}_0\|, \|P^{-1}_0\|\|P_1\|^2,\|P^{-1}_0\|\|Q_0\|^2 \in \mathscr L^1_{\mathrm{loc}}(\mathbb R_+)$.
The main goal in this paper is to study of the deficiency index of the minimal operator $L_0$ generated by expression $l[f]$ in $\mathscr L^2_n(\mathbb R_+)$ in terms of the matrix functions $P$, $Q$, and $R$ ($P_0$, $Q_0$, and $P_1$). The obtained results are applied to differential operators generated by expressions of the form
$$l[f]=-f"+\sum_{k=1}^{+\infty}\mathscr H_k\delta(x-x_{k})f,$$
where $x_k$, $k=1,2,…$, is an increasing sequence of positive numbers, with $\lim_{k\to +\infty}x_k=+\infty$, $\mathscr H_k$ is a number Hermitian matrix of order $n$, and $\delta(x)$ is the Dirac $\delta$-function.

Keywords: Sturm–Liouville operator, deficiency index, Hermitian matrix-function, Jacobi matrix, Cauchy–Bunyakovskii inequality, quasiderivative, quasidifferential equation.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-11-0075414-01-31136-ìîë à14-01-0034915-31-50259 Ministry of Education and Science of the Russian Federation ÌÊ-3941.2015.1

DOI: https://doi.org/10.4213/mzm10854

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English version:
Mathematical Notes, 2016, 99:2, 290–303

Bibliographic databases:

UDC: 517.983.35+517.929.2

Citation: K. A. Mirzoev, T. A. Safonova, “On the Deficiency Index of the Vector-Valued Sturm–Liouville Operator”, Mat. Zametki, 99:2 (2016), 262–277; Math. Notes, 99:2 (2016), 290–303

Citation in format AMSBIB
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• https://doi.org/10.4213/mzm10854
• http://mi.mathnet.ru/eng/mz/v99/i2/p262

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This publication is cited in the following articles:
1. I. N. Braeutigam, “Limit-point criteria for the matrix Sturm-Liouville operator and its powers”, Opuscula Math., 37:1 (2017), 5–19
2. A. Konstantinov, O. Konstantinov, “Sturm-Liouville operators with matrix distributional coefficients”, Methods Funct. Anal. Topol., 23:1 (2017), 51–59
3. V. S. Budyka, M. M. Malamud, A. Posilicano, “To the spectral theory of one-dimensional matrix Dirac operators with point matrix interactions”, Dokl. Math., 97:2 (2018), 115–121
4. K. A. Mirzoev, I. N. Broitigam, “O defektnykh chislakh operatorov, porozhdennykh yakobievymi matritsami s operatornymi elementami”, Algebra i analiz, 30:4 (2018), 1–26
5. I. N. Braeutigam, K. A. Mirzoev, “Asymptotics of Solutions of Matrix Differential Equations with Nonsmooth Coefficients”, Math. Notes, 104:1 (2018), 150–155
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