On the deficiency index of the vector Sturm-Liouville operator

Let $R_{+}:=[0,+\mathrm{\infty })$. Assume that $n\times n$ ($n\in \mathcal{N}$) matrix functions $P,Q$ and $R$ are defined on the set $R_{+}$, $P(x)$ is non-degenerate, $P(x)$ and $Q(x)$ are Hermitian matrices when $x\in R_{+}$ and the elements of the matrix functions $P^{-1}$, $Q$ and $R$ are measurable on $R_{+}$ and integrable on each closed subinterval of this set. In this paper we study operators generated by formal expressions \begin{equation*} \label{trivial} l[f]=-(P(f^{\prime}-Rf))^{\prime}-R^P(f^{\prime}-Rf)+Qf, \end{equation} in the space ${\mathcal{L}}_n^2(R_{+})$ and, as a special case, operators generated by expressions of the form \begin{equation*} \label{2} l[f]=-(P_0f^{\prime})^{\prime}+i((Q_0f)^{\prime}+Q_0f^{\prime})+P^{\prime}1f, \end{equation*} where derivatives are understood in the sense of distributions and $P_0,Q_0$ and $P_1$ are $n\times n$ Hermitian matrix functions with Lebesgue measurable elements, such that $P_0^{-1}$ exists and $|P_0 |, |P^{-1 }0 |, | P^{-1}0| |P_1|^2,$ $|P^{-1}0| | Q_0|^2 \in L^1 {loc} (R+) $. The main aim of this paper is the study of the deficiency index of the minimal operator $L_0$ generated by the expression $l[f]$ in ${\mathcal{L}}_n^2(R_{+})$ in terms of matrix-valued functions $P,Q$ and $R$ ($P_0,Q_0$ and $P_1$). The obtained results are applied to the differential operators generated by \begin{equation*} \label{p2} l[f]=-f^{\prime\prime}+ \sum\limits{k=1}^{+\infty} {\mathcal H}k\delta(x-x{k})f, \end{equation*} where $x_k$ ($k=1,2,\dots$) is an increasing sequence of positive numbers and $ \lim\limits {k \to +\infty} x_k = +\infty $,$ $\mathcal{H}$_k $isa$n \times n$numericalHermitianmatrixand$ \delta(x) $isDirac$\delta $ - function.