On the deficiency index of the vector Sturm-Liouville operator

Let $R_+: = [ 0 , +\infty)$. 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 $\mathcal {L}^2_n(R_+)$ 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 $P^{- 1}_0$ 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 $\mathcal{L}^2_n(R_+)$ 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 $x_k$ ($k = 1,2, \ldots$) is an increasing sequence of positive numbers and $\lim\limits_ {k \to +\infty} x_k = +\infty$, $\mathcal{H}_k$ is a $n \times n$ numerical Hermitian matrix and $\delta(x)$ is Dirac $\delta$ - function.