On the deficiency index of the vector Sturm-Liouville operator

TL;DR

This paper investigates the deficiency index of the minimal operator generated by vector Sturm-Liouville expressions with matrix coefficients, providing conditions and applications to differential operators with delta interactions.

Contribution

It characterizes the deficiency index of vector Sturm-Liouville operators with matrix coefficients in terms of matrix functions, extending classical results to more general matrix-valued cases.

Findings

Derived criteria for the deficiency index based on matrix functions.

Applied results to operators with delta interactions at discrete points.

Extended classical scalar results to matrix-valued differential operators.

Abstract

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.