Limit-point criteria for the matrix Sturm-Liouville operator and its powers

Abstract

We consider matrix Sturm-Liouville operators generated by the formal expression $l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,$ in the space $L^2_n(I)$, $I:=[0, \infty)$. Let the matrix functions $P:=P(x)$, $Q:=Q(x)$ and $R:=R(x)$ of order $n \in \mathbb{N}$ be defined on $I$, $P$ is a nondegenerate matrix, $P$ and $Q$ are Hermitian matrices for $x \in I$ and the entries of the matrix functions $P^{-1}$, $Q$ and $R$ are measurable on $I$ and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices $P$, $Q$ and $R$ that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by $l^k[y]$ $k \in \mathbb{N}$. In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.