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 \(n\) (\(n \in \mathbb{N}\)) 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.