Opuscula Math. 37, no. 1 (2017), 5-19
http://dx.doi.org/10.7494/OpMath.2017.37.1.5

Opuscula Mathematica

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

Irina N. Braeutigam

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.

Keywords: quasi-derivative, quasi-differential operator, matrix Sturm-Liouville operator, deficiency numbers, distributions.

Mathematics Subject Classification: 34L05, 34B24, 47E05.

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• Irina N. Braeutigam
• Northern (Arctic) Federal University named after M. V. Lomonosov, Severnaya Dvina Emb. 17, Arkhangelsk, 163002, Russia
• Communicated by Alexander Gomilko.
• Revised: 2016-09-21.
• Accepted: 2016-09-25.
• Published online: 2016-12-14.