The singular Sturm–Liouville operators with nonsmooth potentials in a space of vector functions

This paper deals with the Sturm-Liouville operators generated on the semi-axis by the differential expression $l[y]=-(y'-Py)'-P(y'-Py)-P^2y$, where $'$ is a derivative in terms of the theory of distributions and $P$ is a real-valued symmetrical matrix with elements $p_{ij}\in L^2_{loc}(R_+)$ ($i,j=1,2,\dots,n$). The minimal closed symmetric operator $L_0$ generated by this expression in the Hilbert space $\mathcal L^2_n(R_+)$ is constructed. Sufficient conditions of minimality and maximality of deficiency numbers of the operator $L_0 $ in terms of elements of a matrix $P$ are presented. Moreover, it is established, that the condition of maximality of deficiency numbers of the operator $L_0 $ (in the case when elements of the matrix $P$ are step functions with an infinite number of jumps) is equivalent to the condition of maximality of deficiency numbers of the operator generated by a generalized Jacobi matrix in the space $l^2_n$.