On a second-order matrix differential operator

Abstract

We consider the second-order matrix differential operator

$$N = \left( {\begin{array}{*{20}c} { - \frac{d}{{dx}}\left( {p_0 \frac{d}{{dx}}} \right) + p_1 } \\ r \\ \end{array} \begin{array}{*{20}c} r \\ { - \frac{d}{{dx}}\left( {q_0 \frac{d}{{dx}}} \right) + q_1 } \\ \end{array} } \right)$$

determined by the expression Nφ, [0 ⩽x < ∞), where$\phi = \left( {\begin{array}{*{20}c} U \\ V \\ \end{array} } \right)$. It has been proved that if p₀, q₀, p₁, q₁,r satisfy certain conditions, then N is in the limit point case at ∞. It has been also shown that certain differential operators in the Hilbert space L² of vectors, generated by the operator N, are symmetric and self-adjoint.