On the Deficiency Index of the Vector-Valued Sturm-Liouville Operator

摘要

Let R_+ := [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R_+ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R_+ and the elements of the matrix functions P~(-1), Q, and R are measurable on R_+ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n~2(R_+) by formal expressions of the form l[f] = -(P(f' - Rf))' - R~*P(f' - Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = -(P_0f')' + i((Q_0f)' + Q_0f') + P_1~'f, where everywhere the derivatives are understood in the sense of distributions and P_0, Q_0, and P_1 are Hermitian matrix functions of order n with Lebesgue measurable elements such that P_0~(-1) exists and ||P_0||, ||P_0~(-1)||, ||P_0~(-1)||||P_1||~2, ||P_0~(-1)||||P_0||~2 ∈ L_(loc)~1(R_+) The main goal in this paper is to study of the deficiency index of the minimal operator L_0 generated by expression l[f] in L_n~2(R_+) in terms of the matrix functions P, Q, and R (P_0, Q_0, and P_1). The obtained results are applied to differential operators generated by expressions of the form l[f] = -f" + +∞ Σ k=1 H_kδ(x-x_k)f, where x_k, k = 1,2,..., is an increasing sequence of positive numbers, with lim_(k→+∞) x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.