This paper deals with the vector differential operators generated by vectorial differential expression $Au(x)=\sum\limits^n_{k=0}(-1)^n(P_k(x)u^{(k)}(x))^{(k)},$ $x\in [0,+\infty)$. First, we obtain two vector inequality in Lemma 2.1 and Lemma 2.2, by using operator decomposition theorem, when the coefficient matrix $P_k(x),$ $k=0,1,\cdots,n$ is an $m\times m$ order real symmetric positive definite matrix and an order real symmetric positive definite diagonal matrix respectively, the dispersion of the spectrum of the class of higher order self-adjoint vector differential operators is studied,some sufficient conditions for the spectrum of this kind of operators to be discrete are obtained; The second, in the special case, the vector differential operator with only two terms $Au(x)=-(P(x)u^{(n)}(x))^{(n)}+Q(x)u(x),$ $u(x)\in C^\infty_0((0,\infty),C^m),x\in [0,+\infty)$ is discussed, the smallest operator generated in its self-adjoint domain is the self-adjoint operator, the sufficient and necessary condition for the spectrum of the kind of operator to be discrete is given; The third, by applying this conclusion to vector-valued Sturm-Liouville operators and vector-valued Schrodinger operators, the necessary and sufficient conditions for spectral dispersion of these two types of operators are obtained. The last, the $2n$-th-order mono-term self-adjoint vector differential operator is considered, The necessary and sufficient condition that the spectrum of this kind of operator is discrete is obtained.
Wish all the best wishes for you