This paper deals with the vector differential operators generated by vectorial differential expression Au(x)=n∑k=0(−1)n(Pk(x)u(k)(x))(k), x∈[0,+∞). First, we obtain two vector inequality in Lemma 2.1 and Lemma 2.2, by using operator decomposition theorem, when the coefficient matrix Pk(x), k=0,1,⋯,n is an m×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)∈C∞0((0,∞),Cm),x∈[0,+∞) 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.
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