Vector Sturm-liouville Differential Equation Eigenvalue Multiplicity

In this paper,we study the multiplicities of vectorial Sturm-Liouville differential equations .First,we find a sufficient condition for which a differential equation of the dimension n=2 can only possess finitely many eigenvalues which have the multiplicity 2 ,when the coefficients of the boundary conditions are diagonal matrixs. Comparing with the results of Chao-Liang Shen , Chung-Tsun Shieh and Chuan-Fu Yang, our results are generalization.We find a sufficient condition on two differential equations of the dimension n=2 which possess finitely many eigenvalues in comman.This result is also right for the dimension n > 2. Finally we find a boundary n_q depending on Q(x), such that the eigenvalues of the equation with index exceeding n_q are all simple.