2 INVERSE EIGENVALUE PROBLEMS FOR VECTORIAL STURM-LIOUVILLE EQUATIONS

Let Q(x) be a continuous n x n symmetric Jacobi matrix-valued even fun ction on [-1, 1]. It is shown that if each element in the Dirichlet sp ectrum of -d(2)/dx(2) + Q(x) has multiplicity n, then there exists a s caler-valued function p(x) such that Q(x) = p(x)I-n. This result is us ed to investigate vectorial Hill's operators with symmetric Jacobi mat rix-valued potential functions, a theorem similar to the Borg theorem for scalar Hill's operators is proved.