Some inverse spectral problems for vectorial Sturm-Liouville equations

Let Q(x) be a smooth n-by-n real symmetric matrix-valued function defined in the interval [0,1]. For the n-dimensional vectorial Sturm-Liouville operator L_Q = -d²/dx² + Q(x), we prove that if Q(x) is an even function in the interval [0,1], i.e. Q(1-x) = Q(x), and each of the Dirichlet eigenvalues of the operator L_Q on the interval [0,1] is of multiplicity n, then Q(x) is of the form q(x)I_n, where q(x) is a scalar-valued function. For the case n = 2, we investigate the inverse problem of the unique determination of the potential function in the situation when some spectra of the operator subject to certain different boundary conditions at zero and at unity are known. We prove that five spectra determine the potential function of a two-dimensional vectorial Sturm-Liouville operator uniquely. Besides we also consider the problem of the unique determination of Q(x) in the situation where Q(x) is prescribed in a proper subinterval of [0,1], and some spectra of the operator restricted on certain proper subintervals of [0,1] are given. For the cases n = 1 and 2, some theorems related to this problem are proved.