Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval

In this paper, the vectorial Sturm-Liouville operator L Q = - d 2 d x 2 + Q ( x ) is considered, where Q(x) is an integrable m × m matrix-valued function defined on the interval [0,π] The authors prove that m 2+1 characteristic functions can determine the potential function of a vectorial Sturm-Liouville operator uniquely. In particular, if Q(x) is real symmetric, then m ( m + 1 ) 2 + 1 characteristic functions can determine the potential function uniquely. Moreover, if only the spectral data of self-adjoint problems are considered, then m 2 + 1 spectral data can determine Q(x) uniquely.