We investigate the two-dimensional vectorial Sturm-Liouville
equation with eigenparameter dependent boundary conditions. Under the assumption that Q(x) is nonnegative definite, we prove that the eigenvalues of the two-dimensional vectorial Sturm-Liouville equation are real, and the algebraic multiplicity of an eigenvalue of the problem as a zero of the characteristic function is equal to its geometric multiplicity.
By the theory of Hadamard's factorization, we also prove that the characteristic function is uniquely determined by the spectral set of the equation. Moreover, we consider the inverse problem of the equation that how many spectral sets can determine the potential function Q(x) uniquely, and find that three spectral sets is necessary for us to determine the potential function Q(x) uniquely..

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