Abstract: We investigate the -dimensional vectorial Sturm-Liouville equation
with eigenparameter dependent boundary conditions
Under the assumption that is nonnegative definite in [0,], we prove that the eigenvalues of the -dimensional vectorial Sturm-Liouville equation are real.
For the case , we show that 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 for the equation, i.e., how many spectral sets can determine the potential function uniquely, and we find that three spectral sets are necessary for us to determine the potential function uniquely.
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