| This passage mainly discuss multiplicity of eigenvalues of the vectorial Sturm-Liouville problems. First we prove that under certain conditions, when n≥2, the vectorial Sturm-Liouville differential equationpossess finitely many eigenvalues which have geometric multiplicity n. Then the passage received the following conclusions, for the case n = 2, we find a sufficient condition on the potential function Q(x),and a bound m_Q depending on Q(x), such that the eigenvalues of the equation with index exceeding m_Q are all simple. These results are applied to find some sufficient conditions which imply that the spectra of two potential equations have finitely many elements in common, and an estimate of the number of elements in the intersection of two spectra is provided. Finally, it described the relation of the algebraic multiplicity and the geometric multiplicity of the vectorial Sturm-Liouville problems, i.e., if the geometric multiplicity of eigenvalues of the vectorial Sturm-Liouville problems is 2, then the algebraic multiplicity also is 2. |
