On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equation and some related spectral problems

Source abstract

We prove that under certain conditions, a vectorial Sturm- Liouville differential equation of dimension n ≥ 2 n \geq 2 can only possess finitely many eigenvalues which have multiplicity n n . For the case n = 2 n=2 , we find a sufficient condition on the potential function Q ( x ) Q(x) , and a bound m Q m_Q depending on Q ( x ) Q(x) , such that the eigenvalues of the equation with index exceeding m Q m_Q are all simple. These results are applied to find some sufficient conditions which imply that the spectra of two potential equations, or two string equations, have finitely many elements in common, and an estimate of the number of elements in the intersection of two spectra is provided.