On the multiplicity of the eigenvalues of the vectorial Sturm-Liouville equation

Abstract

Let $ Q(x) $ be a continuous $ m\times m $ real symmetric matrix-valued function defined on $ [0,1] $, and denote the Sturm-Liouville operator $ -\frac{d^2}{dx^2}+Q(x) $ as $ L_Q $ with $ Q(x)$ as its potential function. In this paper we prove that for each Dirichlet eigenvalue $ \lambda_* $ of $L_Q$, the geometric multiplicity of $ \lambda_* $ is equal to its algebraic multiplicity. Applying this result, we get a necessary and sufficiently condition such that each Dirichlet eigenvalue of $ L_Q $ is of multiplicity $ m $.