Abstract:

The $m$-dimensional vectorial Sturm-Liouville problem with Dirichlet boundary conditions on $(0,1)$ is studied. We firstly discuss the relationship between the matrix-valued potential and the multiplicities of eigenvalues. We prove that if the multiplicities of eigenvalues of $\int_{0}^{1}Q(x){\rm d}x$ are at most $k$ $(1\leq k\leq m-1)$, with finitely many exceptions, the multiplicities of eigenvalues of the vectorial problem are also at most $k$. Then, the inverse nodal problem is investigated with a different method. We show that if there exists an infinite eigenfunctions sequence $\{y_{n_{j},r}(x,\lambda_{n_{j},r})\}_{j=1}^{\infty }$ which are all vectorial functions of type $(CZ)$, then $Q$ is simultaneously diagonalizable.

Key words: Vectorial Sturm-Liouville problems, Multiplicities, Estimation of eigenvalues, Inverse nodal problem