Some eigenvalue problems for vectorial Sturm-Liouville equations with eigenparameter dependent boundary conditions

Abstract:

We investigate the $n$-dimensional vectorial Sturm-Liouville equation \[ \vec {y}\ ''(x)+[\lambda ^2 I_n -Q(x)]\vec {y}(x)=\vec {0} \] with eigenparameter dependent boundary conditions \[ \vec {y}(0)=\vec {0}, A\vec {y}\ ’(\pi )+\lambda \vec {y}(\pi )=\vec {0}.\] Under the assumption that $Q(x)$ is nonnegative definite in [0,$\pi$], we prove that the eigenvalues of the $n$-dimensional vectorial Sturm-Liouville equation are real.

For the case $n=2$, we show that the algebraic multiplicity of an eigenvalue of the problem as a zero of the characteristic function \[ \omega _A(\lambda ;Q)=\det [AY’(\pi ;\lambda ^2;Q)+\lambda Y(\pi ;\lambda ^2;Q)]\] is equal to its geometric multiplicity. By the theory of Hadamard’s factorization, we also prove that the characteristic function $\omega _A(\lambda ;Q)$ 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 $Q(x)$ uniquely, and we find that three spectral sets are necessary for us to determine the potential function $Q(x)$ uniquely.