On the existence of nonsimple real eigenvalues for general Sturm-Liouville problems

Abstract:

The Sturm-Liouville eigenvalue problem $- y'' + q(t)y = \lambda r(t)y$, $t \in [a, b]$, where $y$ is required to satisfy a pair of homogeneous separated boundary conditions at $t = a$, $t = b$ is considered when no sign restrictions are imposed upon the coefficients $q$, $r$. It will be shown that the general eigenvalue problem above can admit at most finitely many nonsimple real eigenvalues (in some cases none at all). Moreover, we will show by means of an example that nonsimple real eigenvalues may occur in the case when each of $q$ and $r$ changes sign in $(a, b)$ and under Dirichlet boundary conditions.