Dependence of Eigenvalues of a Class of Higher-Order Sturm-Liouville Problems on the Boundary

Abstract EN

We show that the eigenvalues of a class of higher-order Sturm-Liouville problems depend not only continuously but also smoothly on boundary points and that the derivative of the n th eigenvalue as a function of an endpoint satisfies a first order differential equation.

In addition, we prove that as the length of the interval shrinks to zero all 2k th-order Dirichlet eigenvalues march off to plus infinity; this is also true for the first (i.e., lowest) eigenvalue.