Dependence of Eigenvalues of Sturm–Liouville Problems on the Boundary

The eigenvalues of Sturm–Liouville (SL) problems depend not only continuously but smoothly on boundary points. The derivative of thenth eigenvalue as a function of an endpoint satisfies a first order differential equation. This for arbitrary (separated or coupled) self-adjoint regular boundary conditions. In addition, as the length of the interval shrinks to zero all higher eigenvalues march off to plus infinity. This is also true for the first (i.e., lowest) Dirichlet eigenvalue but not for the lowest Neumann eigenvalue. The latter has a finite limit.