Singular Sturm-Liouville Problems: The Friedrichs Extension and Comparison of Eigenvalues

Abstract

A new characterization of singular self-adjoint boundary conditions for Sturm-Liouville problems is given. These are an exact parallel of the regular case. They are given explicitly in terms of principal and non-principal solutions. The special nature of the Friedrichs extension is clearly apparent and highlighted. Inequalities among the eigenvalues of different boundary conditions, separated and coupled, are obtained. Most of all we want to stress the method of proof. It is based on a very elementary transformation which transforms any singular non-oscillatory limit-circle endpoint into a regular one.