Characterizing degenerate Sturm-Liouville problems

Abstract

Consider the Dirichlet eigenvalue problem associated with the real two-term weighted Sturm-Liouville equation -(p(x)y')' = λr(x)y on the finite interval [a, b]. This eigenvalue problem will be called degenerate provided its spectrum fills the whole complex plane. Generally, in degenerate cases the coefficients p(x), r(x) must each be sign indefinite on [a, b]. Indeed, except in some special cases, the quadratic forms induced by them on appropriate spaces must also be indefinite. In this note we present a necessary and sufficient condition for this boundary problem to be degenerate. Some extensions are noted.