Characterizing degenerate Sturm-Liouville problems

Consider the Dirichlet eigenvalue problem associated with the real two-term weighted Sturm-Liouville equation $$-(p(x)y')' = lambda r(x)y$$ 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.