Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 130, pp. 1-8.
Title: Characterizing degenerate Sturm-Liouville problems
Author: Angelo B. Mingarelli (Carleton Univ., Ottawa, Ontario, Canada)
Abstract:
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.
Submitted August 19, 2004. Published November 12, 2004.
Math Subject Classifications: 34B24, 34L05.
Key Words: Sturm-Liouville theory; eigenvalues; degenerate operators;
spectral theory; Dirichlet problem.