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.