Geodesic
Characterizing degenerate Sturm-Liouville problems
Electronic journal of differential equations, Tome 2004 (2004)
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.
Classification : 34B24, 34L05
Keywords: Sturm-Liouville theory, eigenvalues, degenerate operators, spectral theory, Dirichlet problem 
@article{EJDE_2004__2004__a15,
     author = {Mingarelli,  Angelo B.},
     title = {Characterizing degenerate {Sturm-Liouville} problems},
     journal = {Electronic journal of differential equations},
     year = {2004},
     volume = {2004},
     zbl = {1076.34025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a15/}
}
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Mingarelli,  Angelo B. Characterizing degenerate Sturm-Liouville problems. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a15/