Geodesic
Characterizing degenerate Sturm-Liouville problems
Electronic Journal of Differential Equations, Tome 2004 (2004).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 
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     author = {Mingarelli, Angelo B.},
     title = {Characterizing degenerate {Sturm-Liouville} problems},
     journal = {Electronic Journal of Differential Equations},
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     year = {2004},
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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__a115/