Electron. J. Diff. Eqns., Vol. 2004(2004), No. 130, pp. 1-8.

Characterizing degenerate Sturm-Liouville problems

Angelo B. Mingarelli

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.

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Angelo B. Mingarelli
School of Mathematics and Statistics
Carleton University
Ottawa, Ontario, Canada, K1S 5B6
email: amingare@math.carleton.ca

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