Angelo B. Mingarelli
Consider the Dirichlet eigenvalue problem associated with the real two-term weighted Sturm-Liouville equation
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 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
Ottawa, Ontario, Canada, K1S 5B6
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