Electron. J. Diff. Eqns., Vol. 2004(2004), No. 130, pp. 18.
Characterizing degenerate SturmLiouville problems
Angelo B. Mingarelli
Abstract:
Consider the Dirichlet eigenvalue problem associated with the real
twoterm weighted SturmLiouville 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: SturmLiouville 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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