Consider the Dirichlet eigenvalue problem associated with the real two-term weighted Sturm-Liouville equation -(p(x)y′)′ = λ(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.

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Keywords Degenerate operators, Dirichlet problem, Eigenvalues, Spectral theory, Sturm-Liouville theory
Journal Electronic Journal of Differential Equations
Mingarelli, A. (2004). Characterizing degenerate Sturm-Liouville problems. Electronic Journal of Differential Equations, 2004, 1–8.