2007 | OriginalPaper | Buchkapitel
A Necessary Aspect of the Generalized Beals Condition for the Riesz Basis Property of Indefinite Sturm-Liouville Problems
verfasst von : Andreas Fleige
Erschienen in: Operator Theory in Inner Product Spaces
Verlag: Birkhäuser Basel
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For the Sturm-Liouville eigenvalue problem −
f
″ =
γrf
[−1, 1] with Dirichlet boundary conditions and with an indefinite weight function
r
changing it’s sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space
L
|
r
|
2
[−1, 1]. In the nineties the sufficient so called generalized one hand Beals condition was found for this Riesz basis property. Now using a new criterion of Parfyonov we show that already the old approach gives rise to a necessary and sufficient condition for the Riesz basis property under certain additional assumptions.