The Critical Point Infinity Associated with Indefinite Sturm–Liouville Operators

Andreas Fleige ()

Chapter 25 in Operator Theory, 2026, pp 635-670 from Springer

Abstract: Abstract Consider the indefinite Sturm–Liouville problem − f ′′ = λrf $$-f^{\prime \prime } = {\lambda } r f$$ on [ −1 , 1 ] $$[-1,1]$$ with Dirichlet boundary conditions and with a real weight function r ∈ L 1 [ −1 , 1 ] $$r \in L^1[-1,1]$$ changing its sign. The question is studied whether or not the eigenfunctions form a Riesz basis of the Hilbert space L | r | 2 [ −1 , 1 ] $$L^2_{|r|}[-1,1]$$ or, equivalently, ∞ $$\infty $$ is a regular critical point of the associated definitizable operator in the Kreı̆n space L r 2 [ −1 , 1 ] $$L^2_r[-1,1]$$ . This question is also related to other subjects of mathematical analysis like half range completeness, interpolation spaces, HELP-type inequalities, regular variation, and Kato’s representation theorems for non-semibounded sesquilinear forms. The eigenvalue problem can be generalized to arbitrary self-adjoint boundary conditions, singular endpoints, higher order, higher dimension, and signed measures. This chapter tries to give an overview over the so far known results in this area.

Keywords: Sturm-Liouville problem; Krein space; Definitizable operator; Critical point; Eigenfunction expansion; Riesz basis; Half range completeness; HELP inequality; Similarity problem; Kato’s representation theorems (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_44

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DOI: 10.1007/978-3-032-16356-1_44

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