The Critical Point Infinity Associated with Indefinite Sturm–Liouville Problems

Andreas Fleige ()

Chapter 17 in Operator Theory, 2015, pp 395-429 from Springer

Abstract: Abstract Consider the indefinite Sturm–Liouville problem − f ′ ′ = λ r f $$-f^{{\prime\prime}} = \lambda rf$$ on [−1, 1] with Dirichlet boundary conditions and with a real weight function r ∈ 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] or, equivalently, ∞ is a regular critical point of the associated definitizable operator in the Kreĭn space L r 2[−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. The present paper tries to give an overview over the so far known results in this area.

Keywords: Hilbert Space; Definitizable Operator; General Boundary Condition; Liouville Problem; Weyl Function (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0667-1_44

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DOI: 10.1007/978-3-0348-0667-1_44

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