The Critical Point Infinity Associated with Indefinite Sturm–Liouville Operators

Consider the indefinite Sturm–Liouville problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with a real weight function r ∈ L1[−1, 1] changing its sign. The question is studied whether or not the eigenfunctions form a Riesz basis of the Hilbert space \(L^2_{|r|}[-1,1]\) or, equivalently, ∞ is a regular critical point of the associated definitizable operator in the Kreı̆n space \(L^2_r[-1,1]\) . 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.