The Critical Point Infinity Associated with Indefinite Sturm–Liouville Problems

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.