The Domains of Self-Adjoint Extensions of a Schrödinger Operator

Abstract

A basic problem in the analysis of formally self-adjoint differential expressions is to characterise the self-adjoint operators, in an appropriate Hilbert space, associated with these expressions. In a paper of 1963 (Quart. J. Math. Oxford, 14(1963), M-45), Everitt gave connected proofs of the characterisations of the domains associated with the ordinary differential expression L, where L Ψ (x) = -(p(x) Ψ ' (x)) ' + q(x) Ψ (x) (0 ≤ x < ∞), with p, q real-valued and p(x) > 0 for all x ≥ 0. The nature of these domains depends on whether L is limit-point or limit-circle, in the sense of Weyl, at infinity. It is natural to look for similar characterisations for the corresponding partial differential operators. Although a complete characterisation seems unknown, Everittis results prompt the investigation of certain domains and indicate tests to apply to particular examples. This lecture presented the contents of the paper with the same title, by D P Goodall and I M Michael, published shortly after the conference in the Journal of the London Mathematical Society, Second Series, Volume 7 (1973), 265-271.