Rank One Perturbations with Infinitesimal Coupling

We consider a positive self-adjoint operator A and formal rank one perturbations B = A + α(φ, ·)φ, where φ ∈ H−2(A) but φ ∉ H−1 (A), with Hs(A) the usual scale of spaces. We show that B can be defined for such φ and what are essentially negative infinitesimal values of α. In a sense we will make precise, every rank one perturbation is one of three forms: (i) φ ∈ H−1(A), α ∈ R; (ii) φ ∈ H−1, α = ∞; or (iii) the new type we consider here.