Rank One Perturbations at Infinite Coupling

We discuss rank one perturbations Aα = A + α(φ,·)φ, α ∈R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (Aα - z)−1 φ) = ∫ dμα(x)/(x - z). We prove there is a measure dρ∞ which is the weak limit of (1 + α2) dμα(x) as α → ∞. If φ is cyclic for A, then A∞, the strong resolvent limit of Aα, is unitarily equivalent to multiplication by x on L2(R, dρ∞). This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case.