Rank One Perturbations at Infinite Coupling

Abstract

We discuss rank one perturbations A_α = A + α(φ,·)φ, α ∈R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (A_α - z)^(−1) φ) = ∫ dμ_α(x)/(x - z). We prove there is a measure dρ_∞ which is the weak limit of (1 + α^2) dμ_α(x) as α → ∞. If φ is cyclic for A, then A_∞, the strong resolvent limit of A_α, is unitarily equivalent to multiplication by x on L^2(R, dρ_∞). This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case.