Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization

Abstract

Abstract. For an operator, A, with cyclic vector ϕ, we study A + λP where P is the rank one projection onto multiples of ϕ. If [α, β] ⊂ spec(A) and A has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense G δ of λ's. §1. IntroductionThe subject of rank one perturbations of self-adjoint operators and the closely related issue of the boundary condition dependence of Sturm-Liouville operators on [0, ∞) has a long history. We're interested here in the connection with BorelStieltjes transforms of measures (Im z > 0):

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