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) andA has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. The 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 Borel-Stieltjes transforms of measures (Im z>0):

Citations