Point spectrum and mixed spectral types for rank one perturbations

We consider examples A_ λ = A + λ(ϕ, •)ϕ of rank one perturbations with ϕ a cyclic vector for A. We prove that for any bounded measurable set B ⊂ I, an interval, there exist A, ϕ so that {E ∈ I | some A_ λ has E as an eigenvalue} agrees with B up to sets of Lebesgue measure zero. We also show that there exist examples where A_ λ has a.c. spectrum [0,1] for all λ, and for sets of λ's of positive Lebesgue measure, A_ λ also has singular continuous spectrum in [0,1].