Point Spectrum and Mixed Spectral Types for Rank One Perturbations

We consider examples Aλ = 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 exists A,φ so that {E ∈ I | some Aλ has E as an eigenvalue} agrees with B up to sets of Lebesgue measure zero. We also show that there exist examples where Aλ has a.c. spectrum [0,1] for all λ, and for sets of λ’s of positive Lebesgue measure, Aλ also has point spectrum in [0, 1], and for a set of λ’s of positive Lebesgue measure, Aλ also has singular continuous spectrum in [0, 1].