POINT SPECTRUM AND MIXED SPECTRAL TYPES FOR RANK-ONE PERTURBATIONS

We consider examples A lambda = A + lambda(phi,.)phi of rank one pertu rbations with phi a cyclic vector for A. We prove that for any bounded measurable set B subset of I, an interval, there exist A, phi so that {E is an element of I\ some A(lambda) has E as an eigenvalue) agrees with B up to sets of Lebesgue measure zero. We also show that there ex ist examples where A(lambda) has a.c. spectrum [0,1] for all lambda, a nd for sets of lambda's of positive Lebesgue measure, A(lambda) also h as point spectrum in [0, 1], and for a set of lambda's of positive Leb esgue measure, A(lambda) also has singular continuous spectrum in [0, 1].