Point spectrum and mixed spectral types for rank one perturbations

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 Lebe...

Spectral Analysis of Rank One Perturbations and Applications

A review or Lhe general lheory of ~df-lVljoillt operatoro or lhe form A + on where n is nlll].;: one is presellled, Applicl1liu,,~ ine!llde proofs of loealizatiun fur S'chr(idingcr operators. resl111s un illVClse Spt'ct.n11 t.\'J()ory, and examples of opcn1lorH wilh singular cOljt.inl1ou~ ~peclruj)).

Rank One Perturbations, Approximations, and Selfadjoint Extensions

of a semibounded selfadjoint operator A are studied with the help of distribution theory. It is shown that such perturbations can be defined for finite values of : even if the element . does not belong to H&1(A). Approximations of the rank one perturbations are constructed in the strong operator topology. It is proven that rank one H&2 perturbations can be defined uniquely for the homogeneous o...

Singular Continuous Spectrum under Rank One Perturbations and Localization for Random Hamiltonians

We consider a selfadjoint operator, A , and a selfadjoint rank-one projection, P, onto a vector, 9, which is cyclic for A. In terms of the spectral measure dp;, we give necessary and sufficient conditions for A + A P to have empty singular continuous spectrum or to have only point spectrum for a.e. A. We apply these results to questions of localization in the oneand multi-dimensional Anderson m...

Rank one perturbations of self-adjoint operators