Decompositions of Singular Continuous Spectra of $$\mathcal{H}_{ - 2} $$ -class Rank One Perturbations

Abstract

The decomposition theory for the singular continuous spectrum of rank one singular perturbations is studied. A generalization of the well-known Aronszajn-Donoghue theory to the case of decompositions with respect to α-dimensional Hausdorff measures is given and a characterization of the supports of the α-singular, α-absolutely continuous, and strongly α-continuous parts of the spectral measure of \(\mathcal{H}_{ - 2} \) - class rank one singular perturbations is given in terms of the limiting behaviour of the regularized Borel transform.