Decompositions of singular continuous spectra of H-2-class rank one perturbations

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 alpha-dimensional Hausdorff measures is given and a characterization of the supports of the alpha-singular, alpha-absolutely continuous, and strongly alpha-continuous parts of the spectral measure of H-2 - class rank one singular perturbations is given in terms of the limiting behaviour of the regularized Borel transform.