Stability of singular spectral types under decaying perturbations

We look at invariance of a.e. boundary condition spectral behavior under perturbations, W, of half-line, continuum or discrete Schrödinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported Wʹs to suitable short-range W. We also discuss invariance of the local Hausdorff dimension of spectral measures under such perturbations.