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 Schr\"odinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported $W$'s to suitable short-range $W$. We also discuss invariance of the local Hausdorff dimension of spectral measures under such perturbations.