Absence of continuous spectral types for certain nonstationary random models

We consider continuum random Schr\"odinger operators of the type $H_{\omega} = -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of $-\Delta +V_0$. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (``random tube'') in arbitrary dimension.