Regularity of the integrated density of states in the continuous spectrum

In this paper we show that spectral measures of the Laplacian on \(\ell ^2({\mathbb {Z}}^d)\) ${\ell }^2({\mathbb{Z}}^d)$ are smooth in some regions of its spectrum, a result that extends to parts of the absolutely continuous spectrum of some random perturbations of it. The spectral measures considered are associated with dense sets of vectors.