Dynamical structure factor in disordered systems

We study the spectral width as a function of the external momentum for the dynamical structure factor of a disordered harmonic solid, considered as a toy model for supercooled liquids and glasses. In the contexts of both the single-link coherent potential approximation and a single-defect approximation, two different regimes are clearly identified: if the density of states at zero energy is zero, the usual [Formula Presented] law is recovered for small momentum. On the contrary, if the disorder induces a nonvanishing density of states at zero energy, a linear behavior is obtained. The dynamical structure factor is numerically calculated in lattices as large as [Formula Presented] and satisfactorily agrees with the analytical computations.