Density of states and critical resistivity of strongly disordered systems

Recently we proposed that the anomalous diffusion at short length scales associated with localization is responsible for the unusual sensitivity of the high-$T_c$ superconductors to disorder. We showed that the Coulomb pseudopotential in superconductors is a universal function of $\frac{\rho }{{\rho }_c}$ where $\rho$ is the resistivity and ${\rho }_c$ is a critical resistivity characteristic of the system, and obtained relatively small values of ${\rho }_c$ from the experimental $T_c-\mathrm{vs}-\rho$ curves. In the present paper we show that in the strong-disorder region in three dimensions the density of states is also a universal function of the same parameter, so that in the absence of any microscopic calculation of ${\rho }_c$, a comparison of the density of states and $T_c$ as function of disorder provides a crucial test of the theory. We find very good agreement with existing data on granular Al. We also find that the density of states has a logarithmic energy dependence in this region, a result independently obtained by Lee. In addition, we make quantitative predictions about where and when to expect this logarithmic correction to be important, and comment on why it has not been observed yet.