Spectral correlations from the metal to the mobility edge

We have studied numerically the spectral correlations in a metallic phase and at the metal-insulator transition. We have calculated directly the two-point correlation function of the density of states R(s,s{'}). In the metallic phase, it is well described by the random-matrix theory. We also find numerically the diffusive corrections for the number variance 〈{δ}n²(s)〉 predicted by Al'tshuler and Shklovskii {} {}{}. At the transition, at small energy scales, R(s-s{'}) starts linearly, with a slope larger than in a metal. At large separations {}s-s{'}{}{}1, it is found to decrease as a power law R(s,s{'}){~}-c/{}s-s{'}^2-γ with c{~}0.041 and {γ}{~}0.83, in good agreement with recent microscopic predictions. At the transition, we have also calculated the form factor K ̃ \~{}{}(t), Fourier transform of R(s-s{'}). At large s, the number variance contains two terms 〈{δ}n²(s)〉=B〈n〉^γ+2{π}K ̃ \~{}{}(0)〈n〉, where K ̃ \~{}{}(0) is the limit of the form factor for t{→}0.