Scattering Approach to Anderson Localization

We develop a novel approach to the Anderson localization problem in a d-dimensional disordered sample of dimension L x Md-1. Attaching a perfect lead with the cross section Md-1 to one side of the sample, we derive evolution equations for the scattering matrix and the Wigner-Smith time delay matrix as a function of L. Using them one obtains the Fokker-Planck equation for the distribution of the proper delay times and the evolution equation for their density at weak disorder. The latter can be mapped onto a nonlinear partial differential equation of the Burgers type, for which a complete analytical solution for arbitrary L is constructed. Analyzing the solution for a cubic sample with M = L in the limit L -> infinity, we find that for d < 2 the solution tends to the localized fixed point, while for d > 2 to the metallic fixed point, and provide explicit results for the density of the delay times in these two limits.