Scattering approach to Anderson localisation

We develop a novel approach to the Anderson localisation problem in a $d$-dimensional disordered sample of dimension $L\times M^{d-1}$. Attaching a perfect lead with the cross-section $M^{d-1}$ 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 non-linear partial differential equation of the Burgers type, for which a complete analytical solution for arbitrary $L$ is constructed. Analysing the solution for a cubic sample with $M=L$ in the limit $L\to \infty$, we find that for $d<2$ the solution tends to the localised 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.