Abstract
We consider a system of non-interacting electrons moving on a regular lattice in the presence of an on-site random potential. With only minimal algebraic assumptions we derive a set of nonlinear self-consistent equations for vertex functions from which the dynamical conductivity can be calculated from the standard Kubo formula. In the limit of weak disorder our equations reproduce the self-consistent equations for the conductivity obtained by Vollhardt and Wolfle and suggest the existence of an Anderson transition from extended (finite conductivity) to localised (zero conductivity) states in three dimensions with localised states only in one and two dimensions. For illustrative purposes we evaluate some auxiliary expressions analytically for the Lloyd model
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