Statistical properties of one-point Green functions in disordered systems and critical behavior near the Anderson transition

We investigate the statistics of local Green functions $G(E, x, x) = \langle x |(E - \hat{H})^{-1}|x\rangle$, in particular of the local density of states ρ∝Im G(E, x, x), with the Hamiltonian $\hat{H}$ describing the motion of a quantum particle in a d-dimensional disordered system. Corresponding distributions are related to a function which plays the role of an order parameter for the Anderson metal-insulator transition. When the system can be described by a nonlinear σ-model, the distribution is shown to possess a specific "inversion" symmetry. We present an analysis of the critical behavior near the mobility edge that follows from the abovementioned relations. We explain the origin of the non-power-like critical behavior obtained earlier for effectively infinite-dimensional models. For any finite dimension d < ∞ the critical behavior is demonstraied to be of the conventional power-law type wilh d = ∞ playing the rote of an upper critical dimension