Existence of mobility edges in Anderson's model for random lattices

Anderson's theory of localization is critically reviewed and extended with particular emphasis on some controversial aspects. It is shown that when the randomness exceeds a certain critical value, all the eigenstates become localized in agreement with Anderson's original result. When the randomness is less than this critical value, the tails of a band consist of localized states. The character of the states changes sharply from localized to extended at mobility edges, in agreement with the Mott-CFO (Cohen-Fritzsche-Ovshinsky) model. As the randomness increases, the mobility edges move inwards into the band and they coincide at Anderson's critical value of the randomness. A criterion is developed which, under certain conditions, imposes upper limits on the extent of the portions of the energy spectrum consisting of extended states. These conditions are fulfilled exactly in the case of a Lorentzian distribution of single-site energies and approximately within the framework of any single-site approximation. Thus in the Lorentzian case upper bounds are obtained for the positions of the mobility edges and the critical value of the randomness for which Anderson's transition takes place. These results are in agreement with the Mott-CFO model.