Interactions and the Anderson transition

We examine the effects of the electron-electron interaction on the Anderson transition. It is shown that the dimensionality of the system and the range of the interaction are crucial in determining the decay properties of a single-particle citation. For a long-range interaction we find that the appropriate one-electron excitations, when localized, decay via a ${(\epsilon -\mu )}^{\frac{1}{d}}$ law where ($\epsilon -\mu$) is the energy above the Fermi energy and $d$ is the dimensionality. At finite temperatures this becomes a $T^{\frac{1}{d}+1\mathrm{}}$ law. The single-particle excitations are bound for short-range forces. The conditions for the persistence of the Anderson transition are presented in terms of the nature of the "$m$-basis" (that in which the Green's function is diagonal) and the convergence of a series for the renormalized self-energy.