Eigenfunctions in one-dimensional disordered systems. I. Formalism

Abstract

Within the framework of Anderson's model for disordered lattices, an integral equation for the joint probability distribution of certain quantities directly related to $\mathrm{Re}G$ and $\mathrm{Im}G$ is obtained: $G$ is the Green's function of the system. The properties of this probability distribution are examined and physically interpreted. Finally convenient expressions for transport-related averages of the type ${〈{G}^{*}G〉}_{\mathrm{av}}$ are obtained. $〈{G}^{*}G〉$ provides detailed information about the eigenfunctions.