A rigorous replica trick approach to Anderson localization in one dimension

Abstract

LetH=−Δ+V onl2(ℤ), whereV(x),x∈ℤ, are i.i.d.r.v.'s, and letG L (x,y;E+iη)= 〈x|(HL−(E+iη))−1|y〉, whereH L denotes the operatorH restricted to {−L, −L+1,...,L} with Dirichlet boundary conditions. We use a supersymmetric replica trick to prove that $$E(|G_L (0,x; E + i\eta )|^2 ) \leqq K\eta ^{ - 2} \exp \{ - m|\log \eta |^{ - \sigma } |x|\} $$ for somem>0, σ>0,K<∞, uniformly inL andE. This estimate, together with the usual necessary estimate on the density of states, implies zero conductivity and gives exponential localization by the Frohlich, Martinelli, Scoppola, and Spencer method.