A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model

Abstract

LetH=−Δ+V onl 2(ℤ), whereV(x),x∈ℤ, are i.i.d.r.v.'s with common probability distributionv. Leth(t)=∫e −itv dv(v) and letk(E) be the integrated density of states. It is proven: (i) Ifh isn-times differentiable withh (j)(t)=O((1+|t|)−α) for some α>0,j=0, 1, ...,n, thenk(E) is aC n function. In particular, ifv has compact support andh(t)=O((1+|t|)−α) with α>0, thenk(E) isC ∞. This allowsv to be singular continuous. (ii) Ifh(t)=O(e −α|t|) for some α>0 thenk(E) is analytic in a strip about the real axis. The proof uses the supersymmetric replica trick to rewrite the averaged Green's function as a two-point function of a one-dimensional supersymmetric field theory which is studied by the transfer matrix method.