Smoothness of the density of states in the Anderson model on a one- dimensional strip. (English) Zbl 0635.60077

Let \(H=-2^{-1}\Delta +V\) on \(l^ 2({\mathcal D}_ m)\), where \({\mathcal D}_ m=Z\times \{1,...,m\}\) and V(x), \(x\in {\mathcal D}_ m\), are i.i.d. r.v.’s with common probability distribution \(\mu\). Let \(h(t)=\int e^{-itv}d\mu (v)\) and let N(E) be the integrated density of states. The folowing are proven.
(i) If h is of class \(C^ d\) on \([0,+\infty)\) with \(4d\geq 3m^ 2-m+4\) with \(h^{(j)}(t)= O((1+| t|)^{-\alpha})\) for some \(\alpha >1/2\) and all \(j=0,1,...,d\), then N(E) is of class \(C^{[d+1/2]}.\)
(ii) If h is of class \(C^{(1/4)(3m^ 2-m+4)}\) on \([0,+\infty)\) with \(h^{(j)}(t)= O((t+| t|)^{-\alpha})\) for some \(\alpha >1/2\) and all \(j=0,1,.., 4^{-1}(3m\) \(2-m+4)\), and \(h^ j(t)= O(e^{-b| t|})\) for \(j=0,1\) and some \(b>0\), then \(N(E)\) has an analytic extension to 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 supersymmetric field theory which is studied by the transfer matrix method.
Reviewer: A.Klein


60H25 Random operators and equations (aspects of stochastic analysis)
81P20 Stochastic mechanics (including stochastic electrodynamics)
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