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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

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