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Smoothness of the density of states in the Anderson model at high disorder. (English) Zbl 0644.60057
We study the differentiability properties of the density of states, $$N_{\epsilon}(E)$$, of the Anderson Hamiltonian $$H_ 2=(-\epsilon /2)\Delta +V$$ on $$\ell^ 2({\mathbb{Z}}^ d)$$; here $$\Delta$$ is the discrete Laplacian on $${\mathbb{Z}}^ d$$ and $$V=V(x)$$ is a diagonal matrix whose elements are independent random variables with a common distribution N. $$\epsilon$$ is a small parameter (this corresponds to “large disorder”).
The main result proven is that, provided h(t), the characteristic function of N, is $$C^{(\infty)}$$ and for all its derivatives, $$h^{(i)}(t)$$, $$(1+t)h^{(i)}(t)$$ is bounded for $$t\geq 0$$, then for $$\epsilon$$ small enough (but nonzero!) $$N_{\epsilon}(E)$$ is $$C^{(\infty)}$$ in E on any compact interval. For the most interesting case where N is the uniform distribution this implies that the density of states is smooth on $${\mathbb{R}}$$, for small $$\epsilon$$.
Technically, these results are first proven for the resolvent of $$H_{\epsilon}$$, which in turn is expressed through the supersymmetric replica trick [see e.g. the third author, F. Martinelli and the last author, ibid. 106, 623-633 (1986; Zbl 0614.60098)] in terms of a functional integral. For the averaged Green’s function this reads $G_{\epsilon}(E)=i\int {\bar \psi}(0)\psi (0)\prod_{x}h(\Phi^ 2(x))e^{i\epsilon \sum_{<xy>}\Phi (x)\cdot \Phi (y)}D \Phi.$ This expression is then analyzed using convergent cluster expansions.
Reviewer: A.Bovier

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