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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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