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The supersymmetric replica trick and smoothness of the density of states for random Schrödinger operators. (English) Zbl 0709.60105
Operator theory, operator algebras and applications, Proc. Summer Res. Inst., Durham/NH (USA) 1988, Proc. Symp. Pure Math. 51, Pt. 1, 315-331 (1990).
[For the entire collection see Zbl 0699.00027.]
The author considers the Anderson model, which describes the motion of an electron on a crystal with impurities. The model is given by the random Schrödinger operator \(H=-2^{-1}\Delta +V\) on \(l^ 2(Z^ d)\), where \(\Delta\) is the finite difference Laplacian, and V(x), \(x\in Z^ d\), are independent, identically distributed random variables. For finite \(\Lambda \subset Z^ d\) let \(H_{\Lambda}\) be the restriction of H to \(l^ 2(\Lambda)\) with Dirichlet boundary conditions. Then the integrated density of states N(E) is defined by the formula \[ N(E)=\lim_{\Lambda \to Z^ d}| \Lambda |^{-1}\#\{\text{eigenvalues of } H_{\Lambda}\leq E\}. \] The author presents the supersymmetric approach for proving smoothness of N(E). The method uses the expression of the averaged Green’s function as a two-point function of a supersymmetric field theory, which is obtained by means of a supersymmetric replica trick. To prove smoothness in one dimension a transfer matrix method is used and in arbitrary dimension a cluster expansion.
Reviewer: S.Pogosian

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics