The integrated density of states for some random operators with nonsign definite potentials
The authors investigate the integrated density of states (IDS) for random Anderson-type additive and multiplicative perturbations of deterministic background operators with nonsign definite potentials. One of the features is that for such operators the single-site potential does not have a fixed sign. In fact, the unperturbed background medium in the multiplicative case is described by a divergence form operator \[ H_0^M = - C_0 \rho_0^{1/2} \nabla \cdot \rho_0^{-1} \nabla \rho_0^{-1/2} C_0, \] where \(\rho_0\) and \(C_0\) are positive functions that describe the unperturbed density and sound velocity, respectively. They are assumed to be sufficiently regular so that \(C_0^{\infty} (\mathbb R^d)\) is an operator core for \(H_0^M\). While, the unperturbed background medium in the additive case is described by a Schrödinger operator \(H_0\) given by \(H_0^A = ( - i \nabla - A)^2 + W\), where \(A\) is a vector potential with \(W = W_+ - W_-\) and \(W_+\) (resp. \(W_-\)) is a member of the local Kato (resp. Kato) class. The local perturbation in the Anderson-type Alloy model is defined by the following random potentials \[ V_{\Lambda}(x) = \sum_{ i \in \widetilde{\Lambda}} \lambda_i(\omega) u_i ( x - i - \xi_i(\omega')), \] where \(\widetilde{\Lambda}\) denotes the lattice points in the region \(\Lambda\), and \(\xi_i\)'s are uniformly bounded random variables modeling thermal vibrations, and \(\lambda_i\)'s are random variables compactly supported in a neighborhood of the origin. The following hypotheses are assumed: (H1) The selfadjoint operator \(H_0^X\) (for \(X=A\) or \(M\)) is essentially selfadjoint on \(C_0^{\infty}(\mathbb R^d)\) and is semi-bounded and has an open spectral gap. (H2) \(H_0^X\) is locally compact for \(X=A\) or \(M\). (H3) The single-site potential \(u_k\) is continuous and compactly supported, and \(\sum_k |u_k |< C_V\) for any bounded \(\Lambda \subset \mathbb R^d\), and the family \(\{ \|u_k \|_{\infty} \}_{k \in Z^d}\) is also uniformly bounded. (H4) The conditional probability distribution of \(\lambda_0\), conditioned on \(\lambda_0^{\perp} \equiv \{ \lambda_i \mid i \not= 0 \}\), is absolutely continuous with respect to Lebesgue measure, and its density \(h_0\) has compact support and is locally absolutely continuous. The authors prove that, under a few additional assumptions on the operator \(H_0^X\), random variables and single-site potentials, the IDS of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and also at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner type estimate with the correct volume dependence, and the proof is greatly due to the \(L^p\)-theory of the spectal shift function for \(p \geq 1\) [cf. \textit{J. M. Combes, P. D. Hislop} and \textit{S. Nakamura}, Commun. Math. Phys. 218, No. 1, 113--130 (2001)] and the vector field methods of \textit{F. Klopp} [Commun. Math. Phys. 167, No. 3, 553--569 (1995; Zbl 0820.60044)] and the Feshbach projection method. Finally, the application of this result to Schrödinger operators with random magnetic fields and to band-edge localization is discussed as well.
scientific article; zbMATH DE number 1888340
scientific article