Wegner estimate for a nonsign definite generalized alloy type potential
The author considers a Schrödinger operator of the form NEWLINE\[NEWLINEH = -\Delta + V(x), \qquad V(x) = \sum_{i\in\mathbb{N}} F_i\;u(x-\xi_i),NEWLINE\]NEWLINE where \(\{F_i\}\) is a sequence of independent and identically distributed (IID) random variables and \(\{\xi_i\}\) is a homogeneous Poisson point process independent from \({F_i}\). One important feature of the model is that the continuous and compactly supported function \(u\) is not assumed to have a fixed sign. The main result is a Wegner estimate for the finite-box self-adjoint approximation of \(H\), that is an upper bound on the probability that the spectrum of the operator contains an element in a small neighbourhood of an energy level, see Theorem 1.1. Almost-optimal volume and regularity exponents are obtained. As a corollary, one can derive a strong Hilbert-Schmidt dynamical localization, see Corollary 1.1.NEWLINENEWLINESection 1 introduces the model and contains the results, as well as a historical account on Wegner estimates. In particular, \textit{P. D. Hislop} and \textit{F. Klopp} [J. Funct. Anal. 195, No. 1, 12--47 (2002; Zbl 1013.60046)] obtained a similar estimate with optimal volume exponent and almost-optimal regularity exponent for a model with somehow different impurities (\(\xi_i = i + Y_i\), \(i\in \mathbb{N}\), where the \(Y_i\)'s are a.s.\ uniformly bounded). Section 2 contains the proof of the main theorem, which uses the method in Hislop and Klopp.
scientific article; zbMATH DE number 6368446
scientific article