Cwikel and Quasi-Szego Type Estimates for Random Operators

Abstract

ABSTRACT. We consider Schrodinger operators with nonergodic random potentials. Specifically, we are interested in eigenvalue estimates and estimates of the entropy for the absolutely continuous part of the spectral measure. We prove that increasing oscillations in the potential at infinity have the same effect on the properties of the spectrum as the decay of the potential.Recall the Rozenbljum-Cwikel-Lieb estimate ([5], [20], [19], [27]) for the number N (V ) of negative eigenvalues of −∆ − V (x):The potential V in this estimate must decay in order to make the integral converge. We are going to study potentials that either decay slower than L d/2 -potentials or do not decay at all. Instead of decay, our theorems require some oscillation of V at infinity. There is an additional disadvantage of our results in that they give an estimate for the number of eigenvalues below arbitrary negative number −γ, which can not be taken equal to zero. In order to estimate the amount of negative spectrum one has to combine our main result with the Laptev-Weidl approach (see Theorem (1.2)). We also study the conditions on V which guarantee the convergence of certain eigenvalue sums. The classical Lieb-Thirring estimate for the eigenvalue sum |λ j | γ holds for all potentials from L d/2+γ , even for the worst ones. Our goal is to show that the probability to meet a "bad" potential is zero. It means that for a typical potential one can expect a better behaviour of the negative spectrum.Denote by F the Fourier transform understood as a unitary operator inwhere a is the multiplication operator by a function denoted by the same letter and V is the multiplication operator by a potential V . First, we need to state the Birman-Schwinger principle, which can be formulated as follows:Proposition 0.1. Let H = −∆ − V be the Schrödinger operator with a real potential V and let a(ξ) = (ξ 2 + γ) −1/2 . Then the number of eigenvalues 1

Approximation of random dynamical systems with discrete time by stochastic differential equations: I. Theory

“…The second part of the paper will be devoted to the transition from localization (pure point spectrum) to the absolutely continuous spectrum in the spirit of the classical paper [6] and recent publications on the fast oscillating piecewise constant potentials [7]- [8].…”