Abstract. For an operator, A, with cyclic vector ϕ, we study A + λP where P is the rank one projection onto multiples of ϕ. If [α, β] ⊂ spec(A) andA has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. The subject of rank one perturbations of self-adjoint operators and the closely related issue of the boundary condition dependence of Sturm-Liouville operators on [0, ∞) has a long history. We’re interested here in the connection with Borel-Stieltjes transforms of measures (Im z>0):

In this article, we prove exponential localization for wide classes of Schrödinger operators, including those with magnetic fields, at the edges of unperturbed spectral gaps. We assume that the unperturbed operator H 0 has an open gap I 0 j (B \Gamma ; B+ ). The random potential is assumed to be Andersontype with independent, identically distributed coupling constants. The common density may have either bounded or unbounded support. For either case, we prove that there exists an interval of energies in the unperturbed gap for which the almost sure spectrum of the family H ! j H 0 +V ! is dense pure point with exponentially decaying eigenfunctions. We also prove that the integrated density of states is Lipschitz continuous in the unperturbed spectral gap I 0 .

An elementary proof is given of localization for linear operators H=H o +lV, with H o translation invariant, or periodic, and V( . ) a random potential, in energy regimes which for weak disorder (l®0) are close to the unperturbed spectrum s(H o ). The analysis is within the approach introduced in the recent study of localization at high disorder by Aizenman and Molchanov [AM]; the localization regimes discussed in the two works being supplementary. Included also are some general auxiliary results enhancing the method, which now yields uniform exponential decay for the matrix elements <0|P [a,b] e -itH |x> of the spectrally filtered unitary time evolution operators, with [a,b] in the relevant energy range. corrected 7/12/93 Localization at Weak Disorder 2 1. Introduction This work presents an elementary derivation of localization for time evolutions generated by linear operators consisting of a translation invariant, or periodic, part and an added random potential, at energy rang...

This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.

Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. The present work establishes on this basis exponential decay for the modulus of the two-point function, at all temperatures as well as in the ground state, for a Fermi gas within the one-particle approximation. Different implications, in particular for the Integral Quantum Hall Effect, are reviewed.

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong...

this paper is a proxy for what deserves a book or at least a very long review article. In attempting to overview such a vast area in a few pages, I have had to focus on the high points. No proofs are given and I have settled for usually quoting the initial or especially significant papers. I have no doubt that I have left out some important papers, and if so, I ask the forgiveness of the reader (and their authors!).

A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient `Lifshitz tail estimates' on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the "multisca...

We prove that the Anderson Hamiltonian H = \Gamma\Delta + V on the Bethe Lattice has "extended states" for small disorder. More precisely, given any closed interval I contained in the interior of the spectrum of the Laplacian on the Bethe lattice, we prove that for small disorder H has purely absolutely continuous spectrum in I with probability one (i.e., oe ac (H )"I = I and oe pp (H )"I = oe sc (H )"I = ; with probability one), and its integrated density of states is continuously differentiable on the interval I. 1 INTRODUCTION The Bethe lattice (or Cayley tree), B , is an infinite connected graph with no closed loops and a fixed degree (number of nearest neighbors) at each vertex (site or point). The degree is called the coordination number and the connectivity, K, is one less the coordination number. The distance between two sites x and y will be denoted by d(x; y) and is equal to the length of the shortest path connecting x and y. 1991 Mathematics Subject Classification...

Abstract. We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients αn = �L ℓ=1 Cℓbn ℓ + O((b∆) n) where ∆ < 1 and |bℓ | = b < 1. 1.