Results 1  10
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95
Localization Near Band Edges For Random Schrödinger Operators
, 1997
"... 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 And ..."
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Cited by 90 (13 self)
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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 invitation to random Schrödinger operators
, 2007
"... 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 m ..."
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Cited by 83 (8 self)
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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.
Localization Bounds for an Electron Gas
, 1998
"... 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 ..."
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Cited by 68 (9 self)
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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.
Localization for random perturbations of periodic Schrödinger operators
, 1998
"... We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A s ..."
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Cited by 67 (19 self)
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We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schrödinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.
The Integrated Density Of States For Some Random Operators With Nonsign Definite Potentials
 J. Funct. Anal
, 2001
"... We study the integrated density of states of random Andersontype additive and multiplicative perturbations of deterministic background operators for which the singlesite potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random v ..."
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Cited by 63 (7 self)
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We study the integrated density of states of random Andersontype additive and multiplicative perturbations of deterministic background operators for which the singlesite potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states 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 at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L function for p 1 [9], and the vector field methods of [20]. We discuss the application of this result to Schrödinger operators with random magnetic fields and to bandedge localization.
Moment Analysis for Localization in Random Schrödinger Operators
, 2005
"... We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the ..."
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Cited by 63 (14 self)
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We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonancediffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the LifshitzKrein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weakL¹ estimate concerning the boundaryvalue distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
Dynamical Localization for Discrete and Continuous Random Schrödinger Operators
 Commun. Math. Phys
, 1997
"... We show for a large class of random Schrodinger operators H! on ` 2 ( ) and on L 2 ( ) that dynamical localization holds, i.e. that, with probability one, for a suitable energy interval I and for q a positive real, sup t r q (t) j sup t ! PI (H! )/ t ; jXj q PI (H! )/ t ? ! 1: Here ..."
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Cited by 63 (11 self)
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We show for a large class of random Schrodinger operators H! on ` 2 ( ) and on L 2 ( ) that dynamical localization holds, i.e. that, with probability one, for a suitable energy interval I and for q a positive real, sup t r q (t) j sup t ! PI (H! )/ t ; jXj q PI (H! )/ t ? ! 1: Here / is a function of sufficiently rapid decrease, / t = e \GammaiH ! t / and PI (H! ) is the spectral projector of H! corresponding to the interval I. The result is obtained through the control of the decay of the eigenfunctions of H! and covers, in the discrete case, the Anderson tightbinding model with Bernouilli potential (dimension = 1) or singular potential ( ? 1), and in the continuous case Anderson as well as random Landau Hamiltonians. 1 Introduction We show for a large class of random Schrodinger operators H! on ` 2 ( ) and on L 2 ( ) that dynamical localization holds, i.e. that, with probability one, for a 1 suitable energy interval I and q ? 0, sup t r q (t)...
An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators, eprint arXiv:mathph/0605029v2
"... et Institut Universitaire de France We prove that the integrated density of states (IDS) of random Schrödinger operators with Andersontype potentials on L 2 (R d), for d ≥ 1, is locally Hölder continuous at all energies with the same Hölder exponent 0 < α ≤ 1 as the conditional probability measu ..."
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Cited by 60 (15 self)
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et Institut Universitaire de France We prove that the integrated density of states (IDS) of random Schrödinger operators with Andersontype potentials on L 2 (R d), for d ≥ 1, is locally Hölder continuous at all energies with the same Hölder exponent 0 < α ≤ 1 as the conditional probability measure for the singlesite random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The singlesite potential u ∈ L ∞ 0 (R d) must be nonnegative and compactlysupported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle. We also prove analogous continuity results for the IDS of random Andersontype perturbations of the Landau Hamiltonian in twodimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures.
A characterization of the Anderson metalinsulator transport transition
 Duke Math. J
"... We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong... ..."
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Cited by 58 (19 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
Multiscale analysis implies strong dynamical localization
, 1999
"... We prove that a strong form of dynamical localization follows from a variable energy multiscale analysis. This abstract result is applied to a number of models for wave propagation in disordered media. ..."
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Cited by 50 (7 self)
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We prove that a strong form of dynamical localization follows from a variable energy multiscale analysis. This abstract result is applied to a number of models for wave propagation in disordered media.