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34
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 80 (12 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 .
Localization for random perturbations of periodic Schrödinger operators
 RANDOM OPER. STOCHASTIC EQUATIONS
, 1996
"... We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d 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) ..."
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Cited by 59 (20 self)
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We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d 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 Schrodinger 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.
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 51 (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.
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 44 (6 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.
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 41 (17 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
Anderson Localization for Random Schrödinger Operators with Long Range Interactions
 COMM. MATH. PHYS
, 1998
"... We prove pure point spectrum at all band edges for Schrödinger Operators with a periodic potential plus a random potential of the form V ! (x) = P q i (!)f(x \Gamma i) where f decays at infinity like jxj \Gammam for m ? 3d resp. m ? 2d depending on the regularity of f . Eigenfunctions are shown ..."
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Cited by 38 (20 self)
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We prove pure point spectrum at all band edges for Schrödinger Operators with a periodic potential plus a random potential of the form V ! (x) = P q i (!)f(x \Gamma i) where f decays at infinity like jxj \Gammam for m ? 3d resp. m ? 2d depending on the regularity of f . Eigenfunctions are shown to decay more rapidly than every inverse polynomial. The random variables q i are supposed to be independent and identically distributed. We suppose that their distribution has a bounded density of compact support.
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 34 (6 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.
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 measure ..."
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Cited by 32 (9 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.
Hölder Continuity Of The Integrated Density Of States For Some Random Operators At All Energies
, 2002
"... We prove that the integrated density of states of random Schrödinger operators with Andersontype potentials on L ), for d ≥ 1, is locally Hölder continuous at all energies. The singlesite potential u must be nonnegative and compactly supported, and the distribution of the random variable must ..."
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Cited by 31 (8 self)
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We prove that the integrated density of states of random Schrödinger operators with Andersontype potentials on L ), for d ≥ 1, is locally Hölder continuous at all energies. The singlesite potential u must be nonnegative and compactly supported, and the distribution of the random variable must be absolutely continuous with a bounded, compactly supported density. We also prove this result for random Andersontype perturbations of the Landau Hamiltonian in twodimensions under a rational flux condition.
Localization for onedimensional, continuum, BernoulliAnderson models
 Duke Math. J
"... We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, continuum, Andersontype models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two ..."
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Cited by 28 (12 self)
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We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, continuum, Andersontype models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported singlesite perturbations of a periodic background which we use to verify the necessary hypotheses of multiscale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs. 1.