Results 1  10
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16
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.
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 27 (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.
Localization of one dimensional, continuum, BernoulliAnderson models
"... Abstract. We use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional, 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 ..."
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Cited by 14 (4 self)
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Abstract. We use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional, 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 single site 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.
Bounds on the Spectral Shift Function and the Density of States
 COMMUN. MATH. PHYS.
, 2005
"... We study spectra of Schrödinger operators on R d. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values µn of the difference of the semigroups as n →∞and deduce bounds ..."
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Cited by 13 (6 self)
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We study spectra of Schrödinger operators on R d. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values µn of the difference of the semigroups as n →∞and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger operators. The single site potential u is assumed to be nonnegative and of compact support. The distributions of the random coupling constants are assumed to be Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.
INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS
, 2003
"... We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators ..."
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Cited by 12 (2 self)
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We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schrödinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.
Localization for the Schrödinger operator with a Poisson random potential
, 2006
"... We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localizat ..."
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Cited by 10 (2 self)
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We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.
Spectral Theory of Sparse Potentials
, 2000
"... We give a number of results concerning dierent possible spectral types for Schrodinger operators with sparse potentials. These potentials are in between stationary (e.g., random) potentials and the short range potentials familiar from scattering theory. They decay at in nity in some averaged s ..."
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Cited by 8 (2 self)
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We give a number of results concerning dierent possible spectral types for Schrodinger operators with sparse potentials. These potentials are in between stationary (e.g., random) potentials and the short range potentials familiar from scattering theory. They decay at in nity in some averaged sense, however in such a way that there is enough \space" for surprising spectral properties.
A Wegner estimate for multiparticle random Hamiltonians
"... Abstract. We prove a Wegner estimate for a large class of multiparticle Anderson Hamiltonians on the lattice. These estimates will allow us to prove Anderson localization for such systems. A detailed proof of localization will be given in a subsequent paper. 1. ..."
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Cited by 3 (0 self)
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Abstract. We prove a Wegner estimate for a large class of multiparticle Anderson Hamiltonians on the lattice. These estimates will allow us to prove Anderson localization for such systems. A detailed proof of localization will be given in a subsequent paper. 1.
Anderson localization and Lifshits tails for random surface potentials, Preprint 2004
"... ABSTRACT. We consider Schrödinger operators on L 2 (R d) with a random potential concentrated near the surface R d1 × {0} ⊂ R d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet ..."
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Cited by 3 (2 self)
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ABSTRACT. We consider Schrödinger operators on L 2 (R d) with a random potential concentrated near the surface R d1 × {0} ⊂ R d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tail relies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of
Dynamical localization for continuum random surface models By
"... Abstract. We prove Anderson localization and strong dynamical localization for random surface models in R d. 1. Introduction, the model, and the results. Spectral and scattering theory for mathematical models of rough surfaces has attracted considerable interest in recent years, as witnessed in [2, ..."
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Cited by 2 (0 self)
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Abstract. We prove Anderson localization and strong dynamical localization for random surface models in R d. 1. Introduction, the model, and the results. Spectral and scattering theory for mathematical models of rough surfaces has attracted considerable interest in recent years, as witnessed in [2, 3, 7–13, 16]. One of the reasons is that these models exhibit a metalinsulator transition. This transition is expected for typical random models in dimensions three and higher but, unfortunately, a