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36
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 50 (7 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.
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 38 (12 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.
Explicit Finite Volume Criteria For Localization In Continuous Random Media And Applications
 GEOM. FUNCT. ANAL
, 2003
"... We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application yields localization for Anderson Hamiltonians on th ..."
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Cited by 23 (11 self)
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We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application yields localization for Anderson Hamiltonians on the continuum at the bottom of the spectrum in an interval of size O() for large , where stands for the disorder parameter. A more sophisticated application proves localization for twodimensional random Schrödinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance O( B ) from the Landau levels, where B is the strength of the magnetic field.
Linear Response Theory for Magnetic Schrödinger Operators in Disordered Media
, 2004
"... We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators whe ..."
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Cited by 17 (9 self)
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We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the wellknown KuboStreda formula for the quantum Hall conductivity at zero temperature.
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.
Multiscale analysis and localization of random operators
 In Random Schrodinger operators: methods, results, and perspectives. Panorama & Synthèse, Société Mathématique de
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Dynamical delocalization in random Landau Hamiltonians
, 2004
"... We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disorderedbroadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, n ..."
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Cited by 11 (5 self)
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We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disorderedbroadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field or the disorder goes to zero.
Generalized eigenvaluecounting estimates for the anderson model
, 2008
"... Abstract. We show how spectral averaging for rank one perturbations, combined with a consequence of the minmax principle, can be used to derive generalized eigenvaluecounting estimates for the Anderson model. In particular, we present a simple and transparent proof of Minami’s estimate for the pro ..."
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Cited by 10 (5 self)
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Abstract. We show how spectral averaging for rank one perturbations, combined with a consequence of the minmax principle, can be used to derive generalized eigenvaluecounting estimates for the Anderson model. In particular, we present a simple and transparent proof of Minami’s estimate for the probability of two or more eigenvalues in an interval. Our estimates generalize Minami’s estimate and its extensions to n eigenvalues in two ways: we allow for singular measures and for n arbitrary intervals. As an application, we derive new results about the multiplicity of eigenvalues and Mott’s formula for the acconductivity when the single site probability distribution is Hölder continuous. 1.