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55
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...
Internal Lifshits Tails For Random Perturbations Of Periodic Schrödinger Operators
"... . Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove tha ..."
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Cited by 41 (7 self)
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. Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove that, at the edge of a gap of H that is not filled in for H! , the integrated density of states of H! has a Lifshits tail behaviour if and only if the integrated density of states of H is nondegenerate. R' esum' e. Soient H un op'erateur de Schrodinger \Gammap'eriodique agissant sur L 2 (R d ), V un potentiel positif et (! fl ) fl2Z d une famille de variables al'eatoires i.i.d positives. Consid'erons l'op'erateur de Schrodinger al'eatoire H! = H + V! o`u V! (x) = X fl2Z d ! fl V (x \Gamma fl). On montre que, au bord d'une lacune spectrale de H qui n'est pas combl'e pour H! la densit'e d"etats int'egr'ee de H! a un comportement asymptotique de Lifshits si et seulement si la densit'e d"...
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.
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 36 (18 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.
FiniteVolume FractionalMoment Criteria for Anderson Localization
, 2000
"... 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 ..."
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Cited by 32 (4 self)
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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 twodimensional Fermi gases. We present a family of finitevolume 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...
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.
Wegner estimates and localization for continuum Anderson models with some singular distributions
 Arch. Math. (Basel
, 1998
"... We give a simple geometric proof of Wegner's estimate which leads to a variety of new results on localization for multidimensional random operators. Introduction One of the most important topics in the mathematical theory of disordered solids is localization by which one understands the phenomeno ..."
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Cited by 21 (7 self)
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We give a simple geometric proof of Wegner's estimate which leads to a variety of new results on localization for multidimensional random operators. Introduction One of the most important topics in the mathematical theory of disordered solids is localization by which one understands the phenomenon that states are confined to a finite region in space. This is in sharp contrast to the case of ordered media where states travel to infinity and leave any finite region as time goes to infinity. Mathematically, localization is most commonly described by the occurence of pure point spectrum with exponentially decreasing eigenfunctions for the hamiltonian in question. For Anderson models, i.e. models of the form H(!) = H 0 + X i2\Gamma q i (!)f(\Delta \Gamma i) the general scheme of proof is by now quite well understood. Here e.g. H 0 = \Gamma\Delta +V 0 with \Gammaperiodic V 0 describes a medium with periodicity lattice \Gamma and the sum describes impurities by a random perturbation ...
The integrated density of states for random Schrödinger operators
"... We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed i ..."
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Cited by 20 (1 self)
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We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems
Spectral Localization by Gaussian Random Potentials in MultiDimensional Continuous Space
, 2000
"... this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multidimensional Euclidean space ..."
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Cited by 19 (4 self)
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this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multidimensional Euclidean space