Results 1  10
of
18
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 ..."
Abstract

Cited by 68 (9 self)
 Add to MetaCart
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.
Spreading Of Wave Packets In The Anderson Model On The Bethe Lattice
 Comm. Math. Phys
, 1996
"... The spreading of wave packets evolving under the Anderson Hamiltonian on the Bethe Lattice is studied for small disorder. The mean square distance travelled by a particle in a time t is shown to grow as t 2 for large t. 1 INTRODUCTION The Anderson model [6] gives a description of the motion of a ..."
Abstract

Cited by 24 (6 self)
 Add to MetaCart
(Show Context)
The spreading of wave packets evolving under the Anderson Hamiltonian on the Bethe Lattice is studied for small disorder. The mean square distance travelled by a particle in a time t is shown to grow as t 2 for large t. 1 INTRODUCTION The Anderson model [6] gives a description of the motion of a quantummechanical electron in a crystal with impurities. It is given by the random Schrodinger operator H = 1 2 \Delta + V on ` 2 (L ) ; (1.1) where L is either Z d or the Bethe lattice B (same as Cayley tree  an infinite connected graph with no closed loops and a fixed number K + 1 of nearest neighbors at each vertex (K 2, so B is not the line R ); the distance between two sites x and y in B will be denoted by d(x; y) and is equal to the length of the shortest path connecting x and y.) The (centered) Laplacian \Delta is defined by (\Deltau)(x) = X y u(y) ; (1.2) To appear in Communications in Mathematical Physics. y 1991 Mathematics Subject Classification. Primary 82B44. ...
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 ..."
Abstract

Cited by 24 (4 self)
 Add to MetaCart
this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multidimensional Euclidean space
Dynamical Localization for Unitary Anderson Models
, 903
"... This paper establishes dynamical localization properties of certain families of unitary random operators on the ddimensional lattice in various regimes. These operators are generalizations of onedimensional physical models of quantum transport and draw their name from the analogy with the discrete ..."
Abstract

Cited by 16 (11 self)
 Add to MetaCart
(Show Context)
This paper establishes dynamical localization properties of certain families of unitary random operators on the ddimensional lattice in various regimes. These operators are generalizations of onedimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMVmatrices in the theory of orthogonal polynomials on the unit circle. We implement the method of AizenmanMolchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization. These results complete the analogy with the selfadjoint case where dynamical localization is known to be true in the same three regimes. partially supported through MSU New Faculty Grant 07IRGP1192. partially supported through USNSF grant DMS0653374
ANDERSON LOCALIZATION FOR A CLASS OF MODELS WITH A SIGNINDEFINITE SINGLESITE POTENTIAL VIA FRACTIONAL MOMENT METHOD
"... A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the signindefinite singlesite potential, which is howev ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
(Show Context)
A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the signindefinite singlesite potential, which is however signdefinite at the boundary of its support. For this class of Anderson operators we establish a finitevolume criterion which implies that above mentioned the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e. g. at spectral boundaries which satisfy ’Lifshitz tail estimates’ on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.
Surface States and Spectra
, 2000
"... Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and ex ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on #(H 0 ) with probability one. To prove this result we combine AizenmanMolchanov theory with techniques of scattering theory.
A note on fractional moments for the onedimensional continuum Anderson model
 J. Math. Anal. Appl
"... Abstract. We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the onedimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the onedimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any singlesite distribution with bounded, compactly supported density. 1.
QUANTUM HARMONIC OSCILLATOR SYSTEMS WITH DISORDER
"... Abstract. We study manybody properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zerovelocity LiebRobinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective oneparticle Hamil ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We study manybody properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zerovelocity LiebRobinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective oneparticle Hamiltonian. We show how stateoftheart techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of oneparticle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective oneparticle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models. 1.
EXPONENTIAL DECAY OF GREEN’S FUNCTION FOR ANDERSON MODELS ONZWITH SINGLESITE POTENTIALS OF FINITE SUPPORT
, 903
"... Abstract. One of the fundamental results in the theory of localisation for discrete Schrödinger operators with random potentials is the exponential decay of Green’s function. In this note we provide a new variant of this result in the onedimensional situation for signchanging singlesite potentials ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. One of the fundamental results in the theory of localisation for discrete Schrödinger operators with random potentials is the exponential decay of Green’s function. In this note we provide a new variant of this result in the onedimensional situation for signchanging singlesite potentials with arbitrary finite support using the fractional moment method. 1.
Random Unitary Models and their Localization Properties
 In Entropy & the Quantum II, Contemporary Mathematics
"... This paper aims at presenting a few models of quantum dynamics whose description involves the analysis of random unitary matrices for which dynamical localization has been proven to hold. Some models come from physical approximations leading to effective descriptions of the dynamics of certain rando ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
This paper aims at presenting a few models of quantum dynamics whose description involves the analysis of random unitary matrices for which dynamical localization has been proven to hold. Some models come from physical approximations leading to effective descriptions of the dynamics of certain random systems that are popular in condensed matter theoretical physics,