69 research outputs found

### Persistence of Anderson localization in Schr\"odinger operators with decaying random potentials

We show persistence of both Anderson and dynamical localization in
Schr\"odinger operators with non-positive (attractive) random decaying
potential. We consider an Anderson-type Schr\"odinger operator with a
non-positive ergodic random potential, and multiply the random potential by a
decaying envelope function. If the envelope function decays slower than
$|x|^{-2}$ at infinity, we prove that the operator has infinitely many
eigenvalues below zero. For envelopes decaying as $|x|^{-\alpha}$ at infinity,
we determine the number of bound states below a given energy $E<0$,
asymptotically as $\alpha\downarrow 0$. To show that bound states located at
the bottom of the spectrum are related to the phenomenon of Anderson
localization in the corresponding ergodic model, we prove: (a) these states are
exponentially localized with a localization length that is uniform in the decay
exponent $\alpha$; (b)~ dynamical localization holds uniformly in $\alpha$

### New characterizations of the region of complete localization for random Schr\"odinger operators

We study the region of complete localization in a class of random operators
which includes random Schr\"odinger operators with Anderson-type potentials and
classical wave operators in random media, as well as the Anderson tight-binding
model. We establish new characterizations or criteria for this region of
complete localization, given either by the decay of eigenfunction correlations
or by the decay of Fermi projections. (These are necessary and sufficient
conditions for the random operator to exhibit complete localization in this
energy region.) Using the first type of characterization we prove that in the
region of complete localization the random operator has eigenvalues with finite
multiplicity

### Characterization of the Anderson metal-insulator transition for non ergodic operators and application

We study the Anderson metal-insulator transition for non ergodic random
Schr\"odinger operators in both annealed and quenched regimes, based on a
dynamical approach of localization, improving known results for ergodic
operators into this more general setting. In the procedure, we reformulate the
Bootstrap Multiscale Analysis of Germinet and Klein to fit the non ergodic
setting. We obtain uniform Wegner Estimates needed to perform this adapted
Multiscale Analysis in the case of Delone-Anderson type potentials, that is,
Anderson potentials modeling aperiodic solids, where the impurities lie on a
Delone set rather than a lattice, yielding a break of ergodicity. As an
application we study the Landau operator with a Delone-Anderson potential and
show the existence of a mobility edge between regions of dynamical localization
and dynamical delocalization.Comment: 36 pages, 1 figure. Changes in v2: corrected typos, Theorem 5.1
slightly modifie

### Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries

Devices exhibiting the integer quantum Hall effect can be modeled by
one-electron Schroedinger operators describing the planar motion of an electron
in a perpendicular, constant magnetic field, and under the influence of an
electrostatic potential. The electron motion is confined to unbounded subsets
of the plane by confining potential barriers. The edges of the confining
potential barrier create edge currents. In this, the first of two papers, we
prove explicit lower bounds on the edge currents associated with one-edge,
unbounded geometries formed by various confining potentials. This work extends
some known results that we review. The edge currents are carried by states with
energy localized between any two Landau levels. These one-edge geometries
describe the electron confined to certain unbounded regions in the plane
obtained by deforming half-plane regions. We prove that the currents are stable
under various potential perturbations, provided the perturbations are suitably
small relative to the magnetic field strength, including perturbations by
random potentials. For these cases of one-edge geometries, the existence of,
and the estimates on, the edge currents imply that the corresponding
Hamiltonian has intervals of absolutely continuous spectrum. In the second
paper of this series, we consider the edge currents associated with two-edge
geometries describing bounded, cylinder-like regions, and unbounded,
strip-like, regions.Comment: 68 page

### Localization for a matrix-valued Anderson model

We study localization properties for a class of one-dimensional,
matrix-valued, continuous, random Schr\"odinger operators, acting on
L^2(\R)\otimes \C^N, for arbitrary $N\geq 1$. We prove that, under suitable
assumptions on the F\"urstenberg group of these operators, valid on an interval
$I\subset \R$, they exhibit localization properties on $I$, both in the
spectral and dynamical sense. After looking at the regularity properties of the
Lyapunov exponents and of the integrated density of states, we prove a Wegner
estimate and apply a multiscale analysis scheme to prove localization for these
operators. We also study an example in this class of operators, for which we
can prove the required assumptions on the F\"urstenberg group. This group being
the one generated by the transfer matrices, we can use, to prove these
assumptions, an algebraic result on generating dense Lie subgroups in
semisimple real connected Lie groups, due to Breuillard and Gelander. The
algebraic methods used here allow us to handle with singular distributions of
the random parameters

### Spectral and Localization Properties for the One-Dimensional Bernoulli Discrete Dirac Operator

A 1D Dirac tight-binding model is considered and it is shown that its
nonrelativistic limit is the 1D discrete Schr?odinger model. For random
Bernoulli potentials taking two values (without correlations), for typical
realizations and for all values of the mass, it is shown that its spectrum is
pure point, whereas the zero mass case presents dynamical delocalization for
specific values of the energy. The massive case presents dynamical localization
(excluding some particular values of the energy). Finally, for general
potentials the dynamical moments for distinct masses are compared, especially
the massless and massive Bernoulli cases.Comment: no figure; 24 pages; to appear in Journal of Mathematical Physic

### Localization Bounds for Multiparticle Systems

We consider the spectral and dynamical properties of quantum systems of $n$
particles on the lattice $\Z^d$, of arbitrary dimension, with a Hamiltonian
which in addition to the kinetic term includes a random potential with iid
values at the lattice sites and a finite-range interaction. Two basic
parameters of the model are the strength of the disorder and the strength of
the interparticle interaction. It is established here that for all $n$ there
are regimes of high disorder, and/or weak enough interactions, for which the
system exhibits spectral and dynamical localization. The localization is
expressed through bounds on the transition amplitudes, which are uniform in
time and decay exponentially in the Hausdorff distance in the configuration
space. The results are derived through the analysis of fractional moments of
the $n$-particle Green function, and related bounds on the eigenfunction
correlators

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