18 research outputs found
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
Local Wegner and Lifshitz tails estimates for the density of states for continuous random Schr\"odinger operators
We introduce and prove local Wegner estimates for continuous generalized
Anderson Hamiltonians, where the single-site random variables are independent
but not necessarily identically distributed. In particular, we get Wegner
estimates with a constant that goes to zero as we approach the bottom of the
spectrum. As an application, we show that the (differentiated) density of
states exhibits the same Lifshitz tails upper bound as the integrated density
of states.Comment: Revised with new titl
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
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
at infinity, we prove that the operator has infinitely many
eigenvalues below zero. For envelopes decaying as at infinity,
we determine the number of bound states below a given energy ,
asymptotically as . 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 ; (b)~ dynamical localization holds uniformly in
On the Joint Distribution of Energy Levels of Random Schroedinger Operators
We consider operators with random potentials on graphs, such as the lattice
version of the random Schroedinger operator. The main result is a general bound
on the probabilities of simultaneous occurrence of eigenvalues in specified
distinct intervals, with the corresponding eigenfunctions being separately
localized within prescribed regions. The bound generalizes the Wegner estimate
on the density of states. The analysis proceeds through a new multiparameter
spectral averaging principle
Generalized eigenvalue-counting estimates for the Anderson model
We generalize Minami's estimate for the Anderson model and its extensions to
eigenvalues, allowing for arbitrary intervals and arbitrary single-site
probability measures with no atoms. As an application, we derive new results
about the multiplicity of eigenvalues and Mott's formula for the
ac-conductivity when the single site probability distribution is H\"older
continuous.Comment: Minor revisio
An Improved Combes-Thomas Estimate of Magnetic Schr\"{o}dinger Operators
In the present paper, we prove an improved Combes-Thomas estimate, i.e., the
Combes-Thomas estimate in trace-class norms, for magnetic Schr\"{o}dinger
operators under general assumptions. In particular, we allow unbounded
potentials. We also show that for any function in the Schwartz space on the
reals the operator kernel decays, in trace-class norms, faster than any
polynomial.Comment: 25 pages, some errors correcte
Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential
We consider non-ergodic magnetic random Sch\"odinger operators with a bounded
magnetic vector potential. We prove an optimal Wegner estimate valid at all
energies. The proof is an adaptation of the arguments from [Kle13], combined
with a recent quantitative unique continuation estimate for eigenfunctions of
elliptic operators from [BTV15]. This generalizes Klein's result to operators
with a bounded magnetic vector potential. Moreover, we study the dependence of
the Wegner-constant on the disorder parameter. In particular, we show that
above the model-dependent threshold , it is
impossible that the Wegner-constant tends to zero if the disorder increases.
This result is new even for the standard (ergodic) Anderson Hamiltonian with
vanishing magnetic field