37 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
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
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 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 . We prove that, under suitable
assumptions on the F\"urstenberg group of these operators, valid on an interval
, they exhibit localization properties on , 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
Localization Bounds for Multiparticle Systems
We consider the spectral and dynamical properties of quantum systems of
particles on the lattice , 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 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 -particle Green function, and related bounds on the eigenfunction
correlators
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
Understanding the Random Displacement Model: From Ground-State Properties to Localization
We give a detailed survey of results obtained in the most recent half decade
which led to a deeper understanding of the random displacement model, a model
of a random Schr\"odinger operator which describes the quantum mechanics of an
electron in a structurally disordered medium. These results started by
identifying configurations which characterize minimal energy, then led to
Lifshitz tail bounds on the integrated density of states as well as a Wegner
estimate near the spectral minimum, which ultimately resulted in a proof of
spectral and dynamical localization at low energy for the multi-dimensional
random displacement model.Comment: 31 pages, 7 figures, final version, to appear in Proceedings of
"Spectral Days 2010", Santiago, Chile, September 20-24, 201
Exponential dynamical localization for the almost Mathieu operator
We prove that the exponential moments of the position operator stay bounded
for the supercritical almost Mathieu operator with Diophantine frequency
The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower
bounds for its fractal dimension in the large coupling regime. These bounds
show that as , converges to an explicit constant (). We also discuss
consequences of these results for the rate of propagation of a wavepacket that
evolves according to Schr\"odinger dynamics generated by the Fibonacci
Hamiltonian.Comment: 23 page
Widths of the Hall Conductance Plateaus
We study the charge transport of the noninteracting electron gas in a
two-dimensional quantum Hall system with Anderson-type impurities at zero
temperature. We prove that there exist localized states of the bulk order in
the disordered-broadened Landau bands whose energies are smaller than a certain
value determined by the strength of the uniform magnetic field. We also prove
that, when the Fermi level lies in the localization regime, the Hall
conductance is quantized to the desired integer and shows the plateau of the
bulk order for varying the filling factor of the electrons rather than the
Fermi level.Comment: 94 pages, v2: a revision of Sec. 5; v3: an error in Sec. 7 is
corrected, major revisions of Sec. 7 and Appendix E, Sec. 7 is enlarged to
Secs. 7-12, minor corrections; v4: major revisions, accepted for publication
in Journal of Statistical Physics; v5: minor corrections, accepted versio