65 research outputs found
On two-particle Anderson localization at low energies
We prove exponential spectral localization in a two-particle lattice Anderson
model, with a short-range interaction and external random i.i.d. potential, at
sufficiently low energies. The proof is based on the multi-particle multi-scale
analysis developed earlier by Chulaevsky and Suhov (2009) in the case of high
disorder. Our method applies to a larger class of random potentials than in
Aizenman and Warzel (2009) where dynamical localization was proved with the
help of the fractional moment method
Wegner bounds for a two-particle tight binding model
We consider a quantum two-particle system on a d-dimensional lattice with
interaction and in presence of an IID external potential. We establish
Wegner-typer estimates for such a model. The main tool used is Stollmann's
lemma
Simplicity of eigenvalues in the Anderson model
We give a simple, transparent, and intuitive proof that all eigenvalues of
the Anderson model in the region of localization are simple
Multi-Particle Anderson Localisation: Induction on the Number of Particles
This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09}
covering the two-particle Anderson model. Here we establish the phenomenon of
Anderson localisation for a quantum -particle system on a lattice
with short-range interaction and in presence of an IID external potential with
sufficiently regular marginal cumulative distribution function (CDF). Our main
method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS},
\cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an
induction on the number of particles, as was proposed in our earlier manuscript
\cite{CS07}. Similar results have been recently obtained in an independent work
by Aizenman and Warzel \cite{AW08}: they proposed an extension of the
Fractional-Moment Method (FMM) developed earlier for single-particle models in
\cite{AM93} and \cite{ASFH01} (see also references therein) which is also
combined with an induction on the number of particles.
An important role in our proof is played by a variant of Stollmann's
eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved
earlier in \cite{C08}, admits a straightforward extension covering the case of
multi-particle systems with correlated external random potentials: a subject of
our future work. We also stress that the scheme of our proof is \textit{not}
specific to lattice systems, since our main method, the MSA, admits a
continuous version. A proof of multi-particle Anderson localization in
continuous interacting systems with various types of external random potentials
will be published in a separate papers
The repulsion between localization centers in the Anderson model
In this note we show that, a simple combination of deep results in the theory
of random Schr\"odinger operators yields a quantitative estimate of the fact
that the localization centers become far apart, as corresponding energies are
close together
Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip
The Bethe Strip of width is the cartesian product \B\times\{1,...,m\},
where \B is the Bethe lattice (Cayley tree). We prove that Anderson models on
the Bethe strip have "extended states" for small disorder. More precisely, we
consider Anderson-like Hamiltonians \;H_\lambda=\frac12 \Delta \otimes 1 + 1
\otimes A + \lambda \Vv on a Bethe strip with connectivity , where
is an symmetric matrix, \Vv is a random matrix potential, and
is the disorder parameter. Given any closed interval , where
and are the smallest and largest
eigenvalues of the matrix , we prove that for small the random
Schr\"odinger operator has purely absolutely continuous spectrum
in with probability one and its integrated density of states is
continuously differentiable on the interval
Quantum site percolation on amenable graphs
We consider the quantum site percolation model on graphs with an amenable
group action. It consists of a random family of Hamiltonians. Basic spectral
properties of these operators are derived: non-randomness of the spectrum and
its components, existence of an self-averaging integrated density of states and
an associated trace-formula.Comment: 10 pages, LaTeX 2e, to appear in "Applied Mathematics and Scientific
Computing", Brijuni, June 23-27, 2003. by Kluwer publisher
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
Localization criteria for Anderson models on locally finite graphs
We prove spectral and dynamical localization for Anderson models on locally
finite graphs using the fractional moment method. Our theorems extend earlier
results on localization for the Anderson model on \ZZ^d. We establish
geometric assumptions for the underlying graph such that localization can be
proven in the case of sufficiently large disorder
Perturbative analysis of disordered Ising models close to criticality
We consider a two-dimensional Ising model with random i.i.d. nearest-neighbor
ferromagnetic couplings and no external magnetic field. We show that, if the
probability of supercritical couplings is small enough, the system admits a
convergent cluster expansion with probability one. The associated polymers are
defined on a sequence of increasing scales; in particular the convergence of
the above expansion implies the infinite differentiability of the free energy
but not its analyticity. The basic tools in the proof are a general theory of
graded cluster expansions and a stochastic domination of the disorder
- …