570 research outputs found
Generic Continuous Spectrum for Ergodic Schr"odinger Operators
We consider discrete Schr"odinger operators on the line with potentials
generated by a minimal homeomorphism on a compact metric space and a continuous
sampling function. We introduce the concepts of topological and metric
repetition property. Assuming that the underlying dynamical system satisfies
one of these repetition properties, we show using Gordon's Lemma that for a
generic continuous sampling function, the associated Schr"odinger operators
have no eigenvalues in a topological or metric sense, respectively. We present
a number of applications, particularly to shifts and skew-shifts on the torus.Comment: 14 page
On certain other sets of integers
We show that if A is a subset of {1,...,N} containing no non-trivial
three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).Comment: 29 pp. Corrected typos. Added definitions for some non-standard
notation and remarks on lower bound
Expansion in perfect groups
Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an
integer q, denote by Ga_q the subgroup of Ga consisting of the elements that
project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q
with respect to the generating set S form a family of expanders when q ranges
over square-free integers with large prime divisors if and only if the
connected component of the Zariski-closure of Ga is perfect.Comment: 62 pages, no figures, revision based on referee's comments: new ideas
are explained in more details in the introduction, typos corrected, results
and proofs unchange
Wegner estimate for discrete alloy-type models
We study discrete alloy-type random Schr\"odinger operators on
. Wegner estimates are bounds on the average number of
eigenvalues in an energy interval of finite box restrictions of these types of
operators. If the single site potential is compactly supported and the
distribution of the coupling constant is of bounded variation a Wegner estimate
holds. The bound is polynomial in the volume of the box and thus applicable as
an ingredient for a localisation proof via multiscale analysis.Comment: Accepted for publication in AHP. For an earlier version see
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=09-10
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
Estimates for measures of sections of convex bodies
A estimate in the hyperplane problem with arbitrary measures has
recently been proved in \cite{K3}. In this note we present analogs of this
result for sections of lower dimensions and in the complex case. We deduce
these inequalities from stability in comparison problems for different
generalizations of intersection bodies
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
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in
is supported on a set of Hausdorff dimension strictly less than
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of , . By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any , and any | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where denotes the
probability that is the first entrance point of the simple random walk
starting at into . Furthermore, must converge to as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne
Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations
In this paper, we consider Hartree-type equations on the two-dimensional
torus and on the plane. We prove polynomial bounds on the growth of high
Sobolev norms of solutions to these equations. The proofs of our results are
based on the adaptation to two dimensions of the techniques we previously used
to study analogous problems on , and on .Comment: 38 page
- …