A tree-strip is the product of a finite set (graph) with an infinite
tree. For a tree-strip of finite cone type, the tree is of finite cone
type and constructed starting from a root with certain substitution rules.
For a vertex of such a tree one can consider the cone of descendants and
the term 'finite cone type' refers to the fact that there are only
finitely many different
non-isomorphic cones of descendants.
On a certain class of such trees we obtain absolutely continuous
spectrum for the Anderson model for low disorder. The proof is based in
an Implicit Function Theorem in a very abstract Banach space.
The most recent result considers the Fibonacci tree-strip which is quite
special.
For the original set up, an essential assumption needed is the fact that
each vertex has at least 2 children, i.e. the tree can not have short
line segments. This is the key assumption that excludes quasi-one
dimensional Anderson models on strips for which Anderson localization is
known.
The Fibonacci tree, whose number of vertices in the n-th generation
corresponds to the n-th Fibonacci number, violates this assumption. But
with certain modifications this special case can also be treated.
The Fibonacci tree-strip is the first tree-strip where the tree has
short line segments and absolutely continuous spectrum for random
operators could be established.
Non-Gaussian functional integrals arising in Quantum Field Theory are notoriously difficult to define and compute. This applies even to relatively simple models such as Chern–Simons gauge theory, where an exact solution was obtained without a direct evaluation of the functional integral. I will explain how to use supersymmetric localization to reduce in some cases a non-Gaussian functional integral to an ordinary integral. This technique can be used to evaluate ex-
actly some observables in Chern–Simons theory as well as in certain
supersymmetric gauge theories in three dimensions and to test various
duality conjectures concerning such theories.
We consider the Anderson tight binding model with strong disorder and discuss a
Newton method to diagonalize the Hamiltonian. The overall aim is to develop a method
to diagonalize weakly nondiagonal nonmonotonic Hamiltonians.
We study the spectrum of discrete Schrodinger operators with potential given by a primitive invertible substitution sequence (and in fact our results hold for a larger class of potentials). We show this family of operators has a spectrum which is a dynamically defined Cantor set of zero Lebesgue measure. We also show that the Hausdorff dimension of this set depends analytically on the coping constant lambda and tends to 1 as lambda tends to 0. Finally, we show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly.
Ishlinsky Institute for Problems in Mechanics, RAS, Moscow, and Moscow Institute of Physics and Technology, Russia
Time:
Thursday, November 1, 2012 - 2:00pm
Location:
RH 340
Using as examples the Schroedinger equation and the wave equation we show that homogenization of many linear operators with oscillating coefficients could be done in a frame of the adiabatic approximation based on pseudodifferential operators (functions of noncommuting operators) and the Maslov methods. This approach allows one to reproduce well known homogenization results in the other way, but also take into account so-called dispersion effects leading to a change of structure of original equation. We discuss as example the asymptotic of the solution to the Cauchy problem with localized initial data and rapidly oscillating velocity.
This work was done together with J.Bruening, V.Grushin and S.Sergeev.
As, beginning with the famous Hofstadter's butterfly, all
numerical studies of spectral and dynamical quantities related to
quasiperiodic operators are actually performed for their rational
frequency approximants, the questions of continuity upon such
approximation are of fundamental importance. The fact that continuity
issues may be delicate is illustrated by the recently discovered
discontinuity of the Lyapunov exponent for non-analytic potentials.
I will review the subject and then focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences.
This is the first of (likely) two talks, where an almost entire proof will be presented. For understanding most of the talk knowledge of spectral theory should not be necessary and just knowing some basic harmonic analysis should suffice.
David--Semmes conjecture relates Singular Integrals with Geometric Measure Theory. We are in R^d.
If classical singular integrals (of singularity m) are becoming bounded operators after restriction to an m-dimensional set, does this imply that the set is necessarily ``smooth" (for example, is a subset of m-dimensional Lipschitz manifold)? Everybody believed that the answer is positive. It has been proved for only one case: d=2, m=1. This has been done in the combination of papers by Peter Jones, Pertti Mattila, Mark Melnikov, Joan Verdera, Guy David. However, if d>2 the method explored in these papers did not work, and this was a big roadblock in this part of Harmonic Analysis and Geometric Measure Theory. It still is for d>2, m< d-1. But for any dimension d, and m=d-1, Fedja Nazarov, Xavier Tolsa, and myself, we recently answered positively to this question of Guy David and Steven Semmes.
As, beginning with the famous Hofstadter's butterfly, all
numerical studies of spectral and dynamical quantities related to
quasiperiodic operators are actually performed for their rational
frequency approximants, the questions of continuity upon such
approximation are of fundamental importance. The fact that continuity
issues may be delicate is illustrated by the recently discovered
discontinuity of the Lyapunov exponent for non-analytic potentials.
I will review the subject and then focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences.
This is the first of (likely) two talks, where an almost entire proof will be presented. For understanding most of the talk knowledge of spectral theory should not be necessary and just knowing some basic harmonic analysis should suffice.
The CMV matrix is a unitary operator on $\ell^2(\mathbb N)$ that is a central tool in the study of orthogonal polynomials on the unit circle. One may view it as a unitary analogue of the Jacobi matrix. We may extend the CMV matrix to be a unitary operator on $\ell^2(\mathbb Z)$. It is more natural to consider the extended CMV matrix in certain contexts: for example, if we wish to generate CMV matrices dynamically. The extended CMV matrix also plays an important role in the study of quantum random walks.
In this talk, we will discuss a Gordon lemma for the CMV matrix (The Gordon lemma is an important tool in the study of Jacobi matrices, used to rule out the possibility pure point spectrum). We will also discuss some results pertaining to the H\"older-continuity of the spectrum of the extended CMV matrix.