Spectral gaps for quasi-periodic Schrodinger operators with Liouville frequencies II

Speaker: 

Yunfeng Shi

Institution: 

Fudan University

Time: 

Thursday, November 16, 2017 - 12:00am

Location: 

rh 340P

We consider the spectral gaps of quasi-periodic Schrodinger operators with Liouville frequencies. By establishing quantitative reducibility of the associated Schrodinger cocycle,  we show that the size of the spectral gaps decays exponentially. This is a joint work with Wencai Liu. 

Combinatorial properties of simple Toeplitz subshifts

Speaker: 

Daniel Sell

Institution: 

Friedrich-Schiller-Universität Jena

Time: 

Thursday, October 26, 2017 - 2:00pm

Location: 

RH 340P

Toeplitz sequences are constructed from periodic sequences with undetermined positions by successively filling these positions with the letters of other periodic sequences. In this talk we will consider  the class of so called simple Toeplitz sequences. We will describe combinatorial properties, such as the word complexity, of the subshifts that are associated with them. The relation between combinatorial properties of the coding sequences and the Boshernitzan condition will be also discussed.

The Projection of some Random Cantor sets and the Decay Rate of the Favard length.

Speaker: 

Shiwen Zhang

Institution: 

Michigan State

Time: 

Friday, January 12, 2018 - 2:00pm

Location: 

Rh 340N

The Favard length of a set E has a probabilistic interpretation: up to a constant factor, it is the probability that the Buffon's needle, a long line segment dropped at random, hits E. In this talk, we study the Favard length of some random Cantor sets of dimension 1. Replace the unit disc by 4 disjoint sub-discs of radius 1/4 inside. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set D. Let D_n be the n-th generation in the construction, which is comparable to the 4^{-n}-neighborhood of D. We are interested in the decay rate of the Favard length of these sets D_n as n tends to infinity, which is the likelihood (up to a constant) that the Buffon's needle will fall into the 4^{-n}-neighborhood of D. It is well known that the lower bound for such 1-dimensional set is constant multiple of 1/n. We show that the upper bound of the Favard length of D_n is also constant multiple of 1/n in the average sense.

 

Spectral gaps for quasi-periodic Schrodinger operators with Liouville frequencies

Speaker: 

Yunfeng Shi

Institution: 

Fudan University

Time: 

Thursday, November 2, 2017 - 2:00pm

Location: 

RH 340N

Abstract: In this talk, we consider the spectral gaps of quasi-periodic Schrodinger operators with Liouville frequencies. By establishing quantitative reducibility of the associated Schrodinger cocycle,  we show that the size of the spectral gaps decays exponentially. This is a joint work with Wencai Liu. 

A new proof of Anderson localization for the 1-d Anderson model.

Speaker: 

Xhiaowen Zhu

Institution: 

UCI

Time: 

Thursday, October 19, 2017 - 2:00pm

Location: 

RH 340P

We give a new and short proof of spectral localization for the 1-d Anderson model with any disorder. The original proof was given by Carmona-Klein-Martinelli in 1987, based on multi-scale analysis. Our proof is based on the large deviation estimates and positivity and subharmonicity of the Lyapunov exponent. We also show how to improve the estimates to get a uniform and quantitative version that allows us to get the exponential dynamical localization.  It is joint work with S. Jitomirskaya. Complete details of the spectral localization proof will be presented during the talk. 

 

Localization in the droplet spectrum of the random XXZ quantum spin chain

Speaker: 

Abel Klein

Institution: 

UCI

Time: 

Thursday, October 12, 2017 - 2:00pm

Location: 

RH 340P

We study the  XXZ quantum spin chain in  a random field. This model is particle number preserving, which allows  the reduction to an infinite system of discrete many-body random Schrodinger operators.  We exploit this reduction to prove a form of  Anderson localization in the droplet  spectrum of the XXZ quantum spin chain Hamiltonian. This yields a strong form of dynamical exponential clustering for eigenstates  in the droplet spectrum: For any pair of local observables,  the sum of the associated correlators over these states decays exponentially  in the distance between the  local observables. Moreover,  this exponential clustering persists under the time evolution in the  droplet spectrum.

Some refined results on mixed Littlewood conjecture for pseudo-absolute values

Speaker: 

Wencai Liu

Institution: 

UC Irvine

Time: 

Thursday, August 3, 2017 - 2:00pm to 2:50pm

Host: 

Location: 

RH306

The Littlewood conjecture states that $\liminf_{n\to\infty}||n\alpha|| ||n\beta|=0|$ holds for all real numbers $\alpha$ and $\alpha$, where $||\cdot||$ denotes the distance to the nearest integer. There are several other formulations of Littlewood conjecture, including the $p$-adic and mixed Littlewood conjecture. In this talk, I start with an introduction to the history of different versions of Littlewood conjecture.  Then I will present several refined results of mixed Littlewood conjecture for pseudo-absolute values.
Let $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$ be $k$ pseudo absolute sequences and define the  $\mathcal{D}$-adic norm $|\cdot|_{\mathcal{D}}:\N\to \{n_k^{-1}:k\ge 0\}$ by $|{n}|_\mathcal{D} = \min\{ n_k^{-1} : n\in  n_k\Z \}.$
Under some minor condition of $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$,  I set up the criteria of sequence $\psi(n)$ such that for almost every $\alpha$ the inequality
\begin{equation*}
    |n|_{\mathcal{D}_1}|n|_{\mathcal{D}_2}\cdots |n|_{\mathcal{D}_k}||n\alpha||\leq \psi(n)
\end{equation*}
has infinitely many solutions for $n\in\N$. Under some minor condition of the pseudo absolute sequence $\mathcal{D}$, I also show that
for any $\alpha\in\R$, $\liminf_{n\to \infty}n|n|_a|n|_\mathcal{D}\|n\alpha\|=0.$

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