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

Speaker: 

Wencai Liu

Speaker Link: 

https://www.math.uci.edu/~wencail/

Institution: 

UC Irvine

Time: 

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

Host: 

Svetlana Jitomirskaya

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 $\alpha\in\R$, $\liminf_{n\to \infty}n|n|_a|n|_\mathcal{D}\|n\alpha\|=0.$

Anderson localization with degenerate energy levels.

Speaker: 

Rajinder Mavi

Institution: 

MSU

Time: 

Friday, July 7, 2017 - 2:00pm

Location: 

RH 340N

Abstract,
We review Anderson localization via separation of resonances. We
introduce systems with apriori degenerate bare energies at large
distances. We demonstrate a form of Anderson localization. For a
simple system we discuss dynamical behavior between degenerate bare
energies.

Localization and delocalization for two interacting quasiperiodic particles.

Speaker: 

Ilya Kachkovskii

Institution: 

IAS

Time: 

Thursday, July 20, 2017 - 2:00pm

Location: 

rh 306

Abstract: The talk is about several tentative results, joint with J. Bourgain and S. Jitomirskaya. We consider a model of two 1D almost Mathieu particles with a finite range interaction. The presence of interaction makes it difficult to separate the variables, and hence the only known approach is to treat it as a 2D model, restricted to a range of parameters (both frequencies and phases of the particles need to be equal). In the usual 2D approach, a positive measure set of frequency vectors is usually removed, and extra care needs to be taken in order to keep the diagonal frequencies (which is a zero measure set). We show that the localization holds at large disorder for energies separated from zero and from certain values associated to the interaction.

These “forbidden” energies do indeed obstruct the localization. One can easily show that, even in the non-interacting regime, zero can sometimes be an eigenvalue of infinite multiplicity. Moreover, in the case of large coupling at interaction and a special relation between the phases of the particles, we show that the interaction can create a “surface" band of ac spectrum, which can be described by an effective 1D quasiperiodic long range operator.

Discrete Bethe-Sommerfeld Conjecture

Speaker: 

Rui Han

Institution: 

IAS

Time: 

Tuesday, July 25, 2017 - 2:00pm

Location: 

rh 306

 

 

 

In this talk we prove a discrete version of the Bethe-Sommerfeld conjecture. 

Namely, we show that the spectra of multi-dimensional discrete periodic Schr\"odinger operators on ℤd​ lattice with sufficiently small potentials contain at most two intervals. 

Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even. This is based on a joint work with Lana.

 

Resolvent norms in the semiclassical limit

Speaker: 

Kiril Datchev

Speaker Link: 

http://www.math.purdue.edu/~kdatchev/

Institution: 

Purdue University

Time: 

Tuesday, May 30, 2017 - 4:00pm to 5:00pm

Host: 

Hamid Hezari

Location: 

NS2 4201

Abstract: Semiclassical resolvent norms relate dynamics of a particle scattering problem to regularity and decay of waves in a corresponding wave scattering problem. In my talk I will discuss the effect that geometric trapping of particles has on resolvent norms. I will focus in particular on the phenomena of propagation of singularities and quantum tunneling, in the setting of scattering by a compactly supported smooth function in Euclidean space. This talk is based in part on joint works with Long Jin and Andras Vasy.

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