# Nonstationary low-dimensional dynamics

Victor Kleptsyn

IRM de Rennes

## Time:

Thursday, September 26, 2019 - 2:00pm to 3:00pm

## Location:

RH 340 P

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. The distribution at a given site is not assumed to be absolutely continuous (or to contain an absolutely continuous component). We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model.

An important ingredient of the proof is a non-stationary analog of the Furstenberg Theorem on random matrix products,
which is also of independent interest.

This is a joint project with A.Gorodetski.

# Quantitative almost reducibility of quasiperiodic cocycles

Lingrui Ge

## Institution:

Nanjing University

## Time:

Thursday, August 29, 2019 - 2:00pm to 3:00pm

## Location:

RH 306

Abstract: In this talk, I will give a brief introduction to several popular topics in the spectral theory of quasi-periodic Schrodinger operators. I will then talk about several sharp results we get recently on these topics (especially for almost Mathieu operator). Our results are based on quantitative almost reducibility, a method originally proposed by Dinaburg and Sinai. Finally I will explain the key points in developing and refining this method to get optimal results.

# A class of Schrodinger operators with convergent perturbation series

ILYA KACHKOVSKIY

Michigan State

## Time:

Monday, August 12, 2019 - 2:00pm

## Location:

RH 306

Abstract: Rayleigh--Schrodinger perturbation series is one of the main tools of analyzing eigenvalues and eigenvectors of operators in quantum mechanics. The first part of the talk is expository: I will explain a way of representing all terms of the series in terms of graphs with certain structure (similar representations appear in physical literature in various forms). The second part of talk is based on joint work in progress with L. Parnovski and R. Shterenberg. We show that, for a class of lattice Schrodinger operators with unbounded quasiperiodic potentials, one can establish convergence of these series (which is surprising because the eigenvalues are not isolated). The proof is based on the careful analysis of the graphical structure of terms in order to identify cancellations between terms that contain small denominators. The result implies Anderson localization for a class of Maryland-type models on higher-dimensional lattices.

# On the kinetic description of the long-time behavior of dispersive PDE

Zaher Hani

## Institution:

University of Michigan

## Time:

Wednesday, July 3, 2019 - 11:00am to 12:00pm

## Location:

RH 306

Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the "wave kinetic equation". This is the wave-analog of Boltzmann's equation for particle collisions. We shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and explore some of its effective long-time behaviors at time scales that are shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah).

# Non-stationary localized oscillations of an infinite Bernoulli-Euler beam

E.V.Shishkina

## Institution:

IPME, St. Petersburg

## Time:

Thursday, June 20, 2019 - 2:00pm to 3:14pm

RH 306

# Smooth infinite energy solutions to nonlinear Schrodinger equations, II

Wei-Min Wang

CNRS

## Time:

Friday, June 7, 2019 - 4:00pm to 5:00pm

RH 440R

# Equations of Higher Order, II

V. Tkachenko

## Institution:

Ben Gurion University

## Time:

Tuesday, April 30, 2019 - 2:00am to 3:00am

Rh 440R

# Regularity of the Density of States for Random Schrodinger Operators

## Institution:

Ashoka University, India

## Time:

Tuesday, June 25, 2019 - 2:00pm to 2:59pm

## Location:

RH 306

In this talk, a joint work with Dhriti Dolai and Anish Mallick, I will present a proof of smoothness of the density of states for Random Schrodinger operators in any dimension.  We show that the integrated density of states is almost as smooth as the single site distribution of the random potential, in the region of exponential localisation.  The proof relies on the fractional moment bounds on the operator kernels in such energy region.

Our proof also gives a part of the results for the Anderson type models proved by Abel Klein and collaborators more than thirty years ago.

# Fractal properties of the Hofstadter's butterfly and singular continuous spectrum of the critical almost Mathieu operator

S. Jitomirskaya

UCI

## Time:

Thursday, May 16, 2019 - 2:00pm

## Location:

RH 340

Abstract: Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly. It reduces to the critical almost Mathieu family, indexed by phase. We discuss the proof of sungular continuous spectrum for this family for all phases, finishing a program with a long history. We also discuss a recent proof (with I. Krasovsky) of the Thouless' conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator  is bounded by 1/2 for all irrational fluxes.

# Localization and unique continuation on the integer lattice, II

C. Smart

U Chicago

## Time:

Monday, June 3, 2019 - 12:00pm to 2:00pm

This will be a series of technical lectures on my recent work with Jian Ding.  After a brief review of the mathematics of Anderson localization, I will explain our unique continuation result.  To motivate our proof, I will describe the unique continuation result of Buhovski--Logunov--Malinnikova--Sodin for harmonic functions on the integer lattice.   I will then explain how to modify this argument, introducing tools from probability theory, to obtain a unique continuation result for Schrodinger operators on the lattice with random potentials.