A class of Schrodinger operators with convergent perturbation series

Speaker: 

ILYA KACHKOVSKIY

Institution: 

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

Speaker: 

Zaher Hani

Institution: 

University of Michigan

Time: 

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

Host: 

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).

Regularity of the Density of States for Random Schrodinger Operators

Speaker: 

Krishna Maddaly

Institution: 

Ashoka University, India

Time: 

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

Host: 

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

Speaker: 

S. Jitomirskaya

Institution: 

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

Speaker: 

C. Smart

Institution: 

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.

Localization and unique continuation on the integer lattice, I

Speaker: 

C. Smart

Institution: 

U Chicago

Time: 

Monday, June 3, 2019 - 9:00am to 10:00am

 

 

Abstract: 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.

Smooth infinite energy solutions to nonlinear Schrodinger equations

Speaker: 

Weimin Wang

Institution: 

CNRS

Time: 

Thursday, June 6, 2019 - 2:00pm to 2:50pm

Host: 

Location: 

RH440R

We discuss smooth infinite energy solutions to nonlinear
Schrodinger equations on $R^d$. These solutions are periodic in time,
and quasi-periodic in space. This is unlike Moser's solutions, which
are space-time periodic. When d=1, we are moreover able to
construct 2-gap type solutions, i.e., space-time quasi-periodic
solutions with two frequencies each. We use the semi-algebraic geometry
technique introduced by Bourgain.

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