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.

 

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).

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.

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.