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.

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.

Abstract: The class of quasiperiodic operators with unbounded monotone potentials is a natural generalization of the Maryland model. In one dimension, we show that Anderson localization holds at all couplings for a large class of Lipschitz monotone sampling functions. The method is partially based on earlier results joint with S. Jitomirskaya on the bounded monotone case. We also establish that the spectrum is the whole real line. In higher dimensions, we tentatively establish perturbative Anderson localization by showing directly that eigenvalue and eigenfunction perturbation series are convergent. Compared to the previously known KAM localization proof by Bellissard, Lima, and Scoppola, our approach gives explicit diagram-like series for eigenvalues and eigenfunctions, and allows a larger class of potentials. The higher-dimensional results are joint with L. Parnovski and R. Shterenberg.