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.
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.
Wednesday, January 2, 2019 - 2:00pm to Tuesday, January 22, 2019 - 3:00pm
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.
We consider the solution to a tight-binding, periodic Schrödinger equation with a random potential evolving stochastically in time. If the potential evolves according to a stationary Markov process, we obtain a positive, finite diffusion constant for the evolution of the solution. More generally, we show that the square amplitude of the wave packet, after diffusive rescaling, converges to a solution of the heat equation. This a joint work with Jeffrey Schenker and Zak Tilocco.
The classical description of nucleation of cavities in a stretched fluid relies on a one-dimensional Fokker-Planck equation (FPE) in the space of their sizes, with the diffusion coefficient constructed from macroscopic hydrodynamics and thermodynamics, as shown by Zeldovich. When additional variables (e.g., vapor pressure) are required to describe the state of a bubble, a similar approach to construct a diffusion tensor generally works only in the direct vicinity of the thermodynamic saddle point corresponding to the critical nucleus. We show, nevertheless, that “proper” kinetic variables to describe a cavity can be selected, allowing to introduce a diffusion tensor in the entire domain of parameters. In this way, for the first time, complete FPE’s are constructed for viscous volatile and inertial fluids.
We discuss ultracold atomic gas with attractive interactions in a one-dimensional optical lattice. We find that its excitation spectrum displays a quantum soliton band, corresponding to N-particle bound states, and a continuum band of other, mostly extended, states. For a system of a finite size, the two branches are degenerate in energy for weak interactions, while a gap opens above a threshold value for the interaction strength. We find that the interplay between degenerate extended and bound states has important consequences for both static and dynamical properties of the system.