# Perturbative and non-perturbative results for quasiperiodic operators with unbounded monotone potentials.

ILYA KACHKOVSKIY

Michigan State

## Time:

Wednesday, January 2, 2019 - 2:00pm to Tuesday, January 22, 2019 - 3:00pm

## Location:

RH 440R

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.

# Diffusion in the Mean for a Periodic Schrödinger Equation Perturbed by a Fluctuating Potential.

Shiwen Zhang

Michigan State

## Time:

Tuesday, January 22, 2019 - 2:00pm to 3:00pm

## Location:

RH 440R

Abstract:

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.