Georgia Institute of Technology College of Sciences

The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on d-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson local- ization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types. We will consider quasiperiodic operators

(H(x)ψ)n =ε(∆ψ)n +f(x+n·ω)ψn, n∈Zd,

where ∆ is the discrete Laplacian, ω is a vector with rationally independent components, and f is a 1-periodic function on R, monotone on (0,1) with a positive lower bound on the derivative and some additional regularity properties. We will focus on two methods of proving Anderson localization for such operators: a perturbative method based on direct analysis of cancellations in the Rayleigh-Schr ̈odinger perturbation series for arbitrary d, and a non?perturbative method based on the analysis of Green?s functions for d = 1, originally developed by S. Jitomirskaya for the almost Mathieu operator.

The talk is based on joint works with S. Krymskii, L. Parnovski, and R. Shterenberg (per- turbative methods) and S. Jitomirskaya (non-perturbative methods).

Topological insulators are materials that exhibit unique physical properties due to their non-trivial topological order. One of the most notable consequences of this order is the presence of protected edge states as well as closure of bulk spectral gaps, which is known as the bulk-edge correspondence. In this talk, I will discuss the mathematical description of topological insulators and their related spectral properties. The presentation assumes only basic knowledge of spectral theory, and will begin with an overview of Floquet theory, Bloch bundles, and the Chern number. We will then examine the bulk-edge correspondence in topological insulators before delving into our research on closure of bulk spectral gaps for topological insulators with general edges. This talk is based on a joint work with Alexis Drouot.

The Fermi variety plays a crucial role in the study of    periodic operators.  In this talk, I will  first discuss recent works on the irreducibility of  the Fermi variety  for discrete periodic Schr\"odinger  operators.   I will then  discuss the applications to  solve  problems of embedded eigenvalues, isospectrality and quantum ergodicity. 

We shall discuss the quantum dynamics associated with ergodic
Schroedinger operators with singular continuous spectrum. Upper bounds
on the transport moments have been obtained for several classes of
one-dimensional operators, particularly, by Damanik--Tcheremchantsev,
Jitomirskaya--Liu, Jitomirskaya--Powell. We shall present a new method
which allows to recover most of the previous results and also to
obtain new results in one and higher dimensions. The input required to
apply the method is a large-deviation estimate on the Green function
at a single energy. Based on joint work with S. Sodin.

The talk will be online at https://gatech.zoom.us/j/96285037913