Schrödinger operators with Sturmian potentials have been studied extensively, and a central question, whether every gap predicted by gap labeling actually appears in the spectrum, was recently resolved affirmatively by Band, Beckus, and Loewy. We consider two natural generalizations of Sturmian sequences: codings of rotations and quasi-Sturmian sequences. For both classes, we use the Johnson-Schwartzman gap labeling theorem to identify the set of admissible gap labels; in the quasi-Sturmian setting, this gives the first description of the Schwartzman group for these subshifts. For binary codings of rotations, we go further and show that every admissible label is attained by some Schrödinger operator in the associated family. More precisely, for each label predicted by gap labeling, there exists a sampling function for which the corresponding gap is open.

Given a periodic Schrödinger operator with analytic potential and rational frequency α, let S_- denote the intersection of its spectra taken over the phase x in a torus. We show that up to sets of Lebesgue measure zero, S_- associated with α could be obtained asymptotically from S_- associated with rationals approximating α that satisfy certain approximating properties. We will talk more about the proof details. This work is joint with Svetlana Jitomirskaya and Xianzhe Li. This talk is part of the WSMP program.

A key signature of MBL (many-body localization) is the slow rate at which information spreads. In this talk I will describe my recent results with Elgart showing that the infinite random XXZ spin-1/2 chain exhibits slow propagation of information (logarithmic light cone) in any arbitrary but fixed energy interval. The relevant parameter regime, which covers both weak interaction and strong disorder, is determined solely by the energy interval.

I will not assume that the audience is familiar with random spin chains. I will introduce the infinite random XXZ spin-1/2 chain, state the main result, and describe some important ingredients for the proof.

Abstract: In this talk, I will introduce the notions of u-states and u-Gibbs measures, and discuss their relationship for linear cocycles over hyperbolic base dynamics. I will then present applications to Lyapunov exponents, including results on properties such as continuity and large deviations of the Lyapunov exponents.

There are some natural situations in which self-adjoint operators cannot exhibit spectral gaps for topological reasons. Perhaps the most prominent examples come from the theory of topological insulators, where boundaries very generally force the appearance of spectrum inside bulk gaps. Closely related phenomena are spectral flows, where topology can stabilize gap closings of continuous families of self-adjoint operators, leading for example to robust eigenvalue crossings. In many cases, these effects can be understood in a unified way using K-theory for C*-algebras. In this talk, I want to explain the basic mechanism behind such topological gap-filling and illustrate it through some examples.