We will discuss recent results on dynamical localization
for a simple, disordered many-body system: the xy-spin chain.
For the model, with a disordered transversal magnetic field, we prove
dynamical localization. This is expressed in terms of a
zero-velocity Lieb-Robinson bound which holds on (disorder) average.
This is joint work with Gunter Stolz (from the University of Alabama at
Birmingham) and Eman Hamza (from Cairo University in Egypt).

For discrete Schrödinger operators with potential given by a trigonometric polynomial of cosines (called generalized Harper's model), we use the complexified Lyapunov exponent to prove a criterion for subcritical energies in the spectrum and a criterion for supercritical energies. This work was done through the Caltech SURF program, with mentor Christoph Marx.

Homogeneity of closed sets was introduced by Carleson in a 1983 paper which solved the Corona problem on a general class of domains in the complex plane. Recent results of several authors have shed light on the importance of homogeneity from the point of view of inverse spectral theory. I will present some recent work which constructs several large classes of limit periodic operators whose spectra are Carleson-homogeneous Cantor sets.

We will discuss continuum analogues of substitution Hamiltonians -- specifically, we will discuss Schrodinger operators on the real line whose potentials are described by an ergodic subshift over a finite alphabet and a rule that replaces symbols of the alphabet by compactly supported potential pieces. In this setting, the spectrum and the spectral type are almost surely constant, and one can identify the almost sure absolutely continuous spectrum with the Lebesgue essential closure of the set of energies with vanishing Lyapunov exponent. Using this and results of Damanik-Lenz and Klassert-Lenz-Stollmann, we can show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies a precise combinatorial condition due to Boshernitzan. We will discuss the specific case of operators over the Fibonacci subshift in detail. This is joint work with D. Damanik and A. Gorodetski.