Abstract:
 We consider decaying oscillatory perturbations of periodic Schr\"odinger
 operators on the half line. More precisely, the perturbations we study
 satisfy a generalized bounded variation condition at infinity and an $L^p$
 decay condition. We show that the absolutely continuous spectrum is
 preserved, and give bounds on the Hausdorff dimension of the singular part
 of the resulting perturbed measure. Under additional assumptions, we
 instead show that the singular part embedded in the essential spectrum is
 contained in an explicit countable set. Finally, we demonstrate that this
 explicit countable set is optimal. That is, for every point in this set
 there is an open and dense class of periodic Schr\"odinger operators for
 which an appropriate perturbation will result in the spectrum having an
 embedded eigenvalue at that point.

A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520 . . . (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refinements of the AM-GM inequality, known as Maclaurin’s inequalities, relating the 1/kthpowers of the kth elementary symmetric means of n numbers for 1 ≤ k ≤ n. On the left end (when k = n) we have the geometric mean, and on the right end (k = 1) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves f (n) steps away from either extreme. We also study the limiting behavior of such means for quadratic irrational α.

(Joint work with Francesco Cellarosi, Doug Hensley and Steven J. Miller)

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.