Jensen's covering lemma states that either every uncountable set of ordinals is covered by a constructible set of ordinals of the same size or else there is an elementary embedding from the constructible universe to itself. This talk takes up the question of whether there could be an analog of this theorem with constructibility replaced by ordinal definability. For example, we answer a question posed by Woodin: assuming the HOD conjecture and a strongly compact cardinal, there is no nontrivial elementary embedding from HOD to HOD.

We are interested in the nature of the spectrum of the one-dimensional Schr\"odinger operator
$$
  - \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n)
$$
with $F>0$ and two different choices of the coupling constants $\{g_n\}_{n\in \mathbb{Z}}$. In the first model $g_n \equiv \lambda$ and we prove that if $F\in \pi^2 \mathbb{Q}$ then the spectrum is $\mathbb{R}$ and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model $g_n$ are independent random variables with mean zero and variance $\lambda^2$. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is dense pure point if $F < \lambda^2/2$ and purely singular continuous if $F> \lambda^2/2$. Based on joint work with Rupert Frank.