We consider one-dimensional Schrodinger operator with analytic potential
and a single frequency. Two ``local theories'' for such operators have
been developed extensively, and cover small and large potentials.
However, a global picture, which should in particular describe how one
moves from a regime to the other has remained elusive except in the case
of the almost Mathieu operator.

It turns out that, as in the case of the almost Mathieu, energies in the
spectrum can be always separated into three types (subcritical, critical
and supercritical), according to the Lyapunov exponent of the
(complexified) associated cocycle. Our focus is in the study of the
critical locus in the infinite dimensional parameter space.
Our analysis gives a detailed picture for the ``phase transitions''
between subcritical and supercritical regions in the spectrum of typical
operators. One of our tools is a surprising regularity property of the
Lyapunov exponent that emerges from a quantization phenomenon.

The talk will describe recent work, joint with Barry Simon, on how
spectral theoretic ideas can be applied in the study of boundary behavior
of power series.

In particular, we describe how the notions of right-limit and
reflectionless operators from spectral theory can be used to obtain
results for power series with bounded Taylor coefficients.

We recover and (within the class of bounded coefficients) improve
various classical results.

Let $E$ be a homogeneous subset of $\mathbb{R}$ in the
sense of Carleson. Let $\mu$ be a finite positive measure on
$\mathbb{R}$ and $H_\mu(x)$ its Hilbert transform. We prove that
$\lim_{t\to\infty} t|E\cap\{x : |H_\mu(x)|>t\}|=0$ if and only if
$\mu_s(E)=0$, where $\mu_s$ is the singular part of $\mu$.