Consider a Schr"odinger operator on L_2(0,\infty),
and suppose that the potential V is known on an initial interval
[0,N]. We then prove bounds on the spectral measure \rho(I)
of intervals I. This extends (very) classical work of Chebyshev
and Markov on orthogonal polynomials.

We consider the discrete one-dimensional quasi-periodic
Schroedinger operator with potential defined by a Gevrey-class function.
We show - in the perturbative regime - that the operator satisfies
Anderson localization and that the Lyapunov exponent is positive and
continuous for all energies. We also mention a partial nonperturbative
result valid for some particular Gevrey classes. These results extend
some recent work by J. Bourgain, M. Goldstein, W. Schlag to a more general
class of potentials.

This is simultaneously a thesis defense.

Committee: S. Jitomirskaya (chair),
A. Klein
A. Figotin

In this talk we will consider the spectrum of the Almost Mathieu operator, \[ \left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n-1} + b \cos\left(2 \pi n\omega + \phi\right)x_n, \] on $l^2(\mathbb{Z})$. We will show that for $b \ne 0,\pm 2$ and $\omega$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'', for these values of $b$ and $\omega$.

The proof uses a combination of results on reducibility, localization and duality for the Almost Mathieu operator, and its associated eigenvalue equation, sometimes called the Harper equation.

Finally, we will also show that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.

We discuss the trace map associated with the Fibonacci
quasicrystal. While the associated dynamical system has been studied heavily as a real dynamical system, it may also be regarded as a complex dynamical system. We study the stable set and give explicit bounds for the complex approximants. Quantum dynamical consequences of these results will be
explained. This is joint work with Serguei Tcheremchantsev.