We extend some recent results of Lubinsky, Levin, Simon, and Totik
from measures with compact support to spectral measures of
Schr\"odinger operators on the half-line. In particular, we define a
reproducing kernel $S_L$ for Schr\"odinger operators and we use it to
study the fine spacing of eigenvalues in a box of the half-line
Schr\"odinger operator with perturbed periodic potential. We show that
if solutions $u(\xi, x)$ are bounded in $x$ by $e^{\epsilon x}$
uniformly for $\xi$ near the spectrum in an average sense and the
spectral measure is positive and absolutely continuous in a bounded
interval $I$ in the interior of the spectrum with $\xi_0\in I$, then
uniformly in $I$
$$\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)} \rightarrow
\frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},$$ where
$\rho(\xi)d\xi$ is the density of states.
We deduce that the eigenvalues near $\xi_0$ in a large box of size $L$
are spaced asymptotically as $\frac{1}{L\rho}$. We adapt the methods
used to show similar results for orthogonal polynomials.

It is well-known that classical electrodynamics encounters serious
problems at microscopic scales. In the talk I describe a neoclassical
theory of electric charges which is applicable both at macroscopic and
microscopic scales. From a field Lagrangian we derive field equations,
in particular Maxwell equations for EM fields and field equations for
charge distributions. In the nonrelativistic case the charges field
equations are nonlinear Schrodinger equations coupled with EM field
equations. In a macroscopic limit we derive that centers of charge
distributions converge to trajectories of point charges described by
Newton's law of motion with Coulomb interaction and Lorentz forces. In a
microscopic regime a close interaction of two bound charges as in
hydrogen atom is modeled by a nonlinear eigenvalue problem. The critical
energy values of the problem converge to the well-known energy levels of
the linear Schrdinger operator when the free charge size is much larger
than the Bohr radius. The talk is based on a joint work with A. Figotin.

We consider random Dirac operators on a strip of width 2L of the form $J\partial+V$ where J is the $2L \times 2L $ symplectic form and V a hermitian matrix-valued random potential satisfying a time reversal symmetry property.
The operator can be analyzed using transfer matrices. The time reversal symmetry forces the transfer matrices to be in the group $SO^*(2L)$. This leads to symmetry and Kramer's degeneracy for the Lyapunov spectrum which forces two Lyapunov exponents to be zero if L is odd. Adopting a criterion
by Goldsheid and Margulis one proves that these are the only vanishing Lyapunov exponents under sufficient randomness. Adopting Kotani theory one obtains a.c. spectrum of multiplicity two on the whole real line. If moreover the random potential includes i.i.d., a.c. distributed matrix Diracpeaks on a lattice in $\RR$, we can adopt the work of Jaksic and Last to prove that the a.c. spectrum is pure. This is a big contrast to the case where L is even and no Lyapunov exponent vanishes for sufficient randomness. There one expects to get pure
point spectrum using similar techniques as in the one dimensional Anderson model. (joint work with H. Schulz-Baldes)