Normally hyperbolic smooth trapped sets are structurally
stable and occur in many interesting situations: for instance
for Kerr black hole metrics. We show that the corresponding
semiclassical resonances (e.g. quasinormal modes for Kerr
black holes) are separated from the real axis which has
consequences for decay of waves and other phenomena.

The proof is a simple example of techniques
used in the semiclassical study of quantum resonances
and I hope to present it in a self-contained way.

The talk will discuss Jacobi matrices with periodic right
limits and corresponding square-summable variations. In particular, it will illuminate the optimality of a recent theorem of Kaluzhny-Shamis by showing that a recent conjecture of BreuerLastSimon is wrong.

We study the recurrence and ergodicity for the billiard in infinite polygons, either $Z$-periodic or $Z^2$-periodic. In the $Z$-periodic case the results are quite complete. In the more difficult $Z^2$-periodic case we obtain partial results and discuss suggestive examples. This is joint work with J.P. Conze.

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.