We consider the evolution of a tight binding wave packet propagating in a fluctuating potential. If the fluctuations stem from a stationary Markov process satisfying certain technical criteria, we show that the square amplitude of the wave packet, after diffusive rescaling, converges to a superposition of solutions of a heat equation.

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.

In this talk, I consider the problem of finding explicit recursive for- mulas to compute the perturbed eigenvalues and eigenvectors of non- selfadjoint analytic perturbations of matrices with degenerate eigenvalues. Based on some math-physics problems arising from the study of slow light in photonic crystals, we single out a class of perturbations that satisfy what I call the generic condition. It will be shown that for this class of perturbations, the problem mentioned above of finding explicit recursive formulas can be solved. Using these recursive formulas, I will list the first and second order terms for the perturbed eigenvalues and eigenvectors of perturbations belonging to this class.

In 1920 Schrodinger inspired by ideas of de Broglie on the material wave introduced his wave mechanics in which a particle is modeled by a wave packet. As it was pointed out by M. Born the interpretation of a particle by a wave packet has problems: the wave packets must in course of time become dissipated, and on the other hand the description of the interaction of two electrons as a collision of two wave packets in ordinary three-dimensional space lands us in grave difficulties. To address those problems we introduce a concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient - a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is a spatially localized solitary wave which is an exact solution to the Euler-Lagrange equations which are reduced to a certain nonlinear Schrodinger equation. We show that the wave-corpuscle remains spatially localized when it is free or even when it accelerates in a homogeneous electric field. Two or more interacting charges are well defined even when they collide. (joint work with A. Babin)