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)

We consider averages $\kappa$ of spectral measures of rank one
perturbations with respect to a sigma-finite measure $\nu$. It is shownhow various degrees of continuity of $\nu$ with respect to Hausdorff measures are inherited by $\kappa$. This extends Kotani's trick where $\nu$ is simply the Lebesgue measure.

We recapture the orthofermion algebra by means of a q-deformed particle
algebra A_p(q) whose deformed number operator has a finite spectrum
which is given by the deformed integers 0,[1],...[p]

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.