Asymptotic Completeness for Compton Scattering

Scattering in a model of a massive quantum-mechanical particle, an ``electron'', interacting with massless, relativistic bosons, ``photons'', is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of ``photons'' by the ``electron''; but ``electron-positron'' pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed ``electron'' is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one--electron states with propagation estimates for the ``electron'' and the ``photons''.Comment: gap of previous version closed, large parts rewritten, more general results and more comprehensive exposition. 64 pages, 3 figure

Long time motion of NLS solitary waves in a confining potential

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schroedinger equations with a confining, slowly varying external potential, $V(x)$. A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential $V(x)$ over a long time interval.Comment: 42 pages, 2 figure