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

Zooming in on local level statistics by supersymmetric extension of free probability

We consider unitary ensembles of Hermitian NxN matrices H with a confining potential NV where V is analytic and uniformly convex. From work by Zinn-Justin, Collins, and Guionnet and Maida it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by the Voiculescu R-transform, a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner-Efetov supersymmetry method of integration over commuting and anti-commuting variables. This insight leads to a potent new technique for the study of local statistics, e.g., level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.Comment: 38 pages, 3 figures, published version, minor changes in Section

Rayleigh Scattering at Atoms with Dynamical Nuclei

Scattering of photons at an atom with a dynamical nucleus is studied on the subspace of states of the system with a total energy below the threshold for ionization of the atom (Rayleigh scattering). The kinematics of the electron and the nucleus is chosen to be non-relativistic, and their spins are neglected. In a simplified model of a hydrogen atom or a one-electron ion interacting with the quantized radiation field in which the helicity of photons is neglected and the interactions between photons and the electron and nucleus are turned off at very high photon energies and at photon energies below an arbitrarily small, but fixed energy (infrared cutoff), asymptotic completeness of Rayleigh scattering is established rigorously. On the way towards proving this result, it is shown that, after coupling the electron and the nucleus to the photons, the atom still has a stable ground state, provided its center of mass velocity is smaller than the velocity of light; but its excited states are turned into resonances. The proof of asymptotic completeness then follows from extensions of a positive commutator method and of propagation estimates for the atom and the photons developed in previous papers. The methods developed in this paper can be extended to more realistic models. It is, however, not known, at present, how to remove the infrared cutof

Mean-field dynamics of fermions with relativistic dispersion

We extend the derivation of the time-dependent Hartree-Fock equation recently obtained by Benedikter et al. ["Mean-field evolution of fermionic systems," Commun. Math. Phys. (to be published)] to fermions with a relativistic dispersion law. The main new ingredient is the propagation of semiclassical commutator bounds along the pseudo-relativistic Hartree-Fock evolution. (C) 2014 AIP Publishing LLC

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'

Effective Dynamics of Extended Fermi Gases in the High-Density Regime

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases

Magnetism and the Weiss Exchange Field-A Theoretical Analysis Motivated by Recent Experiments

The huge spin precession frequency observed in recent experiments with spin-polarized beams of hot electrons shot through magnetized films is interpreted as being caused by Zeeman coupling of the electron spins to the so-called Weiss exchange field in the film. The microscopic origin of exchange interactions and of large mean exchange fields, leading to different types of magnetic order, is elucidated. A microscopic derivation of the equations of motion of the Weiss exchange field is presented. Novel proofs of the existence of phase transitions in quantum XY-models and antiferromagnets, based on an analysis of the statistical distribution of the exchange field, are presente