Asymptotic Completeness for Rayleigh Scattering

TL;DR

This paper proves asymptotic completeness for Rayleigh scattering in non-relativistic atomic models with a positive infrared cutoff, showing that excited states relax to the ground state with photons propagating to infinity.

Contribution

It establishes asymptotic completeness for Rayleigh scattering in a broad class of atomic models, including cases with massless photons and infrared cutoffs.

Findings

Spectrum below ionization threshold is purely continuous except for the ground state.

Asymptotic completeness holds for models with point and absolutely continuous spectra.

Results apply to models with infrared cutoff or massive photons.

Abstract

It is expected that the state of an atom or molecule, initially put into an excited state with an energy below the ionization threshold, relaxes to a groundstate by spontaneous emission of photons which propagate to spatial infinity. In this paper, this picture is established for a large class of models of non-relativistic atoms and molecules coupled to the quantized radiation field, but with the simplifying feature that an (arbitrarily tiny, but positive) infrared cutoff is imposed on the interaction Hamiltonian. This result relies on a proof of asymptotic completeness for Rayleigh scattering of light on an atom. We establish asymptotic completeness of Rayleigh scattering for a class of model Hamiltonians with the features that the atomic Hamiltonian has point spectrum coexisting with absolutely continuous spectrum, and that either an infrared cutoff is imposed on the interaction Hamiltonian or photons are treated as massive particles. We show that, for models of massless photons, the spectrum of the Hamiltonian strictly below the ionization threshold is purely continuous, except for the groundstate energy.