Christian Gerard
On the scattering theory of massless Nelson models
(959K, Postcript)
ABSTRACT. We study the scattering theory for
a class of non-relativistic quantum field theory models
describing a confined non-relativistic atom interacting with a
relativistic bosonic field. We construct invariant spaces
$\cH_{\c}^{\pm}$
which are defined in terms of propagation properties for large times
and which consist of states containing a finite number of bosons in
the region $\{|x|\geq \c t\}$ for $t\to \pm \infty$. We show the
existence of asymptotic fields and we prove that the associated
asymptotic CCR
representations preserve the spaces $\cH_{\c}^{\pm}$ and induce on
these spaces representations of Fock type. For these induced
representations, we prove the property of {\em geometric asymptotic
completeness}, which gives a characterization of the vacuum states in
terms of propagation properties. Finally we show that a positive
commutator estimate imply the {\em asymptotic completeness} property,
ie the fact that the vacuum states of the induced representations
coincide with the bound states of the Hamiltonian.