On the scattering theory of massless Nelson models

We study the scattering theory for a class of non-relativistic quantum field theory models describing a confined non-relativistic atom interacting with a massless relativistic bosonic field. We construct invariant spaces H-c(+/-) 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\ greater than or equal to ct} for t --> +/-infinity. We show the existence of asymptotic fields and we prove that the associated asymptotic CCR representations preserve the spaces H-c(+/-) and induce on these spaces representations of Fock type. For these induced representations, we prove the property of 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 asymptotic completeness property, i.e. the fact that the vacuum states of the induced representations coincide with the bound states of the Hamiltonian.