01-103 Christian Gerard
On the scattering theory of massless Nelson models (959K, Postcript) Mar 19, 01
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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.

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