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hal-00290672, version 1

## Spectral and scattering theory for some abstract QFT Hamiltonians

Christian Gérard 1, Annalisa Panati 123

(2008-06)

Abstract: We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form $H=\d\G(\omega)+ V$ acting on the bosonic Fock space $\G(\ch)$, where $\omega$ is a massive one-particle Hamiltonian acting on $\ch$ and $V$ is a Wick polynomial $\Wick(w)$ for a kernel $w$ satisfying some decay properties at infinity. We describe the essential spectrum of $H$, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the {\em asymptotic completeness} of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of $H$. As a consequence $H$ is unitarily equivalent to a collection of second quantized Hamiltonians.

• 1:  Laboratoire de Mathématiques d'Orsay (LM-Orsay)
• CNRS : UMR8628 – Université Paris XI - Paris Sud
• 2:  Département de Mathématiques (DP)
• Université Sud Toulon Var
• 3:  Centre de Physique Théorique (CPT)
• CNRS : FR2291 – Université de Provence - Aix-Marseille I – Université de la Méditerranée - Aix-Marseille II – Université Sud Toulon Var
• Domain : Physics/Mathematical Physics
Mathematics/Mathematical Physics

• hal-00290672, version 1
• oai:hal.archives-ouvertes.fr:hal-00290672
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• Submitted on: Thursday, 26 June 2008 10:05:11
• Updated on: Tuesday, 4 October 2011 19:20:24