Spectral and scattering theory for some abstract QFT Hamiltonians

TL;DR

This paper develops a spectral and scattering theory framework for a class of bosonic quantum field theory Hamiltonians, establishing their essential spectrum, Mourre estimates, asymptotic fields, and proving asymptotic completeness.

Contribution

It introduces an abstract class of bosonic QFT Hamiltonians and proves their spectral properties, scattering theory, and asymptotic completeness, linking them to Fock space representations.

Findings

Description of the essential spectrum of H

Proof of Mourre estimate outside thresholds

Establishment of asymptotic completeness

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.