C. G rard, A. Panati
Spectral and scattering theory for some abstract QFT
Hamiltonians
(658K, pdf)
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.