Asymptotic Completeness in Quantum Field Theory: Translation Invariant Nelson Type Models Restricted to the Vacuum and One-Particle Sectors

TL;DR

This paper proves asymptotic completeness for a class of translation invariant quantum field models, like Nelson and Polaron, focusing on vacuum and one-particle sectors, by analyzing spectral properties and establishing Mourre estimates.

Contribution

It establishes the first proof of asymptotic completeness for these models restricted to specific sectors, using spectral analysis and geometric methods.

Findings

Spectral structure of fiber Hamiltonians determined

Mourre estimate established for these models

Asymptotic completeness proved for vacuum and one-particle sectors

Abstract

Time-dependent scattering theory for a large class of translation invariant models, including the Nelson and Polaron models, restricted to the vacuum and one-particle sectors is studied. We formulate and prove asymptotic completeness for these models. The translation invariance imply that the Hamiltonians considered are fibered with respect to the total momentum. On the way to asymptotic completeness we determine the spectral structure of the fiber Hamiltonians, establish a Mourre estimate and derive a geometric asymptotic completeness statement as an intermediate step.