  We prove the convergence of Araki-Haag detectors in any Haag-Kastler quantum
field theory with an upper and lower mass gap. We cover the case of a single
Araki-Haag detector on states of bounded energy, which are selected from the
absolutely continuous part of the energy-momentum spectrum sufficiently close
to the lower boundary of the multi-particle spectrum. These states essentially
encompass those states in the multi-particle spectrum lying below the
three-particle threshold. In our proof, we draw on insights from proofs of
asymptotic completeness in quantum mechanics. Notably, we apply Mourre's
conjugate operator method for the first time within the framework of
Haag-Kastler quantum field theory. Furthermore, we discuss applications of our
findings for the problem of asymptotic completeness in local relativistic
quantum field theory.
