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Towards asymptotic completeness of two-particle scattering in local relativistic QFT

We consider the problem of existence of asymptotic observables in local relativistic theories of massive particles. Let $\ti p_1$ and $\ti p_2$ be two energy-momentum vectors of a massive particle and let $\De$ be a small neighbourhood of $\ti p_1+\ti p_2$. We construct asymptotic observables (two-particle Araki-Haag detectors), sensitive to neutral particles of energy-momenta in small neighbourhoods of $\ti p_1$ and $\ti p_2$. We show that these asymptotic observables exist, as strong limits of their approximating sequences, on all physical states from the spectral subspace of $\De$. Moreover, the linear span of the ranges of all such asymptotic observables coincides with the subspace of two-particle Haag-Ruelle scattering states with total energy-momenta in $\De$. The result holds under very general conditions which are satisfied, for example, in $\la ϕ^4_2$. The proof of convergence relies on a variant of the phase-space propagation estimate of Graf.