Asymptotic completeness of long-range \(N\)-body quantum systems. (English) Zbl 0844.47005

Asymptotic completeness, in the 2-body case, says that all states of the system fall into two categories: bound states and scattering states. The proof normally makes use of the classical phase space and the Enss method. In the \(N\)-body case, there may also be bound states for subsystems while the remaining constituents asymptotically evolve as free particles. For general \(N\) and short-range potentials, the proof of the existence of all wave operators and of asymptotic completeness was first given by Sigal and Soffer. Then Graf was able to replace phase-space analysis by a new type of analysis in configuration space. In a previous paper and in the present work, Dereziński extends Graf’s method to the long-range case. For the construction of wave operators pertaining to each cluster of particles one has to modify the cluster hamiltonian appropriately in order to cope with the long-range interaction. One of the main results, proven in this article, is the existence of the “asymptotic velocities” \(P_\pm\) for \(t\to \pm \infty\). For the Enss condition (\(\mu> \sqrt{3}-1\), \(\mu\) is some exponent describing the decay of interactions at spatial infinity), Dereziński proves asymptotic completeness using the modified wave operators.


47A40 Scattering theory of linear operators
81U10 \(n\)-body potential quantum scattering theory
47N50 Applications of operator theory in the physical sciences
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