## 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.

### MSC:

 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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