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A uniqueness theorem for the $$N$$-body Schrödinger equation and its applications. (English) Zbl 0813.35068
Ikawa, Mitsuru (ed.), Spectral and scattering theory. Proceedings of the Taniguchi international workshop, held at Sanda, Hyogo, Japan. Basel: Marcel Dekker. Lect. Notes Pure Appl. Math. 161, 63-84 (1994).
This paper is mainly devoted to the study of the uniqueness of a solution of the $$N$$-body Schrödinger operator $H - \lambda : = - \sum^{N- 1}_{i=1} \Delta_ i + \sum_{i<j} V_{ij} (x^ i-x^ j) - \lambda$ living (after removing of the center of mass) in the space $$L^{2,s} (X)$$ where $$X = \{(x^ 1, \dots, x^ N)$$; $$\sum m_ i x^ i = 0\}$$ and $$L^{2,s} (X)$$ is the standard weighted space $$L^ 2 (X;(1 + | x |)^ s)$$. Under suitable assumptions on the interaction, the author proves uniqueness for $$s>-1/2$$ and for $$\lambda$$ in the continuous spectrum but not in the point spectrum or the set of thresholds. The author analyzes also in this context the limiting absorption principle and the radiation conditions.
For the entire collection see [Zbl 0798.00016].
Reviewer: B.Helffer (Paris)

MSC:
 35P25 Scattering theory for PDEs 35Q40 PDEs in connection with quantum mechanics 81U05 $$2$$-body potential quantum scattering theory