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A generalization of the radiation condition of Sommerfeld for $$N$$-body Schrödinger operators. (English) Zbl 0811.35107
In 1912 A. Sommerfeld showed that a solution of the Helmholtz equation $$(\Delta + \lambda) u = f$$ is unique if it satisfies the radiation condition $$u_ \pm = O(r^{-1})$$, $$(\partial/ \partial r \neq i \sqrt \lambda) u_ \pm = o(r^{-1})$$ for the outgoing (sign+) and incoming (sign–) waves. Later this statement was made mathematically rigorous and extended to general elliptic operators.
The aim of the paper is to propose a formulation of the radiation condition for the $$N$$-body Schrödinger operator and to prove a generalization of the uniqueness theorem of Sommerfeld. The main theorem of the paper roughly means that for the $$N$$-body problem the operators in $$R^ k_ \mp (\pm \sqrt {a(\lambda)})$$ play the role of $$\partial/ \partial r \mp i \sqrt \lambda$$. Here $$R^ k_ \mp (a)$$ is an algebra of pseudo-differential operators with symbols $$p(x, \xi)$$ satisfying the following conditions $| \partial^ m_ x \partial^ n_ \xi p(x,\xi) | = C_{n,m} \bigl (I + | x |^ 2 \bigr)^{-m/2} \bigl( I + | \xi |^ 2 \bigr)^{-n/2},\;0 \leq m,n \leq k,\;\inf_{x, \xi} \pm x \cdot \xi/ \bigl( I + | x |^ 2 \bigr)^{I/2}.$ The function of the spectral parameter $$a(\lambda)$$ is $$\inf (\lambda - t$$; $$t < \lambda$$, $$t \in \Lambda)$$ where $$\Lambda$$ means a set of thresholds of the $$N$$-body Schrödinger operator.

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 81V70 Many-body theory; quantum Hall effect 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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