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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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