标题： 格林函数和完备性；重新审视$3$体问题 显示英文标题

标题： Green functions and completeness; the $3$-body problem revisited

摘要： Within the class of Dereziński-Enss pair-potentials which includes Coulomb potentials a stationary scattering theory for $N$-body systems was recently developed \cite {Sk1}. In particular the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy, and this holds without imposing any a priori decay condition on channel eigenstates. In this paper we improve for the case of $3$-body systems on the known \emph{弱连续性} properties in that we show that all non-threshold energies are \emph{静止完全} in this case, resolving a conjecture from \cite {Sk1} in the special case $N=3$. 一个结果是，上述散射量在所有非阈值能量下都依赖于能量参数\emph{强连续地}，因此不仅如之前所证明的那样几乎处处成立（对于任意的$N$）。 另一个结果是，在任何这样的能量下散射矩阵都是么正的。 作为旁证，我们给出了长程两体势的$3$体系统渐近完备性的独立平稳证明。 这是对已知的时间依赖性证明\cite{De, En}的一种替代方法。 显示英文摘要

摘要： Within the class of Derezi{\'n}ski-Enss pair-potentials which includes Coulomb potentials a stationary scattering theory for $N$-body systems was recently developed \cite {Sk1}. In particular the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy, and this holds without imposing any a priori decay condition on channel eigenstates. In this paper we improve for the case of $3$-body systems on the known \emph{weak continuity} properties in that we show that all non-threshold energies are \emph{stationary complete} in this case, resolving a conjecture from \cite {Sk1} in the special case $N=3$. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies, hence not only almost everywhere as previously demonstrated (for an arbitrary $N$). Another consequence is that the scattering matrix is unitary at any such energy. As a side result we give an independent stationary proof of asymptotic completeness for $3$-body systems with long-range pair-potentials. This is an alternative to the known time-dependent proofs \cite{De, En}.