论文标题
绿色功能和完整性；重新审视$ 3 $的身体问题
Green functions and completeness; the $3$-body problem revisited
论文作者
论文摘要
在包括库仑电势的derezi {} Skiens Poptentials的类别中，最近开发了$ n $ body Systems的固定散射理论\ cite {sk1}。特别是在任何非阈值能量下都定义了波浪和散射矩阵以及受限制的波浪运算符，并且在渠道本征态施加任何先验衰减条件的情况下，这是在任何非阈值能量下的定义。在本文中，我们在已知的\ emph {弱连续性}属性上的$ 3 $体系统改进，因为在这种情况下，我们表明所有非阈值能量都是\ emph {sentary complete}，在这种情况下，从\ cite {sk1}中解析了\ cite {sk1}的个ugenture {sk1}。结果是，上述散射量取决于\ emph {非常连续}在所有非阈值能量下的能量参数，因此不仅如前所述（对于任意$ n $）如前所述。另一个结果是，散射矩阵在任何这样的能量上都是统一的。作为一方面的结果，我们为$ 3 $体系的系统带有远程配对的独立静态证明。这是已知的时间相关证明\ cite {de，en}的替代方法。
Within the class of Derezi{ń}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}.
