We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schrödinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. We are able to give a conditional dichotomy of the global existence and singularity below the excited energy in Theorem 4, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function \(K_\omega \). The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For \(N\ge 3\), we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of \(K_\omega \).

中文翻译：

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强力N体问题的整体存在性和奇异性

我们在非线性色散方程（例如Klein-Gordon和Schrödinger方程）中使用基态和激发态的思想来描述在某些能量约束下具有强力的N体问题的解。实际上，N体问题的相对平衡与PDE中的孤子具有相似的作用。我们介绍了N体问题的基态和激发能。我们可以在定理4的激发能之下给出全局存在和奇点的条件二分法，其证明似乎是原始且简单的。此二分法由阈值函数\（K_ \ omega \）的符号给出。在这个新的视角下对两体问题的表征是无条件的，并且与PDE中的结果非常相似。对于\（N \ ge 3 \），我们将对表征进行一些改进，特别是检查\（K_ \ omega \）的符号存在无限过渡的情况。