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We provide new insight into the analysis of N-body problems by studying acompactification $M_N$ of $\mathbb{R}^{3N}$ that is compatible with theanalytic properties of the $N$-body Hamiltonian $H_N$. We show that ourcompactification coincides with the compactification introduced by Vasy usingblow-ups in order to study the scattering theory of N-body Hamiltonians andwith a compactification introduced by Georgescu using $C^*$-algebras. Inparticular, the compactifications introduced by Georgescu and by Vasy coincide(up to a homeomorphism that is the identity on $\mathbb{R}^{3N}$). Our resulthas applications to the spectral theory of $N$-body problems and to somerelated approximation properties.  For instance, results about the essential spectrum, the resolvents, and thescattering matrices of $H_N$ (when they exist) may be related to the behaviornear $M_N\setminus \mathbb{R}^{3N}$ (i.e. "at infinity") of their distributionkernels, which can be efficiently studied using our methods. Thecompactification $M_N$ is compatible with the action of the permutation group$S_N$, which allows to implement bosonic and fermionic (anti-)symmetryrelations. We also indicate how our results lead to a regularity result for theeigenfunctions of $H_N$.

A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications