  We provide new insight into the analysis of N-body problems by studying a
compactification $M_N$ of $\mathbb{R}^{3N}$ that is compatible with the
analytic properties of the $N$-body Hamiltonian $H_N$. We show that our
compactification coincides with the compactification introduced by Vasy using
blow-ups in order to study the scattering theory of N-body Hamiltonians and
with a compactification introduced by Georgescu using $C^*$-algebras. In
particular, the compactifications introduced by Georgescu and by Vasy coincide
(up to a homeomorphism that is the identity on $\mathbb{R}^{3N}$). Our result
has applications to the spectral theory of $N$-body problems and to some
related approximation properties.
  For instance, results about the essential spectrum, the resolvents, and the
scattering matrices of $H_N$ (when they exist) may be related to the behavior
near $M_N\setminus \mathbb{R}^{3N}$ (i.e. "at infinity") of their distribution
kernels, which can be efficiently studied using our methods. The
compactification $M_N$ is compatible with the action of the permutation group
$S_N$, which allows to implement bosonic and fermionic (anti-)symmetry
relations. We also indicate how our results lead to a regularity result for the
eigenfunctions of $H_N$.
