  We prove regularity estimates in weighted Sobolev spaces for the
$L^2$-eigenfunctions of Schr\"odinger type operators whose potentials have
inverse square singularities and uniform radial limits at infinity. In
particular, the usual $N$-body Hamiltonians with Coulomb-type singular
potentials are covered by our result: in that case, the weight is
$\delta_{\mathcal{F}}(x) := \min \{ d(x, \bigcup \mathcal{F}), 1\}$, where
$d(x, \bigcup \mathcal{F})$ is the usual euclidean distance to the union
$\bigcup\mathcal{F}$ of the set of collision planes $\bigcup\mathcal{F}$. The
proof is based on blow-ups of manifolds with corners and Lie manifolds. More
precisely, we start with the radial compactification $\overline{X}$ of the
underlying space $X$ and we first blow-up the spheres $\mathbb{S}_Y \subset
\mathbb{S}_X$ at infinity of the collision planes $Y \in \bigcup\mathcal{F}$ to
obtain the Georgescu-Vasy compactification. Then we blow-up the collision
planes $\bigcup\mathcal{F}$. We carefully investigate how the Lie manifold
structure and the associated data (metric, Sobolev spaces, differential
operators) change with each blow-up. Our method applies also to higher order
differential operators, to certain classes of pseudodifferential operators, and
to matrices of scalar operators.
