  We prove regularity estimates for the eigenfunctions of Schr\"odinger type
operators whose potentials have inverse square singularities and uniform radial
limits at infinity. In particular, the usual $N$-body operators are covered by
our result; in that case, the weight is in terms of the (euclidean) distance to
the collision planes. The technique of proof is based on blow-ups and Lie
manifolds. More precisely, we first blow-up the spheres at infinity of the
collision planes to obtain the Georgescu-Vasy compactification and then we
blow-up the collision planes. 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
operators and matrices of scalar operators.
