We prove regularity estimates in weighted Sobolev spaces for the L-2-eigenfunctions of Schrodinger-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(F)(x):= min{d(x, boolean OR F), 1}, where d(x, boolean OR F) is the usual Euclidean distance to the union boolean OR F of the set of collision planes F. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification X of the underlying space X and we first blow up the spheres S-Y subset of S-X at infinity of the collision planes Y is an element of F to obtain the Georgescu-Vasy compactification. Then, we blow up the collision planes 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.