A regularity result for the bound states of N -body Schrödinger operators: blow-ups and Lie manifolds

We prove regularity estimates in weighted Sobolev spaces for the L^2 -eigenfunctions of Schrödinger-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 , where is the usual Euclidean distance to the union 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 ⊂𝕊_X at infinity of the collision planes Y ∈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.