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

Abstract

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 {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 {mathcal {F}} to obtain the Georgescu–Vasy compactification. Then, we blow up the collision planes {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.