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

Abstract

AbstractWe prove regularity estimates in weighted Sobolev spaces for the $$L^2$$ 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 "Equation missing", where "Equation missing" is the usual Euclidean distance to the union "Equation missing" of the set of collision planes $${\mathcal {F}}$$ F . The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification $${\overline{X}}$$ X ¯ of the underlying space X and we first blow up the spheres $${\mathbb {S}}_Y \subset {\mathbb {S}}_X$$ S Y ⊂ S X at infinity of the collision planes $$Y \in {\mathcal {F}}$$ Y ∈ F to obtain the Georgescu–Vasy compactification. Then, we blow up the collision planes $${\mathcal {F}}$$ 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.