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We show that the space introduced by Vasy in order to construct a pseudodifferential calculus adapted to the N -body problem can be obtained as the primitive ideal spectrum of one of the N -body algebras considered by Georgescu. In the process, we provide an alternative description of the iterated blow-up space of a manifold with corners with respect to a clean semilattice of adapted submanifolds (i.e. p-submanifolds). Since our constructions and proofs rely heavily on manifolds with corners and their submanifolds, we found it necessary to clarify the various notions of submanifolds of a manifold with corners. CONTENTSIntroduction 1. Manifolds with corners and their submanifolds 1.1. Manifolds with corners 1.2. The boundary and boundary faces 1.3. Submanifolds of manifolds with corners 2. The blow-up for manifolds with corners 2.1. Definition of the blow-up and its smooth structure 2.2. Exploiting the local structure of the blow-up 2.3. Cleanly intersecting families and liftings 3. The graph blow-up 3.1. Definition of the graph blow-up 3.2. Disjoint submanifolds 4. Iterated blow-ups 4.1. Definition of the iterated blow-up 4.2. Clean semilattices 4.3. The pair blow-up lemma 5. Applications to the N -body problem 5.1. Spherical compactifications 5.2. Quotients and compactifications 5.3. Induced maps on C * -algebras 5.4. Identification of the Georgescu and Vasy spaces Appendix A. Proper maps Appendix B. Submanifold Criteria References B.A. has been partially supported by SPP 2026 (Geometry at infinity) and the SFB 1085 (Higher Invariants), both funded by the DFG (German Science Foundation). J.M. and V.N. have been partially supported by ANR-14-CE25-0012-01 (SINGSTAR) funded by ANR (French Science Foundation).

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study N -body type Hamiltonians with interactions. More precisely, let Y ⊂ X be a linear subspace of a finite dimensional Euclidean space X, and v Y be a continuous function on X/Y that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the formwhere the subspaces Y ⊂ X belong to some given family S of subspaces. Georgescu and Nistor have considered the case when S consists of all subspaces Y ⊂ X, and Nistor and the authors considered the case when S is a finite semi lattice and Georgescu generalized these results to any families. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where S is any family of subspaces also, and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of X introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.

We 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.

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