A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications
by
Bernd Ammann, Jérémy Mougel, Victor Nistor


A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications (.pdf)

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Abstract

We provide new insight into the analysis of N-body problems by studying a compactification MN of ℝ3N that is compatible with the analytic properties of the N-body Hamiltonian HN. We show that our compactification coincides with the compactification introduced by Vasy using blow-ups in order to study the scattering theory of N-body Hamiltonians and with a compactification introduced by Georgescu using C*-algebras. Furthermore, we also provide a third description of the compactification as a submanifold of a product of elementary blowups. Our results allow many applications to the spectral theory of N-body problems and to some related approximation properties. For instance, results about the essential spectrum, the resolvents, and the scattering matrices (when they exist) of HN may be related to the behavior at infinity on MN of their distribution kernels, which can be efficiently studied by blow-up methods. The compactification MN is compatible with the action of the permutation group which allows to implement bosonic and fermionic (anti-)symmetry relations. We also obtain a regularity result for the eigenfunctions of HN.

Notes about Version 1 (versus Version 2)

The current version is strongly modified compared to the first version. In particular, it contains many more comments towards applications, as desired by the journal. Also the title is slightly changed.

Abstract of Version 1

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.
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The Paper was written on 21.10.2019
Last update 16.01.2021