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arXiv · 1910.10656

A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications

Bernd Ammann·Jeremy Mougel·Victor Nistor
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Abstract

We provide new insight into the analysis of N-body problems by studying a compactification $M_N$ of $\mathbb{R}^{3N}$ that is compatible with the analytic properties of the $N$-body Hamiltonian $H_N$. 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. In particular, the compactifications introduced by Georgescu and by Vasy coincide (up to a homeomorphism that is the identity on $\mathbb{R}^{3N}$). Our result has 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 of $H_N$ (when they exist) may be related to the behavior near $M_N\setminus \mathbb{R}^{3N}$ (i.e. "at infinity") of their distribution kernels, which can be efficiently studied using our methods. The compactification $M_N$ is compatible with the action of the permutation group $S_N$, which allows to implement bosonic and fermionic (anti-)symmetry relations. We also indicate how our results lead to a regularity result for the eigenfunctions of $H_N$.

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Bernd Ammann, Jeremy Mougel, Victor Nistor. 2019-10-23. A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications. https://doi.org/10.1007/s00023-021-01109-1

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