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

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 $C^{\ast }$-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 $H_N$.

In version 2 several application towards physics were added, according to the wishes of the journal. The numbering has changed. In version 3 several formulations were improved, however the mathematical content is unchanged. Again the numbering has changed. The article was accepted in Annales Henri Poincare