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

TL;DR

This paper introduces a new compactification of the N-body configuration space that unifies previous approaches and enhances the analysis of spectral properties, scattering, and symmetry in N-body quantum problems.

Contribution

It establishes that Georgescu and Vasy's compactifications are equivalent, providing a unified framework for studying N-body Hamiltonians and their spectral and scattering properties.

Findings

Unified compactification coincides with Georgescu and Vasy's methods.

Applications to spectral theory, including essential spectrum and resolvents.

Compatibility with symmetry groups enables analysis of bosonic and fermionic states.

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 $H_N$.