A Comparison of the Georgescu and Vasy Spaces Associated to the N-Body Problems and Applications

We provide new insight into the analysis of N-body problems by studying a compactification M-N of R-3N that is compatible with the analytic properties of the N-body Hamiltonian H-N. We show that our compactification coincides with a 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 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\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.

Automated summary

Our study provides new insights into the analysis of N-body problems through a compactification method compatible with the analytic properties of the N-body Hamiltonian. We demonstrate that this compactification aligns with previous work by Vasy and Georgescu, and has applications in spectral theory and approximation properties related to the behavior near infinity of distribution kernels. Our results also lead to a regularity result for the eigenfunctions of the N-body Hamiltonian.