Zusammenfassung
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 ...
Zusammenfassung
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
-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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