A comparison of the Georgescu and Vasy spaces associated to the \(N\)-body problems and applications. (English) Zbl 1514.81284
Summary: 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 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 \({\mathbb{R}}^{3N})\). 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 \(H_N\).
MSC:
| 81V70 | Many-body theory; quantum Hall effect |
| 54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |
| 35B44 | Blow-up in context of PDEs |
| 81U10 | \(n\)-body potential quantum scattering theory |
| 46L05 | General theory of \(C^*\)-algebras |
| 47A10 | Spectrum, resolvent |
| 81U20 | \(S\)-matrix theory, etc. in quantum theory |
| 20B05 | General theory for finite permutation groups |
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