Exhaustive families of representations of C∗C^*-algebras associated to NN-body Hamiltonians with asymptotically homogeneous interactions

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study $N$-body type Hamiltonians with interactions. More precisely, let $Y$ be a linear subspace of a finite dimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$ that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form $H = - \Delta + \sum_{Y \in S} v_Y$, where the subspaces $Y$ belong to some given family S of subspaces. We prove results on the spectral theory of the Hamiltonian when $S$ is any family of subspaces and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of $X$ introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.Comment: 5 page

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 $\mathbb{R}^{3N}$ 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. Furthermore, we also provide a third description of the compactification as a submanifold of a product of elementary blowups. Our results allow many 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 (when they exist) of $H_N$ may be related to the behavior at infinity on $M_N$ of their distribution kernels, which can be efficiently studied by blow-up methods. The compactification $M_N$ is compatible with the action of the permutation group which allows to implement bosonic and fermionic (anti-)symmetry relations. We also obtain a regularity result for the eigenfunctions of $H_N$.Comment: In version 2 several application towards physics were added, according to the wishes of the journal. The numbering has change