Exhaustive families of representations of C⁎-algebras associated with N-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⊂X be a linear subspace of a finite-dimensional Euclidean space X, and vY 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=−Δ+∑Y∈SvY, where the subspaces Y⊂X belong to some given family S of subspaces. Georgescu and Nistor have considered the case when S consists of all subspaces Y⊂X, and Nistor and coauthors considered the case when S is a finite semilattice and Georgescu generalized these results to any family. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where S is any family of subspaces also, and extend those results to other operators affiliated to a larger algebra of pseudodifferential operators associated with 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.