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We continue the analysis of algebras introduced by Georgescu, Nistor andtheir coauthors, in order to study $N$-body type Hamiltonians withinteractions. More precisely, let $Y$ be a linear subspace of a finitedimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$that has uniform homogeneous radial limits at infinity. We consider, in thispaper, Hamiltonians of the form $H = - \Delta + \sum_{Y \in S} v_Y$, where thesubspaces $Y$ belong to some given family S of subspaces. We prove results onthe spectral theory of the Hamiltonian when $S$ is any family of subspaces andextend those results to other operators affiliated to a larger algebra ofpseudo-differential operators associated to the action of $X$ introduced byConnes. In addition, we exhibit Fredholm conditions for such ellipticoperators. We also note that the algebras we consider answer a question ofMelrose and Singer.

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