  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.
