  We overview some of our recent results on the essential spectrum of N-body
Hamiltonians with potentials defined by functions that have radial limits at
infinity. The results extend the HVZ theorem which describes the essential
spectrum of usual N-body Hamiltonians. The proof is based on a careful study of
algebras generated by potentials and their cross-products. We also describe the
topology on the spectrum of these algebras, thus extending to our setting a
result of A. Mageira. Our techniques apply to more general classes of
potentials associated to translation invariant algebras of bounded uniformly
continuous functions on a finite dimensional vector space.
