On the Essential Spectrum of N-Body Systems with Asymptotically Homogeneous of Order Zero Interactions

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.