ON THE ESSENTIAL SPECTRUM OF N-BODY HAMILTONIANS WITH ASYMPTOTICALLY HOMOGENEOUS INTERACTIONS

We determine the essential spectrum of Hamiltonians with N-body type interactions that have radial limits at infinity, which extends the classical HVZ-theorem for potentials that tend to zero at infinity. Let E(X) be the algebra generated by functions of the form v ○ π Y, where Y ⊂ X is a subspace, π Y : X → X/Y is the projection, and v : X/Y → C is continuous with uniform radial limits at infinity. We consider Hamiltonians affiliated to E(X) := E(X) ⋊ X. We determine the characters of E(X) and then we describe the quotient of E(X)/K with respect to the ideal of compact operators, which in turn gives a formula for the essential spectrum of any self-adjoint operator affiliated to E(X).