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 ε (X) be the algebra generated by functions of the form v o π_γ, where Y ⊂ X is a subspace, π_γ: X → X/Y is the projection, and v: X/Y → C is continuous with uniform radial limits at infinity. We consider Hamiltonians affiliated to ε(X) := ε (X) × X. We determine the characters of ε (X) and then we describe the quotient of ε(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 ε (X).