On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

Abstract

We determine the essential spectrum of N-body Hamiltonians with 2-body (or, more generally, k-body) potentials that have radial limits at infinity. The classical N-body Hamiltonians appearing in the well known HVZ-theorem are a particular case of this type of potentials corresponding to zero limits at infinity. Our result thus extends the HVZ-theorem that describes the essential spectrum of the usual N-body Hamiltonians. More precisely, if the configuration space of the system is a finite dimensional real vector space X, then let e(X) be the C *-algebra of functions on X generated by the algebras C(X/Y), where Y runs over the set of all linear subspaces of X and C(X/Y) is the space of continuous functions on X/Y that have radial limits at infinity. This is the algebra used to define the potentials in our case, while in the classical case the C(X/Y) are replaced by C0 (X/Y). The proof of our main results is based on the study of the structure of the algebra e(X), in particular, we determine its character space and the structure of its cross-product e(X) := e(X) ⋊ X by the natural action τ of X on e(X). Our techniques apply also to more general classes of Hamiltonians that have a many-body type structure. We allow, in particular, potentials with local singularities and more general behaviours at infinity. We also develop general techniques that may be useful for other operators and other types of questions, such as the approximation of eigenvalues.