A REFINED HVZ-THEOREM FOR ASYMPTOTICALLY HOMOGENEOUS INTERACTIONS AND FINITELY MANY COLLISION PLANES

We study algebras associated to N -body type Hamiltonians with interactions that are asymptotically homogeneous at infinity on a Euclidean space X. More precisely, let Y subset of X be a linear subspace and nu(y) be a continuous function on X/Y that has uniform homogeneous radial limits at infinity. We consider in this paper Hamiltonians of the form H = -Delta + Sigma(Y is an element of S) nu(y), where the subspaces Y subset of X belong to some given, semi -lattice S of subspaces of X. Georgescu and Nistor have considered the case when S consists of all subspaces Y subset of X (in a paper to appear in Journal of Operator Theory). As in that paper, we also consider more general Hamiltonians affiliated to a suitable cross -product algebra epsilon(s) (X) (sic) X. A first goal of this note is to see which results of that paper carry through to the case S finite and, for the ones that do not, what is their suitable modification. While the results on the essential spectra of the resulting Hamiltonians and the affiliation criteria carry through, the spectra of the corresponding algebras are quite different. Identifying these spectra may have implications for regularity of eigenvalues and numerical methods. Our results also shed some new light on the results of Georgescu and Nistor in the aforementioned paper and, in general, on the theory developed by Georgescu and his collaborators. For instance, we show that, in our case, the closure is not needed in the union of the spectra of the limit operators. We also give a quotient topology description of the topology on the spectrum of the graded N -body C*-algebras introduced by Georgescu.