Exhaustive families of representations of C⁎-algebras associated with N-body Hamiltonians with asymptotically homogeneous interactions

Abstract

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study N -body type Hamiltonians with interactions. More precisely, let Y ⊂ X be a linear subspace of a finite dimensional Euclidean space X, and v Y be a continuous function on X/Y that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the formwhere the subspaces Y ⊂ X belong to some given family S of subspaces. Georgescu and Nistor have considered the case when S consists of all subspaces Y ⊂ X, and Nistor and the authors considered the case when S is a finite semi lattice and Georgescu generalized these results to any families. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where S is any family of subspaces also, and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of X introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.

A comparison of the Georgescu and Vasy spaces associated to the N-body problems and applications

“…These actions are easy to obtain at the level of spectra of C * -algebras or for the graphfamily blow-up, but more difficult to obtain geometrically using iterated blow-ups. In particular, this explains the answer to the question of Melrose and Singer [17] provided in [19].…”