On the Essential Spectrum of N-Body Systems with Asymptotically Homogeneous of Order Zero Interactions

TL;DR

This paper extends the HVZ theorem to N-body Hamiltonians with potentials having radial limits at infinity, analyzing their essential spectrum through algebraic and topological methods.

Contribution

It introduces new techniques to study the essential spectrum of N-body systems with asymptotically homogeneous interactions, broadening the class of potentials analyzed.

Findings

Extended HVZ theorem for new classes of potentials

Described the topology on the spectrum of associated algebras

Applied techniques to translation invariant algebras of functions

Abstract

We overview some of our recent results on the essential spectrum of N-body Hamiltonians with potentials defined by functions that have radial limits at infinity. The results extend the HVZ theorem which describes the essential spectrum of usual N-body Hamiltonians. The proof is based on a careful study of algebras generated by potentials and their cross-products. We also describe the topology on the spectrum of these algebras, thus extending to our setting a result of A. Mageira. Our techniques apply to more general classes of potentials associated to translation invariant algebras of bounded uniformly continuous functions on a finite dimensional vector space.