On the Essential Spectrum of N-Body Hamiltonians with Asymptotically Homogeneous Interactions

TL;DR

This paper extends the HVZ-theorem by characterizing the essential spectrum of N-body Hamiltonians with asymptotically homogeneous interactions using $C^*$-algebra techniques, applicable to systems with complex behavior at infinity.

Contribution

It introduces a $C^*$-algebra framework to analyze the essential spectrum of N-body Hamiltonians with radial limits at infinity, generalizing previous results.

Findings

Determined the characters of the algebra $\\mathcal{E}(X)$.

Provided a formula for the essential spectrum of operators affiliated to the crossed product algebra.

Extended the HVZ-theorem to a broader class of interactions with asymptotic homogeneity.

Abstract

We determine the essential spectrum of Hamiltonians with N-body type interactions that have radial limits at infinity. This extends the HVZ-theorem, which treats perturbations of the Laplacian by potentials that tend to zero at infinity. Our proof involves $C^*$-algebra techniques that allows one to treat large classes of operators with local singularities and general behavior at infinity. In our case, the configuration space of the system is a finite dimensional, real vector space $X$, and we consider the $C^*$-algebra $\mathcal{E}(X)$ of functions on $X$ generated by functions of the form $v\circ\pi_Y$, where $Y$ runs over the set of all linear subspaces of $X$, $\pi_Y$ is the projection of $X$ onto the quotient $X/Y$, and $v:X/Y\to\mathbb{C}$ is a continuous function that has uniform radial limits at infinity. The group $X$ acts by translations on $\mathcal{E}(X)$, and hence the crossed product $\mathscr{E}(X) := \mathcal{E}(X)\rtimes X$ is well defined; the Hamiltonians that are of interest to us are the self-adjoint operators affiliated to it. We determine the characters of $\mathcal{E}(X)$. This then allows us to describe the quotient of $\mathscr{E}(X)$ 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 $\mathscr(X)$.