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

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 $v\circπ_Y$, where $Y$ runs over the set of all linear subspaces of $X$, $π_Y$ 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 $\mathscr(X)$.