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 $C^{\ast }$-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 $C^{\ast }$-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 $\mathcal{E}(X):=\mathcal{E}(X)⋊X$ 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 $\mathcal{E}(X)$ 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 $(X)$.

arXiv comment33 pages. In version 2, the affiliation criteria have been improved and the introduction has been modified in a substantial way. In version 3, a table of contents has been added