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We determine the essential spectrum of Hamiltonians with N-body typeinteractions that have radial limits at infinity. This extends the HVZ-theorem,which treats perturbations of the Laplacian by potentials that tend to zero atinfinity. Our proof involves $C^*$-algebra techniques that allows one to treatlarge classes of operators with local singularities and general behavior atinfinity. In our case, the configuration space of the system is a finitedimensional, 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 atinfinity. The group $X$ acts by translations on $\mathcal{E}(X)$, and hence thecrossed product $\mathscr{E}(X) := \mathcal{E}(X)\rtimes X$ is well defined;the Hamiltonians that are of interest to us are the self-adjoint operatorsaffiliated to it. We determine the characters of $\mathcal{E}(X)$. This thenallows us to describe the quotient of $\mathscr{E}(X)$ with respect to theideal of compact operators, which in turn gives a formula for the essentialspectrum of any self-adjoint operator affiliated to $\mathscr(X)$.

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