This paper contains the full proofs of the results announced in the previous authors’ paper (see [C. R., Math., Acad. Sci. Paris 352, No. 12, 1023–1027 (2014; Zbl 1432.70029)]), as well as several extensions of those results.
Authors’ 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 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 \({\mathscr {E}}(X)\).
A typical example is given by the operator \({h(P) +\sum_Y V_Y}\), where \(h: X^*\to [0,\infty)\) is continuous and satisfies \({h(k)\to\infty}\) as \(| k| \to\infty\), \(P=-i\nabla\) is the momentum observable. The potential \(\Sigma_Y V_Y\) is a finite sum over linear subspaces \(Y\subset X\), where \(V_Y\) is a bounded Borel function with uniform radial limits \(\lim\limits_{r\to\infty}V_Y(ra)\).