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Journal of Operator Theory

Volume 77, Issue 2,  Spring  2017  pp. 333-376.

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

Authors:  Vladimir Georgescu (1) and Victor Nistor (2)
Author institution: (1) Departement de Mathematiques, Universite de Cergy-Pontoise, 95000 Cergy-Pontoise, France
(2) Departement de Mathematiques, Universite de Lorraine, 57045 Metz, France and Institute of Mathematics of the Romanian Academy, P.O. BOX 1-764, 014700 Bucharest, Romania

Summary:  We determine the essential spectrum of Hamiltonians with $N$-body type interactions that have radial limits at infinity, which extends the classical HVZ-theorem for potentials that tend to zero at infinity. Let $\mathcal{E}(X)$ be the algebra generated by functions of the form $v\circ\pi_Y$, where $Y \subset X$ is a subspace, $\pi_Y: X \to X/Y$ is the projection, and $v:X/Y\to\mathbb{C}$ is continuous with \textit{uniform radial limits at infinity.} We consider Hamiltonians affiliated to $\rond{E}(X) := \mathcal{E}(X)\rtimes X$. We determine the characters of $\mathcal{E}(X)$ and then we describe the quotient of $\rond{E}(X)/\mathcal{K}$ 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 $\rond{E}(X)$.

Keywords:  self-adjoint operator, essential spectrum, compact operator, $C^*$-algebra, limit operator, character, radial compatification, $N$-body problem

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