# 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 $\rond{E}(X)$.

**DOI: **http://dx.doi.org/10.7900/jot.2016apr08.2115

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

Contents
Full-Text PDF