Title: On the essential spectrum of elliptic differential operators,
Abstract: Let $\mathcal{A}$ be a $C^*$-algebra of bounded uniformly continuous
functions on $X=\mathbb{R}^d$ such that $\mathcal{A}$ is stable under
translations and contains the continuous functions that have a limit at
infinity. Denote $\mathcal{A}^\dagger$ the boundary of $X$ in the character
space of $\mathcal{A}$. Then the crossed product
$\mathscr{A}=\mathcal{A}\rtimes X$ of $\mathcal{A}$ by the natural action of
$X$ on $\mathcal{A}$ is a well defined $C^*$-algebra and to each operator
$A\in\mathscr{A}$ one may naturally associate a family of bounded operators
$A_\varkappa$ on $L^2(X)$ indexed by the characters
$\varkappa\in\mathcal{A}^\dagger$. We show that the essential spectrum of $A$
is the union of the spectra of the operators $A_\varkappa$. The applications
cover very general classes of singular elliptic operators.