# On the essential spectrum of elliptic differential operators

## TL;DR

This paper characterizes the essential spectrum of elliptic differential operators using $C^*$-algebra techniques, showing it as a union of spectra of associated boundary operators, applicable to broad classes of singular elliptic operators.

## Contribution

It introduces a $C^*$-algebra framework to analyze the essential spectrum of elliptic operators, extending previous methods to more general singular cases.

## Key findings

- Essential spectrum equals union of spectra of boundary operators.
- Framework applies to broad classes of singular elliptic operators.
- Provides a new perspective using crossed product $C^*$-algebras.

## 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.