# Essential spectrum of elliptic systems of pseudo-differential operators on $L^2(\mathbb{R}^N)\oplus L^2(\mathbb{R}^N)$

## TL;DR

This paper characterizes the essential spectrum of certain non-self-adjoint elliptic pseudo-differential operator systems on $L^2(R^N) imes L^2(R^N)$, extending previous results and applying to fluid dynamics.

## Contribution

It provides an analytic description of the essential spectrum for non-self-adjoint mixed-order elliptic systems, generalizing Wong's earlier work.

## Key findings

- Established the essential spectrum for a class of elliptic systems.
- Extended Wong's result to non-self-adjoint systems.
- Applied the theory to fluid film dynamics.

## Abstract

Inspired by a result of Wong (Commun. Partial Differ. Equ. 13(10):1209-1221, 1988), we establish an analytic description of the essential spectrum of non-self-adjoint mixed-order systems of pseudo-differential operators on $L^2(\mathbb{R}^N) \oplus L^2(\mathbb{R}^N)$ that are uniformly Douglis-Nirenberg elliptic with positive-order diagonal entries. We apply our result to a problem arising in the dynamics of falling liquid films.