Essential spectrum of non-self-adjoint singular matrix differential operators

TL;DR

This paper characterizes the essential spectrum of non-self-adjoint singular matrix differential operators in a Hilbert space, identifying points originating from ellipticity breakdown or singularities at infinity.

Contribution

It provides an analytic description of the essential spectrum for a class of non-self-adjoint matrix differential operators under specific coefficient assumptions.

Findings

Essential spectrum points originate from ellipticity breakdown.

Singularities at infinity also contribute to the essential spectrum.

The spectrum description is explicit and based on operator coefficients.

Abstract

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space $L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ induced by matrix differential expressions of the form \begin{align}\label{abstract:mdo} \left(\begin{array}{cc} \tau_{11}(\,\cdot\,,D) & \tau_{12}(\,\cdot\,,D)\\[3.5ex] \tau_{21}(\,\cdot\,,D) & \tau_{22}(\,\cdot\,,D) \end{array}\right), \end{align} where $\tau_{11}$, $\tau_{12}$, $\tau_{21}$, $\tau_{22}$ are respectively $m$-th, $n$-th, $k$-th and 0 order ordinary differential expressions with $m=n+k$ being even. Under suitable assumptions on their coefficients, we establish an analytic description of the essential spectrum. It turns out that the points of the essential spectrum either have a local origin, which can be traced to points where the ellipticity in the sense of Douglis and Nirenberg breaks down, or they are caused by singularity at infinity.