Essential spectrum of non-self-adjoint singular matrix differential   operators
  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. 
Orif O. Ibrogimov
math.SP