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

Abstract

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space L2(R)⊕L2(R) induced by matrix differential expressions of the form(0.1)(τ11(⋅,D)τ12(⋅,D)τ21(⋅,D)τ22(⋅,D)), where τ11, τ12, τ21, τ22 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.