On the essential spectrum of a class of singular matrix differential operators. II. Weyl's limit circles for the Hain-Lüst operator whenever quasi-regularity conditions are not satisfied

Abstract

The essential spectrum of the singular matrix differential operator of mixed order determined by the operator matrix (Equation Presented) is studied. Investigation of the essential spectrum of the corresponding self-adjoint operator is continued but now without assuming that the quasi-regularity conditions are satisfied. New conditions that guarantee that the operator is semi-bounded from below are derived. It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential operator is given by the range range((mρ-β2)/ρx2) in the case where the quasi-regularity conditions are not satisfied. © 2008 The Royal Society of Edinburgh.