The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part I: The General Problem and an Almost Regular Case

Abstract

A system of ordinary differential operators of mixed order on an interval (0,τ0), ro0 > 0, is considered, where the coefficients may be singular at 0. A special case has been dealt with by Kako where the essential spectrum of an operator associated with a linearized magnetohydrodynamic equation was explicitly calculated. In the present first part of the paper we study an almost regular special case which can be treated by the operator theoretical methods developed by Atkinson, Langer, Mennicken and Shkalikov. A closed linear operator is associated with the given system of differential operators and its essential spectrum is explicitly characterized in terms of the coefficients of these differential operators.