The essential spectrum of a system of singular ordinary differential operators of mixed order.: Part III:: A strongly singular case

We consider a system of ordinary differential operators of mixed order on an interval (0, r(0)), r(0) > 0, where some of the coefficients are 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. Generalizations of this problem have been considered by Hardt, Mennicken, Naboko and Faierman, Mennicken and Moller, where in each case some kind of regularity condition was required. The essential spectrum has been calculated explicitly in terms of the coefficient functions of the system; it is always bounded in these cases. Here we consider a class of problems for which the essential spectrum is unbounded. The essential spectrum is explicitly given as the essential spectrum in the limiting case. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.