The essential spectrum of a system of singular ordinary differential operators of mixed order. Part II: The generalization of Kako's problem

A system of ordinary differential equations of mixed order on an interval ( 0, r(0)) is considered, where some coefficients are singular at 0. Special cases have been dealt with by KAKO, where the essential spectrum of an oper ator associated with a linearized MHD model was calculated, and more recent ly by HARDT, MENNICKEN and NABOKO. In both papers this operator is a selfad joint extension of an operator on sufficiently smooth functions. The approa ch in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly def ined as the closure of an operator defined on sufficiently smooth functions . This closed operator can be written as a sum of a selfadjoint operator an d a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of C, and formulas for the calculation of the essential spe ctrum in terms of the coefficients are given.