Some operator models for a linearized equation describing oscillations of plasma
by
Manfred Möller
University of the Witwatersrand, South Africa

Kako [4] calculated the essential spectrum of an operator associated with a linearized magnetohydrodynamic equation describing the oscillations of plasma in an equilibrium configuration in a cylindrical domain. This operator is given by a mixed-order 3×3-system of ordinary differential operators on (0, r₀), where 0 is a singular endpoint. In different approaches, Hardt, Mennicken and Naboko [3] as well as Faierman, Mennicken, Möller [1, 2] generalized the results and gave detailed proofs. These results concern the so-called almost regular case and the much more involved quasi regular case. In the quasi regular case, an additional part of the essential spectrum may occur; in Kako's case, it is contained in the `regular' part of the essential spectrum.

In this talk, I will review these result and show some extensions to singular cases.

References

1

Faierman, M., Mennicken, R., Möller, M., The essential spectrum of a system of singular ordinary differential operators of mixed order; Part I: The general problem and an almost regular case, Math. Nachr. 208 (1999), 101-115.

2

Faierman, M., Mennicken, R., Möller, M., The essential spectrum of a system of singular ordinary differential operators of mixed order; Part II: The generalization of Kako's problem, Math. Nachr. 209 (2000), 55-81.

3

Hardt, V., Mennicken, M., Naboko, S., Math. Nachr. 205 (1999), 19-68.

4

Kako, T., Essential spectrum of linearized operator for MHD plasma in  cylindrical region, J. Appl. Math. Phys. (ZAMP) 38 (1987), 433-449.

Date received: June 14, 2002