Systems of Singular Differential Operators of Mixed Order and Applications to 1 - Dimensional MHD Problems

Abstract

In some weighted L2 vector space we study a symmetric semibounded operator IL'0 which is given by a 3 x 3 system of ordinary differential operators on an intervall [0, ro] with a singularity at r = 0 (see (0.1)). This system can be considered as a “smooth” perturbation of a more specific physical model describing the oscillations of plasma in an equilibrium configuration in a cylindrical domain (see (1.12)). This perturbation is smooth in the sense that in the system under study in comparison with the physical model only the smooth parts of the coefficients are changed conserving all types of singularities. It is the goal of this paper to construct a suitable selfadjoint extension IL of the symmetric operator IL'0 (and its closure IL0) and to determine the essential spectrum of this extension. The essential spectrum consists of two bands (which may overlap) if we exclude the singularities by considering the system on an interval (r1,ro] with 0 < r1 < ro. In the corresponding physical model these bands are called Alfen spectrum and slow magnetosonic spectrum. It is shown that the singularity in 0 generates additional components of the essential spectrum which under specific conditions, as in the case of the physical model, “disappear” in the two bands known from the “regular” case (r1, ro) with r1 0.