Systems of singular differential operators of mixed order and applicationsto 1-dimensional MHD problems

In some weighted L-2 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 interval [0, r(0)] with a singularity at r = 0 (see (0.1)). This sys tem can be considered asa "smooth" perturbation of a more specific physical model describing the oscillations of plasma in an equilibrium configuratio n 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 self adjoint extension IL of the symmetric operator IL'(0) land its closure IL0) and to determine the essential spectrum of this extension. The essential s pectrum consists of two bands (which may overlap) if we exclude the singula rities by considering the system on an interval [r(1), r(0)] with 0 < r(1) < r(0). In the corresponding physical model these bands are called Alfven s pectrum and slow magnetosonic spectrum. It is shown that the singularity in 0 generates additional components of the essential spectrum which under sp ecific conditions, as in the case of the physical model, "disappear" in the two bands known from the "regular" case [r(1), r(0)] with r(1) SE arrow 0.