The essential spectrum of an operator relative to the stability of a toroidal plasma

The essential spectrum of the self-adjoint Hilbert space operator of the linearized equations of ideal magnetohydrodynamics which determines equilibrium stability is analyzed for the case of a toroidally confined plasma. The essential spectrum relative to the magnetohydrodynamic and kinetic energy equations is defined in terms of an isolated eigenvalue of finite multiplicity, and the solution of the eigenvalue problem of a system of two second-order ordinary differential equations with periodic boundary conditions is shown to correspond to the essential spectrum.