In relation to the algebraic (non-exponential) behavior of vortex fluctuations observed in non-neutral plasma, galactic phenomena and atmospheric dynamics, the linear stability analysis is receiving careful reconsideration. This temporally complicated behavior stems from the non-Hermitian (non-selfadjoint) property of governing equation, and has been overlooked in the normal modes analysis. In this paper, algebraic behavior of fluctuations is analyzed for a slab equilibrium of ideal MHD without flow.

In general, stability of plasma is analyzed by solving the linearized MHD equations (simultaneous partial differential equations), which is represented by an evolution equation for the fluctuation part of velocity (v), magnetic field (b) and pressure (p). In many cases, the perturbation fields (v,b,p) in the vicinity of the equilibrium are reduced to a displacement vector (ξ) and the linearized equation is written as ∂²ξ/∂t²=Hξ where H is known as a selfadjoint operator. Since the spectral resolution of selfadjoint operator is mathematically possible, we can assume the exponential behavior (e^-iωt) of eigenmodes. The MHD stability analysis is conventionally based on the dispersion relation which determines if the imaginary part of ω is positive or not.

However, since the special initial conditions are assumed in the above reduction process, the
spectrum of the original system for (v,b,p) does not perfectly coincide with that of H. In this paper, we will show that the original system is non-selfadjoint by considering a slab equilibrium as an simple example. When the generator is non-selfadjoint, degenerate spectra may cause algebraic growth of fluctuations even if ω is real (resonance between eigenmodes), and the dispersion relation is not sufficient to predict the stability of fluctuations. Since the spectral resolution of infinite dimensional non-selfadjoint operator is mathematically unsolved problem, we must solve the initial value problem. The difficulty of non-Hermitian system originates from continuous spectrum, and degenerate continuous spectra yield much more spatially and temporally complicated behavior than degenerate point spectra.