The spectrum of an ideal linear magnetohydrodynamic model with translational symmetry

Abstract

We consider an ideal linear magnetohydrodynamic model with translational symmetry and investigate the spectral properties of the force operator A_k with a fixed longitudinal wavenumber k. We establish that the essential spectrum of A_k consists of two bounded segments (slow magnetosonic and Alfvén continuum). Next, we show that the isolated eigenvalues of A_k do not accumulate to the tips of the Alfvén continuum. Further, we obtain detailed information about the asymptotics of the eigenvalues of A_k which lie to the right of the essential spectrum and tend to +∞. At last, in the case when the two segments of the essential spectrum of A_k are disjoint, we find that, generically, the isolated eigenvalues of A_k accumulate to the right tip of the slow magnetosonic continuum, and calculate the first asymptotic term of the corresponding infinite eigenvalue sequence.