An MHD spectral theory approach to Jeans' magnetized gravitational
  instability
  In this paper, we revisit the governing equations for linear
magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized,
plane-parallel, self-gravitating slab. Our approach allows for fully
non-uniformly magnetized slabs, which deviate from isothermal conditions, such
that the well-known Alfv\'en and slow continuous spectra enter the description.
We generalize modern MHD textbook treatments, by showing how self-gravity
enters the MHD wave equation, beyond the frequently adopted Cowling
approximation. This clarifies how Jeans' instability generalizes from hydro to
magnetohydrodynamic conditions without assuming the usual Jeans' swindle
approach. Our main contribution lies in reformulating the completely general
governing wave equations in a number of mathematically equivalent forms,
ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian
formulation linked to coupled harmonic oscillators, up to a convenient matrix
differential form. The latter allows us to derive analytically the
eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an
example we give the exact closed form dispersion relations for the
hydrodynamical p- and Jeans-unstable modes, with the latter demonstrating how
the Cowling approximation modifies due to a proper treatment of self-gravity.
The various reformulations of the MHD wave equation open up new avenues for
future MHD spectral studies of instabilities as relevant for cosmic filament
formation, which can e.g. use modern formal solution strategies tailored to
solve coupled Sturm-Liouville or harmonic oscillator problems.
