Irregular singularity of the magneto-rotational instability in a Keplerian disk.

Magneto-rotational instabilities in a Keplerian disk are studied via both the eigenvalue and initial-value approaches on the basis of the incompressible magnetohydrodynamics model. The center of the Keplerian disk is an irregular singularity for the eigenfunctions of the magneto-rotational instabilities. This singularity yields continuous eigenvalues (growth rates in the unstable regime and frequencies in the stable regime). The eigenfunctions belonging to the continuum are square-integrable and are not orthogonal with each other. Physical implications of such a continuum and the eigenfunctions are rather complex, because of the ``non- Hermitian'' nature of the rotating plasma system. Invoking the Laplace transform, as well as numerical simulations, interesting long-term behavior of the instability, or the slow change of the mode structure in the exponentially growing phase, has been found.