On the spectrum of the magnetohydrodynamic mean field α²-dynamo operator

The existence of magnetohydrodynamic mean-field α²-dynamos with spherically symmetric, isotropic helical turbulence function α is related to a non-self-adjoint spectral problem for a coupled system of two singular second oder ordinary differential equations. We establish global estimates for the eigenvalues of this system in terms of the profile α and its derivative α'. They allow us to formulate an anti-dynamo theorem and a non-oscillation theorem. The conditions of these theorems, which again involve α and α', must be violated in order to reach overcritical and oscillatory regimes, respectively.