Some aspects of a Krein space related perturbation theory for MHD α²-dynamos and the resonant unfolding of diabolical points

The spectrum of the spherically symmetric α²-dynamo is studied in the case of idealized boundary conditions. Starting from the exact analytical solutions of models with constant α-profiles a perturbation theory and a Galerkin technique are developed in a Krein-space approach. With the help of these tools a very pronounced α-resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh. Non-oscillatory as well as oscillatory dynamo regimes are obtained. A Fourier component based estimation technique is developed for obtaining the critical α-profiles at which the eigenvalues enter the right spectral half-plane with non-vanishing imaginary components (at which overcritical oscillatory dynamo regimes form). Finally, Fréchet derivative (gradient) based methods are developed, suitable for further numerical investigations of Krein-space related setups like MHD α²-dynamos or models of PT-symmetric quantum mechanics.