Krein space related perturbation theory for MHD α 2−dynamos and 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 halfplane 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. PACS numbers: 02.30.Tb, 91.25.Cw, 11.30.Er, 02.40.Xx AMS classification scheme numbers: 47B50, 46C20, 47A11, 32S05 Submitted to: J. Phys. A: Math. Gen.