New results on the spectrum of the MHD α²-dynamo and on Jordan algebra related canonical structures of PT-symmetric matrices

The talk consists of two parts.

The results reported in the first part concern some unexpected properties of the spectrum of the α²-dynamo operator. In a work from February 2006 we investigated the unfolding of diabolical points in the spectrum of a dynamo operator with idealized boundary conditions (BCs). Similar to the Hamiltonians of PT-symmetric Quantum Mechanics this operator is selfadjoint in a Krein space. For constant α-profiles α₀ its spectrum λ[α₀] forms a mesh-like structure in the (α₀,Re λ)-plane with diabolical points (degeneration points of geometric and algebraic multiplicity two) at the nodes of the mesh. Under inhomogeneous perturbations of the α-profile the spectral mesh undergoes deformations which are accompanied by a resonant unfolding of the diabolical points. It was shown that each Fourier component of an α-profile α(x) induces a selective unfolding of diabolical points located on a specific parabola in the (α₀,Re λ)-plane. In recent research we extended this analysis to dynamo operators with realistic boundary conditions. Unexpectedly, it was found that the parabolas related to the resonant unfolding of diabolical points in models with idealized BCs coincide exactly with the spectral branches of dynamo operators with realistic BCs. We discuss some first aspects of the underlying mechanism responsible for this coincidence and indicate some of its possible implications.

In the second part of the talk, canonical representations of Jordan normal forms of PT-symmetric matrices are presented. These canonical representations are derived with the help of Jordan-algebra based conjugations which map Jordan blocks and Jordan chains into corresponding explicitly PT-symmetric matrices and their eigenvectors and associated vectors.