Bundle stratification of PT-symmetric 4×4 matrix systems and fourth-order spectral branch points of MHD α²-dynamos

The spectra of PT-symmetric quantum systems are, in general, highly complicated complex varieties over the parameter spaces of these systems. Parameter space regions of exact PT-symmetry with purely real eigenvalues adjoin other regions with broken PT-symmetry and pairwise complex conjugate eigenvalues. The boundaries between different regions correspond to (exceptional) hypersurfaces in the parameter space and have certain system defined codimensions. Hypersurfaces (strata) of lower codimension may intersect at sub-hypersurfaces (sub-strata) of higher codimension forming in this way a hierarchical structure of dimensionally nested hypersurfaces of increasing codimension (and increasing "exceptionality"), i.e. they form a so called stratified manifold.

In case of matrix systems, the most general parameter space can be identified with the matrix space itself. Endowing this space with a natural fiber bundle structure, regarding matrix orbits (equivalence classes) over a given Jordan structure as fibers, the hierarchical hypersurface structure of the spectral variety can be studied with the help of nested Jordan structures. As result, one arrives at a so called bundle stratification.

We perform such a bundle stratification explicitly for a PT-symmetric 4×4 matrix systems and use the obtained results to comparatively analyze the spectral structure of the spherically symmetric α²-dynamo of magnetohydrodynamics in the vicinity of a fourth-order branch point.