The MHD α²-dynamo: operator pencil, pseudo-Hermiticity, level crossings, spectral phase transitions

The lecture is splitted into three parts. In its first part, a brief introduction is presented into the underlying physics of the dynamo effect of magnetohydrodynamics (MHD), in order to make clear the motivation of the subsequent mathematical considerations. Paleomagnetic evidences for fields reversals of the geomagnetic field are discussed, as well as the results of the Riga dynamo experiment. Special emphasis is laid on the planned next generation dynamo experiments.

In the second part of the lecture, the basic operator-theoretic properties of the 2×2 operator matrix of the spherically symmetric α²-dynamo are discussed: its pseudo-Hermiticity in the case of idealized boundary conditions (with corresponding representation of the operator in a Krein space as indefinite generalization of a Hilbert space), and the general lack of symmetry and Hermiticity for physically realistic boundary conditions. A brief summary is given of the results of Ref. [1] on testing the operator struture on its compatibility with an operator intertwining technique as descriptive tool for possibly existing isospectral classes of dynamo operators. A corresponding no-go theorem is discussed for first-order differential operators as intertwiners.

Subject of the third part of the lecture are recent results on spectral phase transitions. Such phase transtions from non-oscillating dynamo regimes to oscillating regimes occur at two-fold degenerate level-crossing points of the spectrum (exceptional points), where two real-valued branches of the spectrum cross and leave the real axis to develop as pair of complex conjugate branches. Some general properties of the spectrum in the vicinity of the exceptional points are discussed. For this purpose, the linear eigenvalue problem of the 2×2 operator matrix is reformulated as equivalent eigenvalue problem of an associated quadratic operator pencil. Special emphasis is laid on the differences of exceptional points for quantum mechanical operators in Hilbert spaces and for pseudo-Hermitian operators in Krein spaces.

The lecture is completed by a discussion of open operator theoretic problems with relevance for the planned next generation dynamo experiments.

[1] Günther U. and Stefani F., Isospectrality of spherical MHD dynamo operators: pseudo-Hermiticity and a no-go
theorem, J. Math. Phys. 44, 2003, 3097, math-ph/0208012.