The pseudo-Hermiticity of MHD dynamo operators

The presentation provides a short introduction to MHD dynamos and to the pseudo-Hermiticity properties of the matrix differential operator of the spherically symmetric α²-dynamo.
In the first part of the lecture, the underlying physics is described starting from a discussion of the homopolar disk dynamo, of field reversals of the Earth's magnetic field and of the dynamo experiments in Riga and Karlsruhe. Furthermore, a sketchy outline of the helicity based meanfield technique for the MHD induction equation is presented and the closing of the dynamo cycle is discussed.
In the second part, attention is paid to the derivation of the matrix differential operator for the spherically symmetric α²-dynamo and its associated quadratic operator pencil. Boundary conditions, pseudo-Hermiticity and Krein-space features are used to heuristically explain the level-crossing properties of the spectrum. Along the lines of [1], the derivation of a constructive no-go theorem for isospectral dynamo configurations is described and interesting open problems are listed.

[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.