A no-go theorem for isospectral alpha^2-dynamos

The data analysis of current dynamo experiments and the design of new experiments led to an increasing interest in the spectral properties of MHD dynamos. In particular, the huge amount of multichannel data measured with high spectral resolution at large magnetic sensor arrays outside the liquid sodium facilities calls for the development of tomographic methods for these experiments. In general, such methods would allow for a rough reconstruction of velocity and alpha profiles inside a facility from the magnetic field data measured outside the facility. Due to the fact that viable tomographic methods are crucially based on a correct interpretation of the measured data, one has first to deeply understand the spectral properties of the experimental setup and to clarify the uniqueness of the corresponding inverse problem.

As step in this direction and in order to keep the calculations at the beginning as simple as possible, in [1] the uniqueness of spectral data was studied numerically for the toy model of a spherically symmetric alpha^2-dynamo. The results, which were obtained by combining a spectral forward solver with an inverse problem solver, indicated a possibly existing isospectrality of spherically symmetric alpha^2-dynamos with different alpha-profiles alpha_1(r) and alpha_2(r) and a corresponding non-uniqueness of the inverse problem.

The present contribution is based on Ref. [2] and reports analytical results on the isospectrality problem of spherically symmetric alpha^2-dynamo operators. First, it is shown that the 2 x 2 operator matrix of the alpha^2-dynamo possesses a fundamental (canonical) symmetry J, so that the operator itself can be naturally described as J-pseudo-Hermitian operator in a Krein space (a Hilbert space with an additional indefinite inner product structure). Then this symmetry is used to extend the operator intertwining techniques of supersymmetric quantum mechanics to the operator matrix of the alpha^2-dynamo. The intertwiners are assumed as first-order matrix differential operators with coefficients which are highly constrained by a system of nonlinear matrix differential equations. The (hidden) symmetries of this equation system are analyzed by transforming it into a set of constrained and interlinked matrix Riccati equations. As next step, the structure of the spherically symmetric alpha^2-dynamo operator is tested on its compatibility with the considered intertwining ansatz and a no-go theorem is derived which states that the structure is not compatible with an intertwining ansatz based on first-order differential operators as intertwiners. This implies, that other, more powerful techniques should be used for a clarification of the isospectrality and uniqueness problem of the dynamo operator. Finally, the linear eigenvalue problem for the 2 x 2 dynamo operator matrix is transformed to the equivalent eigenvalue problem for the associated quadratic operator pencil. From this quadratic pencil a general functional equation is derived, which will be useful in future research (extending [3]) for discriminating oscillating dynamo regimes from non-oscillating regimes and for localizing the degenerate spectral intersection points where transitions from one regime to the other occur.

[1] F. Stefani and G. Gerbeth, Astron. Nachr. 321, (2000), 235, astro-ph/0010090; Phys. Earth Planet. Inter. 128, (2001), 109.

[2] U. Günther and F. Stefani, J. Math. Phys. 44, (2003), 3097, math-ph/0208012.

[3] F. Stefani and G. Gerbeth, Phys. Rev. E 67, (2003), 027302, astro-ph/0210412.