Perturbation theory for non-selfadjoint operator matrices and its application to the MHD α²-dynamo

We consider singular boundary value problems for linear non-selfadjoint m-th order N×N matrix differential operators on the interval (0,1]∋ x. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend on the spectral parameter λ as well as on a vector-valued parameter distribution p(x). For simplicity, we restrict our attention to setups with a (boundary) singular point (x=0) of regular (Fuchsian) type.

Based on a bi-orthogonal solution ansatz, we study perturbations of simple and multiple eigenvalues under small variations of the parameter distribution p(x). Special attention is paid to perturbations of semi-simple multiple eigenvalues (diabolical points; characterized by coinciding geometric and algebraic multiplicity and corresponding diagonal spectral decomposition of the operator) as well as to perturbations of non-derogatory eigenvalues (branch points, exceptional points; with non-trivial Jordan structures in the spectral decomposition) with one eigenvector (geometric multiplicity one) and several associated vectors forming a Keldysh chain of length equal to the algebraic multiplicity of the eigenvalue. Explicit formulae describing the bifurcation of the eigenvalues are derived.

As application, the general technique is utilized for the investigation of the spectral properties of spherically symmetric MHD α²-dynamos. Specifically, we provide an analytical description for the occurrence of spectral branch points under transitions from idealized boundary conditions to physically realistic boundary conditions. Furthermore, we develop a gradient technique with respect to the α-profile α(x) of the dynamo which allows us to search for most efficient α-perturbations to trigger magnetic field reversals.