Third order spectral branch points in Krein space related setups: PT-symmetric matrix toy model, MHD α²-dynamo, and extended Squire equation

The spectra of self-adjoint operators in Krein spaces are known to possess real sectors as well as sectors of pair-wise complex conjugate eigenvalues. Transitions from one spectral sector to the other are a rather generic feature and they usually occur at exceptional points of square root branching type. For certain parameter configurations two or more such exceptional points may happen to coalesce and to form a higher order branch point. We study the coalescence of two square root branch points semi-analytically for a PT-symmetric 4×4 matrix toy model and illustrate numerically its occurrence in the spectrum of the 2×2 operator matrix of the magnetohydrodynamic α²-dynamo and of an extended version of the hydrodynamic Squire equation.