The MHD alpha^2-dynamo, Z_2-graded pseudo-Hermiticity, level crossings and exceptional points of branching type

TL;DR

This paper investigates the spectral behavior of the alpha^2-dynamo operator, focusing on level crossings, exceptional points, and the mathematical structures underlying these phenomena, using numerical analysis and a simplified toy model.

Contribution

It introduces a Z_2-graded pseudo-Hermitian framework to analyze spectral branching and exceptional points in the alpha^2-dynamo, highlighting new mathematical insights.

Findings

Identification of exceptional points and diabolic points in the dynamo spectrum

Analysis of SU(1,1) symmetry and Krein space structure

Characterization of eigenvalue multiplicities at degeneracies

Abstract

The spectral branching behavior of the 2x2 operator matrix of the magneto-hydrodynamic alpha^2-dynamo is analyzed numerically. Some qualitative aspects of level crossings are briefly discussed with the help of a simple toy model which is based on a Z_2-graded-pseudo-Hermitian 2x2 matrix. The considered issues comprise: the underlying SU(1,1) symmetry and the Krein space structure of the system, exceptional points of branching type and diabolic points, as well as the algebraic and geometric multiplicity of corresponding degenerate eigenvalues.