MHD α2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

2005/01/31 by Uwe Guenther, Uwe Günther, Frank Stefani +1
Mathematics · Physics and Astronomy · #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #astro-ph #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1063/1.1915293

published as J.Math.Phys. 46 (2005) 063504 · 21 pages, LaTeX2e, 10 figures, minor improvements, references added, to appear in J. Math. Phys

arxiv created 2005/03/21 · openalex publication_date 2005/05/13 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

It is shown that the α2-dynamo of magnetohydrodynamics, the hydrodynamic Squire equation as well as an interpolation model of PT-symmetric quantum mechanics are closely related as spectral problems in Krein spaces. For the α2-dynamo and the PT-symmetric model the strong similarities are demonstrated with the help of a 2×2 operator matrix representation, whereas the Squire equation is reinterpreted as a rescaled and Wick-rotated PT-symmetric problem. Based on recent results on the Squire equation the spectrum of the PT-symmetric interpolation model is analyzed in detail and the Herbst limit is described as spectral singularity.

Citations

• Isospectrality of spherical MHD dynamo operators: Pseudo-hermiticity and a no-go theorem
• Fragile PT-symmetry in a solvable model
• The MHD α2-Dynamo, Z2-Graded Pseudo-Hermiticity, Level Crossings and Exceptional Points of Branching Type

Related

• Isospectrality of spherical MHD dynamo operators: Pseudo-hermiticity and a no-go theorem
• The MHD α2-Dynamo, Z2-Graded Pseudo-Hermiticity, Level Crossings and Exceptional Points of Branching Type
• Aspects of the ODE/IM correspondence
• The ODE/IM correspondence and PT-symmetric quantum mechanics
• On a spectral problem in magnetohydrodynamics and its relevance for the geodynamo