Isospectrality of spherical MHD dynamo operators: pseudo-Hermiticity and a no-go theorem

Abstract

The isospectrality problem is studied for the operator of the spherical hydromagnetic alpha^2-dynamo. It is shown that this operator is formally pseudo-Hermitian (J-symmetric) and lives in a Krein space. Based on the J-symmetry, an operator intertwining Ansatz with first-order differential intertwining operators is tested for its compatibility with the structure of the alpha^2-dynamo operator matrix. An intrinsic structural inconsistency is obtained in the set of associated matrix Riccati equations. This inconsistency is interpreted as a no-go theorem which forbids the construction of isospectral alpha^2-dynamo operator classes with the help of first-order differential intertwining operators.

13 pages, LaTeX2e, improved references, to appear in J. Math. Phys

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• Solvability and PT-symmetry in a double-well model with point interactions
• Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation
• Quantum catastrophes: a case study
• Experiments in PT-symmetric quantum mechanics
• Determining role of Krein signature for 3D Arnold tongues of oscillatory dynamos
• Third order spectral branch points in Krein space related setups: PT-symmetric matrix toy model, MHD alpha^2-dynamo, and extended Squire equation
• Propagation of Electromagnetic Waves in Linear Media and Pseudo-Hermiticity
• A globally diagonalizable alpha^2-dynamo operator, SUSY QM and the Dirac equation
• Non linear pseudo-bosons versus hidden Hermiticity
• The MHD alpha^2-dynamo, Z_2-graded pseudo-Hermiticity, level crossings and exceptional points of branching type
• PT-symmetric regularizations in supersymmetric quantum mechanics
• On the spectrum of the magnetohydrodynamic mean-field alpha^2-dynamo operator
• Dynamics of charged fluids and 1/L perturbation expansions
• Inverse Problems in Magnetohydrodynamics: Theoretical and Experimental Aspects