Intertwiners of pseudoHermitian 2×2blockoperator matrices and a nogo theorem for isospectral MHD dynamo operators
Intertwiners of pseudoHermitian 2×2blockoperator matrices and a nogo theorem for isospectral MHD dynamo operators
Günther, U.
PseudoHermiticity as a generalization of usual Hermiticity is a rather common feature of (differential) operators emerging in various physical setups. Examples are Hamiltonians of PT and CPTsymmetric quantum mechanical systems [1] as well as the operator of the spherically symmetric α²dynamo [2] in magnetohydrodynamics (MHD). In order to solve the inverse spectral problem for these operators, appropriate uniqueness theorems should be obtained and possibly existing isospectral configurations should be found and classified. As a step toward clarifying the isospectrality problem of dynamo operators, we discuss an intertwining technique for ηpseudoHermitian 2×2blockoperator matrices with secondorder differential operators as matrix elements. The intertwiners are assumed as firstorder matrix differential operators with coefficients which are highly constrained by a system of nonlinear matrix differential equations. We analyze the (hidden) symmetries of this equation system, transforming it into a set of constrained and interlinked matrix Riccati equations. Finally, we test the structure of the spherically symmetric MHD α²dynamo operator on its compatibility with the considered intertwining ansatz and derive a nogo theorem.
[1] Bender C.M. and Boettcher S., Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 1998, 5243, physics/9712001; Znojil M., PTsymmetric harmonic oscillators, Phys. Lett. A259, 1999, 220, quantph/9905020; Bender C.M., Brody D.C. and Jones H.F., Complex extension of quantum mechanics, Phys. Rev. Lett. 89, 2002, 270401, quantph/0208076.
[2] Günther U. and Stefani F., Isospectrality of spherical MHD dynamo operators: pseudoHermiticity and a nogo theorem,
J. Math. Phys. 44, 2003, 3097, mathph/0208012.
Keywords: MHD; dynamo; discrete symmetry; Krein space; Supersymmetry; operator intertwining technique; matrix Riccati equation

Lecture (Conference)
Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" June 2329, 2003, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (Kiev), Ukraine  Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine 50(2004, 780787
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