Intertwiners of pseudo-Hermitian 2×2-block-operator matrices and a no-go theorem for isospectral MHD dynamo operators

Pseudo-Hermiticity 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 CPT-symmetric quantum mechanical systems [1] as well as the operator of the spherically symmetric α²-dynamo [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 η-pseudo-Hermitian 2×2-block-operator matrices with second-order differential operators as matrix elements. The intertwiners are assumed as first-order 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 α²-dynamo operator on its compatibility with the considered intertwining ansatz and derive a no-go theorem.

[1] Bender C.M. and Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 1998, 5243, physics/9712001; Znojil M., PT-symmetric harmonic oscillators, Phys. Lett. A259, 1999, 220, quant-ph/9905020; Bender C.M., Brody D.C. and Jones H.F., Complex extension of quantum mechanics, Phys. Rev. Lett. 89, 2002, 270401, quant-ph/0208076.

[2] Günther U. and Stefani F., Isospectrality of spherical MHD dynamo operators: pseudo-Hermiticity and a no-go theorem,
J. Math. Phys. 44, 2003, 3097, math-ph/0208012.