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

Abstract:
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, 2] as well as the operator of the spherically symmetric a²-dynamo [3] 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 J-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 constraint by a system of nonlinear matrix differential equations. We analyze the (hidden) symmetries of this equation system, transforming it into a set of constraint and interlinked matrix Riccati equations.

Finally, we test the structure of the spherically symmetric MHD a²-dynamo operator on its compatibility with the considered intertwining ansatz and derive a no-go theorem.

The talk extends the results of [3].

References

1. C. M. Bender,  D. C. Brody, H. F. Jones,  ''Must a hamiltonian be hermitian?'', (2003), hep-th/0303005.
2. A. Mostafazadeh, ''Pseudo-Hermiticity and generalized PT- and CPT-symmetries'',  J. Math. Phys. 44, (2003), 974-989, math-ph/0209018.
3. U. Günther, F. Stefani, ''Isospectrality of spherical MHD dynamo operators: pseudo-Hermiticity and a no-go theorem'', J. Math. Phys. (2003), to appear, math-ph/0208012.