Krein space related physics: PT-symmetric Quantum Mechanics, MHD α²-dynamos, planar Couette flows, ...

Operator theoretic structures are discussed which underlie PT-symmetric Quantum Mechanics (PTSQM), the spherically symmetric α²-dynamo of magnetohydrodynamics (MHD) and the plane Couette flow of hydrodynamics (described by the Squire equation). Mathematically, the three types of models are closely related as spectral problems in Krein spaces − Hilbert spaces with an indefinite metric structure.
In contrast to the purely real spectrum of self-adjoint operators in "usual" Hilbert spaces, the spectrum of self-adjoint operators in Krein spaces consists, in general, of two types of spectral sectors: sectors with purely real eigenvalues and other sectors with pairwise complex conjugate eigenvalues. Transitions between different sectors occur at exceptional points of square root branching type. Knowing the boundaries of the sectors in parameter space one would know, e.g., the boundaries of the physical sectors of PTSQM (with exact/unbroken PT-symmetry) or of the oscillatory regimes of α²-dynamos.
The underlying Krein space related structure of the different physical setups indicates a possible fruitful interplay of these models in handling their technical and conceptual aspects on a unified footing. We make some aspects of such an interplay explicit and demonstrate its usefulness on the example of a PT-symmetric interpolation model and the Squire equation of hydrodynamics. This allows us to gain a deeper insight into the specifics of the Herbst limit.

The talk is based on material from math-ph/0501069 (which is to appear in J. Math. Phys.) and extends it.