Krein space related physics (I+II)

Physical models with anti-linear symmetries can often be described by differential operators self-adjoint in suitably chosen Krein spaces.
In the first lecture, we briefly comment on the spectral properties of some specific operators self-adjoint in Krein spaces and related effects:
1) the operator of the Bender-Boettcher model of PT Quantum Mechanics and its historical background in the 2D Ising model, the Lee-Yang model, Yang-Lee edge singularities, conformal field theory and the theory of phase transitions
2) the operator of the hydrodynamic Squire equation, its scaling behavior and mapping to the operator of the Bender-Boettcher model of PT Quantum Mechanics,
3) the cusp-type spectral properties in the vicinity of third-order exceptional points (algebraic branch points),
4) the unfolding of higher-order exceptional points of the spectrum of Hamiltonians in PT-symmetric Bose-Hubbard models described with the help of Puiseux series expansions and Newton polygon techniques.
The second lecture comprises the following three main subjects:
1) the eigenvector isotropization in the vicinity of exceptional points (algebraic branch points) of the spectra of parameter dependent operators and matrices, and underlying Lie group structures of such isotropizations. For simple toy model matrix Hamiltonians we demonstrate the structural analogy to Lorentz boost transformations of chiral spinors and the naturally emerging SO(N,C) group structure of these boosts. It will be shown that normalization divergencies of the eigenvectors can be simply resolved via projective extensions and the use of different affine charts of the corresponding projective spaces. For gauged PT-symmetric systems we demonstrate the occurrence of Lie triple systems (ternary Lie algebraic structures) as well as of a hidden Clifford algebra.
2) We briefly explain the basic features of the so-called quantum brachistochrone problem for Hamiltonians self-adjoint in Hilbert spaces and in Krein spaces and demonstrate their interrelation geometrically in terms of contraction-dilation maps in projective Hilbert spaces and via positive operator-valued measures (POVMs) and Naimark dilation.
3) Finally, we briefly comment on recent experimental findings in PT-symmetric (i.e. Krein-space related) physics, especially in optical wave-guide systems and microwave cavities.