A non-Hermitian complex symmetric 2x2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended Quantum Mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity
The existence of magnetohydrodynamic mean-field alpha^2-dynamos with spherically symmetric, isotropic helical turbulence function alpha is related to a non-self-adjoint spectral problem for a coupled system of two singular second order ordinary differential equations. We establish global estimates for the eigenvalues of this system in terms of the turbulence function alpha and its derivative alpha. They allow us to formulate an anti-dynamo theorem and a non-oscillation theorem. The conditions of these theorems, which again involve alpha and alpha, must be violated in order to reach supercritical or oscillatory regimes.
The quantum mechanical brachistochrone system with PT-symmetric Hamiltonian is Naimark dilated and reinterpreted as subsystem of a Hermitian system in a higher-dimensional Hilbert space. This opens a way to a direct experimental implementation of the recently hypothesized PT-symmetric ultra-fast brachistochrone regime of [C. M. Bender et al, Phys. Rev. Lett. {bf 98}, 040403 (2007)] in an entangled two-spin system.
The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. Interpreting this specific setup as subsystem of a larger Hermitian system, we find non-unitary operator equivalence classes (conjugacy classes) as natural ingredient which contain at least one Dirac-Hermitian representative. With the help of a geometric analysis the compatibility of the vanishing passage time solution of a PTS brachistochrone with the Anandan-Aharonov lower bound for passage times of Hermitian brachistochrones is demonstrated.
