Projective Hilbert space structures at exceptional points and their extension to line bundles over spectral Riemann surfaces

A non-Hermitian complex symmetric 2×2 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. The EP-related quantum brachistochrone problem of PT-symmetrically extended Quantum Mechanics is discussed. Finally, aspects of a smooth globally defined line bundle structure over spectral Riemann surfaces are sketched.

The talk is partially based on arXiv:0704.1291v2 [math-ph] and includes newer findings going beyond this e-print.