Projective Hilbert space structures at exceptional points

Abstract

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.

20 pages; discussion extended, refs added; bug corrected

References in corpus (10)

• Faster than Hermitian Quantum Mechanics
• Coupling of eigenvalues of complex matrices at diabolic and exceptional points
• Rabi Oscillations at Exceptional Points in Microwave Billiards
• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator
• Phase rigidity and avoided level crossings in the complex energy plane
• Wavefunction statistics in open chaotic billiards
• Why dynamos are prone to reversals
• A return to observability near exceptional points in a schematic PT-symmetric model
• Shot noise and transport in small quantum cavities with large openings
• Third order spectral branch points in Krein space related setups: PT-symmetric matrix toy model, MHD alpha^2-dynamo, and extended Squire equation

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