On projective Hilbert space structures at exceptional points
On projective Hilbert space structures at exceptional points
Günther, U.; Rotter, I.; Samsonov, B.
We consider a nonHermitian complex symmetric 2×2 matrix toy model to study projective Hilbert space structures in the vicinity of exceptional points (EPs). After Puiseuxexpanding the biorthogonal eigenvectors of a diagonalizable matrix in terms of the root vectors at the EP we resolve the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EPlimit 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, EPrelated aspects of PTsymmetrically extended Quantum Mechanics are discussed and the zero limit of the optimal passage time in nonHermitian quantum brachistochrone problems is identified as an EPrelated artifact.
Keywords: exceptional points; branch points; projective Hilbert space; geometric phase; singularities; PTsymmetric Quantum Mechanics; quantum brachistochrone problem

Invited lecture (Conferences)
Manybody open quantum systems: From atomic nuclei to quantum dots, 14.18.05.2007, Trento, Italy
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Publ.Id: 9663