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 non-Hermitian complex symmetric 2×2 matrix toy model to study projective Hilbert space structures in the vicinity of exceptional points (EPs). After Puiseux-expanding the bi-orthogonal 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 EP-limit 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 the zero limit of the optimal passage time in non-Hermitian quantum brachistochrone problems is identified as an EP-related artifact.

Keywords: exceptional points; branch points; projective Hilbert space; geometric phase; singularities; PT-symmetric Quantum Mechanics; quantum brachistochrone problem

  • Invited lecture (Conferences)
    Many-body open quantum systems: From atomic nuclei to quantum dots, 14.-18.05.2007, Trento, Italy

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Publ.-Id: 9663