Projective Hilbert space structures near exceptional points and the quantum brachistochrone
Projective Hilbert space structures near exceptional points and the quantum brachistochrone
Günther, U.; Samsonov, B.
The talk consists of two parts. In the first part, a brief overview of projective Hilbert space structures related to exceptional points (EPs) is presented. The apparent contradiction between operator (matrix) perturbation schemes related to rootvector expansions and expansions in terms of eigenvectors for diagonal spectral decompositions is projectively resolved. In the second part of the talk, the gained insight is used for a geometric analysis of the brachistochrone problem for nonHermitian as well as for PTssymmetric/pseudoHermitian quantum mechanical systems. The passage time contraction for nonHermitian Hamiltonians compared to Hermitian ones is attributed to a distance contraction in projective Hilbert space due to nonunitary evolution. In the limiting case when a parameter dependent Hamiltonian approaches an EP in its spectral decomposition the distance between the coalescing eigenvectors vanishes and with it the passage time of the brachistochrone.
Keywords: Quantum Mechanics; exceptional point; Hilbert space; projective space; PTsymmetry; nonHermitian operator; quantum brachistochrone; passage time

Lecture (Conference)
6th International Workshop on PseudoHermitian Hamiltonians in Quantum Physics, 16.18.07.2007, London, United Kingdom
Permalink: https://www.hzdr.de/publications/Publ10304
Publ.Id: 10304