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 root-vector 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 non-Hermitian as well as for PT-ssymmetric/pseudo-Hermitian quantum mechanical systems. The passage time contraction for non-Hermitian Hamiltonians compared to Hermitian ones is attributed to a distance contraction in projective Hilbert space due to non-unitary 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; PT-symmetry; non-Hermitian operator; quantum brachistochrone; passage time

  • Lecture (Conference)
    6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, 16.-18.07.2007, London, United Kingdom

Permalink: https://www.hzdr.de/publications/Publ-10304
Publ.-Id: 10304