  The dynamics of open quantum systems is determined by avoided and true
crossings of eigenvalue trajectories of a non-Hermitian Hamiltonian. The phases
of the eigenfunctions are not rigid so that environmentally induced
spectroscopic redistribution processes may take place and a dynamical phase
transition may occur. Due to the formal equivalence between the quantum
mechanical Schr\"odinger equation and the optical wave equation in PT symmetric
lattices, the dynamics of the system is determined also in this case by avoided
and true crossings of eigenvalue trajectories of the non-Hermitian Hamiltonian.
In contrast to the eigenvalues characterizing an open quantum system, the
eigenvalues describing the PT symmetric optical lattice are real as long as the
influence of the environment (lattice) onto the optical wave equation is small.
In the regime of avoided level crossings, the symmetry is destroyed, the
eigenvalues become complex and a dynamical phase transition occurs similar as
in the open quantum system. The redistribution processes in the regime of
avoided level crossings allow to design systems with desired properties in a
broad parameter range.
