  Information on quantum systems can be obtained only when they are open (or
opened) in relation to a certain environment. As a matter of fact, realistic
open quantum systems appear in very different shape. We sketch the theoretical
description of open quantum systems by means of a projection operator formalism
elaborated many years ago, and applied by now to the description of different
open quantum systems. The Hamiltonian describing the open quantum system is
non-Hermitian. Most studied are the eigenvalues of the non-Hermitian
Hamiltonian of many-particle systems embedded in one environment. We point to
the unsolved problems of this method when applied to the description of
realistic many-body systems. We then underline the role played by the
eigenfunctions of the non-Hermitian Hamiltonian. Very interesting results
originate from the fluctuations of the eigenfunctions in systems with gain and
loss of excitons. They occur with an efficiency of nearly 100\%. An example is
the photosynthesis.
