  We consider different properties of small open quantum systems coupled to an
environment and described by a non-Hermitian Hamilton operator. Of special
interest is the non-analytical behavior of the eigenvalues in the vicinity of
singular points, the so-called exceptional points (EPs), at which the
eigenvalues of two states coalesce and the corresponding eigenfunctions are
linearly dependent from one another. The phases of the eigenfunctions are not
rigid in approaching an EP and providing therewith the possibility to put
information from the environment into the system. All characteristic properties
of non-Hermitian quantum systems hold true not only for natural open quantum
systems that suffer loss due to their embedding into the continuum of
scattering wavefunctions. They appear also in systems coupled to different
layers some of which provide gain to the system. Thereby gain and loss,
respectively, may be fixed inside every layer, i.e. characteristic of it.
