  The Hamilton operator of an open quantum system is non-Hermitian. Its
eigenvalues are, generally, complex and provide not only the energies but also
the lifetimes of the states of the system. The states may couple via the common
environment of scattering wavefunctions into which the system is embedded. This
causes an {\it external mixing} (EM) of the states. Mathematically, EM is
related to the existence of singular (the so-called exceptional) points (EPs).
The eigenfunctions of a non-Hermitian operator are biorthogonal, in contrast to
the orthogonal eigenfunctions of a Hermitian operator. A quantitative measure
for the ratio between biorthogonality and orthogonality is the phase rigidity
of the wavefunctions. At and near an EP, the phase rigidity takes its minimum
value. The lifetimes of two nearby eigenstates of a quantum system bifurcate
under the influence of an EP. At the parameter value of maximum width
bifurcation, the phase rigidity approaches the value one, meaning that the two
eigenfunctions become orthogonal. However, the eigenfunctions are externally
mixed at this parameter value. The S-matrix and therewith the cross section do
contain, in the one-channel case, almost no information on the EM of the
states. The situation is completely different in the case with two (or more)
channels where the resonance structure is strongly influenced by the EM of the
states and interesting features of non-Hermitian quantum physics are revealed.
We provide numerical results for two and three nearby eigenstates of a
non-Hermitian Hamilton operator which are embedded in one common continuum and
influenced by two adjoining EPs. The results are discussed. They are of
interest for an experimental test of the non-Hermitian quantum physics as well
as for applications.
