  The eigenvalues of a non-Hermitian Hamilton operator are complex and provide
not only the energies but also the lifetimes of the states of the system. They
show a non-analytical behavior at singular (exceptional) points (EPs). The
eigenfunctions 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. When the parameters are tuned to the point of maximum width bifurcation,
the phase rigidity suddenly increases up to its maximum value. This means that
the eigenfunctions become almost orthogonal at this point. This unexpected
result is very robust as shown by numerical results for different classes of
systems. Physically, it causes an irreversible stabilization of the system by
creating local structures that can be described well by a Hermitian Hamilton
operator. Interesting non-trivial features of open quantum systems appear in
the parameter range in which a clustering of EPs causes a dynamical phase
transition.
