In part I, the formalism for the description of open quantum systems (that are embedded into a common well-defined environment) by means of a non-Hermitian Hamilton operator $ch$ is sketched. Eigenvalues and eigenfunctions are parametrically controlled. Using a 2$times$2 model, we study the eigenfunctions of $ch$ at and near to the singular exceptional points (EPs) at which two eigenvalues coalesce and the corresponding eigenfunctions differ from one another by only a phase. In part II, we provide the results of an analytical study for the eigenvalues of three crossing states. These crossing points are of measure zero. Then we show numerical results for the influence of a nearby (third) state onto an EP. Since the wavefunctions of the two crossing states are mixed in a finite parameter range around an EP, three states of a physical system will never cross in one point. Instead, the wavefunctions of all three states are mixed in a finite parameter range in which the ranges of the influence of different EPs overlap. We may relate these results to dynamical phase transitions observed recently in different experimental studies. The states on both sides of the phase transition are non-analytically connected.
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