Nearby states in non-Hermitian quantum systems II: Three and more states
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Using the formalism for the description of open quantum systems by means of a non-Hermitian Hamilton operator, we study the occurrence of dynamical phase transitions as well as their relation to the singular exceptional points (EPs). First, 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.