Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato's Exceptional Points

For a given operator D(t) of an observable in theoretical parity-time symmetric quantum physics (or for its evolution-generator analogues in the experimental gain-loss classical optics, etc.) the instant tcritical of a spontaneous breakdown of the parity-time alias gain-loss symmetry should be given, in the rigorous language of mathematics, the Kato’s name of an “exceptional point”, tcritical = t(EP). In the majority of conventional applications the exceptional point (EP) values are not real. In our paper, we pay attention to several exactly tractable toy-model evolutions for which at least some of the values of t(EP) become real. These values are interpreted as “instants of a catastrophe”, be it classical or quantum. In the classical optical setting the discrete nature of our toy models might make them amenable to simulations. In the latter context the instant of Big Bang is mentioned as an illustrative sample of possible physical meaning of such an EP catastrophe in quantum cosmology.