Confluences of exceptional points and a systematic classification of quantum catastrophes

Specific quantum phase transitions of our interest are assumed associated with the fall of a closed, unitary quantum system into its exceptional-point (EP) singularity. The physical realization of such a "quantum catastrophe" (connected, typically, with an instantaneous loss of the diagonalizability of the corresponding parameter-dependent Hamiltonian $H(g)$) depends, naturally, on the formal mathematical characteristics of the EP, i.e., in essence, on its so called algebraic multiplicity $N$ and geometric multiplicity $K$. In our paper we assume that both of them are finite, and we illustrate and discuss, using several solvable toy models, some of the most elementary mechanisms of the EP-merger realization of the process of the transition $g \to g^{(EP)}$.