Mathematical and Physical Meaning of the Crossings of Energy Levels in {\mathscr {PT}}-Symmetric Systems

Unavoided crossings of the energy levels due to a variation of a real parameter are studied. It is found that after the quantum system in question passes through one of its energy-crossing points alias Kato’s exceptional points (EP), its physical interpretation may dramatically change even when the crossing energies themselves do not complexify. The anomalous physical phase-transition mechanism of the change is revealed, attributed to the EP-related mathematics and illustrated via several exactly solvable matrix toy models.