The role of exceptional points in quantum systems

In the present paper, first the mathematical basic properties of the exceptional points are discussed. Then, their role in the description of real physical quantum systems is considered. Most interesting value is the phase rigidity of the eigenfunctions which varies between 1 (for distant non-overlapping states) and 0 (at the exceptional point where the resonance states completely overlap). This variation allows the system to incorporate environmentally induced effects. In the very neighborhood of an exceptional point, the system can be described well by a conventional nonlinear Schr\"odinger equation. In the regime of overlapping resonances, a dynamical phase transition takes place to which all states of the system contribute: a few short-lived resonance states are aligned to the scattering states of the environment by trapping the other states. The trapped resonance states show chaotic features. Due to the alignment of a few states with the states of the environment, observable values may be enhanced. The dynamical phase transition allows us to understand some experimental results which remained puzzling in the framework of conventional Hermitian quantum physics. The effects caused by the exceptional points in physical systems allow us to manipulate them for many different applications.