  In the Feshbach projection operator (FPO) formalism the whole function space
is divided into two subspaces. One of them contains the wave functions
localized in a certain finite region while the continuum of extended scattering
wave functions is involved in the other subspace. The Hamilton operator of the
whole system is Hermitian, that of the localized part is, however,
non-Hermitian. This non-Hermitian Hamilton operator $H_{\rm eff}$ represents
the core of the FPO method in present-day studies. It gives a unified
description of discrete and resonance states. Furthermore, it contains the time
operator. The eigenvalues $z_\lambda$ and eigenfunctions $\phi_\lambda$ of
$H_{\rm eff}$ are an important ingredient of the $S$ matrix. They are energy
dependent. The phases of the $\phi_\lambda$ are, generally, nonrigid. Most
interesting physical effects are caused by the branch points in the complex
plane. On the one hand, they cause the avoided level crossings that appear as
level repulsion or widths bifurcation in approaching the branch points under
different conditions. On the other hand, observable values are usually enhanced
and accelerated in the vicinity of the branch points. In most cases, the theory
is time asymmetric. An exception are the ${\cal PT}$ symmetric bound states in
the continuum appearing in space symmetric systems due to the avoided level
crossing phenomenon in the complex plane. In the paper, the peculiarities of
the FPO method are considered and three typical phenomena are sketched: (i) the
unified description of decay and scattering processes, (ii) the appearance of
bound states in the continuum and (iii) the spectroscopic reordering processes
characteristic of the regime with overlapping resonances.
