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.
In the Feshbach projection operator formalism, resonance as well as decay phenomena are described by means of the complex eigenvalues and eigenfunctions of the non-Hermitian Hamilton operator $H_{rm eff}$ that appears in an intermediate stage of the formalism. The formalism can be applied for the description of isolated resonances as well as for resonances in the overlapping regime. Time asymmetry is related to the time operator which is a part of $H_{rm eff}$. An expression for the decay rates of resonance states is derived. For isolated resonance states $lambda$, this expression gives the fundamental relation $tau_lambda = hbar / Gamma_lambda$ between life time and width of a resonance state. A similar relation holds for the average values obtained for narrow resonances superposed by a smooth background term. In the cross over between these two cases (regime of overlapping resonances), the decay rate decreases monotonously as a function of increasing time.
Recently, the quantum brachistochrone problem is discussed in the literature by using non-Hermitian Hamilton operators of different type. Here, it is demonstrated that the passage time is tunable in realistic open quantum systems due to the biorthogonality of the eigenfunctions of the non-Hermitian Hamilton operator. As an example, the numerical results obtained by Bulgakov et al. for the transmission through microwave cavities of different shape are analyzed from the point of view of the brachistochrone problem. The passage time is shortened in the crossover from the weak-coupling to the strong-coupling regime where the resonance states overlap and many branch points (exceptional points) in the complex plane exist. The effect can {it not} be described in the framework of standard quantum mechanics with Hermitian Hamilton operator and consideration of $S$ matrix poles.
