A powerful method for the description of open quantum systems is the Feshbach projection operator (FPO) technique. In this formalism, the whole function space is divided into two subspaces that are coupled with one another. One of the subspaces 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 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 λ and eigenfunctions φ λ of H eff are an important ingredient of the S matrix. They are energy dependent. The phases of the φ λ 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 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.

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In this paper we consider the evolution of an open quantum mechanical system which together with its environment forms a closed system. The numerical calculations are performed for the nuclear system at low as well as at high level density. In each case, the relevant modes are discussed and compared to the results of the standard methods developed for their description. The influence of the respective remaining modes is compared with existing experimental data. The transition from the resonance reaction mechanism at low level density to the direct reaction mechanism at high level density takes place at a stochasticity threshold. The many-body properties are conserved, at high level density, in long-lived traps, but the spectroscopic information is lost. The evolution to the thermal equilibrium takes place via the formation of quantum chaos in accordance with the second law of thermodynamics. The evolution is accompanied, in the open system, by the formation of a new order with less degrees of freedom. These modes are far from thermal equilibrium. They screen the long-lived modes which are near to thermal equilibrium.

This Report on Progress explores recent advances in our theoretical and experimental understanding of the physics of open quantum systems (OQSs). The study of such systems represents a core problem in modern physics that has evolved to assume an unprecedented interdisciplinary character. OQSs consist of some localized, microscopic, region that is coupled to an external environment by means of an appropriate interaction. Examples of such systems may be found in numerous areas of physics, including atomic and nuclear physics, photonics, biophysics, and mesoscopic physics. It is the latter area that provides the main focus of this review, an emphasis that is driven by the capacity that exists to subject mesoscopic devices to unprecedented control. We thus provide a detailed discussion of the behavior of mesoscopic devices (and other OQSs) in terms of the projection-operator formalism, according to which the system under study is considered to be comprised of a localized region (Q), embedded into a well-defined environment (P ) of scattering wavefunctions (with Q + P = 1). The Q subspace must be treated using the concepts of nonHermitian physics, and of particular interest here is: the capacity of the environment to mediate a coupling between the different states of Q; the role played by the presence of exceptional points (EPs) in the spectra of OQSs; the influence of EPs on the rigidity of the wavefunction phases, and; the ability of EPs to initiate a dynamical phase transition (DPT). EPs are singular points in the continuum, at which two resonance states coalesce, that is where they exhibit a non-avoided