In this study we are interested in properties of the single-particle or quasiparticle resonances in the nuclear average potential. Quantum mechanics deals with numerous types of quasi-stationary states. These states may be classified by the singularities of S-matrix or by the mechanism of their production. One-particle shape resonances are perhaps the simplest long-lived states. The particle is captured via quantum tunneling effect to the strongly attractive inner region of the potential. In this way, a quasi-stationary state is formed which, again through the tunneling effect, may leave this inner region. Similar in nature to the one-particle shape resonances are the one-particle virtual states. For these states, the confining barrier is absent but the potential exhibits large jump at the potential boundary region which, in turn, causes jump in the particle wavelength. As a result, the quasi-stationary state is formed, which slowly penetrates from inner to outer region of the potential. The one-particle shape resonances and the one-particle virtual states are specifically quantum phenomena which have no counterpart in the classical physics.
Another kind of resonance states, the so-called Feshbach resonances, are formed when incoming particle excites many particles in the parent nucleus and is captured in the intermediate state for which direct decay channels are closed. The decay of this state is then proceeding through the series of de-excitations of a parent system to either initial channel or to the state of a total system having a lower positive energy [52]. Detailed microscopic theory of these resonances have been worked out in the case, when the coupling of inner excitation to decay channels is weak [53,54]. This has lead to the formulation of continuum shell model (CSM) [54] and to the description, e.g., of giant resonances as quasi-bound N-particle states embedded in the continuum [55,56]. The CSM was extended recently to the realistic multi-configurational shell model which describes the coupling between the many body wave functions for bound states and the one-particle scattering continuum. This so-called Shell Model Embedded in the Continuum (SMEC), allows for a simultaneous description not only of the shell model bound states and resonances but also of the radiative capture cross-sections [57,58].
Another type of many-channel quasi-stationary states may result due to the near-threshold singularity [59,60]. This can happen if the overlap of wave function in a given channel with other channel wave functions is small, leading to the effective decrease of the channel-channel coupling and, hence, to the long lifetime. This resonance mechanism is common in the CSM [54,57].
Three-particle, near-threshold long-lived states constitute another class of resonance states which is particularly interesting in the context of drip-line physics. In this class, both near-threshold virtual states (S-matrix poles at negative energies on nonphysical sheets) and resonance states (S-matrix poles in the complex plane) can be formed. Long-lived states of this class are possible if there exists near-threshold bound, virtual or resonance state in the two-particle subsystem. Multiple transitions of particles between these states of two-particle subsystem are leading to the appearance of effective, long-range exchange forces in the three-particle system [61,62,63]. An analogous exchange process is also possible in the four-particle systems [64]. The above recollection of most common resonance phenomena in nuclear physics is by no means exhaustive.