4
1 Introduction: From Bound States to the Continuum
neutrons
protons
N-2,Z
N-2,Z
N-2
Z+2
N-2
Z+1
N-2
Z+1
N,Z
N,Z+1
N,Z+2
N-2
Z+2
N
Z
E*
Fig. 1.1 (Color online) Illustration of transition to an open quantum system regime of nuclear
states at higher excitation energies above the lowest particle emission threshold, and in the vicinity
of nucleon driplines. In the open quantum system regime, many-body states in neighboring nuclei
form the domains of correlated states interconnected via the coupling to decay channels
the decay thresholds, and the double-poles (exceptional points) of the scattering
matrix, which are essential ingredients of the configuration mixing, are all neglected
in the traditional shell model. What can be said about the structure of many-body
states in the narrow range of energies around the reaction threshold? Are those
properties independent of any particular realization of the Hamiltonian? Is there
a connection between the branch point singularity at the particle emission threshold
and the appearance of cluster states?
Incompleteness of the shell model description of atomic nucleus has been
noticed very early. For instance, Wigner [25] explained the universal properties of
reaction cross sections at the particle-emission threshold(s) by changing boundary
conditions. Similarly, the change in boundary conditions at the nuclear surface
due to Coulomb wave function distortion in the external region explained relative
displacement of states in the mirror nuclei [26–28]. Fano noticed [29] that the
ordinary perturbation theory is inadequate for the description of resonances because
the continuum states of different configurations coincide in energy exactly. It was
therefore clear that one needs a conceptual revolution to resolve various inconsistencies of the shell model picture of the atomic nucleus. In this context, application
of time-dependent Schrödinger equation for the description of resonances is not an
alternative due to the huge difference of time scales involved in nuclear resonances,
which ranges from ∼10 −22 s for an average passage time of nucleon in a nucleus to
1 Introduction: From Bound States to the Continuum
neutrons
protons
N-2,Z
N-2,Z
N-2
Z+2
N-2
Z+1
N-2
Z+1
N,Z
N,Z+1
N,Z+2
N-2
Z+2
N
Z
E*
Fig. 1.1 (Color online) Illustration of transition to an open quantum system regime of nuclear
states at higher excitation energies above the lowest particle emission threshold, and in the vicinity
of nucleon driplines. In the open quantum system regime, many-body states in neighboring nuclei
form the domains of correlated states interconnected via the coupling to decay channels
the decay thresholds, and the double-poles (exceptional points) of the scattering
matrix, which are essential ingredients of the configuration mixing, are all neglected
in the traditional shell model. What can be said about the structure of many-body
states in the narrow range of energies around the reaction threshold? Are those
properties independent of any particular realization of the Hamiltonian? Is there
a connection between the branch point singularity at the particle emission threshold
and the appearance of cluster states?
Incompleteness of the shell model description of atomic nucleus has been
noticed very early. For instance, Wigner [25] explained the universal properties of
reaction cross sections at the particle-emission threshold(s) by changing boundary
conditions. Similarly, the change in boundary conditions at the nuclear surface
due to Coulomb wave function distortion in the external region explained relative
displacement of states in the mirror nuclei [26–28]. Fano noticed [29] that the
ordinary perturbation theory is inadequate for the description of resonances because
the continuum states of different configurations coincide in energy exactly. It was
therefore clear that one needs a conceptual revolution to resolve various inconsistencies of the shell model picture of the atomic nucleus. In this context, application
of time-dependent Schrödinger equation for the description of resonances is not an
alternative due to the huge difference of time scales involved in nuclear resonances,
which ranges from ∼10 −22 s for an average passage time of nucleon in a nucleus to