1
Introduction: From Bound States
to the Continuum
The paramount complexity of nuclear many-body problem and its great importance
for an understanding of various systems ranging from the subnucleon to astronomical scales makes it to the intellectual challenge of first importance [1]. Complexity
of this problem is the reason why the theory of atomic nucleus does not exist and
one deals with various models, describing selected features of the nuclear manybody problem. As a result, our understanding of the nuclear properties and structure
remains incomplete and in certain aspects incoherent. Let us take an example of
the low-energy excitations in light atomic nuclei, where the states are described
using either the nuclear shell model or the cluster model. Organization of nucleons
according to these two models is quite different. Shell model is based on a concept
of single-particle mean field and associated to it the shell structure. On the other
hand, cluster model builds the wave functions from the correlated substructures,
such as the α-particles. Can such two different pictures of nuclear structure coexist,
and can they be reconciled?
Resonances are one of the most striking phenomena in Nature. They are genuine
intrinsic properties of quantum systems, associated with their natural frequencies,
and describing preferential decays of unbound states. Experimental manifestation
of the resonances is either the sharp peak in the cross section or the exponential
decay of the probability to find the unstable particle. The sharp peaks in the cross
section are characterized by their energy and width, whereas the decay of a particle
is characterized by its energy and lifetime. Resonances should find a satisfactory
formulation in quantum mechanics. However, the standard quantum mechanics
formulated in Hilbert space does not allow the description of state vectors with
exponential growth and exponential decay, such as resonance states, and they are
simply discarded as unphysical. In Hilbert space, the usual procedure to deal with
resonance states is either to extract the trace of resonances from the real-energy
continuum level density or describe the resonances by joining the bound state
solution in the interior region with an asymptotic solution. This does not yield a
© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_1
1
Introduction: From Bound States
to the Continuum
The paramount complexity of nuclear many-body problem and its great importance
for an understanding of various systems ranging from the subnucleon to astronomical scales makes it to the intellectual challenge of first importance [1]. Complexity
of this problem is the reason why the theory of atomic nucleus does not exist and
one deals with various models, describing selected features of the nuclear manybody problem. As a result, our understanding of the nuclear properties and structure
remains incomplete and in certain aspects incoherent. Let us take an example of
the low-energy excitations in light atomic nuclei, where the states are described
using either the nuclear shell model or the cluster model. Organization of nucleons
according to these two models is quite different. Shell model is based on a concept
of single-particle mean field and associated to it the shell structure. On the other
hand, cluster model builds the wave functions from the correlated substructures,
such as the α-particles. Can such two different pictures of nuclear structure coexist,
and can they be reconciled?
Resonances are one of the most striking phenomena in Nature. They are genuine
intrinsic properties of quantum systems, associated with their natural frequencies,
and describing preferential decays of unbound states. Experimental manifestation
of the resonances is either the sharp peak in the cross section or the exponential
decay of the probability to find the unstable particle. The sharp peaks in the cross
section are characterized by their energy and width, whereas the decay of a particle
is characterized by its energy and lifetime. Resonances should find a satisfactory
formulation in quantum mechanics. However, the standard quantum mechanics
formulated in Hilbert space does not allow the description of state vectors with
exponential growth and exponential decay, such as resonance states, and they are
simply discarded as unphysical. In Hilbert space, the usual procedure to deal with
resonance states is either to extract the trace of resonances from the real-energy
continuum level density or describe the resonances by joining the bound state
solution in the interior region with an asymptotic solution. This does not yield a
© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_1
1