314
7 Wave Functions in the Vicinity of Particle-Emission Threshold
states which would resemble the degrees of freedom of the cluster model. Extensive
studies in the shell model embedded in the continuum [24, 25] demonstrated that
the troubling coexistence of cluster model and shell model wave functions in the
description of low-energy excitations in the light nuclei, can be overcome in the
open quantum system formulation of the shell model. These studies allowed to
comprehend the phenomenological assertion of Ikeda et al. [23] and formulate
its generalization, namely that the coupling to a nearby particle or cluster-decay
channel induces particle or cluster correlations in shell model wave functions which
are the imprint of this channel and do not originate from any particular property
of the nuclear forces or any special symmetry of the nuclear many-body problem
[24, 25]. The specific aspects of this generic phenomenon, depending, for example,
on the interactions, are related to the energy and nature of various dissociation
thresholds, and the stability of correlated multi-particle structures in the final state
after the decay.
This generic phenomenon in open quantum systems should be seen also in atomic
physics, molecular physics, or in nanoscience. However, it is in nuclear physics
that those clustering effects are particularly interesting. This is due to the isospin
structure of strong interactions which leads to a large number of different bound
and unbound multi-nucleon systems and hence, to a rich spectrum of different
particle or cluster emission thresholds. This in turn leads to a truly spectacular
richness of correlations in near-threshold many-body states. Universality of the
clustering phenomenon originates from basic properties of the scattering matrix in a
multichannel system [26]. The decay threshold is a branching point of the scattering
matrix. For energies below the lowest particle-decay threshold at energy E 1 , one
finds the analytic phase with a single solution which is regular in the entire space.
Above E 1 and before the next decay channel opens at energy E 2 , the new analytic
phase corresponds to two regular solutions of the scattering problem, etc. Hence,
one obtains a set of analytic phases, each one with a different number of regular
scattering solutions. These phases are separated by decay thresholds (branching
points) which are the non-analytic points of the scattering matrix.
Such a network is illustrated schematically in Fig. 7.1. For energies below the first
decay threshold E12 C→3α , the regular phase consists of elastic scattering solutions.
For E > E12 C→3α and below the second threshold at energy E12 C→ 11 B+p , the
regular phase contains two solutions coupled by unitarity, and the story repeats at
higher excitation energies as more and more channels open up. The structure of
any eigenfunction in this open quantum system depends on the total energy of the
system which in turn determines the environment of decay channels accessible for
couplings. With increasing excitation energy, the open quantum system network
of couplings becomes more and more complex. The flux conservation (unitarity)
implies that the mixing of shell model eigenfunctions changes if a new channel
becomes available. The role of unitarity can also be seen on the example of
spectroscopic factors [12] and radial overlap integrals [27] for near-threshold
eigenfunctions (see a discussion in Sect. 7.4.2.
Microscopic description of states close to the branching point needs minimal
assumptions about degrees of freedom involved but requires the unitary description
7 Wave Functions in the Vicinity of Particle-Emission Threshold
states which would resemble the degrees of freedom of the cluster model. Extensive
studies in the shell model embedded in the continuum [24, 25] demonstrated that
the troubling coexistence of cluster model and shell model wave functions in the
description of low-energy excitations in the light nuclei, can be overcome in the
open quantum system formulation of the shell model. These studies allowed to
comprehend the phenomenological assertion of Ikeda et al. [23] and formulate
its generalization, namely that the coupling to a nearby particle or cluster-decay
channel induces particle or cluster correlations in shell model wave functions which
are the imprint of this channel and do not originate from any particular property
of the nuclear forces or any special symmetry of the nuclear many-body problem
[24, 25]. The specific aspects of this generic phenomenon, depending, for example,
on the interactions, are related to the energy and nature of various dissociation
thresholds, and the stability of correlated multi-particle structures in the final state
after the decay.
This generic phenomenon in open quantum systems should be seen also in atomic
physics, molecular physics, or in nanoscience. However, it is in nuclear physics
that those clustering effects are particularly interesting. This is due to the isospin
structure of strong interactions which leads to a large number of different bound
and unbound multi-nucleon systems and hence, to a rich spectrum of different
particle or cluster emission thresholds. This in turn leads to a truly spectacular
richness of correlations in near-threshold many-body states. Universality of the
clustering phenomenon originates from basic properties of the scattering matrix in a
multichannel system [26]. The decay threshold is a branching point of the scattering
matrix. For energies below the lowest particle-decay threshold at energy E 1 , one
finds the analytic phase with a single solution which is regular in the entire space.
Above E 1 and before the next decay channel opens at energy E 2 , the new analytic
phase corresponds to two regular solutions of the scattering problem, etc. Hence,
one obtains a set of analytic phases, each one with a different number of regular
scattering solutions. These phases are separated by decay thresholds (branching
points) which are the non-analytic points of the scattering matrix.
Such a network is illustrated schematically in Fig. 7.1. For energies below the first
decay threshold E12 C→3α , the regular phase consists of elastic scattering solutions.
For E > E12 C→3α and below the second threshold at energy E12 C→ 11 B+p , the
regular phase contains two solutions coupled by unitarity, and the story repeats at
higher excitation energies as more and more channels open up. The structure of
any eigenfunction in this open quantum system depends on the total energy of the
system which in turn determines the environment of decay channels accessible for
couplings. With increasing excitation energy, the open quantum system network
of couplings becomes more and more complex. The flux conservation (unitarity)
implies that the mixing of shell model eigenfunctions changes if a new channel
becomes available. The role of unitarity can also be seen on the example of
spectroscopic factors [12] and radial overlap integrals [27] for near-threshold
eigenfunctions (see a discussion in Sect. 7.4.2.
Microscopic description of states close to the branching point needs minimal
assumptions about degrees of freedom involved but requires the unitary description