8
1 Introduction: From Bound States to the Continuum
decoherence of wave function(s) comes out naturally as a result of the interference
between localized part of the system and its structureless background.
Application of the Gamow shell model for a description of reaction cross sections
requires its reformulation [57–59]. Indeed, any eigenfunction of the Gamow shell
model is a superposition of Slater determinants , and as such the Gamow shell model
is the tool par excellence for structure studies. However, in this representation,
reaction channels, in particular entrance and exit channel functions, cannot be accurately defined. To describe nuclear reactions, one has to express the Gamow shell
model in the representation of coupled channels which, formally, is equivalent to
the Slater determinant representation. An advantage of such a double representation
is that one may always compare the Gamow shell model eigenvalues calculated in
both representations and in this way check that the important reaction channels are
included.
In the coupled-channels representation of Gamow shell model, the nuclear
structure and nuclear reactions become unified because the same Hamiltonian
and the same many-body approach describes both the discrete part of the energy
spectrum and the reaction cross sections at low excitation energies [57–60]. This
provides the unique opportunity to remove uncertainties in the interpretation of
results of indirect experiments where, for example, the measured cross sections
are interpreted by a model to obtain another quantity of interest. Such indirect
measurements include the reactions of transfer, breakup, or knockout, the Coulomb
excitations, and many others. An example of this kind concerns the interpretation
of the ratio of experimental σ exp and theoretical σ th inclusive one-nucleon removal
cross sections for a large number of projectiles which shows a strong dependence
on the asymmetry of the neutron and proton separation energies [61–63].
Different formulations of the Gamow shell model open a possibility for a
vast number of applications in nuclear physics and related fields. In a core-plusvalence particle approximation [64], one may study spectra and reactions in different
mass regions using effective interactions. The no-core formulation of Gamow shell
model [56, 65] opens a possibility to test the realistic two- and three-nucleon
forces on examples of the resonances in light nuclei. Using interchangeably the
representations of Slater determinant or coupled-channels representations of the
Gamow shell model wave functions, formulated in Jacobi coordinates or in cluster
orbital shell model variables, allows to unify many apparently unrelated aspects of
the nuclear structure and reactions in the well-bound states, weakly bound states,
or the unbound states. Those different rigorous formulations of the Gamow shell
model can be a convenient starting point to derive multiple simpler models to study
consequences of the flux conservation (unitarity) at around reaction thresholds. On
the practical level, the Gamow shell model can provide foundation of the modern
nuclear structure and reaction theory, helping to understand the wealth of data on
nuclear levels, moments, collective excitations, various decays, and different lowenergy cross sections [17].
1 Introduction: From Bound States to the Continuum
decoherence of wave function(s) comes out naturally as a result of the interference
between localized part of the system and its structureless background.
Application of the Gamow shell model for a description of reaction cross sections
requires its reformulation [57–59]. Indeed, any eigenfunction of the Gamow shell
model is a superposition of Slater determinants , and as such the Gamow shell model
is the tool par excellence for structure studies. However, in this representation,
reaction channels, in particular entrance and exit channel functions, cannot be accurately defined. To describe nuclear reactions, one has to express the Gamow shell
model in the representation of coupled channels which, formally, is equivalent to
the Slater determinant representation. An advantage of such a double representation
is that one may always compare the Gamow shell model eigenvalues calculated in
both representations and in this way check that the important reaction channels are
included.
In the coupled-channels representation of Gamow shell model, the nuclear
structure and nuclear reactions become unified because the same Hamiltonian
and the same many-body approach describes both the discrete part of the energy
spectrum and the reaction cross sections at low excitation energies [57–60]. This
provides the unique opportunity to remove uncertainties in the interpretation of
results of indirect experiments where, for example, the measured cross sections
are interpreted by a model to obtain another quantity of interest. Such indirect
measurements include the reactions of transfer, breakup, or knockout, the Coulomb
excitations, and many others. An example of this kind concerns the interpretation
of the ratio of experimental σ exp and theoretical σ th inclusive one-nucleon removal
cross sections for a large number of projectiles which shows a strong dependence
on the asymmetry of the neutron and proton separation energies [61–63].
Different formulations of the Gamow shell model open a possibility for a
vast number of applications in nuclear physics and related fields. In a core-plusvalence particle approximation [64], one may study spectra and reactions in different
mass regions using effective interactions. The no-core formulation of Gamow shell
model [56, 65] opens a possibility to test the realistic two- and three-nucleon
forces on examples of the resonances in light nuclei. Using interchangeably the
representations of Slater determinant or coupled-channels representations of the
Gamow shell model wave functions, formulated in Jacobi coordinates or in cluster
orbital shell model variables, allows to unify many apparently unrelated aspects of
the nuclear structure and reactions in the well-bound states, weakly bound states,
or the unbound states. Those different rigorous formulations of the Gamow shell
model can be a convenient starting point to derive multiple simpler models to study
consequences of the flux conservation (unitarity) at around reaction thresholds. On
the practical level, the Gamow shell model can provide foundation of the modern
nuclear structure and reaction theory, helping to understand the wealth of data on
nuclear levels, moments, collective excitations, various decays, and different lowenergy cross sections [17].