402
9 The Unification of Structure and Reaction Frameworks
A framework going along these lines has been formulated in the context of the
continuum shell model [12–14], where the Hilbert space is decomposed following
the Feshbach projection technique. In this model, the Hilbert space is divided into
orthogonal subspaces Q 0 , Q 1 , . . . containing 0, 1, . . . particles in the scattering
continuum, respectively. All basis states have real energies, contrary to the Gamow
shell model. Despite the real-energy character of the basis of the continuum shell
model, unbound states can still be calculated in this framework. Indeed, coupling
to the continuum generates non-Hermitian components, which are added to the
initial Hermitian Hamiltonian describing discrete states of Q 0 subspace in the closed
quantum system approximation. Consequently, an open quantum system description
of states in Q 0 includes couplings to the environment of decay channels through
the energy-dependent and in general non-Hermitian effective Hamiltonian (see Ref.
[15] and references cited therein). The eigenvalues of the effective Hamiltonian are
real at energies below all decay thresholds and complex above the lowest decay
threshold.
In the past, the continuum shell model was applied with schematic Hamiltonians
[13,14]. In such a formulation, the continuum shell model cannot provide a realistic
description of complex structures in atomic nuclei. To remedy this problem, the
shell model embedded in the continuum has been developed [16–21], which derives
from the continuum shell model. In the shell model embedded in the continuum,
the subspace of discrete states is identified with a standard shell model space, i.e.
one utilizes a standard shell model interaction to generate configuration mixing in
Q 0 . Furthermore, a zero-range interaction is used for the effective nucleon–nucleon
interaction pertaining to continuum coupling [15]. This allowed to reproduce
nuclear spectra of light nuclei, which is impossible using a schematic nuclear
interaction [16–23]. Moreover, due to the coupled-channel representation of the
Hamiltonian in the shell model embedded in the continuum, one can conveniently
calculate various cross sections, such as the elastic/inelastic scattering, radiative
capture, and Coulomb dissociation cross sections, using the S-matrix formalism
[16–20]. Up to now, the shell model embedded in the continuum studies has been
performed in a core + valence nucleons approach, but the model can be easily
generalized into an ab initio no-core formulation as well. Attempts in this direction
have been made by including continuum degrees of freedom on top of the no-core
shell model using the resonating group method [24–33]. Another yet unexplored
path would be the formulation of no-core Gamow shell model in the coupledchannel representation (see Sect. 8).
The calculation of reaction cross sections in an ab initio framework has been
developed in the context of the no-core shell model with the resonating group
method [24, 28–31, 33] and, more recently, in the no-core shell model with
continuum [26–29]. In the no-core shell model with the resonating group method,
target and projectile wave functions are calculated in the no-core shell model
framework in a basis of harmonic oscillator states. In the no-core shell model with
continuum, these wave functions are complemented with a continuum part, taking
into account asymptotic degrees of freedom. The Hamiltonian can then be written in
9 The Unification of Structure and Reaction Frameworks
A framework going along these lines has been formulated in the context of the
continuum shell model [12–14], where the Hilbert space is decomposed following
the Feshbach projection technique. In this model, the Hilbert space is divided into
orthogonal subspaces Q 0 , Q 1 , . . . containing 0, 1, . . . particles in the scattering
continuum, respectively. All basis states have real energies, contrary to the Gamow
shell model. Despite the real-energy character of the basis of the continuum shell
model, unbound states can still be calculated in this framework. Indeed, coupling
to the continuum generates non-Hermitian components, which are added to the
initial Hermitian Hamiltonian describing discrete states of Q 0 subspace in the closed
quantum system approximation. Consequently, an open quantum system description
of states in Q 0 includes couplings to the environment of decay channels through
the energy-dependent and in general non-Hermitian effective Hamiltonian (see Ref.
[15] and references cited therein). The eigenvalues of the effective Hamiltonian are
real at energies below all decay thresholds and complex above the lowest decay
threshold.
In the past, the continuum shell model was applied with schematic Hamiltonians
[13,14]. In such a formulation, the continuum shell model cannot provide a realistic
description of complex structures in atomic nuclei. To remedy this problem, the
shell model embedded in the continuum has been developed [16–21], which derives
from the continuum shell model. In the shell model embedded in the continuum,
the subspace of discrete states is identified with a standard shell model space, i.e.
one utilizes a standard shell model interaction to generate configuration mixing in
Q 0 . Furthermore, a zero-range interaction is used for the effective nucleon–nucleon
interaction pertaining to continuum coupling [15]. This allowed to reproduce
nuclear spectra of light nuclei, which is impossible using a schematic nuclear
interaction [16–23]. Moreover, due to the coupled-channel representation of the
Hamiltonian in the shell model embedded in the continuum, one can conveniently
calculate various cross sections, such as the elastic/inelastic scattering, radiative
capture, and Coulomb dissociation cross sections, using the S-matrix formalism
[16–20]. Up to now, the shell model embedded in the continuum studies has been
performed in a core + valence nucleons approach, but the model can be easily
generalized into an ab initio no-core formulation as well. Attempts in this direction
have been made by including continuum degrees of freedom on top of the no-core
shell model using the resonating group method [24–33]. Another yet unexplored
path would be the formulation of no-core Gamow shell model in the coupledchannel representation (see Sect. 8).
The calculation of reaction cross sections in an ab initio framework has been
developed in the context of the no-core shell model with the resonating group
method [24, 28–31, 33] and, more recently, in the no-core shell model with
continuum [26–29]. In the no-core shell model with the resonating group method,
target and projectile wave functions are calculated in the no-core shell model
framework in a basis of harmonic oscillator states. In the no-core shell model with
continuum, these wave functions are complemented with a continuum part, taking
into account asymptotic degrees of freedom. The Hamiltonian can then be written in