378
8 No-Core Gamow Shell Model
oscillator basis and hence can be separated in center-of-mass and relative coordinates. Continuum coupling is included because of the presence of coupled-channel
basis states, so that one can expand resonance states. However, in this case, the
resolution of continuum in practical applications depends on the choice of reaction
channels.
The Lawson method can be applied in the no-core Gamow shell model as long as
the Gamow shell model space is sufficiently large to generate a many-body ground
state wave function separable in intrinsic and center-of-mass parts. As bound manybody states are well localized, it is possible to project their center-of-mass part on a
harmonic oscillator center-of-mass 0s state, similarly to no-core shell model using
a basis of harmonic oscillator states. In fact, the use of harmonic oscillator and
Berggren bases is equivalent for that matter in the domain of well-bound manybody states. However, because of the very large model spaces involved in no-core
Gamow shell model, it is hardly possible to apply the Lawson method beyond 4 He.
Consequently, it is more convenient to rely on the generalized variational principle
(see Sect. 5.1.4), with which no projection on center-of-mass harmonic oscillator
ground state is needed. In other words, one does not apply the Lawson method, but
uses the fact that the eigenenergy is stationary when enlarging the Gamow shell
model space. The eigenenergy then converges to the exact nuclear binding energy at
the limit of infinite Gamow shell model space because of the intrinsic character of
the Hamiltonian (see Sect. 5.1.4). The generalized variational principle can be also
applied to unbound many-body states, contrary to the Lawson method, where the
projection on a harmonic oscillator center-of-mass 0s state automatically binds the
many-nucleon system. This will be the method of choice for the next applications,
concerning the resonance ground states of 5 He and 4 n.
8.2.2 5 He Unbound Nucleus
The unbound character of 5 He makes its description in a no-core picture challenging. Indeed, both its ground and first excited states are many-body resonances
with sizable widths. The complex-energy formulation of the no-core Gamow shell
model using the Berggren ensemble is then suitable for the calculation of 5 He
many-body states. Indeed, in the Gamow shell model formalism, the energy and
width of 5 He will be obtained from the eigenvalues of the complex-symmetric
Hamiltonian matrix. Contrary to the previous applications, where one used a realenergy basis, one will employ a basis made of complex-energy basis states, which
also includes the 0p 3/2 resonance. The one-body states consist of the bound 0s 1/2
neutron state with an energy −23.290 MeV, the bound 0s 1/2 proton state with an
energy −23.999 MeV and the 0p 3/2 resonance with a real part of energy 1.193 MeV
and a width 1267 keV. The position of this resonance in the complex-energy plane
is: k = (0.277, −0.068) fm −1 . The single-particle basis for protons and neutrons is
generated by a Gamow Hartree–Fock calculation using the N 3 LO V low-k interaction
with Λ = 1.9 fm −1 . The p 3/2 contour is complex and encompasses the 0p 3/2
8 No-Core Gamow Shell Model
oscillator basis and hence can be separated in center-of-mass and relative coordinates. Continuum coupling is included because of the presence of coupled-channel
basis states, so that one can expand resonance states. However, in this case, the
resolution of continuum in practical applications depends on the choice of reaction
channels.
The Lawson method can be applied in the no-core Gamow shell model as long as
the Gamow shell model space is sufficiently large to generate a many-body ground
state wave function separable in intrinsic and center-of-mass parts. As bound manybody states are well localized, it is possible to project their center-of-mass part on a
harmonic oscillator center-of-mass 0s state, similarly to no-core shell model using
a basis of harmonic oscillator states. In fact, the use of harmonic oscillator and
Berggren bases is equivalent for that matter in the domain of well-bound manybody states. However, because of the very large model spaces involved in no-core
Gamow shell model, it is hardly possible to apply the Lawson method beyond 4 He.
Consequently, it is more convenient to rely on the generalized variational principle
(see Sect. 5.1.4), with which no projection on center-of-mass harmonic oscillator
ground state is needed. In other words, one does not apply the Lawson method, but
uses the fact that the eigenenergy is stationary when enlarging the Gamow shell
model space. The eigenenergy then converges to the exact nuclear binding energy at
the limit of infinite Gamow shell model space because of the intrinsic character of
the Hamiltonian (see Sect. 5.1.4). The generalized variational principle can be also
applied to unbound many-body states, contrary to the Lawson method, where the
projection on a harmonic oscillator center-of-mass 0s state automatically binds the
many-nucleon system. This will be the method of choice for the next applications,
concerning the resonance ground states of 5 He and 4 n.
8.2.2 5 He Unbound Nucleus
The unbound character of 5 He makes its description in a no-core picture challenging. Indeed, both its ground and first excited states are many-body resonances
with sizable widths. The complex-energy formulation of the no-core Gamow shell
model using the Berggren ensemble is then suitable for the calculation of 5 He
many-body states. Indeed, in the Gamow shell model formalism, the energy and
width of 5 He will be obtained from the eigenvalues of the complex-symmetric
Hamiltonian matrix. Contrary to the previous applications, where one used a realenergy basis, one will employ a basis made of complex-energy basis states, which
also includes the 0p 3/2 resonance. The one-body states consist of the bound 0s 1/2
neutron state with an energy −23.290 MeV, the bound 0s 1/2 proton state with an
energy −23.999 MeV and the 0p 3/2 resonance with a real part of energy 1.193 MeV
and a width 1267 keV. The position of this resonance in the complex-energy plane
is: k = (0.277, −0.068) fm −1 . The single-particle basis for protons and neutrons is
generated by a Gamow Hartree–Fock calculation using the N 3 LO V low-k interaction
with Λ = 1.9 fm −1 . The p 3/2 contour is complex and encompasses the 0p 3/2