9.4 Calculation of Spectra and Reaction Cross Sections in the Coupled-. . .
459
i.e. Ψ rest (z) → 0 if z → +∞, where z = r + (R − r)e iθ , R is a radius
outside the nuclear zone, and 0 ≤ θ ≤ π/2 is being properly chosen. This
implies that all channel wave functions in |Ψ rest have an outgoing wave function
asymptote.
Having calculated |Ψ rest in a Berggren basis, its calculation in coordinate space
is straightforward. In cases when the equivalent potential method is numerically stable, it has been checked numerically that both Green function and direct integration
methods provide the same solution for |Ψ
J A
M A
. Equation (9.173) also leads to an
additional numerical advantage when many energies have to be considered. Indeed,
the spectrum of ˆ
H − E is the same for all energies. Consequently, it is sufficient
to calculate a convenient representation of ˆ
H only once, which can then be reused
to solve the linear system of Eq. (9.173) at different energies. Indeed, contrary to
the Gamow shell model, where the ˆ
H matrix is sparse, the ˆ
H matrix in the Gamow
coupled-channel formalism is dense, so that solving many linear systems with the
initial ˆ
H matrix is time-consuming. In fact, the typical numerical cost of solving
a dense linear system is about N 3 /3 per energy, where N is the dimension of the
matrix, with, e.g. the standard lower–upper decomposition of the matrix of ˆ
H [68].
Conversely, if the ˆ
H matrix has a diagonal or tridiagonal form, one has to solve
diagonal or tridiagonal linear systems. The numerical cost associated to this method
is about 2N 2 per energy, which is much faster. Consequently, before solving the
linear systems of Eq. (9.173), the matrix ˆ
H will be firstly transformed into diagonal
or tridiagonal form.
9.4
Calculation of Spectra and Reaction Cross Sections
in the Coupled-Channel Gamow Shell Model
One will present in this section few illustrative applications of the coupled-channel
Gamow shell model in physically relevant problems. The chosen examples mainly
include proton scattering and the proton or neutron radiative capture reactions
of interest for understanding the stellar nucleosynthesis. Deuteron capture will
also be presented as an example of scattering reaction involving many-nucleon
projectile.
As seen in Sect. 9.3.3.1, the coupled-channel Gamow shell model can be used to
calculate the bound and resonance eigenstates of the Hamiltonian. Therefore, one
will also present spectra resulting from the diagonalization of the Hamiltonian using
either a basis of coupled channels or basis of Slater determinants (see Chap. 5).
Both approaches will be shown to be numerically equivalent. Indeed, as one sees
in Sect. 9.3.3.1, the bases of Slater determinants and coupled channels span the
same many-body space of bound, resonance, and scattering states of complex
energies.
459
i.e. Ψ rest (z) → 0 if z → +∞, where z = r + (R − r)e iθ , R is a radius
outside the nuclear zone, and 0 ≤ θ ≤ π/2 is being properly chosen. This
implies that all channel wave functions in |Ψ rest have an outgoing wave function
asymptote.
Having calculated |Ψ rest in a Berggren basis, its calculation in coordinate space
is straightforward. In cases when the equivalent potential method is numerically stable, it has been checked numerically that both Green function and direct integration
methods provide the same solution for |Ψ
J A
M A
. Equation (9.173) also leads to an
additional numerical advantage when many energies have to be considered. Indeed,
the spectrum of ˆ
H − E is the same for all energies. Consequently, it is sufficient
to calculate a convenient representation of ˆ
H only once, which can then be reused
to solve the linear system of Eq. (9.173) at different energies. Indeed, contrary to
the Gamow shell model, where the ˆ
H matrix is sparse, the ˆ
H matrix in the Gamow
coupled-channel formalism is dense, so that solving many linear systems with the
initial ˆ
H matrix is time-consuming. In fact, the typical numerical cost of solving
a dense linear system is about N 3 /3 per energy, where N is the dimension of the
matrix, with, e.g. the standard lower–upper decomposition of the matrix of ˆ
H [68].
Conversely, if the ˆ
H matrix has a diagonal or tridiagonal form, one has to solve
diagonal or tridiagonal linear systems. The numerical cost associated to this method
is about 2N 2 per energy, which is much faster. Consequently, before solving the
linear systems of Eq. (9.173), the matrix ˆ
H will be firstly transformed into diagonal
or tridiagonal form.
9.4
Calculation of Spectra and Reaction Cross Sections
in the Coupled-Channel Gamow Shell Model
One will present in this section few illustrative applications of the coupled-channel
Gamow shell model in physically relevant problems. The chosen examples mainly
include proton scattering and the proton or neutron radiative capture reactions
of interest for understanding the stellar nucleosynthesis. Deuteron capture will
also be presented as an example of scattering reaction involving many-nucleon
projectile.
As seen in Sect. 9.3.3.1, the coupled-channel Gamow shell model can be used to
calculate the bound and resonance eigenstates of the Hamiltonian. Therefore, one
will also present spectra resulting from the diagonalization of the Hamiltonian using
either a basis of coupled channels or basis of Slater determinants (see Chap. 5).
Both approaches will be shown to be numerically equivalent. Indeed, as one sees
in Sect. 9.3.3.1, the bases of Slater determinants and coupled channels span the
same many-body space of bound, resonance, and scattering states of complex
energies.