9.3 Computational Methods for Solving the Coupled-Channel Gamow Shell. . .
457
Nevertheless, the considered coupled channels are expected to well reproduce
physical states if all important decay channels are included. Consequently, as this is
the case in practice, one can expect that the eigenergies calculated either from the
diagonalization of ˆ
H with a basis of Slater determinants or with coupled channels
will be close to each another.
u c (r) can be expanded in a coupled-channel Berggren basis as
u c (r) =
n c
a n c u
(n c )
c (r) ,
(9.163)
where the u
(n c )
c (r) functions are eigenstates of ˆ
h = ˆ
T + ˆ
U basis (see Eq. (9.11)) in the
one-nucleon case and of ˆ
H CM (see Eq. (9.90)) in the multi-nucleon (nuclear) case.
The a n c coefficients arise from the diagonalization of ˆ
H with Berggren channel
basis states u
(n c )
c (r). The potentials entering ˆ
H are either finite-ranged or of onebody Coulomb type (see Eqs. (9.23) and (9.121)). Therefore, the matrix elements
associated to the potentials present in ˆ
H can be conveniently calculated with direct
integration using complex scaling (see Sects. 3.3 and 3.4).
Similarly to the matrix problem generated by the Gamow shell model using Slater
determinants, one obtains a complex-symmetric matrix to be diagonalized, whose
eigenstates consist of bound, resonance, and scattering eigenstates. The resonance
eigenstates are embedded among the scattering eigenstates, so one has to use the
overlap method (see Sect. 5.4) to identify them in the coupled-channel Gamow shell
model spectrum of ˆ
H . In order to diagonalize the obtained Hamiltonian matrix,
one uses the Jacobi–Davidson method, which is very successful within the Gamow
shell model (see Sect. 5.7) The diagonalization of the coupled-channel Hamiltonian
matrix is then as convenient as that done in a basis of Slater determinants. The
unification of structure and reaction frameworks is also clear, as Eq. (9.163) can
be used alternatively to expand resonant and scattering wave functions. Energy
spectra calculated with the coupled-channel Gamow shell model will be presented
in Sect. 9.4, with the examples of 7 Li, 7 Be, and 18 Ne.
9.3.3.2 Scattering Solutions of Coupled-Channel Equations
It is convenient to use the Green’s function method to obtain the scattering solution
of Eq. (9.149). Let us begin by introducing a zeroth-order Hamiltonian ˆ
H (0) and its
eigenvector |Ψ (0)
ˆ
H
(0)
= ˆ
t + ˆ
U basis
(nucleon)
= ˆ
T CM + ˆ
U CM
(cluster)
(9.164)
ˆ
H
(0)
|Ψ
(0)
= E |Ψ
(0)
.
(9.165)
More specifically ˆ
H (0) is a matrix with off-diagonal matrix elements equal to
zero (no coupling with other channels), and only one diagonal element is nonzero
457
Nevertheless, the considered coupled channels are expected to well reproduce
physical states if all important decay channels are included. Consequently, as this is
the case in practice, one can expect that the eigenergies calculated either from the
diagonalization of ˆ
H with a basis of Slater determinants or with coupled channels
will be close to each another.
u c (r) can be expanded in a coupled-channel Berggren basis as
u c (r) =
n c
a n c u
(n c )
c (r) ,
(9.163)
where the u
(n c )
c (r) functions are eigenstates of ˆ
h = ˆ
T + ˆ
U basis (see Eq. (9.11)) in the
one-nucleon case and of ˆ
H CM (see Eq. (9.90)) in the multi-nucleon (nuclear) case.
The a n c coefficients arise from the diagonalization of ˆ
H with Berggren channel
basis states u
(n c )
c (r). The potentials entering ˆ
H are either finite-ranged or of onebody Coulomb type (see Eqs. (9.23) and (9.121)). Therefore, the matrix elements
associated to the potentials present in ˆ
H can be conveniently calculated with direct
integration using complex scaling (see Sects. 3.3 and 3.4).
Similarly to the matrix problem generated by the Gamow shell model using Slater
determinants, one obtains a complex-symmetric matrix to be diagonalized, whose
eigenstates consist of bound, resonance, and scattering eigenstates. The resonance
eigenstates are embedded among the scattering eigenstates, so one has to use the
overlap method (see Sect. 5.4) to identify them in the coupled-channel Gamow shell
model spectrum of ˆ
H . In order to diagonalize the obtained Hamiltonian matrix,
one uses the Jacobi–Davidson method, which is very successful within the Gamow
shell model (see Sect. 5.7) The diagonalization of the coupled-channel Hamiltonian
matrix is then as convenient as that done in a basis of Slater determinants. The
unification of structure and reaction frameworks is also clear, as Eq. (9.163) can
be used alternatively to expand resonant and scattering wave functions. Energy
spectra calculated with the coupled-channel Gamow shell model will be presented
in Sect. 9.4, with the examples of 7 Li, 7 Be, and 18 Ne.
9.3.3.2 Scattering Solutions of Coupled-Channel Equations
It is convenient to use the Green’s function method to obtain the scattering solution
of Eq. (9.149). Let us begin by introducing a zeroth-order Hamiltonian ˆ
H (0) and its
eigenvector |Ψ (0)
ˆ
H
(0)
= ˆ
t + ˆ
U basis
(nucleon)
= ˆ
T CM + ˆ
U CM
(cluster)
(9.164)
ˆ
H
(0)
|Ψ
(0)
= E |Ψ
(0)
.
(9.165)
More specifically ˆ
H (0) is a matrix with off-diagonal matrix elements equal to
zero (no coupling with other channels), and only one diagonal element is nonzero