456
9 The Unification of Structure and Reaction Frameworks
renormalized using the C
(0)
b and C
(+)
b integration constants of the previous iteration
(see Sect. 9.3.2.1), so that w(r) is both solution of the local equivalent coupledchannel Schrödinger equation and of the nonlocal initial Schrödinger equation at
convergence.
9.3.3 Representation of Coupled-Channel Equations
in the Berggren Basis
The methods presented in the previous sections are iterative and hence can present
instabilities in cases when channel coupling is too strong. It is then more convenient
to solve the coupled-channel equations using the Berggren basis. For this, one starts
from the A-body scattering state of energy E, which is a solution of the Schrödinger
equation of Eq. (9.1).
In the following, we aim at calculating the u c (r) channel functions of the A-body
state that are decomposed in the basis of reaction channel states using the Berggren
basis representation (see Eqs. (9.19) and (9.112)). Due to the antisymmetry of the
target–projectile composite, the channel functions u c (r) are not orthogonal. In order
to consider a matrix representation of channels, one uses the method described in
Sect. 9.3.1. The u c (r) channel functions are indeed recovered from w c (r) channel
functions at the end of the calculation using ˆ
N −
1
2 as well (see Sect. 9.3.1). In
order to simplify the notation, one will implicitly consider that the c channels
are the orthogonalized channels associated to w c (r) functions, even though they
are in principle linear combinations of the initial channels defining u c (r) channel
functions.
One will now present how to calculate bound, resonance, and scattering states
in a Berggren basis representation. Due to their different asymptotic behaviors,
different numerical methods are used for resonant and scattering states, namely
diagonalization and linear system solution, respectively. They will be detailed in
the next two sections.
9.3.3.1 Bound and Resonance Solutions of Coupled-Channel Equations
One will firstly deal with the diagonalization of ˆ
H (see Eq. (9.1)) in the basis of
channel Berggren states. This diagonalization provides the bound and resonance
states of ˆ
H . In the case of an infinite model space, diagonalizing ˆ
H in a basis
of Slater determinants (see Chap. 5) or with coupled channels are equivalent
approaches. As both bases are complete at the theoretical level, diagonalizing
ˆ
H in a basis of Slater determinants (see Sect. 5) or with coupled-channel basis
should provide with the same bound and resonance eigenstates of ˆ
H . However,
model spaces have to be truncated, and model space truncation done at the level of
channels is not equivalent to that done at the level of configurations (see Chap. 5).
Consequently, the eigenstates of ˆ
H obtained in a basis of Slater determinants or
from the solution of coupled-channel equations will be different in practice.
9 The Unification of Structure and Reaction Frameworks
renormalized using the C
(0)
b and C
(+)
b integration constants of the previous iteration
(see Sect. 9.3.2.1), so that w(r) is both solution of the local equivalent coupledchannel Schrödinger equation and of the nonlocal initial Schrödinger equation at
convergence.
9.3.3 Representation of Coupled-Channel Equations
in the Berggren Basis
The methods presented in the previous sections are iterative and hence can present
instabilities in cases when channel coupling is too strong. It is then more convenient
to solve the coupled-channel equations using the Berggren basis. For this, one starts
from the A-body scattering state of energy E, which is a solution of the Schrödinger
equation of Eq. (9.1).
In the following, we aim at calculating the u c (r) channel functions of the A-body
state that are decomposed in the basis of reaction channel states using the Berggren
basis representation (see Eqs. (9.19) and (9.112)). Due to the antisymmetry of the
target–projectile composite, the channel functions u c (r) are not orthogonal. In order
to consider a matrix representation of channels, one uses the method described in
Sect. 9.3.1. The u c (r) channel functions are indeed recovered from w c (r) channel
functions at the end of the calculation using ˆ
N −
1
2 as well (see Sect. 9.3.1). In
order to simplify the notation, one will implicitly consider that the c channels
are the orthogonalized channels associated to w c (r) functions, even though they
are in principle linear combinations of the initial channels defining u c (r) channel
functions.
One will now present how to calculate bound, resonance, and scattering states
in a Berggren basis representation. Due to their different asymptotic behaviors,
different numerical methods are used for resonant and scattering states, namely
diagonalization and linear system solution, respectively. They will be detailed in
the next two sections.
9.3.3.1 Bound and Resonance Solutions of Coupled-Channel Equations
One will firstly deal with the diagonalization of ˆ
H (see Eq. (9.1)) in the basis of
channel Berggren states. This diagonalization provides the bound and resonance
states of ˆ
H . In the case of an infinite model space, diagonalizing ˆ
H in a basis
of Slater determinants (see Chap. 5) or with coupled channels are equivalent
approaches. As both bases are complete at the theoretical level, diagonalizing
ˆ
H in a basis of Slater determinants (see Sect. 5) or with coupled-channel basis
should provide with the same bound and resonance eigenstates of ˆ
H . However,
model spaces have to be truncated, and model space truncation done at the level of
channels is not equivalent to that done at the level of configurations (see Chap. 5).
Consequently, the eigenstates of ˆ
H obtained in a basis of Slater determinants or
from the solution of coupled-channel equations will be different in practice.