408
9 The Unification of Structure and Reaction Frameworks
and of two diagonal parts in which n ≥ 0 and n ≤ n max , with n = n i =
n f . Explain why it is convenient to have an infinite sum starting from n =
0. Calculate the infinite sum of H if using the completeness properties of the
Berggren basis.
D. Using all previous results, write H if as a function of target states energies,
single-particle energies of Berggren basis states, and Slater determinant expansions of channel states. Conclude that H if can be calculated numerically using
the Gamow shell model formalism.
A convenient way to remove antisymmetry when n > n max is to have matrix
elements αβ|V res − U 0 |γ δ vanished when n α > n max , n β > n max , n γ > n max ,
or n δ > n max . These assumptions are always present in Gamow shell model
calculations as one always uses a finite model space. In the absence of antisymmetry
if n > n max , it is convenient to rewrite the Hamiltonian ˆ
H of Eq. (9.11) introducing
an operator acting only on the target
ˆ
H = ˆ
T + ˆ
U basis + ( ˆ
V res − ˆ
U 0 )
A−1 + [( ˆ
V res − ˆ
U 0 ) − ( ˆ
V res − ˆ
U 0 )
A−1
] ,
(9.12)
where one defines ( ˆ
V res − ˆ
U 0 ) A−1 by its action on the non-antisymmetrized A-body
states |Ψ
J T
T ⊗ |u n mτ
( ˆ
V res − ˆ
U 0 )
A−1 (|Ψ
J T
T ⊗ |u n j mτ = [( ˆ
V res − ˆ
U 0 ) |Ψ
J T
T ] ⊗ |u n j mτ . (9.13)
In this expression, ( ˆ
V res − ˆ
U 0 ) A−1 is the finite-range part of ˆ
V res − ˆ
U 0 restricted to
act on the (A − 1)-body states only. Consequently, one can evaluate matrix elements
of ˆ
T + ˆ
U basis + ( ˆ
V res − U 0 ) A−1 (see Eqs. (9.12), (9.13)) without recurring to the
Slater determinant expansion of Eq. (9.10). Indeed, ˆ
T + ˆ
U basis + ( ˆ
V res − ˆ
U 0 ) A−1 can
be written as a sum of target and projectile Hamiltonians, ˆ
H T + ˆ
H P , with ˆ
H T acting
only on |Ψ
J T
T and ˆ
H P acting only on |u n j τ (see Eq. (9.6))
ˆ
H T = ˆ
T
A−1
+ ( ˆ
U basis )
A−1
+ ( ˆ
V res − ˆ
U 0 )
A−1
(9.14)
ˆ
H P = ˆ
t + ˆ
U basis .
(9.15)
In these expressions, ˆ
T and ˆ
U basis operators have been separated in target (A − 1
superscript) and projectile parts. Eq. (9.10) is calculated in Exercise I. This exercise
is technical, however, and can be omitted in a first reading.
9.1.1.1 Orthogonality of the Composite States with Respect to the Core
States
As the cluster orbital shell model formulation is used in coupled-channel Gamow
shell model, the valence states have to be orthogonal to the core states. For this, one
9 The Unification of Structure and Reaction Frameworks
and of two diagonal parts in which n ≥ 0 and n ≤ n max , with n = n i =
n f . Explain why it is convenient to have an infinite sum starting from n =
0. Calculate the infinite sum of H if using the completeness properties of the
Berggren basis.
D. Using all previous results, write H if as a function of target states energies,
single-particle energies of Berggren basis states, and Slater determinant expansions of channel states. Conclude that H if can be calculated numerically using
the Gamow shell model formalism.
A convenient way to remove antisymmetry when n > n max is to have matrix
elements αβ|V res − U 0 |γ δ vanished when n α > n max , n β > n max , n γ > n max ,
or n δ > n max . These assumptions are always present in Gamow shell model
calculations as one always uses a finite model space. In the absence of antisymmetry
if n > n max , it is convenient to rewrite the Hamiltonian ˆ
H of Eq. (9.11) introducing
an operator acting only on the target
ˆ
H = ˆ
T + ˆ
U basis + ( ˆ
V res − ˆ
U 0 )
A−1 + [( ˆ
V res − ˆ
U 0 ) − ( ˆ
V res − ˆ
U 0 )
A−1
] ,
(9.12)
where one defines ( ˆ
V res − ˆ
U 0 ) A−1 by its action on the non-antisymmetrized A-body
states |Ψ
J T
T ⊗ |u n mτ
( ˆ
V res − ˆ
U 0 )
A−1 (|Ψ
J T
T ⊗ |u n j mτ = [( ˆ
V res − ˆ
U 0 ) |Ψ
J T
T ] ⊗ |u n j mτ . (9.13)
In this expression, ( ˆ
V res − ˆ
U 0 ) A−1 is the finite-range part of ˆ
V res − ˆ
U 0 restricted to
act on the (A − 1)-body states only. Consequently, one can evaluate matrix elements
of ˆ
T + ˆ
U basis + ( ˆ
V res − U 0 ) A−1 (see Eqs. (9.12), (9.13)) without recurring to the
Slater determinant expansion of Eq. (9.10). Indeed, ˆ
T + ˆ
U basis + ( ˆ
V res − ˆ
U 0 ) A−1 can
be written as a sum of target and projectile Hamiltonians, ˆ
H T + ˆ
H P , with ˆ
H T acting
only on |Ψ
J T
T and ˆ
H P acting only on |u n j τ (see Eq. (9.6))
ˆ
H T = ˆ
T
A−1
+ ( ˆ
U basis )
A−1
+ ( ˆ
V res − ˆ
U 0 )
A−1
(9.14)
ˆ
H P = ˆ
t + ˆ
U basis .
(9.15)
In these expressions, ˆ
T and ˆ
U basis operators have been separated in target (A − 1
superscript) and projectile parts. Eq. (9.10) is calculated in Exercise I. This exercise
is technical, however, and can be omitted in a first reading.
9.1.1.1 Orthogonality of the Composite States with Respect to the Core
States
As the cluster orbital shell model formulation is used in coupled-channel Gamow
shell model, the valence states have to be orthogonal to the core states. For this, one