Solutions to Exercises
477
correlations present in the eigenstates of 42 Sc. As the nucleon–nucleon interaction is
typically the strongest when nucleons are coupled to 0 + , it is expected to obtain the
largest discrepancy in the ground state of 42 Sc. Conversely, the eigenstates of highest
spins do not differ significantly in all used calculation schemes. Indeed, one expects
the nucleon–nucleon interaction to be the least binding for these nuclear states.
However, it is difficult to predict quantitatively the energy differences between
exact and approximate calculations.
As a consequence, one has shown that nonresonant target states are necessary
to quantitatively reproduce Gamow shell model eigenvalues when using Gamow
shell model in coupled-channel representation. The importance of nonresonant
target states differs from one eigenstate to another, so that it is difficult to predict
the amount of missing correlations in each eigenstate a priori. One has seen
that it is possible to renormalize the effect of nonresonant targets on energies
phenomenologically via corrective factors in Hamiltonian matrix elements. However, coupled-channel wave functions might change significantly when considering
nonresonant target states, so that other observables as energies could differ.
Solutions to Exercises 1
Exercise I.
A. The Berggren basis expansion of H if (see Eq. (9.10)) is readily obtained from
Eq. (9.6):
H if =
n i n f
u n i (r i ) u n f (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
H | ˆ
A {|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
.
(9.176)
In order to separate H if in several parts according to the values of n i , n f , one
uses the fundamental assumption imposed on H (see Sect. 9.1.1), i.e. that V res
vanishes when n > n max in Eq. (9.12). Therefore, ˆ
H = ˆ
H T + ˆ
H P when acted on
Berggren basis channel states verifying n i > n max or n f > n max . One has also
seen that antisymmetry requirements disappear when n i > n max or n f > n max ,
so that antisymmetrizers ˆ
A can been removed from Eq. (9.176) in this situation.
The demanded expression of H if comes forward by suppressing the ˆ
V res residual
interaction and antisymmetrizers ˆ
A when n i > n max or n f > n max
H if =
n≤nmax
u n (r i ) u n (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f j f τ f |}
J A
M A
| ˆ
H | ˆ
A {|Ψ
J T i
T i
⊗ |u n i j i τ i
J A
M A
+
n i =n f
n i ,n f ≤nmax
u n i (r i ) u n f (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
H | ˆ
A {|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.
477
correlations present in the eigenstates of 42 Sc. As the nucleon–nucleon interaction is
typically the strongest when nucleons are coupled to 0 + , it is expected to obtain the
largest discrepancy in the ground state of 42 Sc. Conversely, the eigenstates of highest
spins do not differ significantly in all used calculation schemes. Indeed, one expects
the nucleon–nucleon interaction to be the least binding for these nuclear states.
However, it is difficult to predict quantitatively the energy differences between
exact and approximate calculations.
As a consequence, one has shown that nonresonant target states are necessary
to quantitatively reproduce Gamow shell model eigenvalues when using Gamow
shell model in coupled-channel representation. The importance of nonresonant
target states differs from one eigenstate to another, so that it is difficult to predict
the amount of missing correlations in each eigenstate a priori. One has seen
that it is possible to renormalize the effect of nonresonant targets on energies
phenomenologically via corrective factors in Hamiltonian matrix elements. However, coupled-channel wave functions might change significantly when considering
nonresonant target states, so that other observables as energies could differ.
Solutions to Exercises 1
Exercise I.
A. The Berggren basis expansion of H if (see Eq. (9.10)) is readily obtained from
Eq. (9.6):
H if =
n i n f
u n i (r i ) u n f (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
H | ˆ
A {|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
.
(9.176)
In order to separate H if in several parts according to the values of n i , n f , one
uses the fundamental assumption imposed on H (see Sect. 9.1.1), i.e. that V res
vanishes when n > n max in Eq. (9.12). Therefore, ˆ
H = ˆ
H T + ˆ
H P when acted on
Berggren basis channel states verifying n i > n max or n f > n max . One has also
seen that antisymmetry requirements disappear when n i > n max or n f > n max ,
so that antisymmetrizers ˆ
A can been removed from Eq. (9.176) in this situation.
The demanded expression of H if comes forward by suppressing the ˆ
V res residual
interaction and antisymmetrizers ˆ
A when n i > n max or n f > n max
H if =
n≤nmax
u n (r i ) u n (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f j f τ f |}
J A
M A
| ˆ
H | ˆ
A {|Ψ
J T i
T i
⊗ |u n i j i τ i
J A
M A
+
n i =n f
n i ,n f ≤nmax
u n i (r i ) u n f (r f ) ˆ
A {{Ψ
J T f
T f
| ⊗ ⊗u n f f j f τ f |}
J A
M A
| ˆ
H | ˆ
A {|Ψ
J T i
T i
⊗ |u n i i j i τ i
J A
M A
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.