Solutions to Exercises
483
where, ∀i ∈ {a, b, c, d}, k
i is a linear momentum of modulus k
i and of angle Ω i ,
and where one has used the equality k
i = k
i . As the k
i linear momenta verify
Eq. (9.96), one has
a k
b | ˆ
V res − ˆ
U 0 |k
c k
d 0 from the results of Sect. 9.2.6.1.
Consequently,
a a m a k
b b m b | ˆ
V res − ˆ
U 0 |k
c c m c k
d d m d B 0. Therefore, the integrand of Eq. (9.106) is negligible when the k
i linear momenta verify
Eq. (9.96).
C. The linear momenta k
a , k
b , k
c , k
d must verify either ∃(i, j) ∈ {a, b, c, d} so
that k
i k max and k
j k max or ∃(i, j) ∈ {a, b, c, d} so that k
i
k max and k
j k max . These two cases have been treated above. Consequently,
a f b | ˆ
V res − ˆ
U 0 |f c f d 0 when Eq. (9.96) is fulfilled by the |f i states. Thus,
using Eqs. (9.104) and (9.105), one obtains that ab| ˆ
V res − ˆ
U 0 |cd 0 for the
real-energy neutron Berggren basis states that verify Eq. (9.96).
Exercise IV.
A. Let us expand H
J A M A
cc
(R CM , R
CM ) with the eigenbasis of ˆ
H CM :
H cc (R CM , R
CM ) =
N CM ,N
CM
H
J A M A
cc
(N CM , N
CM ) U N CM (R CM ) U N
CM
(R
CM ) ,
(9.186)
where
H
J A M A
cc
(N CM , N
CM )
= = ˆ
A {{Ψ
J T
T | ⊗ ⊗N CM L CM J int J P M P |}
J A
M A
| ˆ
H | ˆ
A
{|Ψ
J T
T ⊗ |N
CM L
CM J
int J
P M
P
J A
M A
.
(9.187)
Equation (9.186) can be decomposed in four sums:
H cc (R CM , R
CM )
=
N CM ≤N CMmax
N
CM ≤N CMmax
H
J A M A
cc
(N CM , N
CM ) U N CM (R CM ) U N
CM
(R
CM )
+
N CM ≤N CMmax
N
CM >N CMmax
H
J A M A
cc
(N CM , N
CM ) U N CM (R CM ) U N
CM
(R
CM )
+
N CM >N CMmax
N
CM ≤N CMmax
H
J A M A
cc
(N CM , N
CM ) U N CM (R CM ) U N
CM
(R
CM )
+
N CM >N CMmax
N
CM >N CMmax
H
J A M A
cc
(N CM , N
CM ) U N CM (R CM ) U N
CM
(R
CM ) .
(9.188)