302
6 Physical Applications of the Gamow Shell Model
B η , B
†
η
=
N
q
w q Ω q /2 − ˆ ˜
n q
(2 q − E η )(2 q − E η )
B 0 , B
†
η
=
N
q
w q Ω q /2 − ˆ ˜
n q
2 q − E η
B
†
0 , B
†
η
= 0 .
(6.71)
Exercise II.
A. As at most one pair of particles can occupy the level denoted as q, the sum in
Eq. (6.63) disappears. Equation (6.64) is then immediately obtained.
B. Let us define x i = 2 q − E i , where i = 1, 2. Eq. (6.63) then becomes:
(x 2 − x 1 )x 1 − 2G d q x 2 + 2G (d q + 1) x 1 = 0
(x 1 − x 2 )x 2 − 2G d q x 1 + 2G (d q + 1) x 2 = 0.
Let us define S = x 1 + x 2 and D = x 2 − x 1 . The two previous equations
translate into:
− D
2
+ 2GS = 0
SD − 2G(2d q + 1)D = 0.
As G = 0, D cannot be equal to zero, as otherwise one obtains S = 0, and
hence x 1 = x 2 = 0, so that Eq. (6.63) cannot hold. Thus, the two previous
equations imply that:
S = 2G(2d q + 1)
D
2
= 4G
2 (2d q + 1).
The equality d q = ν q /2 − Ω q /4 = −Ω q /4 holds because the number of
unpaired particles ν q is equal to zero. As one has Ω q ≥ 4, 2d q + 1 ≤ −1, so that
D 2 < 0 and one can pose D = 2iG
|2d q + 1|. Consequently, the expression
for x 1 and x 2 comes forward:
x 1 = G(2d q + 1) − iG
|2d q + 1|
x 2 = G(2d q + 1) + iG
|2d q + 1|.
Equation (6.65) is immediately obtained from the expression for x 1 and x 2 .
C. (i) The mean-field approximation of the pairing model has all its energies
equal to 2 q − 2G d q . This leads to infinities in Eq. (6.63), so that a
6 Physical Applications of the Gamow Shell Model
B η , B
†
η
=
N
q
w q Ω q /2 − ˆ ˜
n q
(2 q − E η )(2 q − E η )
B 0 , B
†
η
=
N
q
w q Ω q /2 − ˆ ˜
n q
2 q − E η
B
†
0 , B
†
η
= 0 .
(6.71)
Exercise II.
A. As at most one pair of particles can occupy the level denoted as q, the sum in
Eq. (6.63) disappears. Equation (6.64) is then immediately obtained.
B. Let us define x i = 2 q − E i , where i = 1, 2. Eq. (6.63) then becomes:
(x 2 − x 1 )x 1 − 2G d q x 2 + 2G (d q + 1) x 1 = 0
(x 1 − x 2 )x 2 − 2G d q x 1 + 2G (d q + 1) x 2 = 0.
Let us define S = x 1 + x 2 and D = x 2 − x 1 . The two previous equations
translate into:
− D
2
+ 2GS = 0
SD − 2G(2d q + 1)D = 0.
As G = 0, D cannot be equal to zero, as otherwise one obtains S = 0, and
hence x 1 = x 2 = 0, so that Eq. (6.63) cannot hold. Thus, the two previous
equations imply that:
S = 2G(2d q + 1)
D
2
= 4G
2 (2d q + 1).
The equality d q = ν q /2 − Ω q /4 = −Ω q /4 holds because the number of
unpaired particles ν q is equal to zero. As one has Ω q ≥ 4, 2d q + 1 ≤ −1, so that
D 2 < 0 and one can pose D = 2iG
|2d q + 1|. Consequently, the expression
for x 1 and x 2 comes forward:
x 1 = G(2d q + 1) − iG
|2d q + 1|
x 2 = G(2d q + 1) + iG
|2d q + 1|.
Equation (6.65) is immediately obtained from the expression for x 1 and x 2 .
C. (i) The mean-field approximation of the pairing model has all its energies
equal to 2 q − 2G d q . This leads to infinities in Eq. (6.63), so that a