278
6 Physical Applications of the Gamow Shell Model
where one has used the operators ˆ
n a , b a , b
†
a . Richardson could derive the exact
solution of the pairing Hamiltonian (see Eq. 6.37) with a discrete set of bound
single-particle levels [117,118]. It was later demonstrated that the Richardson model
can be solved by expressing the pairing Hamiltonian as a linear combination of
integrals of motion [123].
The eigenvalue of the pairing Hamiltonian (6.37) for a given configuration of
unpaired particles ν, can be written as:
˜
E
(K)
=
N pair
i=1
E
(K)
i
+
N
a=1
a ν a
K = 0, 1, . . . , K max ,
(6.38)
where index K enumerates the eigenstates of increasing excitation energy and
K max + 1 is the total number of eigenstates. ˜
E (K) is complex in general. Thus,
R( ˜
E (K) ) = E
(K) is the energy of the K th eigenstate, while −2I ( ˜
E (K) ) = Γ
(K) is
the corresponding width.
The pair energies E i
(K) in Eq. (6.38) are solutions of N pair non-linear coupled
equations:
1 − 2G
N
a
d a
2 a − E
(K)
i
+ 2G
N pair
j =i
1
E
(K)
i
− E
(K)
j
= 0
K = 0, 1, . . . , K max ,
(6.39)
where d a = ν a /2 − Ω a /4.
Several authors attempted to derive an analytical solution to the pairing model
including the continuum. Hasegawa and Kaneko studied effects of single-particle
resonances on pairing correlations [124]. Id Betan attempted to solve Richardson
equations with the real-energy continuum [125]. However, while an approximate
solution of Richardson equations involving the continuum could be derived, no
exact solution of the pairing problem was provided. Obviously, it is possible to
diagonalize the pairing Hamiltonian including the continuum numerically using the
Gamow shell model [28, 109, 126, 127]. However, one can only consider a small
number of valence nucleons in this approach due to computer limitations.
One will formulate the rational Gaudin model in the presence of the continuum
of single-particle states using the Berggren single-particle ensemble [128]. In this
representation, the pairing Hamiltonian reads [129, 130]
ˆ
H =
i∈b,r
i ˆ
n i +
c
L
+
c
k c ˆ
n k c dk c
+ G
i,i ∈b,r
b
†
i b i + G
c,c
L
+
c
b
†
k c
b k
c dk c dk
c
+ G
(i∈b,r),c
L
+
c
b
†
k c
b i + b
†
i b k c
dk c .
(6.40)
6 Physical Applications of the Gamow Shell Model
where one has used the operators ˆ
n a , b a , b
†
a . Richardson could derive the exact
solution of the pairing Hamiltonian (see Eq. 6.37) with a discrete set of bound
single-particle levels [117,118]. It was later demonstrated that the Richardson model
can be solved by expressing the pairing Hamiltonian as a linear combination of
integrals of motion [123].
The eigenvalue of the pairing Hamiltonian (6.37) for a given configuration of
unpaired particles ν, can be written as:
˜
E
(K)
=
N pair
i=1
E
(K)
i
+
N
a=1
a ν a
K = 0, 1, . . . , K max ,
(6.38)
where index K enumerates the eigenstates of increasing excitation energy and
K max + 1 is the total number of eigenstates. ˜
E (K) is complex in general. Thus,
R( ˜
E (K) ) = E
(K) is the energy of the K th eigenstate, while −2I ( ˜
E (K) ) = Γ
(K) is
the corresponding width.
The pair energies E i
(K) in Eq. (6.38) are solutions of N pair non-linear coupled
equations:
1 − 2G
N
a
d a
2 a − E
(K)
i
+ 2G
N pair
j =i
1
E
(K)
i
− E
(K)
j
= 0
K = 0, 1, . . . , K max ,
(6.39)
where d a = ν a /2 − Ω a /4.
Several authors attempted to derive an analytical solution to the pairing model
including the continuum. Hasegawa and Kaneko studied effects of single-particle
resonances on pairing correlations [124]. Id Betan attempted to solve Richardson
equations with the real-energy continuum [125]. However, while an approximate
solution of Richardson equations involving the continuum could be derived, no
exact solution of the pairing problem was provided. Obviously, it is possible to
diagonalize the pairing Hamiltonian including the continuum numerically using the
Gamow shell model [28, 109, 126, 127]. However, one can only consider a small
number of valence nucleons in this approach due to computer limitations.
One will formulate the rational Gaudin model in the presence of the continuum
of single-particle states using the Berggren single-particle ensemble [128]. In this
representation, the pairing Hamiltonian reads [129, 130]
ˆ
H =
i∈b,r
i ˆ
n i +
c
L
+
c
k c ˆ
n k c dk c
+ G
i,i ∈b,r
b
†
i b i + G
c,c
L
+
c
b
†
k c
b k
c dk c dk
c
+ G
(i∈b,r),c
L
+
c
b
†
k c
b i + b
†
i b k c
dk c .
(6.40)