6.5 Inclusion of Continuum Couplings in the Pairing Model
279
Sums over c, c denote summations over different partial waves, from ((, j ) to
(( max , j max ). The energy of a single-particle state c in the nonresonant continuum
reads: c = ¯
h
2 k 2
c /2m, where m is the particle mass, and k c is the associated linear
momentum. The discrete sums run over the real-energy bound single-particle states
and the complex-energy single-particle resonances situated between the contour L +
c
and the real k-axis. The same contour L
+
c((,j ) in the complex k-plane is used for all
partial waves.
For discrete single-particle states (bound states and resonances), the pair creation
(annihilation) operators satisfy the commutator relations in Eq. (6.34). Conversely,
one has for the nonresonant scattering single-particle states [129, 130]:
ˆ
n k c , b
†
k
c
= 2δ(k c − k
c )δ cc b
†
k c
b k c , b
†
k
c
= δ(k c − k
c )δ cc
Ω k c
2
− δ k c k
c
δ cc ˆ
n k c
b
†
k c
, b
†
k
c
= 0 .
(6.41)
The continuum has to be discretized in practical applications. For this, it is
convenient to define new number and pair operators:
ˆ ˜
n q = w q ˆ
n q ; ˜
b
†
q =
√ w q b
†
q = ( ˜
b q )
† ,
(6.42)
where the index q runs over all resonant and discretized scattering states of the
Berggren basis, and w q is the Gaussian weight of the Gauss-Legendre quadrature.
For resonant states, w q = 1.
With this definition, all pair states are normalized to unity and treated in the
same manner independently of their resonant or scattering character [129,130]. The
operators ˆ ˜
n q , ˜
b q , ˜
b
†
q satisfy the same SU(2) commutation relations as the operators
ˆ
n i , b i , b
†
i (see Eq. (6.34)):
ˆ ˜
n q , ˜
b
†
q
= 2δ qq ˜
b
†
q
˜
b q , ˜
b
†
q
= 2δ qq
Ω q
4
−
ˆ ˜
n q
2
˜
b
†
q , ˜
b
†
q
= 0 .
(6.43)
The Hamiltonian of the generalized rational Gaudin model (6.40) can then be
written in terms of the operators ˆ ˜
n q , ˜
b q , ˜
b
†
q :
ˆ
H =
N
q
q ˆ ˜
n q +
N
q,q
G qq ˜
b
†
q ˜
b q ; G qq =
√
w q
√ w q G ,
(6.44)
279
Sums over c, c denote summations over different partial waves, from ((, j ) to
(( max , j max ). The energy of a single-particle state c in the nonresonant continuum
reads: c = ¯
h
2 k 2
c /2m, where m is the particle mass, and k c is the associated linear
momentum. The discrete sums run over the real-energy bound single-particle states
and the complex-energy single-particle resonances situated between the contour L +
c
and the real k-axis. The same contour L
+
c((,j ) in the complex k-plane is used for all
partial waves.
For discrete single-particle states (bound states and resonances), the pair creation
(annihilation) operators satisfy the commutator relations in Eq. (6.34). Conversely,
one has for the nonresonant scattering single-particle states [129, 130]:
ˆ
n k c , b
†
k
c
= 2δ(k c − k
c )δ cc b
†
k c
b k c , b
†
k
c
= δ(k c − k
c )δ cc
Ω k c
2
− δ k c k
c
δ cc ˆ
n k c
b
†
k c
, b
†
k
c
= 0 .
(6.41)
The continuum has to be discretized in practical applications. For this, it is
convenient to define new number and pair operators:
ˆ ˜
n q = w q ˆ
n q ; ˜
b
†
q =
√ w q b
†
q = ( ˜
b q )
† ,
(6.42)
where the index q runs over all resonant and discretized scattering states of the
Berggren basis, and w q is the Gaussian weight of the Gauss-Legendre quadrature.
For resonant states, w q = 1.
With this definition, all pair states are normalized to unity and treated in the
same manner independently of their resonant or scattering character [129,130]. The
operators ˆ ˜
n q , ˜
b q , ˜
b
†
q satisfy the same SU(2) commutation relations as the operators
ˆ
n i , b i , b
†
i (see Eq. (6.34)):
ˆ ˜
n q , ˜
b
†
q
= 2δ qq ˜
b
†
q
˜
b q , ˜
b
†
q
= 2δ qq
Ω q
4
−
ˆ ˜
n q
2
˜
b
†
q , ˜
b
†
q
= 0 .
(6.43)
The Hamiltonian of the generalized rational Gaudin model (6.40) can then be
written in terms of the operators ˆ ˜
n q , ˜
b q , ˜
b
†
q :
ˆ
H =
N
q
q ˆ ˜
n q +
N
q,q
G qq ˜
b
†
q ˜
b q ; G qq =
√
w q
√ w q G ,
(6.44)