280
6 Physical Applications of the Gamow Shell Model
where N is the total number of resonant and discretized continuum singleparticle states. However, Eq. (6.44) cannot be analytically solved in the general
case, even though it closely resembles Eq. (6.37). Indeed, state-dependent pairing
Hamiltonians are not integrable in general. An exception is the hyperbolic model of
Refs. [120, 131], where the Gaussian weights w q are linear functions of the singleparticle energies q . Consequently, one must either approximately solve Eq. (6.44),
or change the commutation relations of Eq. (6.41) for nonresonant scattering states,
thus breaking the SU(2) commutator algebra, in order to obtain an ansatz for an
exact eigenstate [129, 130].
The new normalized operators ˆ ˜
n q and ˜
b
†
q , ˜
b q must be applied when diagonalizing
the Hamiltonian of Eq. (6.44). Indeed, the contour discretization leads not only to
new normalized operators but also to new normalized Slater determinants, because
the action of ˆ ˜
n q , ˜
b
†
q , and ˜
b q is defined as in the discrete case. An approximate
solution for the generalized rational pairing model (6.44) can be derived by
replacing the Kronecker delta by the Dirac delta in the commutator of Eq. (6.41)
for states in the nonresonant continuum [130]:
b k c , b
†
k
c
= 2δ(k c − k
c )δ cc
Ω k c
4
−
ˆ
n k c
2
.
(6.45)
The pair operators ˜
b
†
q ( ˜
b q ) for bound, resonance and discretized scattering states
satisfy
ˆ ˜
n q , ˜
b
†
q
= 2δ qq ˜
b
†
q
˜
b q , ˜
b
†
q
= 2δ qq
Ω q
4
−
ˆ ˜
n q
2w q
˜
b
†
q , ˜
b
†
q
= 0 .
(6.46)
Let us now derive the eigenvalues of the pairing Hamiltonian (6.44) in this
approximation. For this, let us rewrite the eigenstate K (K = 1, 2, . . . , K max ) as
a product of the pair states, similarly to the discrete case:
|Ψ norm =
N pair
η=1
B
†
η;norm |ν ,
(6.47)
where all considered eigenstates and operators implicitly depend on K. In
Eq. (6.47), the pair operators read
B
†
η;norm = c η G
N pair
q
˜
b
†
q
√
w q
2 q − E η
,
(6.48)
6 Physical Applications of the Gamow Shell Model
where N is the total number of resonant and discretized continuum singleparticle states. However, Eq. (6.44) cannot be analytically solved in the general
case, even though it closely resembles Eq. (6.37). Indeed, state-dependent pairing
Hamiltonians are not integrable in general. An exception is the hyperbolic model of
Refs. [120, 131], where the Gaussian weights w q are linear functions of the singleparticle energies q . Consequently, one must either approximately solve Eq. (6.44),
or change the commutation relations of Eq. (6.41) for nonresonant scattering states,
thus breaking the SU(2) commutator algebra, in order to obtain an ansatz for an
exact eigenstate [129, 130].
The new normalized operators ˆ ˜
n q and ˜
b
†
q , ˜
b q must be applied when diagonalizing
the Hamiltonian of Eq. (6.44). Indeed, the contour discretization leads not only to
new normalized operators but also to new normalized Slater determinants, because
the action of ˆ ˜
n q , ˜
b
†
q , and ˜
b q is defined as in the discrete case. An approximate
solution for the generalized rational pairing model (6.44) can be derived by
replacing the Kronecker delta by the Dirac delta in the commutator of Eq. (6.41)
for states in the nonresonant continuum [130]:
b k c , b
†
k
c
= 2δ(k c − k
c )δ cc
Ω k c
4
−
ˆ
n k c
2
.
(6.45)
The pair operators ˜
b
†
q ( ˜
b q ) for bound, resonance and discretized scattering states
satisfy
ˆ ˜
n q , ˜
b
†
q
= 2δ qq ˜
b
†
q
˜
b q , ˜
b
†
q
= 2δ qq
Ω q
4
−
ˆ ˜
n q
2w q
˜
b
†
q , ˜
b
†
q
= 0 .
(6.46)
Let us now derive the eigenvalues of the pairing Hamiltonian (6.44) in this
approximation. For this, let us rewrite the eigenstate K (K = 1, 2, . . . , K max ) as
a product of the pair states, similarly to the discrete case:
|Ψ norm =
N pair
η=1
B
†
η;norm |ν ,
(6.47)
where all considered eigenstates and operators implicitly depend on K. In
Eq. (6.47), the pair operators read
B
†
η;norm = c η G
N pair
q
˜
b
†
q
√
w q
2 q − E η
,
(6.48)