6.5 Inclusion of Continuum Couplings in the Pairing Model
285
using the Newton-Raphson method. One will show that one-pair and two-pair
solutions can be used to devise a starting point for the Newton-Raphson method.
A. Assuming that the degeneracy of the single-particle level q is Ω q = 2, show
that Eq. (6.63) has a unique pair energy solution, which is
E 1 = 2 q − 2Gd q .
(6.64)
B. For a higher degeneracy of single-particle states q (Ω q ≥ 4), show that the
analytical solution of Eq. (6.63) for two pairs of particles on the level q is
E 1 = 2 q − G(2d q + 1) + iG
|2d q + 1|
E 2 = 2 q − G(2d q + 1) − iG
|2d q + 1| .
(6.65)
C. Let us now consider the case of three pairs occupying the same level q for
|G| | 1, hence with Ω q ≥ 6. The solution of Eq. (6.63) can be found numerically with the Newton-Raphson method (see Ref. [132] for computational
details).
(i) Explain why a simple mean-field approximation cannot be used.
(ii) Devise a starting point for the Newton-Raphson method using the results
of A and B.
(iii) Explain why the devised starting point leads in practice to a converging
Newton-Raphson process.
This special symmetry of pair energies is broken if the nonresonant continuum
states are included in the basis. Indeed, continuum states are absent in Eq. (6.63)
but become occupied for finite values of the pairing strength G and hence, the
initial symmetry of pair energies is broken in the course of solving the generalized
Richardson equations. The derivation of the starting solution of the NewtonRaphson procedure in the weak coupling limit is detailed in Exercise II.
The generalized seniority ν is not equal to zero for systems with an odd number
of particles and/or broken pairs. Each configuration is defined by the seniorities ν q ,
equal to the number of unpaired particles occupying the level q. Equations (6.64)
and (6.65) are then used to obtain an initial guess for the pair energies and initiate
the iterative procedure.
As a first test of the approximate rational Gaudin model with the continuum, one
will compare it with an exact Gamow shell model diagonalization of the pairing
Hamiltonian (6.40). One discretizes the contour L +
c using the Gauss-Legendre
quadrature method and builds the single-particle spectrum which is used both in the
generalized Richardson equations (6.39) and in the Gamow shell model. Exact solutions of the constant pairing Hamiltonian (6.40) are obtained by diagonalizing the
Hamiltonian matrix using the Jacobi–Davidson method (see Sect. 5.7). Figure 6.15
compares the approximate energies obtained by solving the generalized Richardson
285
using the Newton-Raphson method. One will show that one-pair and two-pair
solutions can be used to devise a starting point for the Newton-Raphson method.
A. Assuming that the degeneracy of the single-particle level q is Ω q = 2, show
that Eq. (6.63) has a unique pair energy solution, which is
E 1 = 2 q − 2Gd q .
(6.64)
B. For a higher degeneracy of single-particle states q (Ω q ≥ 4), show that the
analytical solution of Eq. (6.63) for two pairs of particles on the level q is
E 1 = 2 q − G(2d q + 1) + iG
|2d q + 1|
E 2 = 2 q − G(2d q + 1) − iG
|2d q + 1| .
(6.65)
C. Let us now consider the case of three pairs occupying the same level q for
|G| | 1, hence with Ω q ≥ 6. The solution of Eq. (6.63) can be found numerically with the Newton-Raphson method (see Ref. [132] for computational
details).
(i) Explain why a simple mean-field approximation cannot be used.
(ii) Devise a starting point for the Newton-Raphson method using the results
of A and B.
(iii) Explain why the devised starting point leads in practice to a converging
Newton-Raphson process.
This special symmetry of pair energies is broken if the nonresonant continuum
states are included in the basis. Indeed, continuum states are absent in Eq. (6.63)
but become occupied for finite values of the pairing strength G and hence, the
initial symmetry of pair energies is broken in the course of solving the generalized
Richardson equations. The derivation of the starting solution of the NewtonRaphson procedure in the weak coupling limit is detailed in Exercise II.
The generalized seniority ν is not equal to zero for systems with an odd number
of particles and/or broken pairs. Each configuration is defined by the seniorities ν q ,
equal to the number of unpaired particles occupying the level q. Equations (6.64)
and (6.65) are then used to obtain an initial guess for the pair energies and initiate
the iterative procedure.
As a first test of the approximate rational Gaudin model with the continuum, one
will compare it with an exact Gamow shell model diagonalization of the pairing
Hamiltonian (6.40). One discretizes the contour L +
c using the Gauss-Legendre
quadrature method and builds the single-particle spectrum which is used both in the
generalized Richardson equations (6.39) and in the Gamow shell model. Exact solutions of the constant pairing Hamiltonian (6.40) are obtained by diagonalizing the
Hamiltonian matrix using the Jacobi–Davidson method (see Sect. 5.7). Figure 6.15
compares the approximate energies obtained by solving the generalized Richardson