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6 Physical Applications of the Gamow Shell Model
6.5.1 Numerical Solution of the Generalized Richardson Equations
Equation (6.39) must be solved numerically due to its non-linear character. For this,
a starting point is firstly devised in the weak coupling limit (G → 0), which is then
evolved iteratively by solving the generalized Richardson equations for increasing
values of G. The solution for pair energies for a given G is updated with the NewtonRaphson method using the solution of the previous step as the new starting point
[134]. However, divergencies occur if at a given pairing constant G, two or more
pair energies are equal to twice a single-particle energy. These divergencies exist
only in numerical calculations, as infinities cancel exactly in Eq. (6.39). Hence,
along with the Newton-Raphson procedure, a numerical method suppressing these
numerical divergencies has to be applied as well.
The initial guess for a solution in the limit G → 0 is determined by solving the
generalized Richardson equations. The expression for pair energies E i in this limit
reads
lim
G→0
E i = 2 q
(6.62)
with i = 1, · · · , N pair and q = 1, · · · , N .
Analytical determination of the pair energies is more difficult if many pairs
occupy the same single-particle level q. For N pair pairs occupying the same singleparticle state of energy q , the starting pair energies E i are obtained by solving the
set of N pair coupled equations:
1 −
2Gd q
2 q − E i
+ 2G
N pair
j =i
1
E i − E j
= 0
i = 1, · · · , N pair .
(6.63)
Notice that the nonresonant continuum states in the weak coupling limit |G| | 1
are not occupied and, hence, the corresponding terms in generalized Richardson
equations are absent. Eq. (6.63) cannot be solved analytically in the general case.
However, pair energies can be given in closed form in the one-pair and two-pair
cases (see Exercise II). If in the limit G → 0 two pairs occupy the same singleparticle state q, then their energies are complex conjugate. For the spectrum of real
single-particle energies, this symmetry of the pair energies at G → 0 is preserved
for any value of G in an iterative procedure of solving equations of the generalized
Richardson pairing model.
Exercise II
One will devise analytical solutions for pair energies in Eq. (6.63) in the one-pair
and two-pair cases. Pair energies in Eq. (6.63) are no longer analytical when one
has more than three pairs, so that pair energy solutions must be found numerically
6 Physical Applications of the Gamow Shell Model
6.5.1 Numerical Solution of the Generalized Richardson Equations
Equation (6.39) must be solved numerically due to its non-linear character. For this,
a starting point is firstly devised in the weak coupling limit (G → 0), which is then
evolved iteratively by solving the generalized Richardson equations for increasing
values of G. The solution for pair energies for a given G is updated with the NewtonRaphson method using the solution of the previous step as the new starting point
[134]. However, divergencies occur if at a given pairing constant G, two or more
pair energies are equal to twice a single-particle energy. These divergencies exist
only in numerical calculations, as infinities cancel exactly in Eq. (6.39). Hence,
along with the Newton-Raphson procedure, a numerical method suppressing these
numerical divergencies has to be applied as well.
The initial guess for a solution in the limit G → 0 is determined by solving the
generalized Richardson equations. The expression for pair energies E i in this limit
reads
lim
G→0
E i = 2 q
(6.62)
with i = 1, · · · , N pair and q = 1, · · · , N .
Analytical determination of the pair energies is more difficult if many pairs
occupy the same single-particle level q. For N pair pairs occupying the same singleparticle state of energy q , the starting pair energies E i are obtained by solving the
set of N pair coupled equations:
1 −
2Gd q
2 q − E i
+ 2G
N pair
j =i
1
E i − E j
= 0
i = 1, · · · , N pair .
(6.63)
Notice that the nonresonant continuum states in the weak coupling limit |G| | 1
are not occupied and, hence, the corresponding terms in generalized Richardson
equations are absent. Eq. (6.63) cannot be solved analytically in the general case.
However, pair energies can be given in closed form in the one-pair and two-pair
cases (see Exercise II). If in the limit G → 0 two pairs occupy the same singleparticle state q, then their energies are complex conjugate. For the spectrum of real
single-particle energies, this symmetry of the pair energies at G → 0 is preserved
for any value of G in an iterative procedure of solving equations of the generalized
Richardson pairing model.
Exercise II
One will devise analytical solutions for pair energies in Eq. (6.63) in the one-pair
and two-pair cases. Pair energies in Eq. (6.63) are no longer analytical when one
has more than three pairs, so that pair energy solutions must be found numerically