120
3 Berggren Basis and Completeness Relations
In Exercise XII, the trapezoidal rule is used for the discretization of the k-contour.
While the trapezoidal rule gives a theoretical ground for the off-diagonal method, it
cannot be applied in practice because it would lead to poor numerical accuracy. In
fact, as seen in Sect. 3.3, the Gauss-Legendre quadrature leads to the most precise
results. For this, it is most precise to take Δk equal to the Gauss-Legendre weight of
the considered linear momentum k of the Berggren basis contour discretized with
Gauss-Legendre quadrature. Note that the off-diagonal method used with GaussLegendre quadrature is empirical and that there is no mathematical demonstration
for this recipe.
Let us consider now the Berggren basis expansion of three proton single-particle
wave functions, namely 1s 1/2 , 0d 5/2 , and 0d 3/2 . The parameters defining the WoodsSaxon potential V WS (r) and the Coulomb potential V Coul (Z, r) (see Eqs. (3.83)
and (3.88)) and Berggren basis contour can be found in Ref. [48]. The basisgenerating Hamiltonian and the Hamiltonian whose matrix is diagonalized differ
only through the value of Z in Eq. (3.88), in which Z (b) = 10 and Z (d) = 8. When
wielding the cut method, the radius R of Eq. (3.90) is R = 75 fm for s 1/2 and d 5/2
partial waves, and R = 35 fm for the d 3/2 partial wave. These values yield the best
precision for the cut method. The Berggren basis contour (see Eq. (3.66)) consists of
three segments in the complex k-plane, delimited by the four points k min = 0 fm −1 ,
k = 0.25-0.1i fm −1 (s 1/2 and d 5/2 partial waves) or k = 0.4-0.39i fm −1 (d 3/2 partial
wave), k = 1 fm −1 , and k max = 4 fm −1 .
Exercise XII
One will demonstrate that the condition (3.96) is exact if the second integral
of Eq. (3.93) is discretized with the trapezoidal rule.
A. Calculate the integral of ln |k − k | for k ∈ [0 : k max ] with the trapezoidal rule,
where k > 0 is fixed. For this, one will separate the [0 : k max ] interval into
two intervals, equal to [0 : k) and (k : k max ]. One will also pose N = k/Δk
and M = (k max − k)/Δk as the respective number of discretized points of
the intervals [0 : k) and (k : k max ], with Δk > 0. The infinite end point
contributions of the trapezoidal rule occurring at k = k in [0 : k) and (k : k max ]
will not be considered.
B. Show that the integral of ln |k − k | for k ∈ [0 : k max ] can be made as precise
as in the regular case of the trapezoidal rule, i.e. up to an error of O(Δk 2 ), by
adding Δk ln[Δk/(2π)] to the integral. Show that the integral of ln |k − k | for
k ∈ [0 : k max ] is the same as the exact integral up to O(Δk 2 ) if one replaces k
by k − Δk/(4π) and k by k + Δk/(4π) if k = k .
C. Show that the replacement of k by k − Δk/(4π) and k by k + Δk/(4π) if
k = k can be also performed to precisely calculate the integrals of functions
of the form f (k, k ) + ln |k − k |, where f (k, k ) is regular and symmetric in
k, k . Conclude that this method can be applied not only to the point-particle
3 Berggren Basis and Completeness Relations
In Exercise XII, the trapezoidal rule is used for the discretization of the k-contour.
While the trapezoidal rule gives a theoretical ground for the off-diagonal method, it
cannot be applied in practice because it would lead to poor numerical accuracy. In
fact, as seen in Sect. 3.3, the Gauss-Legendre quadrature leads to the most precise
results. For this, it is most precise to take Δk equal to the Gauss-Legendre weight of
the considered linear momentum k of the Berggren basis contour discretized with
Gauss-Legendre quadrature. Note that the off-diagonal method used with GaussLegendre quadrature is empirical and that there is no mathematical demonstration
for this recipe.
Let us consider now the Berggren basis expansion of three proton single-particle
wave functions, namely 1s 1/2 , 0d 5/2 , and 0d 3/2 . The parameters defining the WoodsSaxon potential V WS (r) and the Coulomb potential V Coul (Z, r) (see Eqs. (3.83)
and (3.88)) and Berggren basis contour can be found in Ref. [48]. The basisgenerating Hamiltonian and the Hamiltonian whose matrix is diagonalized differ
only through the value of Z in Eq. (3.88), in which Z (b) = 10 and Z (d) = 8. When
wielding the cut method, the radius R of Eq. (3.90) is R = 75 fm for s 1/2 and d 5/2
partial waves, and R = 35 fm for the d 3/2 partial wave. These values yield the best
precision for the cut method. The Berggren basis contour (see Eq. (3.66)) consists of
three segments in the complex k-plane, delimited by the four points k min = 0 fm −1 ,
k = 0.25-0.1i fm −1 (s 1/2 and d 5/2 partial waves) or k = 0.4-0.39i fm −1 (d 3/2 partial
wave), k = 1 fm −1 , and k max = 4 fm −1 .
Exercise XII
One will demonstrate that the condition (3.96) is exact if the second integral
of Eq. (3.93) is discretized with the trapezoidal rule.
A. Calculate the integral of ln |k − k | for k ∈ [0 : k max ] with the trapezoidal rule,
where k > 0 is fixed. For this, one will separate the [0 : k max ] interval into
two intervals, equal to [0 : k) and (k : k max ]. One will also pose N = k/Δk
and M = (k max − k)/Δk as the respective number of discretized points of
the intervals [0 : k) and (k : k max ], with Δk > 0. The infinite end point
contributions of the trapezoidal rule occurring at k = k in [0 : k) and (k : k max ]
will not be considered.
B. Show that the integral of ln |k − k | for k ∈ [0 : k max ] can be made as precise
as in the regular case of the trapezoidal rule, i.e. up to an error of O(Δk 2 ), by
adding Δk ln[Δk/(2π)] to the integral. Show that the integral of ln |k − k | for
k ∈ [0 : k max ] is the same as the exact integral up to O(Δk 2 ) if one replaces k
by k − Δk/(4π) and k by k + Δk/(4π) if k = k .
C. Show that the replacement of k by k − Δk/(4π) and k by k + Δk/(4π) if
k = k can be also performed to precisely calculate the integrals of functions
of the form f (k, k ) + ln |k − k |, where f (k, k ) is regular and symmetric in
k, k . Conclude that this method can be applied not only to the point-particle