116
3 Berggren Basis and Completeness Relations
the slow fall-off of the Coulomb potential, in 1/r, the matrix elements involving
two scattering states u(k, r) and u(k , r) with k ∼ k diverge as ln(k − k ). In the
absence of contour discretization, this divergence poses no theoretical problem as
it is integrable in the complex k-plane. However, if one discretizes the contour, one
obtains infinite matrix elements on the diagonal of the Hamiltonian matrix, because
one has exactly k = k therein. Consequently, these infinite matrix elements have to
be regularized and a precise scheme to calculate the matrix elements of a Coulomb
potential with the Berggren basis has to be devised.
Let us generate the single-particle basis states using a spherical Woods-Saxon
potential and a Coulomb potential proportional to an error function:
V Coul (Z, r) =
C Coul Z erf(3
√
πr/(4R 0 ))
r
,
(3.88)
defined with the Coulomb constant C Coul and the charge acting on the proton Z. R 0
in Eq. (3.88) is the radius of the used Woods-Saxon potential. The radial function
entering the Coulomb potential V Coul (Z, r) is standard and arises from the use of a
Gaussian charge density for the closed core [42–44]. The coefficient 3
√
π/(4R 0 ) is
chosen in order for V Coul (Z, r = 0) to have the same value as that of the Coulomb
potential for a uniformly charged sphere of radius R 0 [45]. For simplicity, the
Hamiltonian which is used to generate the basis and the Hamiltonian to diagonalize
will have the same finite-range part, so that they differ only through their charge Z c .
Exercise IX
One will show the main features of the eigenvector components issued from
the diagonalization of a one-body Hamiltonian with the Berggren basis in a
numerical example.
Run the one-particle diagonalization code for the case described above, using
the potential depths: (V
(B)
0 , V 0 ) = (70, 65), (75, 70), (75, 80) (all in MeV). Plot
the c(k) components in Berggren space as a function of k, similarly to Fig. 3.3.
Explain differences with the example shown in the text, namely the importance of continuum coupling according to the values of V
(B)
0
and V 0 .
Exercise X
In this numerical example, one will point out the differences of the diagonalization of a one-body Hamiltonian when using a Berggren basis generated either
by a Woods-Saxon potential or by a Pöschl-Teller-Ginocchio potential.
A. Run the one-particle diagonalization code in which a basis generated by
a Pöschl-Teller-Ginocchio potential diagonalizes a Woods-Saxon potential.
Explain why one can only consider the expansion of bound states therein.
3 Berggren Basis and Completeness Relations
the slow fall-off of the Coulomb potential, in 1/r, the matrix elements involving
two scattering states u(k, r) and u(k , r) with k ∼ k diverge as ln(k − k ). In the
absence of contour discretization, this divergence poses no theoretical problem as
it is integrable in the complex k-plane. However, if one discretizes the contour, one
obtains infinite matrix elements on the diagonal of the Hamiltonian matrix, because
one has exactly k = k therein. Consequently, these infinite matrix elements have to
be regularized and a precise scheme to calculate the matrix elements of a Coulomb
potential with the Berggren basis has to be devised.
Let us generate the single-particle basis states using a spherical Woods-Saxon
potential and a Coulomb potential proportional to an error function:
V Coul (Z, r) =
C Coul Z erf(3
√
πr/(4R 0 ))
r
,
(3.88)
defined with the Coulomb constant C Coul and the charge acting on the proton Z. R 0
in Eq. (3.88) is the radius of the used Woods-Saxon potential. The radial function
entering the Coulomb potential V Coul (Z, r) is standard and arises from the use of a
Gaussian charge density for the closed core [42–44]. The coefficient 3
√
π/(4R 0 ) is
chosen in order for V Coul (Z, r = 0) to have the same value as that of the Coulomb
potential for a uniformly charged sphere of radius R 0 [45]. For simplicity, the
Hamiltonian which is used to generate the basis and the Hamiltonian to diagonalize
will have the same finite-range part, so that they differ only through their charge Z c .
Exercise IX
One will show the main features of the eigenvector components issued from
the diagonalization of a one-body Hamiltonian with the Berggren basis in a
numerical example.
Run the one-particle diagonalization code for the case described above, using
the potential depths: (V
(B)
0 , V 0 ) = (70, 65), (75, 70), (75, 80) (all in MeV). Plot
the c(k) components in Berggren space as a function of k, similarly to Fig. 3.3.
Explain differences with the example shown in the text, namely the importance of continuum coupling according to the values of V
(B)
0
and V 0 .
Exercise X
In this numerical example, one will point out the differences of the diagonalization of a one-body Hamiltonian when using a Berggren basis generated either
by a Woods-Saxon potential or by a Pöschl-Teller-Ginocchio potential.
A. Run the one-particle diagonalization code in which a basis generated by
a Pöschl-Teller-Ginocchio potential diagonalizes a Woods-Saxon potential.
Explain why one can only consider the expansion of bound states therein.