3.7 Numerical Implementation of the Berggren Completeness Relation
121
Coulomb potential, proportional to 1/r, which was considered in A and B, but
also to the arbitrary Coulomb potential, which becomes proportional to 1/r in
the asymptotic region only.
In Eq. (3.90), one uses in the subtraction and off-diagonal methods the values R
= 15 fm and θ = −3π/4, −π/4, π/4 or 3π/4. Improper integrals of Eq. (3.90) are
indeed guaranteed to converge employing one of these angular values. The Berggren
basis contour used is very similar to that of the cut method, except that k min is
chosen so that F η (k min R) ∼ 10 −5 . The latter condition arises from the very small
values of proton scattering wave functions with k ∼ 0. Indeed, proton scattering
wave functions are very close to regular Coulomb wave functions
√
2/πF η (kr)
when k → 0 (see Eqs. (2.142) and (2.143)). However, complex rotation demands
the use of H
±
η (kr) irregular Coulomb wave functions (see Eq. (3.90)). This leads
to important numerical cancellations, because |H
±
η (kr)| becomes very large when
k → 0, whereas F η (kr) → 0 when k → 0. Consequently, one cannot evaluate
numerically Eq. (3.90) using complex scaling with u(k, r) functions whose k value
is small. On the other hand, the Coulomb matrix element (see Eq. (3.90)) becomes
negligible when k → 0. Due to the extreme smallness of their amplitudes, u(k, r)
functions whose k value is close to zero play virtually no role in the completeness of
Berggren basis. Hence, proton scattering states become important only when k min
become sufficiently large, and the condition above has been shown to mitigate the
numerical instability occurring for k ∼ 0 while yielding precise results.
The precision of the radial wave functions issued from the Hamiltonian diagonalization is evaluated by calculating the root mean square deviation of their real
and imaginary parts with respect to the exact functions, issued from the direct
integration:
rms( =
N
i=1
([u(r i )] − −[u e (r i )])
2
N
(3.97)
rms( =
N
i=1
([u(r i )] − −[u e (r i )])
2
N
.
(3.98)
In these expressions, N is equal to 512, r i = i · (R/N) for 1 ≤ i ≤ N is a set
of uniformly distributed radii in an interval [0 : R], u(r) is the diagonalized wave
function, and u e (r) is the exact wave function.
The root-mean-square deviations (3.97), (3.98) for proton 1s 1/2 , 0d 5/2 , and 0d 3/2
wave functions are illustrated in Fig. 3.4 for all studied cases, that is, the cut method,
the subtraction method and the off-diagonal method. One can see that the cut method
produces very poor results, as it does not even reach the precision acquired with
121
Coulomb potential, proportional to 1/r, which was considered in A and B, but
also to the arbitrary Coulomb potential, which becomes proportional to 1/r in
the asymptotic region only.
In Eq. (3.90), one uses in the subtraction and off-diagonal methods the values R
= 15 fm and θ = −3π/4, −π/4, π/4 or 3π/4. Improper integrals of Eq. (3.90) are
indeed guaranteed to converge employing one of these angular values. The Berggren
basis contour used is very similar to that of the cut method, except that k min is
chosen so that F η (k min R) ∼ 10 −5 . The latter condition arises from the very small
values of proton scattering wave functions with k ∼ 0. Indeed, proton scattering
wave functions are very close to regular Coulomb wave functions
√
2/πF η (kr)
when k → 0 (see Eqs. (2.142) and (2.143)). However, complex rotation demands
the use of H
±
η (kr) irregular Coulomb wave functions (see Eq. (3.90)). This leads
to important numerical cancellations, because |H
±
η (kr)| becomes very large when
k → 0, whereas F η (kr) → 0 when k → 0. Consequently, one cannot evaluate
numerically Eq. (3.90) using complex scaling with u(k, r) functions whose k value
is small. On the other hand, the Coulomb matrix element (see Eq. (3.90)) becomes
negligible when k → 0. Due to the extreme smallness of their amplitudes, u(k, r)
functions whose k value is close to zero play virtually no role in the completeness of
Berggren basis. Hence, proton scattering states become important only when k min
become sufficiently large, and the condition above has been shown to mitigate the
numerical instability occurring for k ∼ 0 while yielding precise results.
The precision of the radial wave functions issued from the Hamiltonian diagonalization is evaluated by calculating the root mean square deviation of their real
and imaginary parts with respect to the exact functions, issued from the direct
integration:
rms( =
N
i=1
([u(r i )] − −[u e (r i )])
2
N
(3.97)
rms( =
N
i=1
([u(r i )] − −[u e (r i )])
2
N
.
(3.98)
In these expressions, N is equal to 512, r i = i · (R/N) for 1 ≤ i ≤ N is a set
of uniformly distributed radii in an interval [0 : R], u(r) is the diagonalized wave
function, and u e (r) is the exact wave function.
The root-mean-square deviations (3.97), (3.98) for proton 1s 1/2 , 0d 5/2 , and 0d 3/2
wave functions are illustrated in Fig. 3.4 for all studied cases, that is, the cut method,
the subtraction method and the off-diagonal method. One can see that the cut method
produces very poor results, as it does not even reach the precision acquired with