92
3 Berggren Basis and Completeness Relations
As the Dirac delta distribution δ(r − r ) of Eq. (3.42) arises from the limit of
sin(K(r − r ))/K for K → +∞ (see Exercise IV), the functions that can be
expanded with Eq. (3.36) are those that possess a Fourier transform.
3.2.2 Completeness Relation of Coulomb Wave Functions
The demonstration of the completeness of Coulomb wave functions can be done
using the analytical properties of confluent hypergeometric functions. Let us
consider potentials for which ≥ 0 and v c ∈ R (see Eq. (2.5)). One will follow the
method originally presented by Mukunda [13] in an attractive Coulomb potential
and adapted to the repulsive Coulomb potential in Ref. [10]. For clarity, one will
also replace confluent hypergeometric functions by Coulomb wave functions.
Let us define an integral similar to (3.29):
I c (k s , K) =
2
π
K
k s
F ,η (kr)F ,η (kr
) dk ,
(3.37)
where K > k s > 0 and where the integration constant C 0 of Eq. (2.6) is equal to
√
2/π. The introduction of k s is necessary as k = 0 is an essential singularity for
Coulomb wave functions [9].
In order to write Eq. (3.37) as an integral in [−K : K], one has to write F ,η (kr )
as a function of H
±
,η (kr ) [13] (see Eq. (2.33)). Consequently, one has to pay
attention to the cut of Coulomb wave functions on the negative real axis. For this,
one considers that k belongs to the lower half of the complex plane with strictly
negative imaginary part. H
−
,η (kr) and H
+
(−kr) are minimal for r → +∞ (see
Eq. (2.32)). As they are solutions of the same differential equation, they are mutually
proportional, so that one can determine their coefficient of proportionality from their
asymptote for r → +∞ (see Eq. (2.64)):
H
+
(−kr) = H
−
,η (kr) exp(−iππ) exp(−πη) .
(3.38)
A similar equation arises from the power series defining F ,η (kr) (see Eq. (2.37)):
F (−kr) = −F ,η (kr) exp(iππ) exp(πη) .
(3.39)
Hence, Eq. (3.37) reads:
I c (k s , K) =
1
iπ
−k s
−K
F ,η (kr)H
+
,η (kr
) dk +
1
iπ
K
k s
F ,η (kr)H
+
,η (kr
) dk ,
(3.40)
where k is considered as |k| exp(iπ) if k < 0 and where Eqs. (3.38) and (3.39) have
been used. As a consequence, F ,η (kr)H
+
,η (kr ) can be considered as an analytic
3 Berggren Basis and Completeness Relations
As the Dirac delta distribution δ(r − r ) of Eq. (3.42) arises from the limit of
sin(K(r − r ))/K for K → +∞ (see Exercise IV), the functions that can be
expanded with Eq. (3.36) are those that possess a Fourier transform.
3.2.2 Completeness Relation of Coulomb Wave Functions
The demonstration of the completeness of Coulomb wave functions can be done
using the analytical properties of confluent hypergeometric functions. Let us
consider potentials for which ≥ 0 and v c ∈ R (see Eq. (2.5)). One will follow the
method originally presented by Mukunda [13] in an attractive Coulomb potential
and adapted to the repulsive Coulomb potential in Ref. [10]. For clarity, one will
also replace confluent hypergeometric functions by Coulomb wave functions.
Let us define an integral similar to (3.29):
I c (k s , K) =
2
π
K
k s
F ,η (kr)F ,η (kr
) dk ,
(3.37)
where K > k s > 0 and where the integration constant C 0 of Eq. (2.6) is equal to
√
2/π. The introduction of k s is necessary as k = 0 is an essential singularity for
Coulomb wave functions [9].
In order to write Eq. (3.37) as an integral in [−K : K], one has to write F ,η (kr )
as a function of H
±
,η (kr ) [13] (see Eq. (2.33)). Consequently, one has to pay
attention to the cut of Coulomb wave functions on the negative real axis. For this,
one considers that k belongs to the lower half of the complex plane with strictly
negative imaginary part. H
−
,η (kr) and H
+
(−kr) are minimal for r → +∞ (see
Eq. (2.32)). As they are solutions of the same differential equation, they are mutually
proportional, so that one can determine their coefficient of proportionality from their
asymptote for r → +∞ (see Eq. (2.64)):
H
+
(−kr) = H
−
,η (kr) exp(−iππ) exp(−πη) .
(3.38)
A similar equation arises from the power series defining F ,η (kr) (see Eq. (2.37)):
F (−kr) = −F ,η (kr) exp(iππ) exp(πη) .
(3.39)
Hence, Eq. (3.37) reads:
I c (k s , K) =
1
iπ
−k s
−K
F ,η (kr)H
+
,η (kr
) dk +
1
iπ
K
k s
F ,η (kr)H
+
,η (kr
) dk ,
(3.40)
where k is considered as |k| exp(iπ) if k < 0 and where Eqs. (3.38) and (3.39) have
been used. As a consequence, F ,η (kr)H
+
,η (kr ) can be considered as an analytic