2.6 Analytical Properties of the Wave Functions
57
The asymptotes of u ± (k, r) and u (s) ± (k, r) can be related one to another using
Eq. (2.169) and properties of H
±
,η (kr) (see Sect. 2.3):
u
(s) ± (k, r) = u
± (k, r) exp [∓i(−η ln(2kr) − + σ (η))] + O(r
−1 ) ,
(2.175)
as u (s) ± (k, r) = exp(±ikr), u ± (k, r) = H
±
,η (kr) (see Eq. (2.138)), and r ≥ R (s) .
Consequently, by calculating the Jost functions J + (k)
(s) and J + (k) at r = R (s)
from Eq. (2.169), and using Eq. (2.175) and the fact that u(k, r) and u (s) (k, r) are
proportional to each other, imply that J + (k)
(s) and J + (k) are mutually proportional
up to a small term equal to O(R (s) −1 ). Equation (2.7) implies as well that:
C
+ (s) e
ikR (s) + C
− (s) e
−ikR (s) = A(C
+ H
+
,η (kR
(s) ) + C
− H
−
,η (kR
(s) )) ,
(2.176)
where A is the coefficient of proportionality between u(k, r) and u (s) (k, r).
Consequently, as R (s) → +∞, Eq. (2.176) becomes:
C
+ (s) e
ikR (s) ± C
− (s) e
−ikR (s) = A C
+ exp
iΣ((, k, R
(s)
1 + O
R
(s) −1
± A C
− exp
−iΣ((, k, R
(s)
1 + O
R
(s) −1
,
where Σ((, k, R (s) ) = kR (s) − η ln(2kR (s) ) − π/2 + σ (η) and where one has used
the continuity of u (k, r) and u (s) (k, r) in r = R (s) .
Scattering wave functions u(k, r) and u (s) (k, r) have to be normalized so that
2πC
+ C
−
= 2πC
+ (s) C
− (s) = 1 .
As will be shown in Sect. 3.1, the latter condition is equivalent to the Dirac delta
normalization. Equation (2.177) then implies that:
A = 1 + O
R
(s) −1
.
(2.177)
Therefore, if wave functions are normalized with a Dirac delta, then u (s) (k, r) →
u(k, r) when R (s) → +∞. An immediate consequence is that C 0
(s)
→ C 0 when
R (s) → +∞ (see Eq. (2.6)).
One will now reconsider the limit k → 0 using a screened potential, that is, with
Eq. (2.2) reducing to u (k, r) + k 2 u(k, r) = 0 for r > R (s) . It is only necessary
to consider v c ≥ 0, as the singularity induced by attractive Coulomb potentials
therein is not as strong as in the repulsive case (see Sect. 3.2.3). For this, let us
introduce the u
(s)
c (k, r) functions, which are normalized regular Coulomb wave
57
The asymptotes of u ± (k, r) and u (s) ± (k, r) can be related one to another using
Eq. (2.169) and properties of H
±
,η (kr) (see Sect. 2.3):
u
(s) ± (k, r) = u
± (k, r) exp [∓i(−η ln(2kr) − + σ (η))] + O(r
−1 ) ,
(2.175)
as u (s) ± (k, r) = exp(±ikr), u ± (k, r) = H
±
,η (kr) (see Eq. (2.138)), and r ≥ R (s) .
Consequently, by calculating the Jost functions J + (k)
(s) and J + (k) at r = R (s)
from Eq. (2.169), and using Eq. (2.175) and the fact that u(k, r) and u (s) (k, r) are
proportional to each other, imply that J + (k)
(s) and J + (k) are mutually proportional
up to a small term equal to O(R (s) −1 ). Equation (2.7) implies as well that:
C
+ (s) e
ikR (s) + C
− (s) e
−ikR (s) = A(C
+ H
+
,η (kR
(s) ) + C
− H
−
,η (kR
(s) )) ,
(2.176)
where A is the coefficient of proportionality between u(k, r) and u (s) (k, r).
Consequently, as R (s) → +∞, Eq. (2.176) becomes:
C
+ (s) e
ikR (s) ± C
− (s) e
−ikR (s) = A C
+ exp
iΣ((, k, R
(s)
1 + O
R
(s) −1
± A C
− exp
−iΣ((, k, R
(s)
1 + O
R
(s) −1
,
where Σ((, k, R (s) ) = kR (s) − η ln(2kR (s) ) − π/2 + σ (η) and where one has used
the continuity of u (k, r) and u (s) (k, r) in r = R (s) .
Scattering wave functions u(k, r) and u (s) (k, r) have to be normalized so that
2πC
+ C
−
= 2πC
+ (s) C
− (s) = 1 .
As will be shown in Sect. 3.1, the latter condition is equivalent to the Dirac delta
normalization. Equation (2.177) then implies that:
A = 1 + O
R
(s) −1
.
(2.177)
Therefore, if wave functions are normalized with a Dirac delta, then u (s) (k, r) →
u(k, r) when R (s) → +∞. An immediate consequence is that C 0
(s)
→ C 0 when
R (s) → +∞ (see Eq. (2.6)).
One will now reconsider the limit k → 0 using a screened potential, that is, with
Eq. (2.2) reducing to u (k, r) + k 2 u(k, r) = 0 for r > R (s) . It is only necessary
to consider v c ≥ 0, as the singularity induced by attractive Coulomb potentials
therein is not as strong as in the repulsive case (see Sect. 3.2.3). For this, let us
introduce the u
(s)
c (k, r) functions, which are normalized regular Coulomb wave