2.6 Analytical Properties of the Wave Functions
53
We will now derive the equivalents of Eq. (2.153) in the upper complex k-plane.
One will have to integrate wave functions in the whole upper complex k-plane (see
Sect. 3.2.1). As u +
app (k, r) is bounded if (k) > 0, u +
app (k, r)(1 + + (k, r)) is a
converging Neumann series solution of Eq. (2.144). However, the iterative process
of Eq. (2.148) fails to produce a solution of Eq. (2.144) when applied to u −
app (k, r),
as the latter function diverges in the upper complex k-plane when (k) → +∞. In
order to counteract this divergence, one will consider the following function:
f app (k, r) = u
+
app (k, r) + e
2ikR u
−
app (k, r) .
(2.164)
One can see that the term depending on u −
app (k, r) is now smaller or comparable to
u +
app (k, r) in Eq. (2.164) when (k) → +∞, because both terms behave as exp(ikr)
and exp(ik(2R −r)), respectively. Moreover, Eq. (2.150) implies that f app (k, r) = 0
if |k| is sufficiently large. Consequently, as f app (k, r) is bounded when (k) →
+∞, its associated Neumann series, denoted as f (k, r) = f app (k, r)(1 + f (k, r)),
converges and is then a solution of Eq. (2.144). As f (k, r) is linearly independent
of u + (k, r) for |k| sufficiently large, u(k, r) and u + (k, r) can be written as a linear
combination of the Neumann series u +
app (k, r)(1 + + (k, r)) and f app (k, r)(1 +
f (k, r)).
Let us now derive the equivalent of Eq. (2.154) in the upper complex k-plane.
The rest function f (k, r) vanishes when r = r 0 . Using Eq. (2.164), one obtains
that u(k, r) can diverge as exp((k)(2R − r)) |k| − 0 −3 when (k) → +∞ and
r 0 ≤ r ≤ R. This is due to the appearance of an additional term in Eq. (2.154), equal
to C 0 /(2i) e −2ikR u +
app (r) (( +
u (k, r) − f (k, r)). However, u(k, r) cannot increase
faster than |C 0 | exp((k) r) when (k) → +∞ (see Exercise XII). Consequently,
one has +
u (k, r)− f (k, r) = O(exp(2ik(R−r)) k −2 ) when (k) → +∞ and r 0 ≤
r ≤ R (see Eqs. (2.148)–(2.151)). Therefore, Eq. (2.154) is still valid in the upper
complex k-plane, except that ±
u (k, r) therein is now a complicated function of the
initial +
u (k, r) and f (k, r) rest terms. Clearly, one still has ±
u (k, r) = O(k −2 )
when |k| → +∞ for this newly defined ±
u (k, r) function.
Because Eq. (2.158) was derived from Eq. (2.154), it is also verified when
(k) → +∞. One can then devise the asymptotic behavior of the constant C ± (k)
(see Eqs. (2.155), (2.156), and (2.158)) when |k| → +∞ in the upper complex
k-plane. One will denote as C
±
1 (k) and C
±
2 (k) the two consecutive terms defining
C ± (k) in Eq. (2.158). One obtains from Eq. (2.158) that:
2iC
±
1 (k) = ±C 0 exp(±i(( − 0 )(π/2))
1 + O(ln(k) k
−1 )
(2.165)
C
±
2 (k) = O(C 0 exp(∓2ikR) ln(k) k
−1 ) .
(2.166)
53
We will now derive the equivalents of Eq. (2.153) in the upper complex k-plane.
One will have to integrate wave functions in the whole upper complex k-plane (see
Sect. 3.2.1). As u +
app (k, r) is bounded if (k) > 0, u +
app (k, r)(1 + + (k, r)) is a
converging Neumann series solution of Eq. (2.144). However, the iterative process
of Eq. (2.148) fails to produce a solution of Eq. (2.144) when applied to u −
app (k, r),
as the latter function diverges in the upper complex k-plane when (k) → +∞. In
order to counteract this divergence, one will consider the following function:
f app (k, r) = u
+
app (k, r) + e
2ikR u
−
app (k, r) .
(2.164)
One can see that the term depending on u −
app (k, r) is now smaller or comparable to
u +
app (k, r) in Eq. (2.164) when (k) → +∞, because both terms behave as exp(ikr)
and exp(ik(2R −r)), respectively. Moreover, Eq. (2.150) implies that f app (k, r) = 0
if |k| is sufficiently large. Consequently, as f app (k, r) is bounded when (k) →
+∞, its associated Neumann series, denoted as f (k, r) = f app (k, r)(1 + f (k, r)),
converges and is then a solution of Eq. (2.144). As f (k, r) is linearly independent
of u + (k, r) for |k| sufficiently large, u(k, r) and u + (k, r) can be written as a linear
combination of the Neumann series u +
app (k, r)(1 + + (k, r)) and f app (k, r)(1 +
f (k, r)).
Let us now derive the equivalent of Eq. (2.154) in the upper complex k-plane.
The rest function f (k, r) vanishes when r = r 0 . Using Eq. (2.164), one obtains
that u(k, r) can diverge as exp((k)(2R − r)) |k| − 0 −3 when (k) → +∞ and
r 0 ≤ r ≤ R. This is due to the appearance of an additional term in Eq. (2.154), equal
to C 0 /(2i) e −2ikR u +
app (r) (( +
u (k, r) − f (k, r)). However, u(k, r) cannot increase
faster than |C 0 | exp((k) r) when (k) → +∞ (see Exercise XII). Consequently,
one has +
u (k, r)− f (k, r) = O(exp(2ik(R−r)) k −2 ) when (k) → +∞ and r 0 ≤
r ≤ R (see Eqs. (2.148)–(2.151)). Therefore, Eq. (2.154) is still valid in the upper
complex k-plane, except that ±
u (k, r) therein is now a complicated function of the
initial +
u (k, r) and f (k, r) rest terms. Clearly, one still has ±
u (k, r) = O(k −2 )
when |k| → +∞ for this newly defined ±
u (k, r) function.
Because Eq. (2.158) was derived from Eq. (2.154), it is also verified when
(k) → +∞. One can then devise the asymptotic behavior of the constant C ± (k)
(see Eqs. (2.155), (2.156), and (2.158)) when |k| → +∞ in the upper complex
k-plane. One will denote as C
±
1 (k) and C
±
2 (k) the two consecutive terms defining
C ± (k) in Eq. (2.158). One obtains from Eq. (2.158) that:
2iC
±
1 (k) = ±C 0 exp(±i(( − 0 )(π/2))
1 + O(ln(k) k
−1 )
(2.165)
C
±
2 (k) = O(C 0 exp(∓2ikR) ln(k) k
−1 ) .
(2.166)