52
2 The Discrete Spectrum and the Continuum
By matching Eqs. (2.140) and (2.154) in r = R, one may derive asymptotic
equations for the C ± constants (see Eq. (2.7)) when k → +∞:
2iC
±
= −C 0
u ±
app (k, R)
H
±
,η (kR)
h
∓
,η (kR) − h
±
0 ,η 0
(Λ(R))
h
+
,η (kR) − h
−
,η (kR)
1 + O
k
−2
+ C 0
u ∓
app (k, R)
H
±
,η (kR)
h
∓
,η (kR) − h
∓
0 ,η 0
(Λ(R))
h
+
,η (kR) − h
−
,η (kR)
1 + O
k
−2
,
(2.158)
where the asymptotic expansions are clearly well defined.
Exercise XII
One will show that u(k, r) cannot grow faster than a given exponential
function for large values of |k| in the complex plane.
A. Show that u(k, r) can be defined from the following integral equation:
u(k, r) = C 0 F 0 ,η 0 (k 0 r) +
r
r 0
g(r, r
) ΔV (r
) u(k, r
) dr
,
(2.159)
where the Green’s function of Eq. (2.2) is used [40]. In this expression, g(r, r )
and ΔV (r) are defined by the following equations:
g(r, r
) =
F 0 ,η 0 (k 0 r)G 0 ,η 0 (k 0 r ) − F 0 ,η 0 (k 0 r )G 0 ,η 0 (k 0 r)
k 0
(2.160)
ΔV (r) = v 0 (r) − v 0 .
(2.161)
B. Show that, for r 0 ≤ r ≤ r ≤ R:
g(r, r
) ΔV (r
)
≤ C e
||(k)|(r−r )
|k|
−1 ,
(2.162)
where |k| is sufficiently large and C > 0 is independent of k, r, and r .
Using the same method and notations as in Exercise I, show from the
previous equations that:
|u
(n+1) (k, r) − u
(n) (k, r)| ≤ 2 |C 0 | e
||(k)|r
C n r n |k| −n
n!
.
(2.163)
Deduce that u(k, r) cannot diverge faster than |C 0 | exp(||(k)|r) when
|k| → +∞ for 0 < r ≤ R.
C. Devise a similar result for u ± (k, r) when |k| → +∞ in the zones of the
complex plane where it diverges.