76
2 The Discrete Spectrum and the Continuum
Thus, applying the Picard algorithm to Eq. (2.221) would provide a e ||(k)|(2R−r)
factor instead of e ||(k)|r as in Eq. (2.220). Consequently, u ± (k, r) functions can
be expected to diverge as e ||(k)|(2R−r) when ∓∓(k) → +∞.
Exercise XIII.
A. If 0 ≤ r ≤ r 0 , then u(k, r) = C 0 F 0 ,η 0 (k 0 r) (see Eq. (2.6)). As one imposes
C 0 = k − 0 −1 C 0 (η 0 ) −1 , therefore C 0 F 0 ,η 0 (k 0 r) is an entire power series in k 2
(see Exercise I). Consequently, u(k, r) is an entire function of k if 0 ≤ r ≤ r 0 .
If r > r 0 , the analyticity of u (n) (k, r) is firstly proved by induction using
Eq. (2.202). For this, one notices that u (n+1) (k, r) is the sum of C 0 F 0 ,η 0 (k 0 r 0 ),
C 0 k 0 F
0 ,η 0
(k 0 r 0 ) (r − r 0 ) and of the r-integral of the analytic function (r −
r )L (k, r )u (n) (k, r ). The domain of analyticity of u (n) (k, r) is clearly identical
to that of C 0 F 0 ,η 0 (k 0 r 0 ), that is, that u (n) (k, r) is an entire function. u (n) (k, r)
converges uniformly to u(k, r) on any finite sector of the complex plane (see
Exercise I). Thus, for n → +∞ by integrating u(k, r) along a closed complex
contour C , one obtains :
C
u(k, r) dk
=
C
u(k, r) − u
(n) (k, r)
dk
≤ L(C ) sup
k∈C
|u
(n) (k, r) − u(k, r)| → 0 ,
(2.222)
where L(C ) is the length of the C contour and the analyticity of u (n) (k, r) has
been used. u(k, r) is then an entire function of k if C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 .
B. One now considers the analytic properties of u ± (k, r) function as a function
of k. It is straightforward to verify that one can follow the same method as for
u(k, r) in A, up to the replacements of r 0 by R and C 0 F 0 ,η 0 (k 0 r) by H
±
,η (kr).
However, the fundamental difference here is that H
±
,η (kr) is not analytic in
general, as it bears a cut on the negative k-axis unless is an integer and η = 0.
Consequently, the domain of analyticity of u ± (k, r) follows that of H
±
,η (kr),
that is, u ± (k, r) bears in general a cut on the negative k-axis, while it is analytic
in the rest of the complex k-plane.
u ± (k, r) is an entire function of k if = 0 and η = 0, as then u ± (k, r) =
exp(±ikr). u ± (k, r) is an analytic function of k everywhere in the complex
plane except in k = 0 if η = 0 and ∈ N ∗ . In this latter case, u ± (k, r) is
indeed a spherical Bessel function of the third kind.
Exercise XIV. One has shown in this chapter that the nonzero poles k n of the Smatrix generated by a real potential are simple. Thus, residues of the S-matrix arise
2 The Discrete Spectrum and the Continuum
Thus, applying the Picard algorithm to Eq. (2.221) would provide a e ||(k)|(2R−r)
factor instead of e ||(k)|r as in Eq. (2.220). Consequently, u ± (k, r) functions can
be expected to diverge as e ||(k)|(2R−r) when ∓∓(k) → +∞.
Exercise XIII.
A. If 0 ≤ r ≤ r 0 , then u(k, r) = C 0 F 0 ,η 0 (k 0 r) (see Eq. (2.6)). As one imposes
C 0 = k − 0 −1 C 0 (η 0 ) −1 , therefore C 0 F 0 ,η 0 (k 0 r) is an entire power series in k 2
(see Exercise I). Consequently, u(k, r) is an entire function of k if 0 ≤ r ≤ r 0 .
If r > r 0 , the analyticity of u (n) (k, r) is firstly proved by induction using
Eq. (2.202). For this, one notices that u (n+1) (k, r) is the sum of C 0 F 0 ,η 0 (k 0 r 0 ),
C 0 k 0 F
0 ,η 0
(k 0 r 0 ) (r − r 0 ) and of the r-integral of the analytic function (r −
r )L (k, r )u (n) (k, r ). The domain of analyticity of u (n) (k, r) is clearly identical
to that of C 0 F 0 ,η 0 (k 0 r 0 ), that is, that u (n) (k, r) is an entire function. u (n) (k, r)
converges uniformly to u(k, r) on any finite sector of the complex plane (see
Exercise I). Thus, for n → +∞ by integrating u(k, r) along a closed complex
contour C , one obtains :
C
u(k, r) dk
=
C
u(k, r) − u
(n) (k, r)
dk
≤ L(C ) sup
k∈C
|u
(n) (k, r) − u(k, r)| → 0 ,
(2.222)
where L(C ) is the length of the C contour and the analyticity of u (n) (k, r) has
been used. u(k, r) is then an entire function of k if C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 .
B. One now considers the analytic properties of u ± (k, r) function as a function
of k. It is straightforward to verify that one can follow the same method as for
u(k, r) in A, up to the replacements of r 0 by R and C 0 F 0 ,η 0 (k 0 r) by H
±
,η (kr).
However, the fundamental difference here is that H
±
,η (kr) is not analytic in
general, as it bears a cut on the negative k-axis unless is an integer and η = 0.
Consequently, the domain of analyticity of u ± (k, r) follows that of H
±
,η (kr),
that is, u ± (k, r) bears in general a cut on the negative k-axis, while it is analytic
in the rest of the complex k-plane.
u ± (k, r) is an entire function of k if = 0 and η = 0, as then u ± (k, r) =
exp(±ikr). u ± (k, r) is an analytic function of k everywhere in the complex
plane except in k = 0 if η = 0 and ∈ N ∗ . In this latter case, u ± (k, r) is
indeed a spherical Bessel function of the third kind.
Exercise XIV. One has shown in this chapter that the nonzero poles k n of the Smatrix generated by a real potential are simple. Thus, residues of the S-matrix arise