136
3 Berggren Basis and Completeness Relations
where the norm of the u(k n , r) bound state appears. The residue of Eq. (3.33) can
then be concisely written in terms of normalized bound states u n (r) at k = k n .
Exercise III. The considered integrals either weakly converge to a Dirac delta
distribution or to zero when K → +∞:
−
1
2π
C K
e
ikx dk =
sin(Kx)
πx
→ δ(x)
(3.103)
+∞
0
f (x)
C K
e
ikx a(k)
k
dk dx
≤
K −1/2
0
f (x)
C K
e
ikx a(k)
k
dk dx
+
+∞
K −1/2
f (x)
C K
e
ikx a(k)
k
dk dx
≤ C ln(K)
π
K −1/2
0
|f (x)| dx +
+∞
K −1/2
|f (x)|
π
0
e
−Kx sin(θ) dθ dx
≤ 2π C ln(K) K
−1/2
+∞
0
|f (x)| dx → 0 .
(3.104)
Equation (3.104) is valid for all smooth test functions f (x). Therefore, it implies
the following weak limit when K → +∞ and x ≥ 0:
C K
e
ikx a(k)
k
dk → 0 .
(3.105)
Exercise IV. In this exercise, considered wave functions u(k, r) can be either
neutron s-states or Coulomb wave functions. Indeed, the presented methods are
very close in both situations, so that they have been combined therein in a single
exercise. Neutron s-states are considered in Sect. 3.2.1 and Coulomb wave functions
in Sect. 3.2.2.
A. Here, one considers 0 < r ≤ r. We also demand that r, r ≤ R when discussing
neutron s-states, while r, r can be chosen independently of R in the case of
point-particle Coulomb potential. For K → +∞, one has in this situation:
I (C K ) =
1
2π
C K
A + (k) C 0 u +
app (k, r) u +
ap p (k, r )
−2i C − (k)
1 + O
k
−2
dk
−
1
2π
C K
B + (k) C 0 u −
app (k, r) u −
app (k, r )
−2i C − (k)
1 + O
k
−2
dk