86
3 Berggren Basis and Completeness Relations
The second integral in (3.14) vanishes when R δ → +∞ as f R δ (Δ k ) =
F (Δ k ) sin (β ab Δ k ln(R δ )) /Δ k verifies Eq. (3.2).
One will now determine the limit of the first integral of I
(1)
F (R δ ) in Eq. (3.14)
when R δ → +∞. For this, one introduces the function G R δ (Δ k ) via:
G R δ (Δ k ) =
F (Δ k ) cos(β ab Δ k ln(R δ )) − F (0)
Δ k
.
(3.15)
Then, I
(1)
F (R δ ) becomes:
I
(1)
F (R δ )
=
C k a C k b
2
F (0)
Δ kmax R δ
Δ k min R δ
sin(x)
x
dx +
Δ kmax
Δ k min
G R δ (Δ k ) sin (Δ k R δ ) dΔ k
,
(3.16)
where the change of variable x = Δ k max R δ has been effected.
Clearly, the first integral of Eq. (3.16) has π as a limit when R δ → +∞. As one
can apply the generalized Riemann-Lebesgue lemma to G R δ (Δ k ) (see Eqs. (3.2)
and (3.15)) the second integral of Eq. (3.16) vanishes when R δ → +∞. Thus, one
can evaluate the weak limit of I ab (R δ ) in Eq. (3.7) for R δ → +∞:
I ab (R δ ) →
π
2
C
2
k a
δ(k a − k b ) ,
(3.17)
and from Eq. (3.4) one obtains:
+∞
0
u(k a , r)u(k b , r) dr =
π
2
C
2
k a
δ(k a − k b ) = 2π C
+
k a
C
−
k a
δ(k a − k b ) . (3.18)
The Dirac delta normalization of u(k, r) thus arises from the following equality:
+∞
0
u(k a , r)u(k b , r) dr = δ(k a − k b ) ⇔ 2π C
+ C
−
= 1 ∀u k ,
(3.19)
where Eq. (2.7) has been used for its derivation.
Another formula equivalent to Eq. (3.19) can be written using the expansion
u(k, r) = C F F ,η (kr) + C G G ,η (kr) for r ≥ R:
+∞
0
u(k a , r)u(k b , r) dr = δ(k a − k b ) ⇔ C
2
F + C
2
G =
2
π
∀u k .
(3.20)
which is immediate from Eqs. (2.33) and (2.34).
The asymptotic expansion of u(k, r) for |k| → +∞ (see Sect. 2.6.2) is
indispensable to demonstrate the completeness of u(k, r) functions (see Sect. 3.2).