3.1 Normalization of Gamow Functions
83
distribution:
+∞
0
u(k a , r)u(k b , r) dr = δ(k a − k b ) ,
(3.1)
where k a > 0 and k b > 0 are linear momenta. Dirac delta normalization will be
shown as equivalent to the normalization condition 2π C + C − = 1, which is then the
same equality both in the charged particle case and the neutral particle case. For this,
it will be demonstrated that the partial overlap I ab (R δ ) in [0 : R δ ] between u(k a , r)
and u(k b , r) weakly converges to a Dirac delta distribution when R δ → +∞. As
R δ enters integrated functions via ln(R δ ), a generalization of the Riemann-Lebesgue
lemma including this dependence will be stated for clarity. This lemma will be used
to prove that all terms not leading to a Dirac delta vanish for R δ → +∞.
Let us consider a differentiable function f R δ (k) defined for k ∈ [k min : k max ],
verifying:
|f R δ (k)| = O
ln
n (R δ )
∀k
k max
k min
|f
R δ
(k)| dk = O
ln
n (R δ )
,
(3.2)
where n ∈ N. Integration by parts provides:
k max
k min
f R δ (k) e
ikR δ dk =
1
iR δ
f R δ (k) e
ikR δ
k max
k min
−
k max
k min
f
R δ
(k) e
ikR δ dk
.
(3.3)
Thus majorizing Eq. (3.3), one finds:
k max
k min
f R δ (k) e
ikR δ dk
≤
1
R δ
|f R δ (k min )| + |f R δ (k max )| +
k max
k min
|f
R δ
(k)| dk
= O
ln
n (R δ )
R δ
→ 0 .
The partial overlap I ab (R δ ) between u(k a , r) and u(k b , r) in [0 : R δ ] arises
directly from Eq. (2.130):
I ab (R δ ) =
R δ
0
u(k a , r)u(k b , r) dr =
u (k b , R δ )u(k a , R δ ) − u (k a , R δ )u(k b , R δ )
k 2
a − k 2
b
.
(3.4)
In order to demonstrate that u(k a , r) and u(k b , r) are orthogonal to each other when
k a = k b , one will show that I ab (R δ ) converges weakly to a Dirac delta when R δ →
+∞. Note that if ka = k b then a priori Eq. (3.4) is undefined, as both its numerator