3.1 Normalization of Gamow Functions
85
The four terms of Eq. (3.7) will be denoted respectively as J
(i)
ab (Δ k , R δ ), i ∈
{1, 2, 3, 4}. Only J
(1)
ab (Δ k , R δ ) will provide a Dirac delta when R δ → +∞. All
other terms will be shown to vanish in a weak sense when R δ → +∞.
In order to show that I ab (R δ ) converges to a Dirac delta, one will use a smooth
test function F (Δ k ) of compact support: Δ k ∈ [Δ k min : Δ k max ], where −k a <
Δ k min < 0 and Δ k max > 0. These conditions are consistent with the requirements
k a > 0 (fixed) and k b = k a + Δ k > 0 (see Eq. (3.8)). One now integrates F (Δ k )
with I ab (R δ ):
I F (R δ ) =
Δ kmax
Δ k min
F (Δ k )I ab (R δ ) dΔ k =
4
i=1
I
(i)
F (R δ ) ,
(3.11)
where
I
(i)
F (R δ ) =
Δ kmax
Δ k min
F (Δ k )J
(i) (Δ k , R δ ) dΔ k ,
(3.12)
with i ∈ [1 : 4]. Integrals I
(i)
F (R δ ) with i ≥ 2 can be written as the real or imaginary
part of
Δ kmax
Δ k min
f
(i)
R δ
(Δ k ) e
iΔ k R δ dΔ k , where:
f
(2)
R δ
(Δ k ) = C k a C k b e
iβ ab Δ k ln(R δ )
cos (f − (k a , k b )) − 1
2Δ k
f
(3)
R δ
(Δ k ) = C k a C k b e
iβ ab Δ k ln(R δ )
sin (f − (k a , k b ))
2Δ k
(3.13)
f
(4)
R δ
(Δ k ) = C k a C k b e
2ik a R δ −i(η ka +η k b ) ln(R δ )
e if + (k a ,k b )
2(k a + k b )
.
One can verify that f
(i)
R δ
(Δ k ) with i ≥ 2 always verifies Eq. (3.2). Therefore, one
can apply the generalized Riemann-Lebesgue lemma to I
(i)
F (R δ ) when i ≥ 2, so
that I
(i)
F (R δ ) → 0 when R δ → +∞.
Let us show that I
(1)
F (R δ ) in Eq. (3.14) vanishes when R δ → +∞. For that
purpose, one will expand the sine function in I
(1)
F (R δ ) in products of sine and cosine
functions:
I
(1)
F (R δ ) =
C k a C k b
2
Δ kmax
Δ k min
F (Δ k )
sin (Δ k R δ )
Δ k
cos (β ab Δ k ln(R δ )) dΔ k
+
C k a C k b
2
Δ kmax
Δ k min
F (Δ k ) cos (Δ k R δ )
sin (β ab Δ k ln(R δ ))
Δ k
dΔ k .
(3.14)