3.2 One-Body Completeness Relation
93
function of k if (k) ≥ 0 and k = 0, up to the eventual presence of the poles [13].
The Cauchy theorem can then be applied if the segments [−K : −k s ] and [k s : K]
are complemented by a small half-circle C k s of radius k s and by a large semicircle
C K of radius K in order to form a closed contour in the upper half-plane.
Let us consider the eventual presence of non-analyticities of F ,η (kr)H
+
,η (kr )
inside the considered closed contour. One can see from Eq. (2.37) that this can occur
only for F ,η (kr) due to C (η), because the remaining power series in Eq. (2.37) is
analytic in the upper complex k-plane. Equation (2.32) implies that H
+
,η (kr) can be
nonanalytic only through σ (η), as 2 F 0 is analytic in the upper part of the complex
k-plane as well [13]. Hence, for F ,η (kr)H
+
,η (kr ) to exhibit a non-analyticity, the
Gamma functions in Eqs. (2.31) and (2.32) must be infinite.
Γ (2 + 2) is always finite for ≥ 0. One can see from Eqs. (2.31), (2.32), (2.35),
and (2.37) that the factor Γ (1 + − iη) cancels out in the product F ,η (kr)H
+
,η (kr )
and that the Γ (1 + + iη) function appears therein. Therefore, non-analyticities
may occur only from Γ (1 + + iη), that is from its poles, occurring if 1 + + iη
is a negative integer. This is impossible if ≥ 0 and v c ≥ 0. Conversely, the Γ (1 +
+ iη) function has an infinite number of poles if v c < 0, which correspond to the
infinite number of bound states of Coulomb attractive potential [14].
This series of bound states converges absolutely as bound state wave functions
behave as O(n −3/2 ) [14] when n → +∞. Therefore, it poses no problem when
k s → 0. Equation (3.40) thus reads:
I c (k s , K) =
1
iπ
C ks
F ,η (kr)H
+
,η (kr
) dk −
1
iπ
C K
F ,η (kr)H
+
,η (kr
) dk .
(3.41)
The main difference with Eq. (3.34) is the integral along the C k s contour. For
k → 0, Eqs. (2.60) and (2.61) imply that F ,η (kr)H
+
,η (kr ) → 0, so that the
integral along the C k s contour vanishes when k s → 0. The limit K → +∞
of the integral of Eq. (3.41) is stated in Exercise IV where the case of Coulomb
wave functions is considered (for this, only consider points A and B in Exercise IV
of Sect. 3.2.1 by using Coulomb wave functions instead of neutron s-states). The
Coulomb completeness relations thus follow:
2
π
+∞
0
F ,η (kr)F ,η (kr
) dk = δ(r − r
) , v c ≥ 0 (3.42)
+∞
n=0
u n (r)u n (r
) +
2
π
+∞
0
F ,η (kr)F ,η (kr
) dk = δ(r − r
) , v c < 0 (3.43)
where u n (r) is a bound state bearing n nodes. The functions that can be expanded
with Eqs. (3.42) and (3.43) are those possessing a Fourier transform, following the
same argument as in Sect. 3.2.1.
93
function of k if (k) ≥ 0 and k = 0, up to the eventual presence of the poles [13].
The Cauchy theorem can then be applied if the segments [−K : −k s ] and [k s : K]
are complemented by a small half-circle C k s of radius k s and by a large semicircle
C K of radius K in order to form a closed contour in the upper half-plane.
Let us consider the eventual presence of non-analyticities of F ,η (kr)H
+
,η (kr )
inside the considered closed contour. One can see from Eq. (2.37) that this can occur
only for F ,η (kr) due to C (η), because the remaining power series in Eq. (2.37) is
analytic in the upper complex k-plane. Equation (2.32) implies that H
+
,η (kr) can be
nonanalytic only through σ (η), as 2 F 0 is analytic in the upper part of the complex
k-plane as well [13]. Hence, for F ,η (kr)H
+
,η (kr ) to exhibit a non-analyticity, the
Gamma functions in Eqs. (2.31) and (2.32) must be infinite.
Γ (2 + 2) is always finite for ≥ 0. One can see from Eqs. (2.31), (2.32), (2.35),
and (2.37) that the factor Γ (1 + − iη) cancels out in the product F ,η (kr)H
+
,η (kr )
and that the Γ (1 + + iη) function appears therein. Therefore, non-analyticities
may occur only from Γ (1 + + iη), that is from its poles, occurring if 1 + + iη
is a negative integer. This is impossible if ≥ 0 and v c ≥ 0. Conversely, the Γ (1 +
+ iη) function has an infinite number of poles if v c < 0, which correspond to the
infinite number of bound states of Coulomb attractive potential [14].
This series of bound states converges absolutely as bound state wave functions
behave as O(n −3/2 ) [14] when n → +∞. Therefore, it poses no problem when
k s → 0. Equation (3.40) thus reads:
I c (k s , K) =
1
iπ
C ks
F ,η (kr)H
+
,η (kr
) dk −
1
iπ
C K
F ,η (kr)H
+
,η (kr
) dk .
(3.41)
The main difference with Eq. (3.34) is the integral along the C k s contour. For
k → 0, Eqs. (2.60) and (2.61) imply that F ,η (kr)H
+
,η (kr ) → 0, so that the
integral along the C k s contour vanishes when k s → 0. The limit K → +∞
of the integral of Eq. (3.41) is stated in Exercise IV where the case of Coulomb
wave functions is considered (for this, only consider points A and B in Exercise IV
of Sect. 3.2.1 by using Coulomb wave functions instead of neutron s-states). The
Coulomb completeness relations thus follow:
2
π
+∞
0
F ,η (kr)F ,η (kr
) dk = δ(r − r
) , v c ≥ 0 (3.42)
+∞
n=0
u n (r)u n (r
) +
2
π
+∞
0
F ,η (kr)F ,η (kr
) dk = δ(r − r
) , v c < 0 (3.43)
where u n (r) is a bound state bearing n nodes. The functions that can be expanded
with Eqs. (3.42) and (3.43) are those possessing a Fourier transform, following the
same argument as in Sect. 3.2.1.