88
3 Berggren Basis and Completeness Relations
functions can be multiplied by 1 + O
ln
2 (k) k −2
in all equations (see Sect. 2.6.2):
u(k, r) = (C F + C G A(k, R)) sin (kR −
1 + O
ln
2 (k) k
−2
+ (C G − C F A(k, R)) cos (kR −
1 + O
ln
2 (k) k
−2
.
(3.26)
Consequently, one obtains similarly to the real case:
C F + C G A(k, R) = C 0 (cos(φ) + A 0 (k, R) sin(φ))
1 + O
ln
2 (k) k
−2
(3.27)
C G − C F A(k, R) = C 0 (sin(φ) − A 0 (k, R) cos(φ))
1 + O
ln
2 (k) k
−2
.
(3.28)
Equation (3.25) follows immediately from Eqs. (3.27) and (3.28) when (k) →
+∞ and (k) is arbitrary. Consequently, the asymptotic expansion of Eq. (3.25) is
valid for complex k verifying |k| → +∞.
3.2
One-Body Completeness Relation
The first demonstration of the one-body completeness relation using the analytical
properties of one-body states has been done by R. Newton [9]. The Newton
completeness relation is effected using the Cauchy theorem in the complex plane, by
closing a segment on the real axis by half a circle in the upper part of the k-complex
plane and using the asymptotic form of u(k, r) functions when k → +∞. While
very effective and intuitive, this demonstration can only be used for the neutron
case, as the S-matrix (see Eq. (2.182)) must be analytical in k in the upper half
plane including the real k-axis. The potentials with a Coulomb asymptotic are thus
prohibited.
In order to remove this restriction, the one-body completeness relation has
been demonstrated starting from a box completeness relation [10], where the onebody states u(k, r) are all bound and form a discrete set of states, so that their
completeness property is standard [11]. However, the limit when the radius of
the box goes to infinity (the continuum limit) demands a careful monitoring of
the u(k, r) states, which become infinitely dense. Moreover, the completeness of
Coulomb wave functions is needed therein to properly treat u(k, r) scattering states
in the vicinity of k = 0.
Hence, one will follow another route, which borrows ideas from both approaches.
Firstly, the Newton completeness relation will be shown for the neutron = 0 case.
Secondly, the completeness of the Coulomb wave functions will be demonstrated.
For this, one will firstly screen the centrifugal and Coulomb parts of the considered
3 Berggren Basis and Completeness Relations
functions can be multiplied by 1 + O
ln
2 (k) k −2
in all equations (see Sect. 2.6.2):
u(k, r) = (C F + C G A(k, R)) sin (kR −
1 + O
ln
2 (k) k
−2
+ (C G − C F A(k, R)) cos (kR −
1 + O
ln
2 (k) k
−2
.
(3.26)
Consequently, one obtains similarly to the real case:
C F + C G A(k, R) = C 0 (cos(φ) + A 0 (k, R) sin(φ))
1 + O
ln
2 (k) k
−2
(3.27)
C G − C F A(k, R) = C 0 (sin(φ) − A 0 (k, R) cos(φ))
1 + O
ln
2 (k) k
−2
.
(3.28)
Equation (3.25) follows immediately from Eqs. (3.27) and (3.28) when (k) →
+∞ and (k) is arbitrary. Consequently, the asymptotic expansion of Eq. (3.25) is
valid for complex k verifying |k| → +∞.
3.2
One-Body Completeness Relation
The first demonstration of the one-body completeness relation using the analytical
properties of one-body states has been done by R. Newton [9]. The Newton
completeness relation is effected using the Cauchy theorem in the complex plane, by
closing a segment on the real axis by half a circle in the upper part of the k-complex
plane and using the asymptotic form of u(k, r) functions when k → +∞. While
very effective and intuitive, this demonstration can only be used for the neutron
case, as the S-matrix (see Eq. (2.182)) must be analytical in k in the upper half
plane including the real k-axis. The potentials with a Coulomb asymptotic are thus
prohibited.
In order to remove this restriction, the one-body completeness relation has
been demonstrated starting from a box completeness relation [10], where the onebody states u(k, r) are all bound and form a discrete set of states, so that their
completeness property is standard [11]. However, the limit when the radius of
the box goes to infinity (the continuum limit) demands a careful monitoring of
the u(k, r) states, which become infinitely dense. Moreover, the completeness of
Coulomb wave functions is needed therein to properly treat u(k, r) scattering states
in the vicinity of k = 0.
Hence, one will follow another route, which borrows ideas from both approaches.
Firstly, the Newton completeness relation will be shown for the neutron = 0 case.
Secondly, the completeness of the Coulomb wave functions will be demonstrated.
For this, one will firstly screen the centrifugal and Coulomb parts of the considered