94
3 Berggren Basis and Completeness Relations
3.2.3 Completeness Relation for the General Case
From the Newton completeness relation (3.36) for neutron = 0 case and the
Coulomb completeness relation of Eq. (3.42), it is possible to demonstrate the
completeness relation of the u(k, r) wave functions generated by the Schrödinger
equation of Eq. (2.2) in the case of repulsive Coulomb potential v c ≥ 0 and ≥ 0.
The completeness relation for an attractive Coulomb potential (v c < 0) will also be
discussed afterwards in this section.
Before one considers the demonstration itself, one will explain why one could
not proceed as in the two previous demonstrations. The problem lies solely in the
C ± constants. Indeed, in order to use the Cauchy theorem, one has to calculate an
integral whose interval of integration is [−K : K] (see Eqs. (3.33) and (3.40)). A
necessary condition for this method to hold is that the integrated function has to bear
a k → −k symmetry (see Eqs. (3.33) and (3.37)). The k → −k symmetry arises
from Eq. (3.32) in the neutron one-body completeness relation for = 0, and from
the analytical properties of Coulomb wave functions with a point-particle Coulomb
potential (see Sect. 3.2.2). However, there is no longer any symmetry between
C ± (k) and C ∓ (−k) in the general case. Indeed, u(k, r) is a complicate linear
combination of H
±
,η (kr) functions, whose coefficients are nonanalytic functions of
k in k = 0. Consequently, Eq. (3.29) is not symmetric in k, even though the u(k, r)
states are by construction.
Thus, instead of using the Cauchy theorem with an integral defined in [−K : K],
one will only consider an integral of u(k, r) functions defined in [0 : K] for which
v c ≥ 0. One will consider screened u (s) (k, r) functions at a radius R (s) (the subscript
(s) will always refer to the screened case) in the following functional of u(k, r) and
u (s) (k, r) functions:
I s (r, r
, R
(s) ) =
n
u n (r)u n (r
) −
n
u
(s)
n (r)u
(s)
n (r
)
+
+∞
0
u(k, r)u(k, r
) − u
(s) (k, r)u
(s) (k, r
)
dk , (3.44)
where r > 0 and r > 0 are fixed radii and R (s) is chosen larger than R. As r
and r are fixed, the radial coordinate that can take arbitrary positive values will
always be denoted as r to avoid confusion. The Schrödinger equation (2.2) is
then the same for u(k, r ) and u (s) (k, r ) if 0 ≤ r ≤ R (s) , which implies that
u(k, r ) ∝ u (s) (k, r ) if 0 ≤ r ≤ R (s) . One also supposes that u(k, r ) and
u (s) (k, r ) are normalized so that 2π C + C − = 2π C + (s) C − (s) = 1. One will
show in the following that I s (r, r , R (s) ) vanishes, which is clearly equivalent to the
completeness of u(k, r ) states as I s (R (s) ) amounts to the difference between two
completeness relations in Eq. (3.44).
Let us consider the limit of the terms involving bound states in Eq. (3.44) when
R (s) → +∞, denoted as δ b (R (s) ). For this, let us show that u
(s)
n (r ) → u n (r )
for R (s) → +∞. Firstly, one notices that the number of bound states u n (r ) and
3 Berggren Basis and Completeness Relations
3.2.3 Completeness Relation for the General Case
From the Newton completeness relation (3.36) for neutron = 0 case and the
Coulomb completeness relation of Eq. (3.42), it is possible to demonstrate the
completeness relation of the u(k, r) wave functions generated by the Schrödinger
equation of Eq. (2.2) in the case of repulsive Coulomb potential v c ≥ 0 and ≥ 0.
The completeness relation for an attractive Coulomb potential (v c < 0) will also be
discussed afterwards in this section.
Before one considers the demonstration itself, one will explain why one could
not proceed as in the two previous demonstrations. The problem lies solely in the
C ± constants. Indeed, in order to use the Cauchy theorem, one has to calculate an
integral whose interval of integration is [−K : K] (see Eqs. (3.33) and (3.40)). A
necessary condition for this method to hold is that the integrated function has to bear
a k → −k symmetry (see Eqs. (3.33) and (3.37)). The k → −k symmetry arises
from Eq. (3.32) in the neutron one-body completeness relation for = 0, and from
the analytical properties of Coulomb wave functions with a point-particle Coulomb
potential (see Sect. 3.2.2). However, there is no longer any symmetry between
C ± (k) and C ∓ (−k) in the general case. Indeed, u(k, r) is a complicate linear
combination of H
±
,η (kr) functions, whose coefficients are nonanalytic functions of
k in k = 0. Consequently, Eq. (3.29) is not symmetric in k, even though the u(k, r)
states are by construction.
Thus, instead of using the Cauchy theorem with an integral defined in [−K : K],
one will only consider an integral of u(k, r) functions defined in [0 : K] for which
v c ≥ 0. One will consider screened u (s) (k, r) functions at a radius R (s) (the subscript
(s) will always refer to the screened case) in the following functional of u(k, r) and
u (s) (k, r) functions:
I s (r, r
, R
(s) ) =
n
u n (r)u n (r
) −
n
u
(s)
n (r)u
(s)
n (r
)
+
+∞
0
u(k, r)u(k, r
) − u
(s) (k, r)u
(s) (k, r
)
dk , (3.44)
where r > 0 and r > 0 are fixed radii and R (s) is chosen larger than R. As r
and r are fixed, the radial coordinate that can take arbitrary positive values will
always be denoted as r to avoid confusion. The Schrödinger equation (2.2) is
then the same for u(k, r ) and u (s) (k, r ) if 0 ≤ r ≤ R (s) , which implies that
u(k, r ) ∝ u (s) (k, r ) if 0 ≤ r ≤ R (s) . One also supposes that u(k, r ) and
u (s) (k, r ) are normalized so that 2π C + C − = 2π C + (s) C − (s) = 1. One will
show in the following that I s (r, r , R (s) ) vanishes, which is clearly equivalent to the
completeness of u(k, r ) states as I s (R (s) ) amounts to the difference between two
completeness relations in Eq. (3.44).
Let us consider the limit of the terms involving bound states in Eq. (3.44) when
R (s) → +∞, denoted as δ b (R (s) ). For this, let us show that u
(s)
n (r ) → u n (r )
for R (s) → +∞. Firstly, one notices that the number of bound states u n (r ) and