90
3 Berggren Basis and Completeness Relations
an integral on [−K : K]:
I (K) =
K
−K
u + (k, r)u(k, r )
2π C − (k)
dk .
(3.33)
Exercise II
Calculate the residues of Eq. (3.33) using Eqs. (2.182) and (2.223).
One can then calculate Eq. (3.33) with the Cauchy theorem by closing the
segment [−K : K] by a half-circle C K of radius K in the upper part of the complex
k-plane. The only possible poles in Eq. (3.33) are situated in the upper half plane and
arise from the bound states u(k, r), whose residues are straightforward to calculate
(see Sect. 2.5 and Exercise II). One has shown in Sect. 2.5.1 that the number of
bound states is finite, so that the number of poles in Eq. (3.33) is finite as well. Note
that one cannot have a zero-energy bound state with neutron s-states (see Sect. 2.5).
Exercise III
In this exercise, we show that k-integrals depending on K have simple limits
when K → +∞, which will be used to demonstrate the Berggren completeness
relation of neutron states.
Calculate the limit K → +∞ of the following two integrals:
−
1
2π
C K
e
ikx dk , x ∈ R
+∞
0
f (x)
C K
e
ikx a(k)
k
dk dx ,
where f (x) is a smooth test function, and |a(k)| ≤ C ln(K) ∀k ∈ C K , with C
independent of K.
Deduce the weak limit of the integral
C K
e
ikx a(k)
k
dk, when K → +∞ and
x ≥ 0.
Exercise IV
One will demonstrate the Berggren completeness relation for the case of
neutron s-states.
A. In order to demonstrate the Berggren completeness relation, one will derive the
asymptotic expansion of Eq. (3.35) when K → +∞. One will firstly consider
that 0 < r ≤ r ≤ R in Eq. (3.35). Using the results of Sect. 2.6.2, write
Eq. (3.35) when K → +∞ as the sum of four integrals involving C 0 , A ± (k),
B ± (k), and u ±
app (k, r). The functions ± (k, r) and ±
u (k, r) (see Sect. 2.6.2)
3 Berggren Basis and Completeness Relations
an integral on [−K : K]:
I (K) =
K
−K
u + (k, r)u(k, r )
2π C − (k)
dk .
(3.33)
Exercise II
Calculate the residues of Eq. (3.33) using Eqs. (2.182) and (2.223).
One can then calculate Eq. (3.33) with the Cauchy theorem by closing the
segment [−K : K] by a half-circle C K of radius K in the upper part of the complex
k-plane. The only possible poles in Eq. (3.33) are situated in the upper half plane and
arise from the bound states u(k, r), whose residues are straightforward to calculate
(see Sect. 2.5 and Exercise II). One has shown in Sect. 2.5.1 that the number of
bound states is finite, so that the number of poles in Eq. (3.33) is finite as well. Note
that one cannot have a zero-energy bound state with neutron s-states (see Sect. 2.5).
Exercise III
In this exercise, we show that k-integrals depending on K have simple limits
when K → +∞, which will be used to demonstrate the Berggren completeness
relation of neutron states.
Calculate the limit K → +∞ of the following two integrals:
−
1
2π
C K
e
ikx dk , x ∈ R
+∞
0
f (x)
C K
e
ikx a(k)
k
dk dx ,
where f (x) is a smooth test function, and |a(k)| ≤ C ln(K) ∀k ∈ C K , with C
independent of K.
Deduce the weak limit of the integral
C K
e
ikx a(k)
k
dk, when K → +∞ and
x ≥ 0.
Exercise IV
One will demonstrate the Berggren completeness relation for the case of
neutron s-states.
A. In order to demonstrate the Berggren completeness relation, one will derive the
asymptotic expansion of Eq. (3.35) when K → +∞. One will firstly consider
that 0 < r ≤ r ≤ R in Eq. (3.35). Using the results of Sect. 2.6.2, write
Eq. (3.35) when K → +∞ as the sum of four integrals involving C 0 , A ± (k),
B ± (k), and u ±
app (k, r). The functions ± (k, r) and ±
u (k, r) (see Sect. 2.6.2)