3.5 Completeness Relation Involving Single-Particle Gamow States
103
can be immediately seen for neutrons, because with z = R + x · e iθ and with
|z| → +∞, the product u + (z) · u − (z) → const × e ikz × e −ikz = const, and
the corresponding integral diverges. In this case, however, it is easy to see that
the integral is in fact a δ-distribution, and it can be calculated by using a discrete
representation of the Dirac δ-function,
δ(k − k
) →
δ k,k
Δk
,
(3.64)
with Δk being the discretization step in k.
3.5
Completeness Relation Involving Single-Particle Gamow
States
As one aims at expanding many-body resonance wave functions, one has to devise
the completeness relation of complex-energy one-body eigenstates. The first step
was to devise a metric with which unbound states can be normalized and matrix
elements involving resonance states can be numerically calculated. This has been
effected in Sects. 3.3 and 3.4. In the following step, the real-energy completeness
relation of Eq. (3.50) will be generalized to complex-energy states, so that the
resonances can enter the completeness relation of one-body eigenstates.
A convenient method for that purpose is to use complex integration, by deforming the integration contour of Eq. (3.50) into the complex k−plane, as shown in
Fig. 3.2. Following the residuum theorem, one obtains:
−
+∞
0
u k (r)u k (r
) dk +
L +
u k (r)u k (r
) dk = 2iπ
k n
Res
u k n (r)u k n (r
)
k=k n
,
(3.65)
where k n are the poles of u k (r)u k (r ) lying between the real axis and the complex
contour. The Berggren completeness relation follows immediately:
n
u n (r)u n (r
) +
L +
u k (r)u k (r
) dk = δ(r − r
) .
(3.66)
In the above equation, u n (r) are the bound states and resonances present between
the real k-axis and the L + contour of complex-energy scattering states defined in
the complex k-plane.
Figure 3.2 illustrates the ingredients entering Eq. (3.66). The resonant states, that
is, the poles of the S matrix, are represented by the dots. They are divided into the
bound, decaying resonances, capturing resonances, and antibound states (see, e.g.,
Refs. [7, 22, 23]). The relation (3.66) involves the bound and resonance states and
the contour L + lying in the fourth quadrant of the complex-k plane. One may notice
103
can be immediately seen for neutrons, because with z = R + x · e iθ and with
|z| → +∞, the product u + (z) · u − (z) → const × e ikz × e −ikz = const, and
the corresponding integral diverges. In this case, however, it is easy to see that
the integral is in fact a δ-distribution, and it can be calculated by using a discrete
representation of the Dirac δ-function,
δ(k − k
) →
δ k,k
Δk
,
(3.64)
with Δk being the discretization step in k.
3.5
Completeness Relation Involving Single-Particle Gamow
States
As one aims at expanding many-body resonance wave functions, one has to devise
the completeness relation of complex-energy one-body eigenstates. The first step
was to devise a metric with which unbound states can be normalized and matrix
elements involving resonance states can be numerically calculated. This has been
effected in Sects. 3.3 and 3.4. In the following step, the real-energy completeness
relation of Eq. (3.50) will be generalized to complex-energy states, so that the
resonances can enter the completeness relation of one-body eigenstates.
A convenient method for that purpose is to use complex integration, by deforming the integration contour of Eq. (3.50) into the complex k−plane, as shown in
Fig. 3.2. Following the residuum theorem, one obtains:
−
+∞
0
u k (r)u k (r
) dk +
L +
u k (r)u k (r
) dk = 2iπ
k n
Res
u k n (r)u k n (r
)
k=k n
,
(3.65)
where k n are the poles of u k (r)u k (r ) lying between the real axis and the complex
contour. The Berggren completeness relation follows immediately:
n
u n (r)u n (r
) +
L +
u k (r)u k (r
) dk = δ(r − r
) .
(3.66)
In the above equation, u n (r) are the bound states and resonances present between
the real k-axis and the L + contour of complex-energy scattering states defined in
the complex k-plane.
Figure 3.2 illustrates the ingredients entering Eq. (3.66). The resonant states, that
is, the poles of the S matrix, are represented by the dots. They are divided into the
bound, decaying resonances, capturing resonances, and antibound states (see, e.g.,
Refs. [7, 22, 23]). The relation (3.66) involves the bound and resonance states and
the contour L + lying in the fourth quadrant of the complex-k plane. One may notice