106
3 Berggren Basis and Completeness Relations
Conversely, let us consider a function f (r) of the form (3.67) where k f belongs
to the lower half plane outside the zone between the real k-axis and the L + contour
of Eq. (3.66). Below, one will show that it is not possible to expand this function
using Eq. (3.66). A simple example of such functions f (r) are the resonances of
the Berggren basis lying below the L + contour. Indeed, they are orthogonal to all
bound states, resonances, and scattering states of the Berggren basis (see Eq. (3.66)),
so that the expansion of f (r) generated by Eq. (3.66) vanishes identically.
For the general case, let us assume that Eqs. (3.71)–(3.73)) are valid. If the L + is
moved toward the real axis, Eq. (3.73) can still be used because the L + contour never
crosses k f while being deformed back to the real axis. The residuum theorem must
be considered similarly as in Eq. (3.71), so that the resonance poles of Eq. (3.71)
must be removed from its sum every time the L + contour reaches them. As a result,
Eq. (3.71) can be reduced to the Newton completeness relation, so that all basis
functions are either bound states or scattering states of real energy. Consequently:
|f (r)| ≤ M
n
|c n | +
L +
|c k | dk
,
(3.74)
where M is a constant for which |u n (r)| ≤ M and |u k (r)| ≤ M and the k-integral
converges because c k = O(k −2 ) due to the analyticity of f (r).
Equation (3.74) cannot be correct because |f (r)| → +∞ for r → +∞.
Consequently, a function bearing an asymptote in e ikr , with k in the lower half plane
outside the zone between the real k-axis and the complex contour, does not belong
to the domain of applicability of the Berggren completeness relation when the L +
contour is fixed. Evidently, that function can still be expanded with a Berggren
completeness relation whose L + contour encompasses its complex momentum k.
3.5.2 Berggren Completeness Relation for Complex Potentials
Up to now one has considered the Berggren completeness relation for eigenstates of
real potentials. However, bound states, resonances, and scattering states generated
by complex potentials also form a complete set of states, with the same properties
as in the case of real potentials. A demonstration of this property can be found using
a formula similar to Eq. (3.44):
I (r, r
, λ) =
n
u n (r)u n (r
) −
n
u
(R)
n (r)u
(R)
n (r
)
+
L +
u(k, r)u(k, r
) − u
(R) (k, r)u
(R) (k, r
)
dk , (3.75)
where the one-body states entering the Berggren completeness generated by a real
potential V R (r) are denoted as u (R) (k, r), and the bound states, resonances and
3 Berggren Basis and Completeness Relations
Conversely, let us consider a function f (r) of the form (3.67) where k f belongs
to the lower half plane outside the zone between the real k-axis and the L + contour
of Eq. (3.66). Below, one will show that it is not possible to expand this function
using Eq. (3.66). A simple example of such functions f (r) are the resonances of
the Berggren basis lying below the L + contour. Indeed, they are orthogonal to all
bound states, resonances, and scattering states of the Berggren basis (see Eq. (3.66)),
so that the expansion of f (r) generated by Eq. (3.66) vanishes identically.
For the general case, let us assume that Eqs. (3.71)–(3.73)) are valid. If the L + is
moved toward the real axis, Eq. (3.73) can still be used because the L + contour never
crosses k f while being deformed back to the real axis. The residuum theorem must
be considered similarly as in Eq. (3.71), so that the resonance poles of Eq. (3.71)
must be removed from its sum every time the L + contour reaches them. As a result,
Eq. (3.71) can be reduced to the Newton completeness relation, so that all basis
functions are either bound states or scattering states of real energy. Consequently:
|f (r)| ≤ M
n
|c n | +
L +
|c k | dk
,
(3.74)
where M is a constant for which |u n (r)| ≤ M and |u k (r)| ≤ M and the k-integral
converges because c k = O(k −2 ) due to the analyticity of f (r).
Equation (3.74) cannot be correct because |f (r)| → +∞ for r → +∞.
Consequently, a function bearing an asymptote in e ikr , with k in the lower half plane
outside the zone between the real k-axis and the complex contour, does not belong
to the domain of applicability of the Berggren completeness relation when the L +
contour is fixed. Evidently, that function can still be expanded with a Berggren
completeness relation whose L + contour encompasses its complex momentum k.
3.5.2 Berggren Completeness Relation for Complex Potentials
Up to now one has considered the Berggren completeness relation for eigenstates of
real potentials. However, bound states, resonances, and scattering states generated
by complex potentials also form a complete set of states, with the same properties
as in the case of real potentials. A demonstration of this property can be found using
a formula similar to Eq. (3.44):
I (r, r
, λ) =
n
u n (r)u n (r
) −
n
u
(R)
n (r)u
(R)
n (r
)
+
L +
u(k, r)u(k, r
) − u
(R) (k, r)u
(R) (k, r
)
dk , (3.75)
where the one-body states entering the Berggren completeness generated by a real
potential V R (r) are denoted as u (R) (k, r), and the bound states, resonances and