118
3 Berggren Basis and Completeness Relations
The cut method is the crudest one, as it simply removes all improper integrals in
Eq. (3.90), leaving only the finite integral on [0 : R]. This removes all singularities
in the Hamiltonian kernel of Eq. (3.90), but at the price of introducing a cutdependence on R which has to be assessed.
The subtraction method is the standard method to treat integrable singularities in
Fredholm kernels [47]. The main idea is to separate the kernel integral into two parts,
one whose integrand is regular and the other singular but analytically integrable. For
this, let us firstly write the Fredholm equations verified by the c k i and c k constants
of Eq. (3.89):
i
c k i Reg
+∞
0
u(k i , r) V Coul (ΔZ, r) u(k i , r) dr
+
L +
c k Reg
+∞
0
u(k
, r) V Coul (ΔZ, r) u(k i , r) dr dk
= (E − e i )c k i
(3.91)
i
c k i Reg
+∞
0
u(k i , r) V Coul (ΔZ, r) u(k, r) dr
+
L +
c k Reg
+∞
0
u(k
, r) V Coul (ΔZ, r) u(k, r) dr dk
= (E − e k )c k ,
(3.92)
where u(k i , r) is a bound or resonance basis state of energy e i , u(k, r) is a scattering
basis state of energy e k , and E is the energy of u eig (r) in Eq. (3.89) to be determined.
All matrix elements of Eq. (3.91) converge so that they are directly calculated with
complex scaling using Eq. (3.58). Conversely, the matrix elements of Eq. (3.92) can
diverge, so that one will apply the subtraction method therein. One will consider
only the integral part of Eq. (3.92), as it is the only one which leads to divergences:
L +
c k Reg
+∞
0
u(k
, r) V Coul (ΔZ, r) u(k, r) dr dk
=
L +
Reg
+∞
0
(c k u k (r) V Coul (ΔZ, r) u k (r)
−c k s k (r)
C Coul ΔZ
r
s k (r)
dr
dk
+c k
L +
Reg
+∞
0
s(k
, r) V Coul (ΔZ, r) s(k, r) dr dk
, (3.93)