22
2 Scattering Theory of Three-Particle System
χ
(±)
ki j
= lim
ε→0
±iεG k (E ± iε)
φ i j
k = 0, 1, 2, 3
and
ψ
(±)
i j
(k) = lim
ε→0
±iεG
(k)
(E ± iε)
φ i j
k = 1, 2, 3
(2.42a, b)
We obtain the scattering state split into the components
ψ
(±)
i j
=
χ
(±)
0i j
+
3
k=1
ψ
(±)
i j
(k)
(2.43)
Using the Green’s operator Eq. (2.39b) on Eq. (2.42b), we have
ψ
(±)
i j
(k) = lim
ε→0
±iε
G k (E ± iε) − G 0 (E ± iε)
+
3
l=1
G 0 (E ± iε) M kl (E ± iε)G (l) (E ± iε)
φ i j
(2.44)
We now use Eq. (2.42) once again to get
ψ
(±)
i j
(k) =
χ
(±)
i j
(k) −
χ
(±)
i j
(0) +
3
l=1
G 0 (E ± iε)M kl (E ± iε)
ψ
(±)
i j
(l)
(2.45)
We have a bound pair in the entrance channel, i.e.; i = 0, we shall have
χ
(±)
i j
(k) = δ ki
φ i j
, k = 0, 1, 2, 3,
(2.46)
For i = 0, the scattering state evolves from a free state |φ 0 we thus get
χ
(±)
k0
=
⎧
⎨
⎩
lim
ε→0
±iεG 0 (E ± iε)|φ 0 = |φ 0 = 0, k = 0
lim
ε→0
±iεG k (E + iε)|φ 0 =
(±)
k |φ 0 , k = 0
(2.47)
where
(±)
i
is the two-particle Moller operator in three-particle space. With
Eqs. (2.45) and (2.46), the Faddeev scattering state equations when there is a bound
pair in the entrance channel ( k = 0) are written as:
ψ
(±)
i j
=
3
k=1
ψ
(±)
i j
(k) ,
where