2.5 The Faddeev Equations
21
we have
G = G 0 + GV G 0
or
G = G 0 + G 0 T G 0
or G = G 0 +
3
i=1
G 0 T
(i) G 0
(2.36, 2.37)
Defining the components
G
(i)
(z) = G 0 (z)T
(i) G 0 (z)
(2.38)
The Green’s operators G
(i)
(z) then satisfy the Faddeev equations
⎛
⎝
G
(1)
G
(2)
G
(3)
⎞
⎠ =
⎛
⎝
G 1 − G 0
G 2 − G 0
G 3 − G 0
⎞
⎠ + G 0
⎛
⎝
0 T 1 T 1
T 2 0 T 2
T 3 T 3 0
⎞
⎠
⎛
⎝
G
(1)
G
(2)
G
(3)
⎞
⎠
or
G
(k)
(z) = G k (z) − G 0 (z) +
3
j=1
G 0 (z)M k j G
(k)
(z),
(2.39a, b)
where M k j is the Faddeev (3 × 3) matrix appearing in Eq. (2.39a).
2.6 Faddeev Equations for Scattering States
Let us now see how the problem of uniqueness is achieved in Faddeev’s theory.
Starting from Eq. (2.18) for the scattering state
ψ
(±
i j
we have
ψ
(±)
i j
= lim
ε→0
±iεG(E ± iε)
φ i j
(2.40)
Using (2.37) and (2.38), we get
ψ
(±)
i j
= lim
ε→0
±iεG 0 (E ± iε)
φ i j
+ lim
ε→0
±iε
3
k=1
G
(k)
φ i j
(2.41)
Introducing the definitions