2.6 Faddeev Equations for Scattering States
23
ψ
(±)
i j
(k) = δ ki
φ i j
+
3
l=1
G 0 (E + iε)M kl (E + i0)
ψ
(±)
i j
(l)
(2.48)
For the case when there are three free particles in the entrance channel, i.e., k =
0 are given by
ψ
(±)
0
= |φ 0 +
3
i=1
ψ
(±)
0
(i) ,
where
ψ
(±)
0
(i) =
ϕ
(±)
(i) +
3
j=1
G 0 (E ± iε)M i j
ψ
(±)
0
( j) ,
with
ϕ
(±)
(i) =
(±)
i |φ 0 − |φ 0 ,
(2.49, 2.50)
which represents a spherical outgoing or incoming wave respectively.
The formulation presented so far is quite general. Let us consider one particular
case which describes a situation in which initially particle 1 is free and is scattered
by the pair (2, 3) which is bound.
Also simplifying the notation, we write the total Hamiltonian
H = H 1 + V 1 , with H 1 = H 0 + V
(1)
= H 0 + V 23 , V 1 = V − V
(1)
where H 1 φ 1n = E 1n φ 1n and the eigen state |ψ 1n =
ψ
(1)
+
ψ
(2)
+
ψ
(3)
With
⎛
⎝
ψ
(1)
ψ
(2)
ψ
(3)
⎞
⎠ =
⎛
⎝
|φ 1n
0
0
⎞
⎠ + G o (z)
⎛
⎝
0 T 1 (z) T 1 (z)
T 2 (z) 0 T 2 (z)
T 3 (z) T 3 (z) 0
⎞
⎠
⎛
⎝
ψ
(1)
ψ
(2)
ψ
(3)
⎞
⎠ ,
(2.51)
the index n contains the additional information on bound states, spin, i-spin, etc.
Similarly, for the break- up channel, we have
|ψ n0 = |φ n0 +
ψ
(1)
n0
+
ψ
(2)
n0
+
ψ
(3)
n0
with (z = E n0 + i0)
⎛
⎜
⎜
⎜
⎜
⎝
ψ
(1)
n0
ψ
(2)
n0
ψ
(3)
n0
⎞
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎝
φ
(1)
n0
− |φ n0
φ
(2)
n0
− |φ n0
φ
(3)
n0
− |φ n0
⎞
⎟
⎟
⎟
⎟
⎠
− G 0
⎛
⎜
⎜
⎜
⎝
0 T 1 (z) T 1 (z)
T 2 (z) 0 T 2 (z)
T 3 (z) T 3 (z) 0
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎝
ψ
(1)
n0
ψ
(2)
n0
ψ
(3)
n0
⎞
⎟
⎟
⎟
⎟
⎠
(2.52)