2.5 The Faddeev Equations
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2.5 The Faddeev Equations
In order to remove these difficulties, Faddeev’s starting point is to extend the equation
for two-body T-matrix to three-particle scattering:
T (z) = V + V G(z) V
(2.27)
This operator equation is the formal analog of the two-particle T-operator
(Eq. 1.44). However, Eq. (2.27) is not as directly related to the scattering cross
section as in the two-particle case. In analogy to Eq. (1.43), we use the equation
G(z) = G 0 (z) + G 0 (z)T (z)G 0 (z)
(2.28)
to write the equation for T (z) in the operator form as
T (z) = V + V G 0 (z)T (z)
= V + T (z)G 0 (z)V
(2.29a, b)
similar to the two-particle case [cf. Eqs. (1.53a) and (1.53b). These integral equations for the operator suffer from the same difficulties as the Lippmann–Schwinger
equation since they have the same kernel.
Faddeev proposed a novel but simple idea to split the T-matrix in Eq. (2.29a) into
three parts:
T (z) = T
(1)
(z) + T
(2)
(z) + T
(3)
(z)
(2.30)
where using Eq. (2.27a), we write
T
(i)
= V i + V i G 0 T (i = 1, 2, 3)
(2.31)
Here we are omitting the z-dependence of the operators. Writing Eq. (2.31)
explicitly for i = 1, we have
T
(1)
= V 1 + V 1 G 0 (T
(1)
+ T
(2)
+ T
(3)
)
(2.32)
Now let us bring the term appearing on the right-hand side of Eq. (2.32) to the
left to get
T
(1)
− V 1 G 0 T
(1)
= (1 − V 1 G 0 )T
(1)
= V 1 + V 1 G 0
T
(2)
+ T
(3)
or (1 − V 1 G 0 )T
(1)
= V 1 + V 1 G 0
T
(2)
+ T
(3)
or T
(1)
= (1 − V 1 G 0 )
−1 V 1 + (1 − V 1 G 0 )
−1 V 1 G 0
T
(2)
+ T
(3)
i.e., T
(1)
= T 1 + T 1 G 0
T
(2)
+ T
(3)
(2.33)