20
2 Scattering Theory of Three-Particle System
where for the two-body T-matrix, we write T 1 = (1 − V 1 G 0 )
−1 V 1 [cf., Eq. (1.53a)].
Equations similar to Eq. (2.31) for T
(2) and T
(3) can also be obtained by using
Eq. (2.29). We thus write the three components in the matrix form as
⎛
⎜
⎝
T
(1)
T
(2)
T
(3)
⎞
⎟
⎠ =
⎛
⎜
⎝
T 1
T 2
T 3
⎞
⎟
⎠ +
⎛
⎜
⎝
0 T 1
T 1
T 2 0
T 2
T 3
T 3
0
⎞
⎟
⎠ G 0
⎛
⎜
⎝
T
(1)
T
(2)
T
(3)
⎞
⎟
⎠
(2.34)
In Eq. (2.34), which we call the Faddeev equation, the new features that have
appeared are: (i) instead of two body potentials, we now have two-particle T-matrices
which are to be understood as operators in three-particle space. The T i ’s enter
into the Faddeev equation as off-shell quantities [see, e.g., Eq. (2.13)]. Thus, we
have got a mathematical formulation where the two-particle scattering amplitudes,
which are more closely related to experiment, enter into the three-particle amplitude containing more information; (ii) the kernel of the three-body T-matrix gets
connected-the dangerous δ-functions no longer appear in the kernel. This can be
checked by iteration of the kernel in Eq. (2.34) as
⎛
⎝
T
(1)
T
(2)
T
(3)
⎞
⎠ =
⎛
⎝
T 1
T 2
T 3
⎞
⎠ +
⎛
⎝
T 1 G 0 (T 2 + T 3 )
T 2 G 0 (T 3 + T 1 )
T 3 G 0 (T 1 + T 2 )
⎞
⎠
+
⎛
⎝
T 1 G 0 (T 2 + T 3 )
T 1 G 0 T 3
T 1 G 0 T 2
T 2 G 0 T 3
T 2 G 0 (T 3 + T 1 )
T 2 G 0 T 1
T 3 G 0 T 2
T 3 G 0 T 1
T 3 G 0 (T 1 + T 2 )
⎞
⎠ G 0
⎛
⎝
T
(1)
T
(2)
T
(3)
⎞
⎠
(2.35)
The iterated terms contain only operator products T i G 0 T j with i = j and therefore the kernel contains only terms where all three particles are connected together.
For example, as shown in Fig. 2.2, the new kernel K
2
12 is just
The work by Faddeev and Lovelace has shown that the operator K
2
i j is compact
and with certain requirements on the interaction potentials and can even be square
integrable for all but physical values of z. The limit as z becomes real has been
investigated in detail by Faddeev who shows that for real z, the fifth power of kernel
is a compact operator in a Banach space. This ensures that the solution of Faddeev
equations is unique.
Having discussed Faddeev equations in terms of T-matrices, similar equations
can also be written for the total Green’s operators or for the wave functions. Thus,
Fig. 2.2 Graphical
representation of a typical
term of the iterated kernel
1
T
3
T
1
2
3
2
12
K
2 Scattering Theory of Three-Particle System
where for the two-body T-matrix, we write T 1 = (1 − V 1 G 0 )
−1 V 1 [cf., Eq. (1.53a)].
Equations similar to Eq. (2.31) for T
(2) and T
(3) can also be obtained by using
Eq. (2.29). We thus write the three components in the matrix form as
⎛
⎜
⎝
T
(1)
T
(2)
T
(3)
⎞
⎟
⎠ =
⎛
⎜
⎝
T 1
T 2
T 3
⎞
⎟
⎠ +
⎛
⎜
⎝
0 T 1
T 1
T 2 0
T 2
T 3
T 3
0
⎞
⎟
⎠ G 0
⎛
⎜
⎝
T
(1)
T
(2)
T
(3)
⎞
⎟
⎠
(2.34)
In Eq. (2.34), which we call the Faddeev equation, the new features that have
appeared are: (i) instead of two body potentials, we now have two-particle T-matrices
which are to be understood as operators in three-particle space. The T i ’s enter
into the Faddeev equation as off-shell quantities [see, e.g., Eq. (2.13)]. Thus, we
have got a mathematical formulation where the two-particle scattering amplitudes,
which are more closely related to experiment, enter into the three-particle amplitude containing more information; (ii) the kernel of the three-body T-matrix gets
connected-the dangerous δ-functions no longer appear in the kernel. This can be
checked by iteration of the kernel in Eq. (2.34) as
⎛
⎝
T
(1)
T
(2)
T
(3)
⎞
⎠ =
⎛
⎝
T 1
T 2
T 3
⎞
⎠ +
⎛
⎝
T 1 G 0 (T 2 + T 3 )
T 2 G 0 (T 3 + T 1 )
T 3 G 0 (T 1 + T 2 )
⎞
⎠
+
⎛
⎝
T 1 G 0 (T 2 + T 3 )
T 1 G 0 T 3
T 1 G 0 T 2
T 2 G 0 T 3
T 2 G 0 (T 3 + T 1 )
T 2 G 0 T 1
T 3 G 0 T 2
T 3 G 0 T 1
T 3 G 0 (T 1 + T 2 )
⎞
⎠ G 0
⎛
⎝
T
(1)
T
(2)
T
(3)
⎞
⎠
(2.35)
The iterated terms contain only operator products T i G 0 T j with i = j and therefore the kernel contains only terms where all three particles are connected together.
For example, as shown in Fig. 2.2, the new kernel K
2
12 is just
The work by Faddeev and Lovelace has shown that the operator K
2
i j is compact
and with certain requirements on the interaction potentials and can even be square
integrable for all but physical values of z. The limit as z becomes real has been
investigated in detail by Faddeev who shows that for real z, the fifth power of kernel
is a compact operator in a Banach space. This ensures that the solution of Faddeev
equations is unique.
Having discussed Faddeev equations in terms of T-matrices, similar equations
can also be written for the total Green’s operators or for the wave functions. Thus,
Fig. 2.2 Graphical
representation of a typical
term of the iterated kernel
1
T
3
T
1
2
3
2
12
K