4
1 Essentials of Non-relativistic Two-Body Scattering Theory
(±)
∗ ψ n
p
= ψ n |
(±)
|
p = ψ n |
p
(±) = 0
(1.10)
because the states ψ n and |
p
(±) are orthogonal belonging to different energies. The
time limit in Eq. (1.6) can be replaced by Euler’s limit
(±)
= s − lim
t→∓∞
exp(i Ht) exp(−i H 0 t)
= s − lim
ε→0
±ε
0
∓∞
exp(±εt) exp(i Ht) exp(−i H 0 t)dt
(1.11)
Note that:
lim
ε→0
0
−∞
dtε exp(εt) f (t) = f (∞)
0
−∞
dx exp(x) = f (∞)
(1.12)
By introducing the Euler limit, the time dependence in the scattering theory can
be eliminated. We shall see in the next section that following this procedure, a timeindependent integral equation which incorporates the boundary condition can be
obtained.
1.4 Resolvent Equation and Lippmann–Schwinger
Equation
We can not carry out the integration in the expression (1.11) for the Moller operator
for the simple reason H 0 and H do not commute with each other, i.e.,
exp(i Ht) exp(−i H 0 t) = exp(i(H − H 0 )t)
But since one can apply the Moller operators on the plane states, we can write
| p
(±)
= lim
ε→0
±ε
0
∓∞
dt exp(±εt) exp(i Ht) exp(−i Et)|
p
= lim
ε→0
±ε
0
∓
dt exp[i(H − E ∓ iε)t]|
p
= lim
ε→0
±iε[E ± iε − H ]
−1
|
p
(1.13)
This enables us to introduce the definition of resolvent or Green’s function: