1.4 Resolvent Equation and Lippmann–Schwinger Equation
7
g 0 ( p, R) = −(4π
2 R)
−1
+∞
−∞
p
sin p
R
p − p 2 d p
or g 0 ( p.R) = −(16π
2 i R)
−1
+∞
−∞
exp(i p
R)
1
p − p
+
1
p + p
d p
−
+∞
−∞
exp(−i p
R)[
1
p − p
+
1
p + p
]d p
(1.26)
Here, we only examine the first integral on the right-hand side of Eq. (1.26) and
analyze this expression by means of contour integration in the complex p
-plane.
Calling the integral
I 1 =
C
exp(i p
R)
1
p − p
+
1
p + p
d p
and choosing the contour C in the upper half plane, Imp
> 0 by a large semi-circle
so that the contribution to the integral from this semi-circle tends to zero as its radius
tends to infinity. Out of the four possible choices of performing the integral along
the real axis, the one that gives us outgoing spherical wave is as shown in Fig. (1.1).
Thus, the expression:
g
(+)
0 ( p, R) = −(4π
2 R)
−1
P
p
sin p
R
p − p 2 d p
= −
exp(i p R)
4π R
or g
(+)
0 ( p; ;
r , − → r
) = −
1
4π
exp{i p
r − − → r
r − − → r
(1.27)
This final expression obtained in Eq. (1.27) is now inserted (with the proviso that
p =
√
2μE) in Eq. (1.22
) to get
Fig. 1.1 Contour integration
on the complex p-plane
+p
-p
Re p
Im p