j f i ¼
Z s
r
dxw x
ð Þf x
ð Þjxi:
ð10:89Þ
In (10.89) we considered f(x) as if it were an expansion coefficient. The following
notation would be reasonable accordingly:
f x
ð Þ xj f
h i:
ð10:90Þ
In (10.90), f(x) can be viewed as coordinate representation of | f i. Thus, from (10.89)
we get
x
0
j f
h i ¼ f x
0
ð Þ ¼
Z s
r
dxw x
ð Þf x
ð Þ x
0
jx
h i:
ð10:91Þ
Alternatively, we have
f x
0
ð Þ ¼
Z s
r
dxf x
ð Þδ x À x
0
ð
Þ:
ð10:92Þ
This comes from a property of the δ function [1] described as
Z s
r
dxf x
ð Þδ x
ð Þ ¼ f 0
ð Þ:
ð10:93Þ
Comparing (10.91) and (10.92), we have
w x
ð Þ x
0
jx
h i ¼ δ x À x
0
ð
Þor x
0
jx
h i ¼
δ x À x
0
ð
Þ
w x
ð Þ
¼
δ x
0
À x
ð
Þ
w x
ð Þ
:
ð10:94Þ
Thus comparing (10.86) and (10.87) and using (10.94), we get
L x G x, y
ð Þ ¼
δ x À y
ð
Þ
w x
ð Þ
:
ð10:95Þ
In a similar manner, we also have
L x
{ g x, y
ð Þ ¼
δ x À y
ð
Þ
w x
ð Þ
:
ð10:96Þ
To arrive at (10.96), we start the discussion assuming an operator (L x
{ )
À1 such that
(L x
{ )
À1
g with gL
{
¼ L
{ g ¼ E.
The function G(x, y) is called a Green’s function and g(x, y) is said to be an adjoint
Green’s function. Handling of Green’s functions and adjoint Green’s functions is
based upon (10.95) and (10.96), respectively. As (10.81) is defined in a domain
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10 Introductory Green’s Functions