1 Introduction 2 Resolvents and Green's Functions

and
|k〉 = φ k (x), (15)
we define
G(x, y; z) = ∑ φ k (x)φ ∗ k(y)
, (16)
k
q k − z
so that G operates on a wave function according to
∫
[G(z)ψ] (x) = d 3 (y)G(x, y; z)ψ(y). (17)
(∞)
Thus G(x, y; z) is the kernel of an integral transform. We may see that this
correspondence is as claimed as follows: Expand
ψ(y) = ∑ l
ψ l φ l (y). (18)
Then
∫
d 3 (y) ∑ φ k (x)φ ∗ k(y) ∑
ψ l φ l (y)
(∞)
k
q k − z
l
= ∑ φ k (x) ∑
∫
ψ l d 3 (y)φ ∗
k
q k − z
k(y)φ l (y)
l
(∞)
= ∑ ψ k φ k (x)
k
q k − z . (19)
This is to be compared with
[G(z)ψ] (x) = ∑ k
= ∑ k
1
q k − z |k〉〈k| ∑ ψ l |l〉
l
ψ k φ k (x)
q k − z . (20)
We have thus demonstrated the representation as an integral transform. Similar
results would apply on other L 2 spaces of wave functions besides L 2 (R 3 ).
Consider now the formal relation:
1
(Q − z)G(z) = (Q − z) = I. (21)
Q − z
Corresponding to this, we have operator: 2
(Q x − z)G(x, y; z) = δ (3) (x − y), (22)
2 We use a subscript x on Q to denote that, if, for example, Q is a differential operator,
the differentiation is on variable x.
3