Environmental Engineering Reference
In-Depth Information
When
n
= 1, eqn (6.1.30) may be written
hωG
1
+
γ
2
G
2
=
A−
γ
0
G
0
,
and the semi-infinite series of equations with
n
≥
1 may be solved in
terms of
G
0
and
D
n
:
γ
0
G
0
)
D
2
1
G
1
=(
A−
=(
A−
γ
0
G
0
)
,
(6
.
1
.
33)
D
1
hω
−
γ
1
γ
2
D
3
/D
2
utilizing (6.1.32) in the last step. In terms of the two infinite continued
fractions (
n
≥
1)
γ
n
(
q
)
z
n
(
q
,ω
)=
hω
−
γ
n
(
q
)
z
n
+1
(
q
,ω
)
(6
.
1
.
34)
γ
−n
(
q
)
z
−n
(
q
,ω
)=
−n−
1
(
q
,ω
)
,
hω
−
γ
−n
(
q
)
z
eqn (6.1.33) may be written
γ
1
G
1
=(
A−
γ
0
G
0
)
z
1
,andinthesameway,
we have
γ
−
1
G
−
1
=(
γ
0
G
0
)
z
−
1
. Introducing these expressions into
(6.1.30), with
n
= 0, we finally obtain
A−
z
1
(
q
,ω
)+
z
−
1
(
q
,ω
)
G
0
(
q
,ω
)=
A
hω
.
(6
.
1
.
35)
γ
0
{
z
1
(
q
,ω
)+
z
−
1
(
q
,ω
)
}−
A similar result was derived by Liu (1980). In this formal solution, there
is no small parameter, except in the high-frequency limit, which allows
a perturbative expansion of
G
0
(
q
,ω
). The infinite continued fraction
determining
z
n
never repeats itself, but it is always possible to find an
n
=
s
such that
z
s
is arbitrary close to, for instance,
z
1
. This property
may be used for determining the final response function when
hω
→
0.
In this limit, we have from (6.1.34):
z
1
=
−
1
/z
2
=
z
3
=
−
1
/z
4
=
···
and, using
z
1
z
s
for
s
even, we get
z
1
=
−
1
/z
1
or
z
1
=
±
i
.At
q
=
0
,
we have by symmetry
z
1
i
, which also has to be valid at
any other
q
. The correct sign in front of the
i
is determined from a
replacement of
ω
by
ω
+
i
,where
is an infinitesimal positive quantity
=
z
=
±
−
1
or, more easily, from the property that Im
G
0
(
q
,ω
)
should have the
opposite sign to
ω
, i.e.
0) =
γ
0
− i
2
γ
0
G
(
q
,ω→
hω.
(6
.
1
.
36
a
)
Since
4
G
(
q
,ω
)+
G
∗
(
q
,
ω
)
,
1
χ
ξξ
(
q
,ω
)=
χ
ηη
(
q
,ω
)=
−
−
we get
1
hω
χ
ξξ
(
q
,ω
→
0) =
(
q
)
+
i
2
.
(6
.
1
.
36
b
)
J
(
Q
)
−J
⊥
A{J
(
Q
)
−J
⊥
(
q
)
}
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