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with boundary condition
α
(1)
0
=0 at
x
3
=
±
d/
2
,
(6.65)
(
j
+1)
β
(1)
im
β
(3)
j
=0 at
x
3
=
j
+1
−
±
d/
2
,
(6.66)
j
α
(1)
j
+1
+
β
(3)
im
α
(3)
j
=0 at
x
3
=
−
±
d/
2
,
(6.67)
j
where
j
=0
,
1
,
2
,...
;
C
k
are the
x
1
z
0
- power expansion coecients of
the operator function
C
(
x
1
,x
3
). The notations
α
(1)
j
−
and
β
(1)
j
,
α
(2)
j
and
β
(2)
j
,
α
(3)
j
and
β
(3
j
are used for the first three, the fourth and the last two elements
of
α
j
and
β
j
, respectively.
A simple but rather a cumbersome analysis of the chain recurrent bound-
ary problems (6.62)-(6.67) shows, that
α
0
=0
,β
0
= 0 (i.e. the solution is
really resonance) only when the boundary problem
⎛
⎝
⎞
⎠
ik
0
√
gg
22
d
d
x
3
e
1
(
x
3
)
e
2
(
x
3
)
=0
,
−
(6.68)
d
d
x
3
ik
0
√
gg
11
ε
⊥
−
e
1
(
−
d/
2) = 0
,e
1
(
d/
2) = 0
,
(6.69)
has a non-trivial solution. Here
g
0
=
g
(
z
0
),
g
i
0
=
g
ik
(
z
0
),
ε
i
0
=
ε
ik
(
z
0
).
Assume that such solution exists at some
z
0
and
ω
=
ω
0
. Using the no-
tation
e
=(
e
1
,e
2
)
tr
for this solution, we obtain
α
0
(
x
3
)
,
β
0
(
x
3
) from (6.63),
(6.64):
α
0
=(
e
1
,e
2
,
0
,
0
,
0)
tr
,
(6.70)
β
0
=(
C
0
e
1
,C
0
e
2
,
im
g
22
g
13
0
g
11
0
ik
0
√
g
0
ε
2
0
e
2
,ik
0
√
g
0
ε
2
0
e
2
,ime
2
)
tr
,
0
g
11
0
−
e
1
−
(6.71)
where
C
0
is a constant determined from the condition of solvability of the
problem in the next approximation.
Let for each
ω
in a vicinity of
ω
0
there is
z
=
z
(
ω
) at which the boundary
problem (6.68), (6.69) has a solution. Taking the inverse function to
z
(
ω
), we
find resonance frequencies
ω
=
ω
r
(
z
) corresponding to the ”complex magnetic
shell”
z
.Letat
z
=
z
0
the derivative
z
=
z
0
d
ω
r
d
z
=0
,
(6.72)
The analysis of the chain of recurrent boundary problems (6.62)-(6.67)
(not given here because of its awkwardness) shows that they are subse-
quently resolvable at the condition (6.72) and determine unambiguously the
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