Theorem 1.
Let
Θ
(
r
,
τ
^
;
k
,
μ
)
and
Φ
(
r
,
τ
^
;
k
,
μ
)
be appropriate combinations of the eigenfunctions solutions of (5) with (6) for
|
τ
^
|
=
|
τ
2
-
ε
|
small such that
(32)
m
+
(
τ
^
)
=
-
lim
r
→
b
Θ
(
r
,
τ
^
;
k
,
μ
)
Φ
(
r
,
τ
^
;
k
,
μ
)
,
m
-
(
τ
^
)
=
-
lim
r
→
0
Θ
(
r
,
τ
^
;
k
,
μ
)
Φ
(
r
,
τ
^
;
k
,
μ
)
,
n
+
(
τ
^
)
=
-
lim
r
→
0
Θ
(
r
,
τ
^
;
k
,
μ
)
Φ
(
r
,
τ
^
;
k
,
μ
)
,
n
-
(
τ
^
)
=
-
lim
r
→
a
Θ
(
r
,
τ
^
;
k
,
μ
)
Φ
(
r
,
τ
^
;
k
,
μ
)
.
Then the solutions to (5) would be
(33)
Ψ
(
r
,
τ
^
;
k
,
μ
)
=
Θ
(
r
,
τ
^
;
k
,
μ
)
+
{
m
+
(
τ
^
)
Φ
(
r
,
τ
^
;
k
,
μ
)
∈
L
2
(
[
ρ
1
*
,
b
]
)
m
-
(
τ
^
)
Φ
(
r
,
τ
^
;
k
,
μ
)
∈
L
2
(
[
0
,
ρ
1
*
]
)
n
+
(
τ
^
)
Φ
(
r
,
τ
^
;
k
,
μ
)
∈
L
2
(
[
ρ
2
*
,
0
]
)
n
-
(
τ
^
)
Φ
(
r
,
τ
^
;
k
,
μ
)
∈
L
2
(
[
a
,
ρ
2
*
]
)
,
where
m
+
,
m
-
,
n
+
,
n
-
are as defined in Section 2.1.
Proof.
For convenience we suppress the
k
,
μ
dependence of
Θ
and
Φ
. We start with the regular solutions
M
k
,
μ
(
2
i
τ
^
1
/
2
r
)
and
W
k
,
μ
(
2
i
τ
^
1
/
2
r
)
of (5) on
I
3
∪
I
4
, where
t
^
=
(
τ
2
-
ε
)
,
k
=
i
z
(
τ
2
-
ε
)
1
/
2
, and
μ
=
l
+
1
/
2
.
Thus we let
(34)
Θ
(
r
,
τ
^
)
=
α
0
+
M
k
,
μ
(
2
i
τ
^
1
/
2
r
)
+
α
1
+
M
k
,
-
μ
(
2
i
τ
^
1
/
2
r
)
,
(35)
Φ
(
r
,
τ
^
)
=
β
0
+
W
k
,
μ
(
2
i
τ
^
1
/
2
r
)
+
β
1
+
W
-
k
,
μ
(
2
i
τ
^
1
/
2
r
)
.
We now proceed to determine
α
0
+
,
α
1
+
,
β
0
+
, and
β
1
+
.
Substituting (34) into (16) we obtain
(36)
Θ
(
ρ
1
*
,
τ
^
)
=
α
0
+
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
α
1
+
M
k
,
-
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
=
1
,
Θ
′
(
ρ
1
*
,
τ
^
)
=
α
0
+
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
2
i
τ
^
1
/
2
ρ
1
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
2
μ
-
2
k
+
1
)
(
2
μ
+
1
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
M
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
1
*
)
}
+
α
1
+
{
(
1
-
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
4
i
τ
^
1
/
2
ρ
1
*
M
k
,
-
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
1
-
2
μ
-
2
k
)
(
1
-
2
μ
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
M
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
1
*
)
}
=
0
.
From (36) we obtain the following:
(37)
α
0
+
=
1
Q
1
{
(
1
-
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
4
i
τ
^
1
/
2
ρ
1
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
1
-
2
μ
-
2
k
)
(
1
-
2
μ
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
M
k
-
1
/
2
,
-
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
1
*
)
}
,
α
1
+
=
-
1
Q
1
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
2
i
τ
^
1
/
2
ρ
1
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
2
μ
-
2
k
+
1
)
(
2
μ
+
1
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
M
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
1
*
)
}
,
where
(38)
Q
1
=
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
×
{
(
1
-
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
4
i
τ
^
1
/
2
ρ
1
*
M
k
,
-
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
1
-
2
μ
-
2
k
)
(
1
-
2
μ
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
M
k
-
1
/
2
,
-
μ
-
1
/
2
(
2
i
τ
^
1
/
2
ρ
1
*
)
}
+
(
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
2
i
τ
^
1
/
2
ρ
1
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
-
M
k
,
-
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
×
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
2
i
τ
^
1
/
2
ρ
1
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
2
μ
-
2
k
+
1
)
(
2
μ
+
1
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
M
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
1
*
)
}
)
.
Similarly, substitution of (35) into (16) yields
(39)
β
0
+
=
1
Q
2
{
(
-
(
1
+
2
μ
+
2
k
)
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
W
-
k
+
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
)
(
1
+
2
μ
+
4
i
τ
^
1
/
2
ρ
1
*
)
4
i
τ
^
1
/
2
ρ
1
*
W
-
k
,
μ
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
-
(
1
+
2
μ
+
2
k
)
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
W
-
k
+
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
)
}
,
β
1
+
=
-
1
Q
2
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
2
i
τ
^
1
/
2
ρ
1
*
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
2
k
-
2
μ
-
1
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
W
k
-
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
}
,
where
(40)
Q
2
=
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
×
{
+
(
-
(
1
+
2
μ
+
2
k
)
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
W
-
k
+
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
)
(
1
+
2
μ
+
4
i
τ
^
1
/
2
ρ
1
*
)
4
i
τ
^
1
/
2
ρ
1
*
W
-
k
,
μ
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
-
(
1
+
2
μ
+
2
k
)
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
W
-
k
+
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
)
}
+
(
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
2
i
τ
^
1
/
2
ρ
1
*
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
-
W
-
k
,
μ
(
-
2
i
τ
^
1
/
2
ρ
1
*
)
×
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
1
*
)
2
i
τ
^
1
/
2
ρ
1
*
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
1
*
)
+
(
2
k
-
2
μ
-
1
)
(
2
i
τ
^
1
/
2
ρ
1
*
)
1
/
2
W
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
1
*
)
}
)
.
Using (17) we may then find
m
+
(
τ
^
)
and
m
-
(
τ
^
)
, respectively, at
r
=
b
and
r
=
0
as follows:
(41)
m
+
(
τ
^
)
=
-
lim
r
→
b
Θ
(
r
,
τ
^
)
Φ
(
r
,
τ
^
)
,
m
-
(
τ
^
)
=
-
lim
r
→
0
Θ
(
r
,
τ
^
)
Φ
(
r
,
τ
^
)
,
where
Θ
and
Φ
are as given in (34) and (35) with
α
j
,
β
j
,
j
=
0,1
as obtained in (37) to (40). From (19) we then have
(42)
Ψ
+
+
(
r
,
τ
^
)
≡
Θ
(
r
,
τ
^
)
+
m
+
(
τ
^
)
Φ
(
r
,
τ
^
)
∈
L
2
(
[
ρ
1
*
,
b
]
)
,
Ψ
+
-
(
r
,
τ
^
)
≡
Θ
(
r
,
τ
^
)
+
m
-
(
τ
^
)
Φ
(
r
,
τ
^
)
∈
L
2
(
[
0
,
ρ
1
*
]
)
.
Therefore, the eigenfunctions for
I
3
∪
I
4
are
(43)
Ψ
+
(
r
,
τ
^
)
=
{
Ψ
+
+
(
r
,
τ
^
)
,
r
∈
[
ρ
1
*
,
b
]
Ψ
+
-
(
r
,
τ
^
)
,
r
∈
[
0
,
ρ
1
*
]
,
which automatically satisfy the boundary conditions at
b
.
Now, for the interval
I
1
∪
I
2
, we let
(44)
Θ
(
r
,
τ
^
)
=
α
0
-
M
k
,
μ
(
2
i
τ
^
1
/
2
r
)
+
α
1
-
M
k
,
-
μ
(
2
i
τ
^
1
/
2
r
)
,
Φ
(
r
,
τ
^
)
=
β
0
-
W
k
,
μ
(
2
i
τ
^
1
/
2
r
)
+
β
1
-
W
-
k
,
μ
(
2
i
τ
^
1
/
2
r
)
.
Proceeding as above we obtain
(45)
α
0
-
=
1
Q
3
{
(
1
-
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
4
i
τ
^
1
/
2
ρ
2
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
+
(
1
-
2
μ
-
2
k
)
(
1
-
2
μ
)
(
2
i
τ
^
1
/
2
ρ
2
*
)
M
k
-
1
/
2
,
-
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
2
*
)
}
,
α
1
-
=
1
Q
3
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
+
(
2
μ
-
2
k
+
1
)
(
1
+
2
μ
)
(
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
×
M
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
2
*
)
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
}
,
where
(46)
Q
3
=
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
×
{
(
1
-
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
4
i
τ
^
1
/
2
ρ
2
*
M
k
,
-
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
+
(
1
-
2
μ
-
2
k
)
(
1
-
2
μ
)
(
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
M
k
-
1
/
2
,
μ
-
1
/
2
(
2
i
τ
^
1
/
2
ρ
2
*
)
}
+
(
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
-
M
k
,
-
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
×
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
+
(
2
μ
-
2
k
+
1
)
(
1
+
2
μ
)
(
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
M
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
2
*
)
}
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
M
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
)
.
Using the same process we find that
(47)
β
0
-
=
1
Q
4
{
(
1
+
2
μ
+
4
i
τ
^
1
/
2
ρ
2
*
)
4
i
τ
^
1
/
2
ρ
2
*
W
-
k
,
μ
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
+
(
-
(
1
+
2
μ
+
2
k
)
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
W
-
k
+
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
)
}
,
β
1
-
=
-
1
Q
4
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
+
(
2
k
-
2
μ
-
1
)
(
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
W
k
-
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
}
,
where
(48)
Q
4
=
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
×
{
(
-
(
1
+
2
μ
+
2
k
)
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
W
-
k
+
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
)
(
1
+
2
μ
+
4
i
τ
^
1
/
2
ρ
2
*
)
4
i
τ
^
1
/
2
ρ
2
*
W
-
k
,
μ
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
+
+
(
-
(
1
+
2
μ
+
2
k
)
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
W
-
k
+
1
/
2
,
μ
+
1
/
2
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
)
}
+
(
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
-
W
-
k
,
μ
(
-
2
i
τ
^
1
/
2
ρ
2
*
)
×
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
+
(
2
k
-
2
μ
-
1
)
(
2
i
τ
^
1
/
2
ρ
2
*
)
1
/
2
W
k
-
1
/
2
,
μ
+
1
/
2
(
2
i
τ
^
1
/
2
ρ
2
*
)
}
{
(
1
+
2
μ
-
4
i
τ
^
1
/
2
ρ
2
*
)
2
i
τ
^
1
/
2
ρ
2
*
W
k
,
μ
(
2
i
τ
^
1
/
2
ρ
2
*
)
)
.
By using (44) to (48) we obtain
(49)
Ψ
-
-
(
r
,
τ
^
)
≡
Θ
(
r
,
τ
^
)
+
n
+
(
τ
^
)
Φ
(
r
,
τ
^
)
∈
L
2
(
[
a
,
ρ
2
*
]
)
Ψ
-
+
(
r
,
τ
^
)
≡
Θ
(
r
,
τ
^
)
+
n
-
(
τ
^
)
Φ
(
r
,
τ
^
)
∈
L
2
(
[
ρ
2
*
,
0
]
)
,
where
(50)
n
+
(
τ
^
)
=
-
lim
r
→
0
Θ
(
r
,
τ
^
)
Φ
(
r
,
τ
^
)
,
n
-
(
τ
^
)
=
-
lim
r
→
a
Θ
(
r
,
τ
^
)
Φ
(
r
,
τ
^
)
.
Therefore, the eigenfunctions for
I
1
∪
I
2
are
(51)
Ψ
-
(
r
,
τ
^
)
=
{
Ψ
-
-
(
r
,
τ
^
)
,
r
∈
[
a
,
ρ
2
*
]
Ψ
-
+
(
r
,
τ
^
)
,
r
∈
[
ρ
2
*
,
0
]
,
and these automatically satisfy the boundary conditions at
a
. From (43) and (51) we find that
(52)
lim
r
→
0
-
Ψ
-
(
r
,
τ
^
)
=
lim
r
→
0
+
Ψ
+
(
r
,
τ
^
)
=
0
.
We then conclude that the eigenfunctions solutions to (5) with the accompanying boundary conditions would be
(53)
Ψ
(
r
,
τ
^
)
=
{
Ψ
+
(
r
,
τ
^
)
,
r
∈
[
a
,
0
]
Ψ
_
(
r
,
τ
^
)
,
r
∈
[
0
,
b
]
.
This completes the proof of the theorem.