1 Introduction
In this work, we consider a star graph G with vertex set
V
=
{
υ
0
,
υ
1
,
…
,
υ
ν
}
and nonhomogeneous
edge set
E
=
{
e
1
,
e
2
,
…
,
e
ν
}
, where
υ
1
,
υ
2
,
…
,
υ
ν
are the boundary vertices,
υ
0
is the interior vertex, and
e
j
=
[
υ
j
,
υ
0
]
for
j
=
1
,
…
,
ν
, where
ν
≥
2
is a positive integer.
We suppose that the length of the edge
e
j
∈
E
is equal to
ℓ
j
, where
ℓ
j
=
ℓ
>
0
,
j
=
1
,
…
,
ν
. We introduce a parameter x for each
edge
e
j
∈
E
,
x
∈
[
0
,
ℓ
j
]
. The following choice of
orientation is convenient for us:
x
=
0
corresponds to the boundary vertices
υ
1
,
υ
2
,
…
,
υ
ν
,
and
x
=
ℓ
j
to the interior vertex
υ
0
.
Let
y
=
{
y
j
(
x
)
}
j
=
1
,
…
,
ν
be a vector function
on G. Consider a second order differential expression
(
L
j
y
j
)
(
x
)
:=
-
y
j
′′
(
x
)
+
q
j
(
x
)
y
j
(
x
)
,
x
∈
(
0
,
π
4
)
∪
(
π
4
,
π
2
)
∪
(
π
2
,
π
)
,
j
=
1
,
…
,
ν
,
where
q
j
(
x
)
,
j
=
1
,
…
,
ν
, are real-valued functions from
L
2
[
0
,
π
]
.
The domain of expressions
L
j
,
j
=
1
,
…
,
ν
, is defined by
D
(
L
j
)
:=
{
y
j
∈
W
2
2
[
(
0
,
π
)
∖
{
π
4
,
π
2
}
]
∩
W
2
0
[
(
0
,
π
)
]
:
y
j
′
∈
A
C
[
0
,
π
]
,
L
j
y
j
∈
L
2
[
0
,
π
]
,
y
j
(
0
)
=
0
,
y
j
(
π
4
+
0
)
=
y
j
(
π
4
-
0
)
,
y
j
′
(
π
4
+
0
)
-
y
j
′
(
π
4
-
0
)
=
α
j
y
j
(
π
4
)
,
y
j
(
π
2
+
0
)
-
y
j
(
π
2
-
0
)
=
β
j
′
y
j
(
π
2
)
,
y
j
′
(
π
2
+
0
)
=
y
j
′
(
π
2
-
0
)
,
y
j
′
(
π
)
=
0
}
.
We study the boundary value problem
L
=
L
(
q
,
h
)
for the
Sturm–Liouville equations on a star graph G with nonhomogeneous edges
(1.1)
(
L
j
y
j
)
(
x
)
=
λ
y
j
(
x
)
,
x
∈
[
0
,
π
]
,
j
=
1
,
…
,
ν
,
λ
=
s
2
,
with the matching conditions
(1.2)
y
1
(
ℓ
1
)
=
y
j
(
ℓ
j
)
,
j
=
2
,
…
,
ν
,
∑
j
=
1
ν
y
j
′
(
ℓ
j
)
=
0
in the interior vertex
υ
0
, and the boundary conditions
(1.3)
y
j
′
(
0
)
-
h
j
y
j
(
0
)
=
0
,
j
=
1
,
…
,
ν
.
In the boundary vertices
υ
1
,
…
,
υ
ν
, where
h
j
are
real,
h
=
{
h
j
}
j
=
1
,
…
,
ν
, and
q
=
{
q
j
(
x
)
}
j
=
1
,
…
,
ν
, is called potential
function on a star graph G.
Suppose
e
j
,
j
=
1
,
…
,
ν
, branches are nonhomogeneous, that is, the discontinuity conditions
(1.4)
{
y
j
(
π
4
+
0
)
=
y
j
(
π
4
-
0
)
=
y
j
(
π
4
)
,
y
j
′
(
π
4
+
0
)
-
y
j
′
(
π
4
-
0
)
=
α
j
y
j
(
π
4
)
,
{
y
j
(
π
2
+
0
)
-
y
j
(
π
2
-
0
)
=
β
j
y
j
′
(
π
2
)
,
y
j
′
(
π
2
+
0
)
=
y
j
′
(
π
2
-
0
)
=
y
j
′
(
π
2
)
are met. Here
α
j
,
β
j
,
j
=
1
,
…
,
ν
, are real and
α
j
>
0
.
Inverse nodal problems on graphs for the Sturm–Liouville operators are
studied in sufficient detail in [3, 10]. The works [7, 2, 4, 6, 12, 13] deal with inverse nodal problems for the Sturm–Liouville operators on star-shaped graphs with equal edges.
The work [9] is related to inverse nodal problems on an equal edge graph with loops.
In particular, the asymptotic expressions of eigenvalues are difficult to find.
For inverse nodal problems on graphs, Currie and Watson [5] proved that the
specification of the spectrum and the set of all nodal points uniquely determine the potential.
This paper is organized as follows. In Section 2, we find asymptotical
formulas of eigenvalues of the problem L. Section 3 is concerned with the
asymptotical formulas of the nodal points and an inverse nodal problem.
2 Properties of the spectrum
Let
C
j
(
x
,
λ
)
,
S
j
(
x
,
λ
)
,
j
=
1
,
…
,
ν
,
be solutions of equation (1.1) under the initial conditions
C
j
(
0
,
λ
)
=
S
j
′
(
0
,
λ
)
=
1
,
C
j
′
(
0
,
λ
)
=
S
j
(
0
,
λ
)
=
0
, and denote by
φ
j
(
x
,
λ
)
,
j
=
1
,
…
,
ν
, the solutions of equation (1.1) satisfying the initial
conditions
φ
j
(
0
,
λ
)
=
1
,
φ
j
′
(
0
,
λ
)
=
h
j
.
For each fixed
x
∈
[
0
,
π
]
, the functions
C
j
(
x
,
λ
)
,
S
j
(
x
,
λ
)
,
C
j
′
(
x
,
λ
)
,
S
j
′
(
x
,
λ
)
,
φ
j
(
x
,
λ
)
,
φ
j
′
(
x
,
λ
)
,
for
j
=
1
,
…
,
ν
, are entire in λ and
φ
j
(
x
,
s
)
=
C
j
(
x
,
s
)
+
h
j
S
j
(
x
,
s
)
.
From [8, 1], one get the following asymptotical formulas as
|
λ
|
→
∞
:
(2.1)
φ
j
(
x
,
λ
)
=
sin
k
x
k
-
cos
k
x
2
k
2
∫
0
x
q
j
(
t
)
𝑑
t
+
o
(
e
|
τ
|
x
k
2
)
,
x
<
π
4
,
(2.2)
φ
j
(
x
,
λ
)
=
sin
k
x
k
-
cos
k
x
2
k
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
+
α
j
cos
k
(
π
2
-
x
)
2
k
2
+
o
(
e
|
τ
|
x
k
2
)
,
π
4
<
x
<
π
2
,
φ
j
(
x
,
λ
)
=
β
j
2
(
cos
k
x
+
cos
k
(
π
-
x
)
)
+
sin
k
x
2
k
(
2
+
β
j
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
j
)
)
(2.3)
+
sin
k
(
π
-
x
)
2
k
(
β
j
2
(
α
j
+
∫
0
π
2
q
j
(
t
)
𝑑
t
-
∫
π
2
x
q
j
(
t
)
𝑑
t
)
)
+
o
(
e
|
τ
|
x
k
)
,
x
>
π
2
,
(2.4)
φ
j
′
(
x
,
λ
)
=
cos
k
x
+
sin
k
x
2
k
∫
0
x
q
j
(
t
)
𝑑
t
+
o
(
e
|
τ
|
x
k
)
,
x
<
π
4
,
(2.5)
φ
j
′
(
x
,
λ
)
=
cos
k
x
+
sin
k
x
2
k
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
j
)
+
α
j
sin
k
(
π
2
-
x
)
2
k
+
o
(
e
|
τ
|
x
k
)
,
π
4
<
x
<
π
2
,
φ
j
′
(
x
,
λ
)
=
β
j
k
2
(
-
sin
k
x
+
sin
k
(
π
-
x
)
)
+
cos
k
x
2
(
2
+
β
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
j
)
)
(2.6)
-
cos
k
(
π
-
x
)
2
(
β
j
2
(
α
j
+
∫
0
π
2
q
j
(
t
)
𝑑
t
-
∫
π
2
x
q
j
(
t
)
𝑑
t
)
)
+
o
(
e
|
τ
|
x
)
,
x
>
π
2
.
Then the solutions of equation (1.1) satisfying the boundary conditions (1.3)
and discontinuity conditions (1.4) are
(2.7)
y
j
(
x
,
s
)
=
H
j
(
s
)
φ
j
(
x
,
s
)
,
where
H
j
(
s
)
are functions independent of x.
Substituting (2.7) into the matching conditions (1.2), we obtain the
characteristic function of the problem L:
(2.8)
Δ
(
λ
)
=
∑
j
=
1
ν
φ
j
′
(
π
,
s
)
∏
j
≠
k
∈
{
1
,
…
,
ν
}
φ
j
(
π
,
s
)
=
∑
j
=
1
ν
[
β
j
2
(
-
k
sin
k
π
+
ω
o
j
cos
k
π
2
-
ω
1
j
2
)
+
o
(
e
|
τ
|
π
)
]
×
∏
j
≠
k
∈
{
1
,
…
,
ν
}
[
β
j
2
(
cos
k
π
+
1
)
+
sin
k
π
2
k
(
2
+
β
j
2
(
∫
0
π
q
j
(
t
)
d
t
+
α
j
)
)
+
o
(
e
|
τ
|
π
k
)
]
,
where
ω
o
j
=
4
β
+
α
j
+
∫
0
π
q
j
(
t
)
𝑑
t
,
ω
1
j
=
α
j
+
2
∫
π
2
π
q
j
(
t
)
𝑑
t
-
∫
0
π
q
j
(
t
)
𝑑
t
.
As defined in [12], if
λ
0
is a zero of
Δ
(
λ
)
,
λ
=
k
2
,
then the function
y
(
x
,
λ
0
)
=
y
(
x
,
k
0
2
)
of the form
y
(
x
,
λ
)
=
{
y
i
(
x
,
λ
)
}
i
=
1
,
…
,
ν
,
y
i
(
x
,
λ
)
=
H
i
(
λ
)
φ
i
(
x
,
λ
)
is an eigenfunction, and
λ
=
k
0
2
is an eigenvalue of problem (1.1)–(1.3).
Conversely, if
λ
0
is an eigenvalue, then the corresponding eigenfunction is of the form
y
(
x
,
λ
)
=
{
y
i
(
x
,
λ
)
}
i
=
1
,
…
,
ν
,
y
i
(
x
,
λ
)
=
H
i
(
λ
)
φ
i
(
x
,
λ
)
,
with
λ
=
λ
0
. Since
y
(
x
,
λ
0
)
≠
0
, the algebraic
system has a nontrivial solution: consequently,
Δ
(
λ
0
)
=
0
.
Let us introduce the auxiliary function
(2.9)
Δ
0
(
λ
)
=
∑
j
=
1
ν
β
j
2
(
-
k
sin
k
π
+
(
2
β
j
+
α
j
2
)
cos
k
π
-
α
j
2
)
∏
j
≠
k
∈
{
1
,
…
,
ν
}
[
β
j
2
(
cos
k
π
+
1
)
+
sin
k
π
2
k
(
2
+
α
j
β
j
2
)
]
=
ν
β
j
2
(
-
k
sin
k
π
+
(
2
β
j
+
α
j
2
)
cos
k
π
-
α
j
2
)
[
β
j
2
(
cos
k
π
+
1
)
+
sin
k
π
2
k
(
2
+
α
j
β
j
2
)
]
ν
-
1
.
Using (2.9) and the Rouche method (see [11]), we deduce that the
function of
Δ
0
(
λ
)
has countably many eigenvalues
s
n
o
=
{
k
n
o
}
∪
{
λ
n
o
}
∪
{
t
n
o
}
.
We arrange the zeros in the following way:
s
n
0
=
k
n
0
=
λ
n
0
=
n
,
n
∈
ℤ
,
Using [10, Lemma 2.1], we can obtain the following proposition.
Lemma 2.1.
The zeros of
Δ
(
s
)
can be enumerated as
(2.10)
t
n
=
t
n
(
0
)
+
o
(
1
)
.
Using Lemma 2.1, we can obtain following result.
Lemma 2.2.
The following asymptotic behavior for the set
{
t
n
}
of zeros of the function
Δ
(
s
)
is true:
(2.11)
s
n
=
n
+
(
-
1
)
n
2
π
ν
n
∑
j
=
1
ν
[
(
-
1
)
n
ω
0
j
-
ω
1
j
]
+
o
(
1
n
)
(2.12)
k
n
=
n
-
(
-
1
)
n
2
π
ν
n
(
2
β
+
(
1
-
(
-
1
)
n
)
α
2
)
∑
j
=
1
ν
[
(
-
1
)
n
ω
0
j
-
ω
1
j
]
+
o
(
1
n
)
(2.13)
λ
n
=
n
-
(
-
1
)
n
2
π
ν
n
∑
j
=
1
ν
[
(
-
1
)
n
ω
0
j
-
ω
1
j
]
+
o
(
1
n
)
for
j
=
1
,
…
,
ν
.
Proof.
Using Rouche’s theorem, for a sufficiently large integer
|
n
|
,
Δ
(
s
)
has exactly one zero in a suitable neighborhood of
s
n
(
0
)
=
n
. Set
(2.14)
s
n
=
n
+
ε
n
,
where
ε
n
=
o
(
1
)
as
|
n
|
→
+
∞
. Substituting (2.14) into
Δ
(
s
)
=
0
, from (2.8), we have
(2.15)
Δ
(
s
n
)
=
∑
j
=
1
ν
[
(
-
1
)
n
+
1
n
sin
ε
n
π
+
(
-
1
)
n
2
w
0
j
cos
ε
n
π
-
w
1
j
2
+
o
(
exp
(
|
τ
n
|
π
)
)
]
×
∏
j
≠
k
∈
{
1
,
…
,
ν
}
[
(
-
1
)
n
β
j
2
cos
ε
n
π
+
β
j
2
+
o
(
1
n
+
ε
n
exp
(
|
τ
n
|
π
)
)
]
=
0
.
From (2.15), we have
sin
ε
n
π
=
o
(
1
n
)
. Using the inversion formula, we get
(2.16)
ε
n
=
(
-
1
)
n
2
π
ν
n
∑
j
=
1
ν
[
(
-
1
)
n
ω
0
j
-
ω
1
j
]
+
o
(
1
n
)
,
Similarly, if
k
n
=
n
+
δ
n
and
λ
n
=
n
+
γ
n
are
written into
Δ
(
s
)
=
0
, then
(2.17)
δ
n
=
-
(
-
1
)
n
2
π
ν
n
(
2
β
+
(
1
-
(
-
1
)
n
)
α
2
)
∑
j
=
1
ν
[
(
-
1
)
n
ω
0
j
-
ω
1
j
]
+
o
(
1
n
)
,
(2.18)
γ
n
=
-
(
-
1
)
n
2
π
ν
n
∑
j
=
1
ν
[
(
-
1
)
n
ω
0
j
-
ω
1
j
]
+
o
(
1
n
)
are obtained.
In this case, if we make use of (2.14) and (2.16), we get (2.11).
∎
3 Inverse nodal problem
Let
(3.1)
y
(
x
,
s
)
=
{
y
j
(
x
,
s
)
}
j
=
1
,
…
,
ν
,
y
j
(
x
,
s
)
=
H
j
φ
j
(
x
,
s
)
,
where
H
j
,
j
=
1
,
…
,
ν
, are constants depending on s.
Then the function
y
(
x
,
s
)
satisfies (1.1), (1.3) and (1.4).
Here we set
s
n
=
s
n
(
1
)
. It follows from (2.11) that
(3.2)
s
n
=
n
+
R
(
ν
)
n
π
+
o
(
1
n
)
,
where
(3.3)
R
(
ν
)
=
(
-
1
)
n
2
ν
∑
j
=
1
ν
[
(
-
1
)
n
ω
0
j
-
ω
1
j
]
.
The eigenfunctions of problem (1.1)–(1.4) have the form
y
n
j
(
x
)
=
φ
j
(
x
,
s
n
)
.
We note that the
φ
j
(
x
,
s
n
)
are real-valued functions.
Substituting (3.2) into (2.1) and (2.2), we obtain the following asymptotic formulas for
|
n
|
→
+
∞
, uniformly in x:
(3.4)
φ
j
(
x
,
s
n
)
=
sin
n
x
n
+
1
n
2
(
x
π
R
(
ν
)
-
1
2
∫
0
x
q
j
(
t
)
𝑑
t
)
cos
n
x
+
o
(
1
n
2
)
,
x
∈
(
0
,
π
4
)
,
φ
j
(
x
,
s
n
)
=
sin
n
x
n
+
1
n
2
(
x
π
R
(
ν
)
-
1
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
)
cos
n
x
(3.5)
+
α
j
2
n
2
cos
n
(
π
2
-
x
)
+
o
(
1
n
2
)
,
x
∈
(
π
4
,
π
2
)
,
φ
j
(
x
,
s
2
m
)
=
sin
2
m
x
2
m
+
1
(
2
m
)
2
(
x
π
R
(
ν
)
-
1
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
j
)
+
α
j
2
(
-
1
)
m
)
cos
2
m
x
(3.6)
+
o
(
1
m
2
)
,
x
∈
(
π
4
,
π
2
)
,
φ
j
(
x
,
s
2
m
-
1
)
=
(
1
2
m
-
1
+
α
j
(
-
1
)
m
2
(
2
m
-
1
)
2
)
sin
(
2
m
-
1
)
x
+
1
(
2
m
-
1
)
2
(
x
π
R
(
ν
)
-
1
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
j
)
)
cos
(
2
m
-
1
)
x
(3.7)
+
o
(
1
m
2
)
,
x
∈
(
π
4
,
π
2
)
,
(3.8)
φ
j
(
x
,
s
n
)
=
β
j
2
(
1
+
(
-
1
)
n
)
cos
n
x
-
1
n
π
R
(
ν
)
A
(
x
)
sin
n
x
+
1
2
n
2
π
R
(
ν
)
B
(
x
)
cos
n
x
+
o
(
1
n
2
)
,
where
(3.9)
A
(
x
)
=
x
-
(
-
1
)
n
(
π
-
x
)
-
π
R
(
ν
)
(
1
+
β
4
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
)
+
(
-
1
)
n
π
R
(
ν
)
β
4
(
α
+
∫
0
π
2
q
j
(
t
)
𝑑
t
-
∫
π
2
x
q
j
(
t
)
𝑑
t
)
,
(3.10)
B
(
x
)
=
x
(
2
+
β
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
)
+
(
-
1
)
n
(
π
-
x
)
β
2
(
α
+
∫
0
π
2
q
j
(
t
)
𝑑
t
-
∫
π
2
x
q
j
(
t
)
𝑑
t
)
.
From the oscillation theorem, it is easy to see that the eigenfunction
φ
j
(
x
,
s
n
)
has exactly
|
n
|
(simple) zeros inside in the interval
(
0
,
ℓ
)
, namely
0
<
x
n
j
1
<
⋯
<
x
n
j
n
<
ℓ
for
n
>
0
,
0
<
x
n
j
0
<
x
n
j
-
1
<
⋯
<
x
n
j
n
+
1
<
ℓ
for
n
<
0
.
The set is
X
:=
{
x
n
j
k
}
n
∈
ℤ
,
j
=
1
,
…
,
ν
,
k
∈
ℤ
=
⋃
j
=
1
ν
X
j
,
where
X
j
=
{
x
n
j
k
:
n
>
N
0
∈
ℕ
,
k
=
1
,
…
,
n
}
∪
{
x
n
j
k
:
n
<
-
N
0
,
k
=
0
,
n
+
1
}
is called the set of nodal points of problem (1.1)–(1.4).
Define
X
t
:=
{
x
2
m
-
t
,
j
k
}
,
t
=
0
,
1
. Clearly,
X
(
0
)
∪
X
(
1
)
=
X
.
The set
X
j
,
j
=
1
,
…
,
ν
, is called nodal point set on the edge
ℓ
j
.
Clearly, the set
X
j
is dense in
(
0
,
ℓ
j
)
for
j
=
1
,
…
,
ν
,
with respect to the eigenvalues
s
n
2
.
Taking (3.4) and (3.5) into account, we obtain the following asymptotic formulas for nodal points
as
|
n
|
→
+
∞
, uniformly in
k
∈
ℤ
:
(3.11)
x
n
j
k
=
k
n
π
+
1
n
2
(
1
2
∫
0
x
q
j
(
t
)
𝑑
t
-
x
π
R
(
ν
)
)
+
o
(
1
n
2
)
,
x
n
j
k
∈
(
0
,
π
4
)
,
(3.12)
x
n
j
k
=
k
n
π
+
1
n
2
(
-
x
π
R
(
ν
)
+
1
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
-
α
2
(
-
1
)
m
)
+
o
(
1
n
2
)
,
n
=
2
m
,
x
n
j
k
∈
(
π
4
,
π
2
)
,
(3.13)
x
n
j
k
=
k
n
π
+
1
n
2
(
-
x
π
R
(
ν
)
+
1
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
)
+
o
(
1
n
2
)
,
n
=
2
m
-
1
,
x
n
j
k
∈
(
π
4
,
π
2
)
,
(3.14)
x
n
j
k
=
k
-
1
2
n
π
+
2
A
(
x
)
R
(
ν
)
n
2
π
β
(
1
+
(
-
1
)
n
)
+
o
(
1
n
3
)
,
x
n
j
k
∈
(
π
2
,
π
)
.
Theorem 3.1.
Fix
t
=
0
,
1
,
j
=
1
,
…
,
ν
, and
x
∈
[
0
,
ℓ
j
]
.
Suppose that
{
x
n
j
k
}
∈
X
j
t
is chosen such that
lim
|
n
|
→
+
∞
x
n
j
k
=
x
.
Then there exists a finite limit
(3.15)
f
j
t
(
x
)
:=
lim
|
n
|
→
+
∞
n
2
(
x
n
j
k
-
k
n
π
)
such that
(3.16)
f
j
t
(
x
)
=
1
2
∫
0
x
q
j
(
t
)
𝑑
t
-
x
π
R
(
ν
)
,
x
∈
[
0
,
π
4
]
,
(3.17)
f
j
t
(
x
)
=
1
2
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
-
x
π
R
(
ν
)
,
x
∈
[
π
4
,
π
2
]
,
g
j
t
(
x
)
:=
lim
|
n
|
→
+
∞
n
2
(
x
n
j
k
-
k
-
1
2
n
π
)
,
g
j
t
(
x
)
:=
2
A
j
t
(
x
)
R
(
ν
)
π
β
(
1
+
(
-
1
)
t
)
,
where
A
j
t
(
x
)
=
x
-
(
-
1
)
t
(
π
-
x
)
-
π
R
(
ν
)
(
1
+
β
4
(
∫
0
x
q
j
(
t
)
𝑑
t
+
α
)
)
+
(
-
1
)
t
π
R
(
ν
)
β
4
(
α
+
∫
0
π
2
q
j
(
t
)
𝑑
t
-
∫
π
2
x
q
j
(
t
)
𝑑
t
)
.
Proof.
If we use the asymptotical formulas (3.11)–(3.14), we get that
(3.18)
n
2
(
x
n
j
k
-
k
n
π
)
=
1
2
∫
0
x
n
j
k
q
j
(
t
)
𝑑
t
-
x
n
j
k
π
R
(
ν
)
+
o
(
1
)
,
(3.19)
n
2
(
x
n
j
k
-
k
n
π
)
=
1
2
(
∫
0
x
n
j
k
q
j
(
t
)
𝑑
t
+
α
)
-
x
n
j
k
π
R
(
ν
)
+
o
(
1
)
,
n
2
(
x
n
j
k
-
k
-
1
2
n
π
)
=
2
A
j
t
(
x
n
j
k
)
R
(
ν
)
π
β
(
1
+
(
-
1
)
t
)
+
o
(
1
n
)
.
Since
lim
|
n
|
→
+
∞
x
n
j
k
=
x
and
lim
|
n
|
→
+
∞
A
j
t
(
x
n
j
k
)
=
A
j
t
(
x
)
,
by (3.18) and (3.19), we conclude that, as
|
n
|
→
+
∞
,
the limits of the left-hand sides of (3.18) and (3.19) exist, and (3.16) and (3.17) hold. Thus,
Theorem 3.1 is proved.
∎
Let us now state a uniqueness theorem and present a contractive procedure for solving the inverse nodal problem.
Theorem 3.2.
Let
X
j
(
0
)
⊂
X
j
be a subset of nodes which is dense in
(
0
,
ℓ
j
)
for
j
=
1
,
…
,
ν
. Then the specification of
⋃
j
=
1
ν
X
j
(
0
)
uniquely determines the potential
q
j
(
x
)
-
〈
q
〉
a.e. on
(
0
,
ℓ
j
)
,
and the coefficients
h
j
,
j
=
1
,
…
,
ν
, of the boundary
conditions. The potentials
q
j
(
x
)
-
〈
q
〉
and the numbers
h
j
can be constructed via the following algorithm:
For
j
=
1
,
…
,
ν
and each
x
∈
[
0
,
ℓ
j
]
, we
choose a sequence
{
x
n
j
k
}
⊂
X
j
(
0
)
such that
lim
|
n
|
→
+
∞
x
n
j
k
=
x
.
From (
3.15
), we find the function
f
j
t
(
x
)
and
calculate values for
f
j
t
(
x
)
at
x
=
0
, i.e.
(3.20)
h
j
=
1
2
f
j
t
(
0
)
,
j
=
1
,
…
,
ν
.
The functions
q
j
(
x
)
can be determined as
(3.21)
q
j
(
x
)
-
〈
q
j
〉
=
2
(
f
j
t
)
′
+
(
-
1
)
t
π
ν
∑
i
=
1
ν
[
(
-
1
)
t
(
4
β
+
α
)
-
α
]
,
x
∈
[
0
,
π
4
]
,
(3.22)
〈
q
j
〉
:=
(
-
1
)
t
π
ν
∑
i
=
1
ν
[
(
1
+
(
-
1
)
t
)
∫
0
π
q
i
(
t
)
𝑑
t
-
2
∫
π
2
π
q
i
(
t
)
𝑑
t
]
,
x
∈
[
0
,
π
4
]
,
(3.23)
q
j
(
x
)
-
〈
q
j
〉
=
2
(
f
j
t
)
′
+
(
-
1
)
t
π
ν
∑
i
=
1
ν
[
(
-
1
)
t
(
4
β
+
α
)
-
α
]
,
x
∈
[
π
4
,
π
2
]
,
(3.24)
〈
q
j
〉
:=
(
-
1
)
t
π
ν
∑
i
=
1
ν
[
(
1
+
(
-
1
)
t
)
∫
0
π
q
i
(
t
)
𝑑
t
-
2
∫
π
2
π
q
i
(
t
)
𝑑
t
]
,
x
∈
[
π
4
,
π
2
]
,
(3.25)
q
j
(
x
)
-
〈
q
j
〉
=
-
2
(
g
j
t
)
′
+
2
(
-
1
)
t
π
β
ν
∑
i
=
1
ν
[
(
-
1
)
t
(
4
β
+
α
)
-
α
]
,
x
∈
[
π
2
,
π
]
,
(3.26)
〈
q
j
〉
:=
2
(
-
1
)
t
π
β
ν
∑
i
=
1
ν
[
(
1
+
(
-
1
)
t
)
∫
0
π
q
i
(
t
)
𝑑
t
-
2
∫
π
2
π
q
i
(
t
)
𝑑
t
]
,
x
∈
[
π
2
,
π
]
.
Proof.
Formulas (3.20) and (3.21) can be derived from (3.17) step by step. We obtain the following reconstruction procedure:
Taking value for
f
j
t
(
x
)
at
x
=
0
yields
h
j
=
1
2
f
j
t
(
0
)
,
j
=
1
,
…
,
ν
.
After the
h
j
are reconstructed, taking the derivatives of the function
g
j
t
(
x
)
,
we obtain (3.21).
∎
