Abstract
In this paper, the vectorial SturmLiouville operator
Keywords:
Inverse spectral problems; SturmLiouville equation1. Introduction
The study on inverse spectral problems for the vectorial SturmLiouville differential equation
on a finite interval is devoted to determine the potential matrix Q(x) from the spectral data of (1.1) with boundary conditions
where λ is the spectral parameter,
where Q_{2}(x) is a real symmetric matrixvalued function defined in the interval [0, π]. Let σ_{D }(Q) denotes the Dirichlet spectrum of (1.3), σ_{ND }(Q) the NeumannDirichlet spectrum of (1.3) and σ_{j }(Q) the spectrum of (1.3) with boundary condition
for j = 1, 2, 3, where
is a real symmetric matrix and {(α_{j}, β_{j}, γ_{j},), j = 1, 2, 3} is linearly independent over ℝ. Then
Theorem 1.1 ([18], Theorem 4.1). Let Q_{2}(x) and
The purpose of this paper is to generalize the above theorem for the case m ≥ 3. The idea we use is the Weyl's matrix for matrixvalued SturmLiouville equation
Some uniqueness theorems for vectorial SturmLiouville equation are obtained in the last section.
2. Main Results
Let
where 0_{m }is the m × m zero matrix,
be two solutions of equation (1.5) so that φ(x, λ) = C(x, λ) + S(x, λ)h and
Then,
Theorem 2.1 ([20], Theorem 1). Let
Also note that from [20], we have
where
where Adj(A) denotes the adjoint matrix of A and det(A) denotes the determinant of A. In the remaining of this section, we shall prove some uniqueness theorems for vectorial
SturmLiouville equations. Let
and B(0, 0) = 0_{m }The characteristic function for this boundary value problem L_{m }(Q, h + B(i, j), H) is
The first problem we want to study is as following:
Problem 1. How many Δ_{ij }(λ) can uniquely determine Q, h and H? where (i, j) = (0, 0) or 1 ≤ i, j ≤ m
To find the solution of Problem 1, we start with the following lemma
Lemma 2.2. Let B(i, j) = [b_{rs}]_{m×m }and Δ_{ij }be defined as above. Then
where φ_{k }(π, λ) is the kth column of φ (π, λ) and S_{k }(π, λ) the kth column of S (π, λ) for k = 1, 2, 3, ..., m.
Proof. Let
Then
□
Next, we shall prove the first main theorem. For simplicity, if a symbol α denotes an object related to L_{m}(Q, h, H), then the symbol
Theorem 2.3. Suppose that
Proof. Since
and
we have that
for each i = 1, ..., m, that is,
By Crammer's rule,
Applying Theorem 2.1, we conclude that
Lemma 2.4. Suppose that h, H are real symmetric matrices and Q(x) is a real symmetric matrixvalued function. Then,
Proof. Let
For λ ∈ ℝ,
This leads to
Now let
Then
and
Since
we have
i.e.,
Definition 2.1. We call L_{m}(h, H, Q) a real symmetric problem if h, H are real symmetric matrices and Q(x) is a real symmetric matrixvalued function.
Corollary 2.5. Let L_{m}(h, H, Q) and
Proof. For λ ∈ ℝ. both
Hence,
From now on, we let L_{m}(Q, h, H) be a real symmetric problem. We would like to know that how many spectral data can determine the problem L_{m}(Q, h, H) if we require all spectral data come from real symmetric problems. Denote
where
associated with some boundary conditions
then
where V (L_{j}) denotes the jth column of (V(L)) for a m × m matrix L. Similarly, we denote Ω_{ij}(λ) the characteristic function of the real symmetric problem
for 1 ≤ i, j ≤ m. For simplicity, we write
Now, we are going to focus on selfadjoint problems. For a selfadjoint problem L_{m}(Q, h, H) all its eigenvalues are real and the geometric multiplicity of an eigenvalue is equal to its algebraic multiplicity. Moreover, if we denote {(λ_{i}, m_{i})}_{i = 1,∞ }the spectral data of L_{m}(Q, h, H) where m_{i }is the multiplicity of the eigenvalue λ_{i }of L_{m}(Q, h, H) then the characteristic function of L_{m}(Q, h, H) is
where C is determined by {(λ_{i}, m_{i})}_{i = 1,∞}. This means that the spectral data determined the corresponding characteristic function.
Theorem 2.6. Assuming that L_{m}(Q, h, H) and
(1)
(2)
are satisfied, then
Proof. Note that for any problem L_{m}(Q, h, H) we have
Similarly,
Moreover, by the assumptions and Lemma 2.4, we have M_{ij}(λ) = M_{ji}(λ) Hence,
(1) Δ_{ij }(λ) = Δ_{ji}(λ) and
(2)
(3)
This implies
The authors want to emphasis that for n = 1, the result is classical; for n = 2, Theorem 2.6 leads to Theorem 1.1. Shen also shows by providing an example that 5 minimal number of spectral sets can determine the potential matrix uniquely (see [18]).
The readers may think that if all Q, h and H are diagonals then L_{m}(Q, h, H) is an uncoupled system. Hence, everything for the operator L_{m}(Q, h, H) can be obtained by applying inverse spectral theory for scalar SturmLiouville equation. Unfortunately, it is not true. We say L_{m}(Q, h, H) diagonal if all Q, h and H are diagonals.
Corollary 2.7. Suppose L_{m}(Q, h, H) and
Proof. Since L_{m}(Q, h, H) and
is diagonal and so is
Moreover,
for k = 1, 2, ..., m. This implies.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.
Footnote
This work was partially supported by the National Science Council, Taiwan, ROC.
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