In this paper, the vectorial Sturm-Liouville operator is considered, where Q(x) is an integrable m × m matrix-valued function defined on the interval [0,π] The authors prove that m²+1 characteristic functions can determine the potential function of a vectorial Sturm-Liouville operator uniquely. In particular, if Q(x) is real symmetric, then characteristic functions can determine the potential function uniquely. Moreover, if only the spectral data of self-adjoint problems are considered, then m² + 1 spectral data can determine Q(x) uniquely.

The study on inverse spectral problems for the vectorial Sturm-Liouville 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, and are in and is an integrable matrix-valued function. We use L_m = L(Q, h, H) to denote the boundary problem (1.1)-(1.2). For the case m = 1, (1.1)-(1.2) is a scalar Sturm-Liouville equation. The scalar Sturm-Liouville equation often arises from some physical problems, for example, vibration of a string, quantum mechanics and geophysics. Numerous research results for this case have been established by renowned mathematicians, notably Borg, Gelfand, Hochstadt, Krein, Levinson, Levitan, Marchenko, Gesztesy, Simon and their coauthors and followers (see [1-9] and references therein). For the case m ≥ 2, some interesting results had been obtained (see [10-20]). In particular, for m = 2 and Q(x) is a two-by-two real symmetric matrix-valued smooth functions defined in the interval [0, π] Shen [18] showed that five spectral data can determine Q(x) uniquely. More precisely speaking, he considered the inverse spectral problems of the vectorial Sturm-Liouville equation:

where Q₂(x) is a real symmetric matrix-valued function defined in the interval [0, π]. Let σ_D (Q) denotes the Dirichlet spectrum of (1.3), σ_ND (Q) the Neumann-Dirichlet 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 Q₂(x) and be two continuous two-by-two real symmetric matrix-valued functions defined on [0, π]. Suppose that and for j = 1, 2, 3, then on [0, π].

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 matrix-valued Sturm-Liouville equation

Some uniqueness theorems for vectorial Sturm-Liouville equation are obtained in the last section.

Let and be two solutions of equation (1.5) which satisfy the initial conditions

where 0_m is the m × m zero matrix, is the m × m identity matrix and δ_ij is the Kronecker symbol. For given complex-valued matrices h and H, we denote

be two solutions of equation (1.5) so that φ(x, λ) = C(x, λ) + S(x, λ)h and which satisfy the boundary conditions

Then, . The matrix is called the Weyl matrix for L_m (Q, h, H). In 2006, Yurko proved that:

Theorem 2.1 ([20], Theorem 1). Let and denote Weyl matrices of the problems L_m (Q, h, H) and separately. Suppose , then , and .

Also note that from [20], we have

where is a matrix solution of equation (1.5) associated with the conditions ψ(π, λ) = I_m and ψ' (π, λ) = -H. It is not difficult to see that both Φ(x, λ) and are meromorphic in λ and the poles of are coincided with the eigenvalues of L_m (Q, h, H). Moreover, we have

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 Sturm-Liouville 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 denotes the analogous object related to .

Theorem 2.3. Suppose that for (i, j) = (0, 0) or 1 ≤ i, j ≤ m then , and .

Proof. Since

and

we have that

for each i = 1, ..., m, that is,

By Crammer's rule,

Applying Theorem 2.1, we conclude that , and . □

Lemma 2.4. Suppose that h, H are real symmetric matrices and Q(x) is a real symmetric matrix-valued function. Then, is real symmetric for all λ ∈ ℝ.

Proof. Let

For λ ∈ ℝ,

This leads to

Now let

Then

and

Since

we have

i.e., is real symmetric for all λ ∈ ℝ. □

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 matrix-valued function.

Corollary 2.5. Let L_m(h, H, Q) and be two real symmetric problems. Suppose that for (i, j) = (0, 0) or 1 ≤ i ≤ j ≤ m, then , and .

Proof. For λ ∈ ℝ. both and are real symmetric. Moreover,

Hence, for λ ∈ ℝ and 1 ≤ i, j ≤ m. This leads to for λ ∈ ℝ. We conclude that and for λ ∈ ℂ. This completes the proof. □

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 Hence, Γ_ij + Γ^ij = I_m. Let Θ_ij(λ) be the characteristic function of the self-adjoint problem

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, then

for 1 ≤ i, j ≤ m. For simplicity, we write

Now, we are going to focus on self-adjoint problems. For a self-adjoint 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 are two real symmetric problems. If the conditions

(1) for (i, j) = (0, 0) or 1 ≤ i ≤ j ≤ m,

(2) for 1 ≤ i < j ≤ m.,

are satisfied, then , and a.e on [0, 1].

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 for 1 ≤ i ≤ j ≤ m,

(2) for i = 0, 1, ..., m,

(3) for 1 ≤ i < j ≤ m.

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 Sturm-Liouville 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 are both diagonals. If for k = 0, 1, ..., m, then , and .

Proof. Since L_m(Q, h, H) and are both diagonals, we know

is diagonal and so is . Hence,

Moreover,

for k = 1, 2, ..., m. This implies. . Applying Theorem 2.1 again, we have , and . □

The authors declare that they have no competing interests.

Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.

This work was partially supported by the National Science Council, Taiwan, ROC.

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