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Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval
Boundary Value Problems volume 2011, Article number: 40 (2011)
Abstract
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.
1. Introduction
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.
2. Main Results
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) andseparately. 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×mand 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 j th 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, anda.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 . □
Footnote
This work was partially supported by the National Science Council, Taiwan, ROC.
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Chang, TH., Shieh, CT. Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval. Bound Value Probl 2011, 40 (2011). https://doi.org/10.1186/1687-2770-2011-40
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DOI: https://doi.org/10.1186/1687-2770-2011-40
Keywords
- Inverse spectral problems
- Sturm-Liouville equation