Abstract
In this paper, the vectorial SturmLiouville operator is considered, where Q(x) is an integrable m × m matrixvalued function defined on the interval [0,π] The authors prove that m^{2}+1 characteristic functions can determine the potential function of a vectorial SturmLiouville 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 selfadjoint problems are considered, then m^{2 }+ 1 spectral data can determine Q(x) uniquely.
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, and are in and is an integrable matrixvalued 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 SturmLiouville equation. The scalar SturmLiouville 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 [19] and references therein). For the case m ≥ 2, some interesting results had been obtained (see [1020]). In particular, for m = 2 and Q(x) is a twobytwo real symmetric matrixvalued 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 SturmLiouville equation:
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 be two continuous twobytwo real symmetric matrixvalued 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 matrixvalued SturmLiouville equation
Some uniqueness theorems for vectorial SturmLiouville 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 complexvalued 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 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 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 matrixvalued 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 matrixvalued 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 selfadjoint 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 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 are two real symmetric problems. If the conditions
(1) for (i, j) = (0, 0) or 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,
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 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 . □
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.
References

Borg, G: Eine Umkehrung der SturmLiouvilleschen Eigenwertaufgabe Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78, 1–96 (1945)

Gesztesy, F, Simon, B: On the determination of a potential from three spectra. In: Differential Operators and Spectral Theory. Am Math Soc Transl Ser 2, pp. 85–92. American Mathematical Society, Providence, RI (1999)

Hochstadt, H: The inverse SturmLiouville problem. Commun Pure Appl Math. 26, 75129 (1973)

Hochstadt, H, Lieberman, B: An inverse SturmLiouville problem with mixed given data. SIAM J Appl Math. 34, 67680 (1978)

Kren, MG: Solution of the inverse SturmLiouville problem. Dokl Akad Nauk SSSR. 76, 214 (1951)

Levitan, BM: Inverse SturmLiouville Problems. VNM, Utrecht (1987)

Levitan, BM, Gasymov, MG: Determination of a differential equation by two of its spectra. Russ Math Surv. 19, 163 (1964). Publisher Full Text

Marchenko, VA: SturmLiouville Operators and Applications. Birkhauser, Basel (1986)

Yurko, VA: Method of spectral mappings in the inverse problem theory. Inverse and IllPosed Problems Series (2002)

Andersson, E: On the Mfunction and BorgMarchenko theorems for vectorvalued SturmLiouville equations. In: J Math Phys. 44(12), 6077–6100

Carlson, R: An inverse problem for the matrix Schrödinger equation. J Math Anal Appl. 267, 564–575 (2002). Publisher Full Text

Chern, HH, Shen, CL: On the ndimensional Ambarzumyan's theorem. In: Inverse Probl. 13(1), 15–18

Clarka, S, Gesztesy, F, Holdenc, H, Levitand, BM: BorgType theorems for matrixvalued Schrödinger operators. J Differ Equ. 167(1), 181–210 (2000). Publisher Full Text

Gesztesy, F, Kiselev, A, Makarov, KA: Uniqueness results for matrixvalued Schrödinger, Jacobi, and DiracType operators. Math Nachr. 239240(1), 103–145 (2002). Publisher Full Text

Jodeit, M, Levitan, BM: The isospectrality problem for the classical SturmLiouville equation. Adv Differ Equ. 22, 297–318 (1997)

Jodeit, M, Levitan, BM: Isospectral vectorvalued SturmLiouville problems. Lett Math Phys. 43, 117–122 (1998). Publisher Full Text

Shen, CL: Some eigenvalue problems for the vectorial Hill's equation. Inverse Probl. 16(3), 749–783 (2000). Publisher Full Text

Shen, CL: Some inverse spectral problems for vectorial SturmLiouville equations. Inverse Probl. 17(5), 1253–1294 (2001). Publisher Full Text

Shieh, CT: Isospectral sets and inverse problems for vectorvalued SturmLiouville equations. Inverse Probl. 23(6), 2457–2468 (2007). Publisher Full Text

Yurko, VA: Inverse problems for the matrix SturmLiouville equation on a finite intervl. Inverse Probl. 22, 1139–1149 (2006). Publisher Full Text