Open Access Research

Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval

Tsorng-Hwa Chang12 and Chung-Tsun Shieh1*

Author Affiliations

1 Department of Mathematics, Tamkang University, No.151, Yingzhuan Rd., Danshui Dist., New Taipei City 25137, Taiwan, PR China

2 Department of Electronic Engineering, China University of Science and Technology, No.245, Academia Rd., Sec. 3, Nangang District, Taipei City 115, Taiwan, PR China

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Boundary Value Problems 2011, 2011:40  doi:10.1186/1687-2770-2011-40


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/40


Received:28 April 2011
Accepted:26 October 2011
Published:26 October 2011

© 2011 Chang and Shieh; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the vectorial Sturm-Liouville operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M1">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M2">View MathML</a> 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.

Keywords:
Inverse spectral problems; Sturm-Liouville equation

1. Introduction

The study on inverse spectral problems for the vectorial Sturm-Liouville differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M3">View MathML</a>

(1.1)

on a finite interval is devoted to determine the potential matrix Q(x) from the spectral data of (1.1) with boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M4">View MathML</a>

(1.2)

where λ is the spectral parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M6">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M8">View MathML</a> is an integrable matrix-valued function. We use Lm = 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M9">View MathML</a>

(1.3)

where Q2(x) is a real symmetric matrix-valued function defined in the interval [0, π]. Let σD (Q) denotes the Dirichlet spectrum of (1.3), σND (Q) the Neumann-Dirichlet spectrum of (1.3) and σj (Q) the spectrum of (1.3) with boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M10">View MathML</a>

(1.4)

for j = 1, 2, 3, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M11">View MathML</a>

is a real symmetric matrix and {(αj, βj, γj,), j = 1, 2, 3} is linearly independent over ℝ. Then

Theorem 1.1 ([18], Theorem 4.1). Let Q2(x) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M12">View MathML</a> be two continuous two-by-two real symmetric matrix-valued functions defined on [0, π]. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M13">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M15">View MathML</a> for j = 1, 2, 3, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M16">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M17">View MathML</a>

(1.5)

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

2. Main Results

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M19">View MathML</a> be two solutions of equation (1.5) which satisfy the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M20">View MathML</a>

where 0m is the m × m zero matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M21">View MathML</a> is the m × m identity matrix and δij is the Kronecker symbol. For given complex-valued matrices h and H, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M22">View MathML</a>

be two solutions of equation (1.5) so that φ(x, λ) = C(x, λ) + S(x, λ)h and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M23">View MathML</a> which satisfy the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M24">View MathML</a>

(2.1)

Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M25">View MathML</a>. The matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M26">View MathML</a> is called the Weyl matrix for Lm (Q, h, H). In 2006, Yurko proved that:

Theorem 2.1 ([20], Theorem 1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M28">View MathML</a> denote Weyl matrices of the problems Lm (Q, h, H) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M29">View MathML</a>separately. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M30">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32">View MathML</a>.

Also note that from [20], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M33">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M34">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M35">View MathML</a> is a matrix solution of equation (1.5) associated with the conditions ψ(π, λ) = Im and ψ' (π, λ) = -H. It is not difficult to see that both Φ(x, λ) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27">View MathML</a> are meromorphic in λ and the poles of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27">View MathML</a> are coincided with the eigenvalues of Lm (Q, h, H). Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M36">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M37">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M38">View MathML</a>

and B(0, 0) = 0m The characteristic function for this boundary value problem Lm (Q, h + B(i, j), H) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M39">View MathML</a>

(2.4)

The first problem we want to study is as following:

Problem 1. How many Δij (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) = [brs]m×m and Δij be defined as above. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M40">View MathML</a>

where φk (π, λ) is the kth column of φ (π, λ) and Sk (π, λ) the kth column of S (π, λ) for k = 1, 2, 3, ..., m.

Proof. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M41">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M42">View MathML</a>

Next, we shall prove the first main theorem. For simplicity, if a symbol α denotes an object related to Lm(Q, h, H), then the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M43">View MathML</a> denotes the analogous object related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M44">View MathML</a>.

Theorem 2.3. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M45">View MathML</a> for (i, j) = (0, 0) or 1 ≤ i, j m then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32">View MathML</a>.

Proof. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M47">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M48">View MathML</a>

we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M49">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M50">View MathML</a>

By Crammer's rule,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M51">View MathML</a>

Applying Theorem 2.1, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32">View MathML</a>. □

Lemma 2.4. Suppose that h, H are real symmetric matrices and Q(x) is a real symmetric matrix-valued function. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M52">View MathML</a>is real symmetric for all λ ∈ ℝ.

Proof. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M53">View MathML</a>

(2.5)

For λ ∈ ℝ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M54">View MathML</a>

This leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M55">View MathML</a>

(2.6)

Now let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M56">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M57">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M58">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M59">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M60">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M61">View MathML</a>is real symmetric for all λ ∈ ℝ. □

Definition 2.1. We call Lm(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 Lm(h, H, Q) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M62">View MathML</a> be two real symmetric problems. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M63">View MathML</a> for (i, j) = (0, 0) or 1 ≤ i j m, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46">View MathML</a>.

Proof. For λ ∈ ℝ. both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M65">View MathML</a> are real symmetric. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M66">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M67">View MathML</a> for λ ∈ ℝ and 1 ≤ i, j m. This leads to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M63">View MathML</a> for λ ∈ ℝ. We conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M67">View MathML</a> for λ ∈ ℂ. This completes the proof. □

From now on, we let Lm(Q, h, H) be a real symmetric problem. We would like to know that how many spectral data can determine the problem Lm(Q, h, H) if we require all spectral data come from real symmetric problems. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M68">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M69">View MathML</a> Hence, Γij + Γij = Im. Let Θij(λ) be the characteristic function of the self-adjoint problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M70">View MathML</a>

(2.7)

associated with some boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M71">View MathML</a>

(2.8)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M72">View MathML</a>

where V (Lj) denotes the jth column of (V(L)) for a m × m matrix L. Similarly, we denote Ωij(λ) the characteristic function of the real symmetric problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M73">View MathML</a> for 1 ≤ i, j m, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M74">View MathML</a>

(2.9)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M75">View MathML</a>

Now, we are going to focus on self-adjoint problems. For a self-adjoint problem Lm(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 {(λi, mi)}i = 1,∞ the spectral data of Lm(Q, h, H) where mi is the multiplicity of the eigenvalue λi of Lm(Q, h, H) then the characteristic function of Lm(Q, h, H) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M76">View MathML</a>

where C is determined by {(λi, mi)}i = 1,∞. This means that the spectral data determined the corresponding characteristic function.

Theorem 2.6. Assuming that Lm(Q, h, H) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M77">View MathML</a> are two real symmetric problems. If the conditions

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M78">View MathML</a>for (i, j) = (0, 0) or 1 ≤ i j m,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M79">View MathML</a>for 1 ≤ i < j m.,

are satisfied, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M16">View MathML</a>a.e on [0, 1].

Proof. Note that for any problem Lm(Q, h, H) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M80">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M81">View MathML</a>

Moreover, by the assumptions and Lemma 2.4, we have Mij(λ) = Mji(λ) Hence,

(1) Δij (λ) = Δji(λ) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M82">View MathML</a> for 1 ≤ i j m,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M83">View MathML</a> for i = 0, 1, ..., m,

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M84">View MathML</a> for 1 ≤ i < j m.

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M85">View MathML</a>

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 Lm(Q, h, H) is an uncoupled system. Hence, everything for the operator Lm(Q, h, H) can be obtained by applying inverse spectral theory for scalar Sturm-Liouville equation. Unfortunately, it is not true. We say Lm(Q, h, H) diagonal if all Q, h and H are diagonals.

Corollary 2.7. Suppose Lm(Q, h, H) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M86">View MathML</a> are both diagonals. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M87">View MathML</a> for k = 0, 1, ..., m, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32">View MathML</a>.

Proof. Since Lm(Q, h, H) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M86">View MathML</a> are both diagonals, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M88">View MathML</a>

is diagonal and so is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M89">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M90">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M91">View MathML</a>

for k = 1, 2, ..., m. This implies. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M30">View MathML</a>. Applying Theorem 2.1 again, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/40/mathml/M32">View MathML</a>. □

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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