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

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

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 L Q = - d 2 d x 2 + Q ( x ) 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 m ( m + 1 ) 2 + 1 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

y + ( λ I m - Q ( x ) ) y = 0 , 0 < x < π , (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

U ( y ) : = y ( 0 ) - h y ( 0 ) = 0 , V ( y ) : = y ( π ) + H y ( π ) = 0 , (1.2)

where λ is the spectral parameter, h = [ h i j ] i , j = 1 , m ¯ and H = [ H i j ] i , j = 1 , m ¯ are in M n ( ) and Q ( x ) = [ Q i j ( x ) ] i , j = 1 , m ¯ 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:

y + ( λ I 2 - Q 2 ( x ) ) y ( x ) = 0 , 0 < x < π , (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

y ( 0 ) - B j y ( 0 ) = y ( π ) = 0 , (1.4)

for j = 1, 2, 3, where

B j = α j β j β j γ j

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 Q ̃ 2 ( x ) be two continuous two-by-two real symmetric matrix-valued functions defined on [0, π]. Suppose that σ D ( Q ) = σ D ( Q ̃ ) , σ N D ( Q ̃ ) = σ N D ( Q ̃ ) and σ j ( Q ) = σ j ( Q ̃ ) for j = 1, 2, 3, then Q ( x ) = Q ̃ ( x ) 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

Y + ( λ I m - Q ( x ) ) Y = 0 , 0 < x < π . (1.5)

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

### 2. Main Results

Let C ( x , λ ) = [ C i j ( x , λ ) ] i , j = 1 , m ¯ and S ( x , λ ) = [ S i j ( x , λ ) ] i , j = 1 , m ¯ be two solutions of equation (1.5) which satisfy the initial conditions

C ( 0 , λ ) = S ( 0 , λ ) = I m , C ( 0 , λ ) = S ( 0 , λ ) = 0 m ,

where 0m is the m × m zero matrix, I m = [ δ i j ] i , j = 1 , m ¯ is the m × m identity matrix and δij is the Kronecker symbol. For given complex-valued matrices h and H, we denote

φ ( x , λ ) = φ i j ( x , λ ) i , j = 1 , m ¯ and Φ ( x , λ ) = Φ i j ( x , λ ) i , j = 1 , m ¯

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

U ( Φ ) = Φ ( 0 ) - h Φ ( 0 ) = I m , V ( Φ ) = Φ ( π ) + H Φ ( π ) = 0 m . (2.1)

Then, M ( λ ) = Φ ( 0 , λ ) . The matrix M ( λ ) = [ M i j ( λ ) ] i , j = 1 , m ¯ is called the Weyl matrix for Lm (Q, h, H). In 2006, Yurko proved that:

Theorem 2.1 ([20], Theorem 1). Let M ( λ ) and M ̃ ( λ ) denote Weyl matrices of the problems Lm (Q, h, H) and L m ( Q ̃ , h ̃ , H ̃ ) separately. Suppose M ( λ ) = M ̃ ( λ ) , then Q ( x ) = Q ̃ ( x ) , h = h ̃ and H = H ̃ .

Also note that from [20], we have

Φ ( x , λ ) = S ( x , λ ) + φ ( x , λ ) M ( λ ) = ψ ( x , λ ) ( U ( ψ ) ) - 1 , (2.2)

M ( λ ) = - ( V ( φ ) ) - 1 V ( S ) = ψ ( 0 , λ ) ( U ( ψ ) ) - 1 (2.3)

where ψ ( x , λ ) = [ ψ i j ( x , λ ) ] i , j = 1 , m ¯ 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 M ( λ ) are meromorphic in λ and the poles of M ( λ ) are coincided with the eigenvalues of Lm (Q, h, H). Moreover, we have

M ( λ ) = - ( V ( φ ) ) - 1 V ( S ) = - Adj ( φ ( π , λ ) + H φ ( π , λ ) ) det ( φ ( π , λ ) + H φ ( π , λ ) ) ( S ( π , λ ) + H S ( π , λ ) ) ,

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 B ( i , j ) = b r s r , s = 1 , m ¯ ,

b r s = 0 , ( r , s ) ( i , j ) , 1 , ( r , s ) = ( i , j ) , 1 i , j m ,

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

Δ i j ( λ ) = det ( V ( φ + S B ( i , j ) ) ) , 1 i , j m   or   ( i , j ) = ( 0 , 0 ) . (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

Δ i j ( λ ) = Δ 00 ( λ ) + det ( A u g m e n t [ φ 1 ( π , λ ) + H φ 1 ( π , λ ) , , S i ( π , λ ) + H S i ( π , λ ) ( j t h   c o l u m n ) , , φ m ( π , λ ) + H φ m ( π , λ ) ] ) ,

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

Y ( x , λ ) = [ C ( x , λ ) + S ( x , λ ) ( h + B ( i , j ) ) ] = [ ( C ( x , λ ) + S ( x , λ ) h ) + S ( x , λ ) B ( i , j ) ] = [ φ ( x , λ ) + S ( x , λ ) B ( i , j ) ]

Then

Δ i j ( λ ) = det ( Y ( π , λ ) + H Y ( π , λ ) ) = det ( ( φ ( π , λ ) + H φ ( π , λ ) ) + ( S ( π , λ ) + H S ( π , λ ) B ( i , j ) ) = det ( ( φ ( π , λ ) + H φ ( π , λ ) ) + [ 0 , S i ( π , λ ) + H S i ( π , λ ) 0 ] ) ( j th column ) = det ( φ ( π , λ ) + H φ ( π , λ ) ) + det ( φ 1 ( π , λ ) + H φ 1 ( π , λ ) , , S i ( π , λ ) + H S i ( π , λ ) ( j th column ) , , φ m ( π , λ ) + H φ m ( π , λ ) ) = Δ 00 ( λ ) + det ( φ m ( π , λ ) + H φ 1 ( π , λ ) , , S i ( π , λ ) + H S i ( π , λ ) ( j th column ) , , φ m ( π , λ ) + H φ m ( π , λ ) ) .

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 α ̃ denotes the analogous object related to L m ( Q ̃ , h ̃ , H ̃ ) .

Theorem 2.3. Suppose that Δ i j ( λ ) = Δ ̃ i j ( λ ) for (i, j) = (0, 0) or 1 ≤ i, j m then Q = Q ̃ , h = h ̃ and H = H ̃ .

Proof. Since

0 m = Φ ( π , λ ) + H Φ ( π , λ )

and

Φ ( x , λ ) = S ( x , λ ) + φ ( x , λ ) M ( λ ) ,

we have that

- ( S ( π , λ ) + H S ( π , λ ) ) e i = ( φ ( π , λ ) + H φ ( π , λ ) ) M ( λ ) e i

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

- ( S i ( π , λ ) + H S i ( π , λ ) ) = ( φ ( π , λ ) + H φ ( π , λ ) ) M i ( λ ) .

By Crammer's rule,

M j i ( λ ) = det ( φ 1 ( π , λ ) + H φ 1 ( π , λ ) , , S i ( π , λ ) + H S i ( π , λ ) ( j th column ) , , φ m ( π , λ ) + H φ m ( π , λ ) ) det ( φ ( π , λ ) + H φ ( π , λ ) ) = Δ 00 ( λ ) Δ i j ( λ ) Δ 00 ( λ ) = Δ ˜ 00 ( λ ) Δ ˜ i j ( λ ) Δ ˜ 00 ( λ ) = M ˜ j i ( λ )   for   1 i , j m .

Applying Theorem 2.1, we conclude that Q = Q ̃ , h = h ̃ and H = H ̃ . □

Lemma 2.4. Suppose that h, H are real symmetric matrices and Q(x) is a real symmetric matrix-valued function. Then, M ( λ ) = - V ( φ ) - 1 V ( S ) is real symmetric for all λ ∈ ℝ.

Proof. Let

U ( x , λ ) = φ ( x , λ ) S ( x , λ ) φ ( x , λ ) S ( x , λ ) . (2.5)

For λ ∈ ℝ,

( S * φ - S * φ ) ( x , λ ) = ( S * φ - S * φ ) ( 0 , λ ) = I m , ( S * S - S * S ) ( x , λ ) = ( S * S - S * S ) ( 0 , λ ) = 0 m , ( φ * φ - φ * φ ) ( x , λ ) = ( φ * φ - φ * φ ) ( 0 , λ ) = 0 m , ( φ * S - φ * S ) ( x , λ ) = ( φ * S - φ * S ) ( 0 , λ ) = I m ,

U - 1 ( x , λ ) = - ( S ) * ( x , λ ) ( S * ) ( x , λ ) φ * ( x , λ ) - ( φ * ) ( x , λ ) . (2.6)

Now let

U 2 ( x , λ ) = I m H 0 I m U ( x , λ ) .

Then

U 2 ( 1 , λ ) = I m H 0 I m U ( 1 , λ ) = V ( φ ) V ( S ) φ ( 1 , λ ) S ( 1 , λ )

and

U 2 1 ( 1 , λ ) = ( [ I m H 0 I m ] U ( 1 , λ ) ) 1 = [ S ( 1 ; λ ) [ V ( S ) ] ( φ ) ( 1 , λ ) [ V ( φ ) ] ] .

Since

U ( x , λ ) U - 1 ( x , λ ) = I 2 m ,

we have

V ( φ ) [ V ( S ) ] * = V ( S ) [ V ( φ ) ] * ,

i.e., M ( λ ) = V ( φ ) - 1 V ( S ) 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 L ( h ̃ , H ̃ , Q ̃ ) be two real symmetric problems. Suppose that Δ i j ( λ ) = Δ ̃ i j ( λ ) for (i, j) = (0, 0) or 1 ≤ i j m, then h = h ̃ , h = H ̃ and Q = Q ̃ .

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

M j i ( λ ) = Δ 0 0 ( λ ) - Δ i j ( λ ) Δ 0 0 ( λ ) = Δ ̃ 0 0 ( λ ) - Δ ̃ i j ( λ ) Δ ̃ 0 0 ( λ ) = M ̃ j i ( λ ) ,  for  1 i j m .

Hence, M i j ( λ ) = M ̃ i j ( λ ) for λ ∈ ℝ and 1 ≤ i, j m. This leads to Δ i j ( λ ) = Δ ̃ i j ( λ ) for λ ∈ ℝ. We conclude that Δ i j ( λ ) = Δ ̃ i j ( λ ) and M i j ( λ ) = M ̃ i j ( λ ) 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

Γ i j = [ e 1 , , 0 ( i th-column ) , , 0 ( j th-column ) , , e m ] , Γ i j = [ 0 , , e i ( i th-column ) , , e j ( i th-column ) , ,0 ] ,

where e i = ( 0 , 0 , , 0 , 1 ( i th-coordiante ) , 0 , , 0 ) t . Hence, Γij + Γij = Im. Let Θij(λ) be the characteristic function of the self-adjoint problem

y + ( λ I m - Q ( x ) ) y = 0 , 0 < x < π (2.7)

associated with some boundary conditions

Γ i j y ( 0 , λ ) - ( Γ i j h + Γ i j ) y ( 0 , λ ) = 0 , y ( π , λ ) + H y ( π , λ ) = 0 , (2.8)

then

Θ i j ( λ ) = det [ V ( φ 1 ) , , V ( S j ) ( i th-column ) , , V ( S i ) ( j th-column ) , , V ( φ m ) ] ,

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 L m ( Q , h + 1 2 ( B ( i , j ) + B ( j , i ) ) , H ) for 1 ≤ i, j m, then

Ω i j ( λ ) = det [ V ( φ 1 ) , , V ( φ i ) + 1 2 V ( S j ) ( i th-column ) , , V ( φ j ) + 1 2 V ( S i ) ( j th-column ) , , V ( φ m ) ] = det [ V ( φ 1 ) , , V ( φ i ) , , V ( φ m ) ] + 1 2 det [ V ( φ 1 ) , , V ( S j ) ( i th-column ) , , V ( φ j ) ( j th-column ) , , V ( φ m ) ] + 1 2 det [ V ( φ 1 ) , , V ( φ i ) ( i th-column ) , , V ( S i ) ( j th-column ) , , V ( φ m ) ] + det [ V ( φ 1 ) , , V ( S j ) ( i th-column ) , , V ( S i ) ( j th-column ) , , V ( φ m ) ] (2.9)

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

Ω 0 0 ( λ ) = det [ V ( φ 1 ) , , V ( φ j ) , , V ( φ m ) ] .

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

Δ ( λ ) = C Π i = 1 ( 1 - λ λ i ) m i

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 L m ( Q ̃ , h ̃ , H ̃ ) are two real symmetric problems. If the conditions

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

(2) Θ i j ( λ ) = Θ ̃ i j ( λ ) for 1 ≤ i < j m.,

are satisfied, then h = h ̃ , H = H ̃ and Q ( x ) = Q ̃ ( x ) a.e on [0, 1].

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

Δ i j ( λ ) = det [ V ( φ 1 ) , , V ( φ i ) ( i th-column ) , , V ( φ j ) + V ( S i ) ( j th-column ) , , V ( φ m ) ] = det [ V ( φ 1 ) , , V ( φ j ) , , V ( φ m ) ] + det [ V ( φ 1 ) , , V ( φ i ) ( i th-column ) , , V ( S i ) ( j th-column ) , , V ( φ m ) ] = Δ 00 ( λ ) + det [ V ( φ 1 ) , , V ( φ i ) ( i th-column ) , , V ( S i ) ( j th-column ) , , V ( φ m ) ] = Δ 00 ( λ ) Δ 00 ( λ ) M j i ( λ ) .

Similarly,

Δ ̃ i j ( λ ) = Δ ̃ 0 0 ( λ ) - Δ ̃ 0 0 ( λ ) M ̃ j i ( λ ) .

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

(1) Δij (λ) = Δji(λ) and Δ ̃ i j ( λ ) = Δ ̃ j i ( λ ) for 1 ≤ i j m,

(2) Δ i i ( λ ) = Ω i i ( λ ) = Ω ̃ i i ( λ ) = Δ ̃ i i ( λ ) for i = 0, 1, ..., m,

(3) Δ i j ( λ ) = Ω i j ( λ ) - Θ i j ( λ ) = Ω ̃ i j ( λ ) - Θ ̃ i j ( λ ) = Δ ̃ i j ( λ ) for 1 ≤ i < j m.

This implies L m ( Q , h , H ) = L m ( Q ̃ , h ̃ , H ̃ ) .

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 L m ( Q ̃ , h ̃ , H ̃ ) are both diagonals. If Δ k k ( λ ) = Δ ̃ k k ( λ ) for k = 0, 1, ..., m, then Q = Q ̃ , h = h ̃ and H = H ̃ .

Proof. Since Lm(Q, h, H) and L m ( Q ̃ , h ̃ , H ̃ ) are both diagonals, we know

M ( λ ) = - Adj ( φ ( π , λ ) + H φ ( π , λ ) ) det ( φ ( π , λ ) + H φ ( π , λ ) ) ( S ( π , λ ) + H S ( π , λ ) )

is diagonal and so is M ̃ ( λ ) . Hence,

M i j ( λ ) = 0  for  i j ,   1 i , j m .

Moreover,

M k k ( λ ) = 1 Δ 00 ( λ ) ( φ 1 ( π , λ ) + H 1 φ 1 ( π , λ ) ( S k ( π , λ ) + H k S k ( π , λ ) ) ( k ) = 1 Δ 00 ( λ ) ( Δ k k ( λ ) Δ 00 ( λ ) ) = Δ 00 ( λ ) Δ k k ( λ ) Δ 00 ( λ ) = Δ ˜ 00 ( λ ) Δ ˜ k k ( λ ) Δ ˜ 00 ( λ ) = M ˜ k k ( λ ) .

for k = 1, 2, ..., m. This implies. M ( λ ) = M ̃ ( λ ) . Applying Theorem 2.1 again, we have Q = Q ̃ , h = h ̃ and H = H ̃ . □

### 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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