Open Access

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

Boundary Value Problems20112011:40

DOI: 10.1186/1687-2770-2011-40

Received: 28 April 2011

Accepted: 26 October 2011

Published: 26 October 2011

Abstract

In this paper, the vectorial Sturm-Liouville operator L Q = - d 2 d x 2 + Q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq1_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq2_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ1_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ2_HTML.gif
(1.2)
where λ is the spectral parameter, h = [ h i j ] i , j = 1 , m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq3_HTML.gif and H = [ H i j ] i , j = 1 , m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq4_HTML.gif are in M n ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq5_HTML.gif and Q ( x ) = [ Q i j ( x ) ] i , j = 1 , m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq6_HTML.gif 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 [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 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ3_HTML.gif
(1.3)
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
y ( 0 ) - B j y ( 0 ) = y ( π ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ4_HTML.gif
(1.4)
for j = 1, 2, 3, where
B j = α j β j β j γ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equa_HTML.gif

is a real symmetric matrix and {(α j , β j , γ j ,), j = 1, 2, 3} is linearly independent over . Then

Theorem 1.1 ([18], Theorem 4.1). Let Q2(x) and Q ̃ 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq7_HTML.gif be two continuous two-by-two real symmetric matrix-valued functions defined on[0, π]. Suppose that σ D ( Q ) = σ D ( Q ̃ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq8_HTML.gif σ N D ( Q ̃ ) = σ N D ( Q ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq9_HTML.gif and σ j ( Q ) = σ j ( Q ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq10_HTML.gif for j = 1, 2, 3, then Q ( x ) = Q ̃ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq11_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ5_HTML.gif
(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 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq12_HTML.gif and S ( x , λ ) = [ S i j ( x , λ ) ] i , j = 1 , m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq13_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equb_HTML.gif
where 0 m is the m × m zero matrix, I m = [ δ i j ] i , j = 1 , m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq14_HTML.gif is the m × m identity matrix and δ ij 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 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equc_HTML.gif
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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq15_HTML.gif which satisfy the boundary conditions
U ( Φ ) = Φ ( 0 ) - h Φ ( 0 ) = I m , V ( Φ ) = Φ ( π ) + H Φ ( π ) = 0 m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ6_HTML.gif
(2.1)

Then, M ( λ ) = Φ ( 0 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq16_HTML.gif. The matrix M ( λ ) = [ M i j ( λ ) ] i , j = 1 , m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq17_HTML.gif is called the Weyl matrix for L m (Q, h, H). In 2006, Yurko proved that:

Theorem 2.1 ([20], Theorem 1). Let M ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq18_HTML.gif and M ̃ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq19_HTML.gif denote Weyl matrices of the problems L m (Q, h, H) and L m ( Q ̃ , h ̃ , H ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq20_HTML.gifseparately. Suppose M ( λ ) = M ̃ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq21_HTML.gif, then Q ( x ) = Q ̃ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq11_HTML.gif, h = h ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gifand H = H ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq23_HTML.gif.

Also note that from [20], we have
Φ ( x , λ ) = S ( x , λ ) + φ ( x , λ ) M ( λ ) = ψ ( x , λ ) ( U ( ψ ) ) - 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ7_HTML.gif
(2.2)
M ( λ ) = - ( V ( φ ) ) - 1 V ( S ) = ψ ( 0 , λ ) ( U ( ψ ) ) - 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ8_HTML.gif
(2.3)
where ψ ( x , λ ) = [ ψ i j ( x , λ ) ] i , j = 1 , m ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq24_HTML.gif 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 M ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq18_HTML.gif are meromorphic in λ and the poles of M ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq18_HTML.gif are coincided with the eigenvalues of L m (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 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equd_HTML.gif
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 ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq25_HTML.gif
b r s = 0 , ( r , s ) ( i , j ) , 1 , ( r , s ) = ( i , j ) , 1 i , j m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Eque_HTML.gif
and B(0, 0) = 0 m The characteristic function for this boundary value problem L m (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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ9_HTML.gif
(2.4)

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, jm

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
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 ) ] ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equf_HTML.gif

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
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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equg_HTML.gif
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 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equh_HTML.gif

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 α ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq26_HTML.gif denotes the analogous object related to L m ( Q ̃ , h ̃ , H ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq27_HTML.gif.

Theorem 2.3. Suppose that Δ i j ( λ ) = Δ ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq28_HTML.gif for(i, j) = (0, 0) or 1 ≤ i, jm then Q = Q ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gifand H = H ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq23_HTML.gif.

Proof. Since
0 m = Φ ( π , λ ) + H Φ ( π , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equi_HTML.gif
and
Φ ( x , λ ) = S ( x , λ ) + φ ( x , λ ) M ( λ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equj_HTML.gif
we have that
- ( S ( π , λ ) + H S ( π , λ ) ) e i = ( φ ( π , λ ) + H φ ( π , λ ) ) M ( λ ) e i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equk_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equl_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equm_HTML.gif

Applying Theorem 2.1, we conclude that Q = Q ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif and H = H ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq23_HTML.gif. □

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq30_HTML.gifis real symmetric for all λ .

Proof. Let
U ( x , λ ) = φ ( x , λ ) S ( x , λ ) φ ( x , λ ) S ( x , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ10_HTML.gif
(2.5)
For λ ,
( 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equn_HTML.gif
This leads to
U - 1 ( x , λ ) = - ( S ) * ( x , λ ) ( S * ) ( x , λ ) φ * ( x , λ ) - ( φ * ) ( x , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ11_HTML.gif
(2.6)
Now let
U 2 ( x , λ ) = I m H 0 I m U ( x , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equo_HTML.gif
Then
U 2 ( 1 , λ ) = I m H 0 I m U ( 1 , λ ) = V ( φ ) V ( S ) φ ( 1 , λ ) S ( 1 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equp_HTML.gif
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 ) ] ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equq_HTML.gif
Since
U ( x , λ ) U - 1 ( x , λ ) = I 2 m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equr_HTML.gif
we have
V ( φ ) [ V ( S ) ] * = V ( S ) [ V ( φ ) ] * , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equs_HTML.gif

i.e., M ( λ ) = V ( φ ) - 1 V ( S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq31_HTML.gifis real symmetric for all λ . □

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 L ( h ̃ , H ̃ , Q ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq32_HTML.gif be two real symmetric problems. Suppose that Δ i j ( λ ) = Δ ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq33_HTML.gif for(i, j) = (0, 0) or 1 ≤ ijm, then h = h ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif, h = H ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq34_HTML.gif and Q = Q ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif.

Proof. For λ . both M ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq18_HTML.gif and M ̃ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq35_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equt_HTML.gif

Hence, M i j ( λ ) = M ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq36_HTML.gif for λ and 1 ≤ i, jm. This leads to Δ i j ( λ ) = Δ ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq33_HTML.gif for λ . We conclude that Δ i j ( λ ) = Δ ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq33_HTML.gif and M i j ( λ ) = M ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq36_HTML.gif for λ . 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
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 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equu_HTML.gif
where e i = ( 0 , 0 , , 0 , 1 ( i th-coordiante ) , 0 , , 0 ) t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq37_HTML.gif Hence, Γ ij + Γ ij = I m . Let Θ ij (λ) be the characteristic function of the self-adjoint problem
y + ( λ I m - Q ( x ) ) y = 0 , 0 < x < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ12_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ13_HTML.gif
(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 ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equv_HTML.gif
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 L m ( Q , h + 1 2 ( B ( i , j ) + B ( j , i ) ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq38_HTML.gif for 1 ≤ i, jm, 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ14_HTML.gif
(2.9)
for 1 ≤ i, jm. For simplicity, we write
Ω 0 0 ( λ ) = det [ V ( φ 1 ) , , V ( φ j ) , , V ( φ m ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equw_HTML.gif
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
Δ ( λ ) = C Π i = 1 ( 1 - λ λ i ) m i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equx_HTML.gif

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 L m ( Q ̃ , h ̃ , H ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq39_HTML.gif are two real symmetric problems. If the conditions
  1. (1)

    Ω i j ( λ ) = Ω ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq40_HTML.gif for (i, j) = (0, 0) or 1 ≤ ijm,

     
  2. (2)

    Θ i j ( λ ) = Θ ̃ i j ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq41_HTML.gif for 1 ≤ i < jm.,

     

are satisfied, then h = h ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif, H = H ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq23_HTML.gifand Q ( x ) = Q ̃ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq11_HTML.gifa.e on [0, 1].

Proof. Note that for any problem L m (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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equy_HTML.gif
Similarly,
Δ ̃ i j ( λ ) = Δ ̃ 0 0 ( λ ) - Δ ̃ 0 0 ( λ ) M ̃ j i ( λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equz_HTML.gif
Moreover, by the assumptions and Lemma 2.4, we have M ij (λ) = M ji (λ) Hence,
  1. (1)

    Δ ij (λ) = Δ ji (λ) and Δ ̃ i j ( λ ) = Δ ̃ j i ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq42_HTML.gif for 1 ≤ ijm,

     
  2. (2)

    Δ i i ( λ ) = Ω i i ( λ ) = Ω ̃ i i ( λ ) = Δ ̃ i i ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq43_HTML.gif for i = 0, 1, ..., m,

     
  3. (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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq44_HTML.gif for 1 ≤ i < jm.

     

This implies L m ( Q , h , H ) = L m ( Q ̃ , h ̃ , H ̃ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq45_HTML.gif

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 L m ( Q ̃ , h ̃ , H ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq46_HTML.gif are both diagonals. If Δ k k ( λ ) = Δ ̃ k k ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq47_HTML.gif for k = 0, 1, ..., m, then Q = Q ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gifand H = H ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq23_HTML.gif.

Proof. Since L m (Q, h, H) and L m ( Q ̃ , h ̃ , H ̃ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq46_HTML.gif 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 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equaa_HTML.gif
is diagonal and so is M ̃ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq48_HTML.gif. Hence,
M i j ( λ ) = 0  for  i j , 1 i , j m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equab_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equac_HTML.gif

for k = 1, 2, ..., m. This implies. M ( λ ) = M ̃ ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq21_HTML.gif. Applying Theorem 2.1 again, we have Q = Q ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif and H = H ̃ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq23_HTML.gif. □

Footnote

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

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Tamkang University
(2)
Department of Electronic Engineering, China University of Science and Technology

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© Chang and Shieh; licensee Springer. 2011

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