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

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

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 ,

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 ( ℂ ) and Q ( x ) = [ Q i j ( x ) ] i , j = 1 , m ¯ is an integrable matrix-valued function. We use L_m = L(Q, h, H) to denote the boundary problem (1.1)-(1.2). For the case m = 1, (1.1)-(1.2) is a scalar Sturm-Liouville equation. The scalar Sturm-Liouville equation often arises from some physical problems, for example, vibration of a string, quantum mechanics and geophysics. Numerous research results for this case have been established by renowned mathematicians, notably Borg, Gelfand, Hochstadt, Krein, Levinson, Levitan, Marchenko, Gesztesy, Simon and their coauthors and followers (see 1 2 3 4 5 6 7 8 9 and references therein). For the case m ≥ 2, some interesting results had been obtained (see 10 11 12 13 14 15 16 17 18 19 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 ,

where Q ₂(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 → ,

for j = 1, 2, 3, where

B j = α j β j β j γ j

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 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 .

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

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 0 _m is the m × m zero matrix, I m = [ δ i j ] i , j = 1 , m ¯ 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 ¯

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 .

Then, M ( λ ) = Φ ( 0 , λ ) . The matrix M ( λ ) = [ M i j ( λ ) ] i , j = 1 , m ¯ 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 ) and M ̃ ( λ ) denote Weyl matrices of the problems L_m (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 ,

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

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) = I_m 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 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 ) ) ,

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) = 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 ) .

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

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 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 ) ]

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 L_m (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 ) .

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 ,

This leads to

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

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 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 ̃ ) 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 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 ] ,

where e i = ( 0 , 0 , … , 0 , 1 ( i th-coordiante ) , 0 , … , 0 ) t . 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

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 ,

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 (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 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 ) ]

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 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

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 ̃ ) 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 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 ) .

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 M_ij (λ) = M_ji (λ) Hence,

(1) Δ _ij (λ) = Δ _ji (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 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 ̃ ) 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 L_m (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 ̃ . □

The authors declare that they have no competing interests.

Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.

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