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

  • Tsorng-Hwa Chang1, 2 and

    Affiliated with

    • Chung-Tsun Shieh1Email author

      Affiliated with

      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 ) http://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 http://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 , http://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 , http://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 ¯ http://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 ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq4_HTML.gif are in M n ( ) http://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 ¯ http://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 , http://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 , http://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 http://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 ℝ. Then

      Theorem 1.1 ([18], Theorem 4.1). Let Q2(x) and Q ̃ 2 ( x ) http://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 ̃ ) , http://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 ̃ ) http://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 ̃ ) http://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 ) http://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 . http://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 ¯ http://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 ¯ http://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 , http://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 ¯ http://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 ¯ http://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 ) http://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 . http://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 ) http://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 ¯ http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq18_HTML.gif and M ̃ ( λ ) http://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 ̃ ) http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq11_HTML.gif, h = h ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gifand H = H ̃ http://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 , http://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 http://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 ¯ http://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 ) http://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 ) http://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 ) ) , http://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 ¯ , http://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 , http://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 ) . http://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 ) ] ) , http://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 ) ] http://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 ) ) . http://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 ̃ http://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 ̃ ) http://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 ) http://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 ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gifand H = H ̃ http://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 ) http://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 ) , http://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 http://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 ) . http://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 . http://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 ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif and H = H ̃ http://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 ) http://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∈ ℝ.

      Proof. Let
      U ( x , λ ) = φ ( x , λ ) S ( x , λ ) φ ( x , λ ) S ( x , λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_Equ10_HTML.gif
      (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 , http://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 ) . http://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 ) . http://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 ) http://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 ) ] ] . http://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 , http://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 ) ] * , http://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 ) http://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 ∈ 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 ̃ ) http://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 ) http://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 ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif, h = H ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq34_HTML.gif and Q = Q ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif.

      Proof. For λ ∈ ℝ. both M ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq18_HTML.gif and M ̃ ( λ ) http://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 . http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq36_HTML.gif for λ ∈ ℝ and 1 ≤ i, jm. This leads to Δ i j ( λ ) = Δ ̃ i j ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq33_HTML.gif for λ ∈ ℝ. We conclude that Δ i j ( λ ) = Δ ̃ i j ( λ ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq36_HTML.gif 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 ] , http://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 . http://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 http://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 , http://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 ) ] , http://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 ) http://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 ) ] http://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 ) ] . http://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 http://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 ̃ ) http://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 ) http://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 ) http://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 ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif, H = H ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq23_HTML.gifand Q ( x ) = Q ̃ ( x ) http://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 ) . http://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 ) . http://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 ) http://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 ) http://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 ) http://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 ̃ ) . http://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 ̃ ) http://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 ) http://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 ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gifand H = H ̃ http://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 ̃ ) http://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 ) ) http://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 ) http://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 . http://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 ) . http://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 ) http://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 ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq29_HTML.gif, h = h ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-40/MediaObjects/13661_2011_Article_85_IEq22_HTML.gif and H = H ̃ http://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

      References

      1. Borg G: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math 1945, 78: 1-96.View ArticleMathSciNet
      2. Gesztesy F, Simon B: On the determination of a potential from three spectra. In: Differential Operators and Spectral Theory. In Am Math Soc Transl Ser 2. Volume 189. American Mathematical Society, Providence, RI; 1999:85-92.
      3. Hochstadt H: The inverse Sturm-Liouville problem. Commun Pure Appl Math 1973, 26: 75129.View ArticleMathSciNet
      4. Hochstadt H, Lieberman B: An inverse Sturm-Liouville problem with mixed given data. SIAM J Appl Math 1978, 34: 67680.View ArticleMathSciNet
      5. Kren MG: Solution of the inverse Sturm-Liouville problem. Dokl Akad Nauk SSSR 1951, 76: 214.
      6. Levitan BM: Inverse Sturm-Liouville Problems. VNM, Utrecht; 1987.
      7. Levitan BM, Gasymov MG: Determination of a differential equation by two of its spectra. Russ Math Surv 1964, 19: 163. 10.1070/RM1964v019n03ABEH001151View ArticleMathSciNet
      8. Marchenko VA: Sturm-Liouville Operators and Applications. Birkhauser, Basel 1986.
      9. Yurko VA: Method of spectral mappings in the inverse problem theory. Inverse and Ill-Posed Problems Series 2002.
      10. Andersson E: On the M-function and Borg-Marchenko theorems for vector-valued Sturm-Liouville equations. J Math Phys 44(12):6077-6100.
      11. Carlson R: An inverse problem for the matrix Schrödinger equation. J Math Anal Appl 2002, 267: 564-575. 10.1006/jmaa.2001.7792View ArticleMathSciNetMATH
      12. Chern H-H, Shen C-L: On the n-dimensional Ambarzumyan's theorem. Inverse Probl 13(1):15-18.
      13. Clarka S, Gesztesy F, Holdenc H, Levitand BM: Borg-Type theorems for matrix-valued Schrödinger operators. J Differ Equ 2000, 167(1):181-210. 10.1006/jdeq.1999.3758View Article
      14. Gesztesy F, Kiselev A, Makarov K-A: Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-Type operators. Math Nachr 2002, 239-240(1):103-145. 10.1002/1522-2616(200206)239:1<103::AID-MANA103>3.0.CO;2-FView ArticleMathSciNet
      15. Jodeit M, Levitan BM: The isospectrality problem for the classical Sturm-Liouville equation. Adv Differ Equ 1997, 22: 297-318.MathSciNet
      16. Jodeit M, Levitan BM: Isospectral vector-valued Sturm-Liouville problems. Lett Math Phys 1998, 43: 117-122. 10.1023/A:1007498010532View ArticleMathSciNetMATH
      17. Shen C-L: Some eigenvalue problems for the vectorial Hill's equation. Inverse Probl 2000, 16(3):749-783. 10.1088/0266-5611/16/3/313View ArticleMATH
      18. Shen C-L: Some inverse spectral problems for vectorial Sturm-Liouville equations. Inverse Probl 2001, 17(5):1253-1294. 10.1088/0266-5611/17/5/303View ArticleMATH
      19. Shieh C-T: Isospectral sets and inverse problems for vector-valued Sturm-Liouville equations. Inverse Probl 2007, 23(6):2457-2468. 10.1088/0266-5611/23/6/011View ArticleMathSciNetMATH
      20. Yurko VA: Inverse problems for the matrix Sturm-Liouville equation on a finite intervl. Inverse Probl 2006, 22: 1139-1149. 10.1088/0266-5611/22/4/002View ArticleMathSciNetMATH

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      © Chang and Shieh; licensee Springer. 2011

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