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tku.edu.tw 1 Department of Mathematics, Tamkang University, No.151, Yingzhuan Rd., Danshui Dist., New Taipei City 25137, Taiwan, PR China Full list of author information is available at

R E S E A R C H Open Access

Uniqueness of the potential function for the

vectorial Sturm-Liouville equation on a finite

interval

Tsorng-Hwa Chang1,2 and Chung-Tsun Shieh1*

* Correspondence: ctshieh@mail.

tku.edu.tw

1 Department of Mathematics,

Tamkang University, No.151,

Yingzhuan Rd., Danshui Dist., New

Taipei City 25137, Taiwan, PR China

Full list of author information is

available at the end of the article

Abstract

In this paper, the vectorial Sturm-Liouville operatorL Q=− d

2

dx2 + 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, thenm(m + 1)

2 + 1characteristic 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+ (λI 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 spec-tral data of (1.1) with boundary conditions

U( y) := y(0)− hy(0) = 0, V(y) := y(π) + Hy(π) = 0, (1:2) wherel is the spectral parameter,h = [h ij]i,j=1,mandH = [H ij]i,j=1,mare inM n(C)and

Q(x) = [Q ij (x)] i,j=1,mis 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 geophy-sics Numerous research results for this case have been established by renowned math-ematicians, 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 parti-cular, for m = 2 and Q(x) is a two-by-two real symmetric matrix-valued smooth func-tions defined in the interval [0, π] Shen [18] showed that five spectral data can

© 2011 Chang and Shieh; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

determine Q(x) uniquely More precisely speaking, he considered the inverse spectral

problems of the vectorial Sturm-Liouville equation:

y+ (λI2− Q2(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 sD(Q) denotes the Dirichlet spectrum of (1.3), sND(Q) the Neumann-Dirichlet

spectrum of (1.3) and sj(Q) the spectrum of (1.3) with boundary condition

for j = 1, 2, 3, where

B j=



α j β j

β j γ j



is a real symmetric matrix and {(aj, bj, gj,), j = 1, 2, 3} is linearly independent overℝ

Then

Theorem 1.1 ([18], Theorem 4.1) Let Q2(x) andQ2(x)be two continuous two-by-two real symmetric matrix-valued functions defined on [0, π] Suppose that

σ ND(Q) = σ ND(σ ND(Q) = σ ND( andσ j (Q) = σ j( for j =1, 2, 3, thenQ(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+ (λI 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

LetC(x, λ) = [C ij (x, λ)] i,j=1,mandS(x, λ) = [S ij (x, λ)] i,j=1,mbe 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 0mis the m × m zero matrix,I m= [δ ij]i,j=1,mis the m × m identity matrix and δijis the Kronecker symbol For given complex-valued matrices h and H, we denote

ϕ(x, λ) =ϕ ij (x, λ)i,j=1,mand 

ij (x, λ)i,j=1,m

be two solutions of equation (1.5) so that (x, l) = C(x, l) + S(x, l)h and

M(λ)which satisfy the boundary conditions



U( (0) m,

Then, The matrixM(λ) = [M ij(λ)] i,j=1,mis 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

h = ˜h,h = ˜handH =  H.

Also note that from [20], we have

M(λ) = ψ(x, λ)(U(ψ))−1, (2:2)

M(λ) = −(V(ϕ))−1V(S) = ψ(0, λ)(U(ψ))−1 (2:3) whereψ(x, λ) = [ψ ij (x, λ)] i,j=1,mis a matrix solution of equation (1.5) associated with the conditionsψ(π, l) = Imandψ’ (π, l) = -H It is not difficult to see that both F(x, l)

andM(λ)are meromorphic in l and the poles ofM(λ)are coincided with the

eigenva-lues of Lm(Q, h, H) Moreover, we have

M(λ) = −(V(ϕ))−1V(S) =−Adj(ϕ(π, λ) + Hϕ(π, λ))

det(ϕ(π, λ) + Hϕ(π, λ)) (S(π, λ) + HS(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

vec-torial Sturm-Liouville equations LetB(i, j) =

b rs



r,s=1,m,

b rs=

0, (r, s) = (i, j),

1, (r, s) = (i, j), 1≤ i, j ≤ m,

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

ij(λ) = det(V(ϕ + SB(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 (l) 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×mandΔijbe defined as above Then

ij(λ) =00(λ) + det(Augment[ϕ

1(π, λ) + Hϕ1(π, λ), ,

(jth column)

Si(π, λ) + HS i(π, λ), , ϕ

m(π, λ) + Hϕ m(π, λ)]),

wherek(π, l) is the kth column of  (π, l) and Sk(π, l) the kth column of S (π, l) 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

ij(λ) = det(Y(π, λ) + HY(π, λ))

= det((ϕ(π, λ) + Hϕ(π, λ)) + (S(π, λ) + HS(π, λ)B(i, j))

= det((ϕ(π, λ) + Hϕ(π, λ)) + [0, Si(π, λ) + HS (jth column) i(π, λ)0])

= det(ϕ(π, λ) + Hϕ(π, λ)) + det(ϕ

1(π, λ) + Hϕ1(π, λ), ,

(jth column)

Si(π, λ) + HS i(π, λ), , ϕ

m(π, λ) + Hϕ m(π, λ))

=00(λ) + det(ϕ

m(π, λ) + Hϕ1(π, λ), ,

(jth column)

Si(π, λ) + HS i(π, λ), , ϕ

m(π, λ) + Hϕ m(π, λ)).

□

Next, we shall prove the first main theorem For simplicity, if a symbol a denotes an object related to Lm(Q, h, H), then the symbol ˜αdenotes the analogous object related

to L m(Q, ˜h,  H)

Theorem 2.3 Suppose that ij(λ) =  ij(λ)for (i, j) = (0, 0) or 1 ≤ i, j ≤ m then

h = ˜h,h = ˜handH =  H.

Proof Since

0m= ( and

M(λ),

we have that

−(S(π, λ) + HS(π, λ))e i= (ϕ(π, λ) + Hϕ(π, λ))M(λ)e i

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

−(S

i(π, λ) + HS i(π, λ)) = (ϕ(π, λ) + Hϕ(π, λ))M i(λ).

By Crammer’s rule,

Mji(λ)

=− det(ϕ

1(π, λ) + Hϕ1(π, λ), , Si(π, λ) + HSi (jth column)(π, λ), , ϕ

m(π, λ) + Hϕm(π, λ))

det(ϕ(π, λ) + Hϕ(π, λ))

=00(λ) − ij(λ)

00(λ)

=00(λ) −  ij(λ)



00(λ)

= Mji(λ) for 1 ≤ i, j ≤ m.

Applying Theorem 2.1, we conclude thatQ =  Q,h = ˜handH =  H.□ Lemma 2.4 Suppose that h, H are real symmetric matrices and Q(x) is a real sym-metric matrix-valued function Then,M(λ) = −V(ϕ)−1V(S)is real symmetric for all l

Î ℝ

Proof Let

U(x, λ) =



ϕ(x, λ) S(x, λ) ϕ(x, λ) S(x, λ)



For lÎ 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

U2(x, λ) =



I m H

0 I m



U(x, λ).

Then

U2(1,λ) =



I m H

0 I m



U(1, λ) =



V( ϕ) V(S) ϕ(1, λ) S(1, λ)



and

U2−1(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 2m,

we have

V( ϕ)[V(S)]∗= V(S)[V( ϕ)]∗, i.e.,M(λ) = V(ϕ)−1V(S)is real symmetric for all lÎ ℝ □ Definition 2.1 We call Lm(h, H, Q) a real symmetric problem if h, H are real sym-metric matrices and Q(x) is a real symsym-metric matrix-valued function

Corollary 2.5 Let Lm(h, H, Q) andL( ˜h, ˜ H, ˜ Q)be two real symmetric problems Sup-pose that ij(λ) = ˜ ij(λ)for(i, j) = (0, 0) or 1≤ i ≤ j ≤ m, thenh = ˜h,h = ˜ HandQ =  Q

Proof For lÎ ℝ bothM(λ)andM(λ)˜ are real symmetric Moreover,

M ji(λ) = 00(λ) − ij(λ)

00(λ)

= ˜00(λ) − ˜ij(λ)

˜00(λ)

= ˜M ji(λ), for 1 ≤ i ≤ j ≤ m.

Hence,M ij(λ) = ˜ M ij(λ)for lÎ ℝ and 1 ≤ i, j ≤ m This leads to ij(λ) = ˜ ij(λ)for

l Î ℝ We conclude that ij(λ) = ˜ ij(λ)andM ij(λ) = ˜ M ij(λ)forl Î ℂ This

com-pletes 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

ij=



e1, , (ith-column)0 , , (jth-column)0 , , e m

 ,

ij=



0, , (ith-column) e i , , (ith-column) e j , , 0

 ,

where e

i= (0, 0, , 0, (ith-coordiante)1 , 0, , 0) t.Hence,Γij+Γij

= Im LetΘij(l) be the characteristic function of the self-adjoint problem

y + (λI m − Q(x))y = 0, 0 < x < π (2:7)

associated with some boundary conditions



ij y(0, ij h + ij )y(0, λ) = 0,

y(π, λ) + Hy(π, λ) = 0, (2:8)

then

 ij(λ) = det[V(ϕ1), , (ith-column) V(S j) , , (jth-column) V(S i) , , V(ϕ m)], where V (Lj) denotes the jth column of (V(L)) for a m × m matrix L Similarly, we denote Ωij(l) the characteristic function of the real symmetric problem

L m (Q, h +12(B(i, j) + B(j, i)), H)for 1≤ i, j ≤ m, then

 ij(λ) = det

⎡

⎢

⎣V(ϕ1), ,

(ith-column)

V(ϕ i) +1

2V(S j), ,

(jth-column)

V(ϕ j) +1

2V(S i), , V(ϕ m)

⎤

⎥

⎦

= det[V( ϕ1), , V(ϕ i), , V(ϕ m)]

+ 1

2det[V( ϕ1), , (ith-column) V(S j) , , (jth-column) V( ϕ j) , , V(ϕ m)]

+ 1

2det[V( ϕ1), , (ith-column) V( ϕ i) , , (jth-column) V(S i) , , V(ϕ m)]

+ det[V( ϕ1), , (ith-column) V(S j) , , (jth-column) V(S i) , , V(ϕ m)]

(2:9)

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

00(λ) = det[V(ϕ1), , 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 {(li, mi)}i = 1, ∞the spectral

data of Lm(Q, h, H) where mi is the multiplicity of the eigenvalue li of Lm(Q, h, H)

then the characteristic function of Lm(Q, h, H) is

(λ) = C∞i=1(1−λ λ

i

)m i

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

Theorem 2.6 Assuming that Lm(Q, h, H) andL m( ˜Q, ˜h, ˜ H)are two real symmetric problems If the conditions

(1) ij(λ) =   ij(λ)for(i, j) = (0, 0) or 1 ≤ i ≤ j ≤ m, (2) ij(λ) =   ij(λ)for1 ≤ i < j ≤ m.,

are satisfied, thenh = ˜h,H =  HandQ(x) =  Q(x)a.e on[0, 1]

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

ij(λ) = det[V(ϕ1), , (ith-column) V( ϕ i) , , V( (jth-column) ϕ j ) + V(S i), , V(ϕ m)]

= det[V( ϕ1), , V(ϕ j), , V(ϕ m)]

+ det[V( ϕ1), , (ith-column) V(ϕ i) , , (jth-column) V(S i) , , V(ϕ m)]

=00(λ) + det[V(ϕ1), , (ith-column) V( ϕ i) , , (jth-column) V(S i) , , V(ϕ m)]

=00(λ) − 00(λ)M ji(λ).

Similarly,

˜ ij(λ) = ˜00(λ) − ˜00(λ) ˜M ji(λ).

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

(1)Δij(l) =Δji(l) and ˜ ij(λ) = ˜ ji(λ)for 1≤ i ≤ j ≤ m, (2) ii(λ) =  ii(λ) =   ii(λ) =  ii(λ)for i = 0, 1, , m, (3) ij(λ) =  ij(λ) −  ij(λ) =   ij(λ) −   ij(λ) =  ij(λ)for 1≤ i < j ≤ m

This impliesL 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, Theo-rem 2.6 leads to TheoTheo-rem 1.1 Shen also shows by providing an example that 5

mini-mal 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

kk(λ) =  kk(λ)for k =0, 1, , m, thenQ =  Q,h = ˜handH =  H

Proof Since Lm(Q, h, H) andL m(Q, ˜h,  H)are both diagonals, we know

M(λ) = −Adj(ϕ(π, λ) + Hϕ(π, λ))

det(ϕ(π, λ) + Hϕ(π, λ)) (S(π, λ) + HS(π, λ))

is diagonal and so isM(λ) Hence,

M ij(λ) = 0 for i = j, 1 ≤ i, j ≤ m.

Moreover,

M kk(λ) = −1

00(λ)(ϕ1(π, λ) + H1ϕ1(π, λ) · · · (Sk(π, λ) + H (k) k S k(π, λ)) · · ·

= −1

00(λ)( kk(λ) − 00(λ))

= 00(λ) − kk(λ)

00(λ)

= 00(λ) −  kk(λ)



00(λ)

= M kk(λ).

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

Footnote

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

Author details

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

Authors ’ contributions

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

manuscript.

Competing interests

The authors declare that they have no competing interests.

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

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□

Next, we shall prove the first main theorem For. .. 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,...

Now let

U2(x, λ) =



I m H