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

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 Chang

and Chung-Tsun Shieh

* 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 operator_LQ=− d

2

dx2 +Q(x)is considered,

whereQ(x) is an integrable m×mmatrix-valued function defined on the interval [0,π] The authors prove thatm2+1 characteristic functions can determine the

potential function of a vectorial Sturm-Liouville operator uniquely. In particular, ifQ(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, thenm2 + 1 spectral data can determineQ(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+ (λIm−Q(x))y= 0, 0<x< π, (1:1)

on a finite interval is devoted to determine the potential matrixQ(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)

wherelis the spectral parameter,h= [hij]i,j=1,m andH= [Hij]i,j=1,m are inMn(C)and Q(x) = [Qij(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 casem= 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 casem≥2, some interesting results had been obtained (see [10-20]). In parti-cular, form= 2 andQ(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

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)

whereQ2(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) andsj(Q) the spectrum of (1.3) with boundary condition

y(0)−Bjy(0) =y(π) =0, (1:4)

for j= 1, 2, 3, where

Bj=

αjβj

βjγj

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

σND(Q) =σND(Q)σND(Q) =σND(Q)andσj(Q) =σj(Q)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 casem≥3. The idea we use is the Weyl’s matrix for matrix-valued Sturm-Liouville equation

Y+ (λIm−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) = [Cij(x,λ)]_i,j=1,m andS(x,λ) = [Sij(x,λ)]_i,j=1,m be two solutions of equation (1.5) which satisfy the initial conditions

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

where 0mis the m×m zero matrix,Im= [δij]i,j=1,mis them×midentity matrix and

δijis the Kronecker symbol. For given complex-valued matriceshand H, we denote ϕ(x,λ) =ϕij(x,λ)_i,j=1,mand(x,λ) =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 (x,λ) =S(x,λ) +ϕ(x,λ)M(λ)which satisfy the boundary conditions

U() =(0)−h(0) =Im,

V() =(π) +H(π) = 0m. (2:1)

Then, M(λ) =(0,λ). The matrixM(λ) = [Mij(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 Lm(Q,h,˜ H)separately. Suppose M(λ) =M(λ), then

Also note that from [20], we have

(x,λ) =S(x,λ) +ϕ(x,λ)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) = [ψij(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 bothF(x,l) andM(λ)are meromorphic inland the poles ofM(λ)are coincided with the eigenva-lues ofLm(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 ofAand 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) =brs_r,s=1,m,

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

1, (r,s) = (i,j), 1≤i,j≤m,

andB(0, 0) = 0mThe characteristic function for this boundary value problemLm(Q,h+

B(i,j),H) is

ij(λ) = det(V(ϕ+SB(i,j))), 1≤i,j≤mor (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[ϕ₁(π,λ) +Hϕ1(π,λ),. . ., (jth column)

S_i(π,λ) +HSi(π,λ),. . .,ϕ_m(π,λ) +Hϕm(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)) +

(jth column)

[0,S_i(π,λ) +HSi(π,λ)0]) = det(ϕ(π,λ) +Hϕ(π,λ)) + det(ϕ₁(π,λ) +Hϕ1(π,λ),. . .,

(jth column)

Si(π,λ) +HSi(π,λ),. . .,ϕm(π,λ) +Hϕm(π,λ)) =00(λ) + det(ϕ_m(π,λ) +Hϕ1(π,λ),. . .,

(jth column)

S_i(π,λ) +HSi(π,λ),. . .,ϕ_m(π,λ) +Hϕm(π,λ)).

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

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

h=h˜,h=h˜andH=H. Proof. Since

0m=(π,λ) +H(π,λ)

and

(x,λ) =S(x,λ) +ϕ(x,λ)M(λ),

we have that

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

for eachi= 1, ...,m, that is,

−(S_i(π,λ) +HSi(π,λ)) = (ϕ(π,λ) +Hϕ(π,λ))Mi(λ).

By Crammer’s rule, Mji(λ)

=−det(ϕ

1(π,λ) +Hϕ1(π,λ),. . .,

(jth column)

Si(π,λ) +HSi(π,λ),. . .,ϕm(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)for1≤i,j≤m.

Applying Theorem 2.1, we conclude thatQ=Q,_h=h˜ and_H=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)−1_{V(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)

. (2:5)

ForlÎℝ, ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(S∗ϕ−S∗ϕ)(x,λ) = (S∗ϕ−S∗ϕ)(0,λ) =Im, (S∗S−S∗S)(x,λ) = (S∗S−S∗S)(0,λ) = 0m, (ϕ∗ϕ−ϕ∗ϕ)(x,λ) = (ϕ∗ϕ−ϕ∗ϕ)(0,λ) = 0m, (ϕ∗S−ϕ∗S)(x,λ) = (ϕ∗S−ϕ∗S)(0,λ) =Im,

This leads to

U−1(x,λ) =

−(S)∗(x,λ) (S∗)(x,λ)

ϕ∗_{(x,λ) −(ϕ}∗_)(x,λ)

Now let

U2(x,λ) =

Im H 0 Im

U(x,λ).

Then

U2(1,λ) =

Im H 0 Im

U(1,λ) =

V(ϕ) V(S)

ϕ(1,λ)S(1,λ)

and

U₂−1(1,λ) = (

ImH 0 Im

U(1,λ))−1=

−S∗(1;λ) [V(S)]∗ (ϕ)∗(1,λ)−[V(ϕ)]∗

.

Since

U(x,λ)U−1(x,λ) =I2m,

we have

V(ϕ)[V(S)]∗=V(S)[V(ϕ)]∗,

i.e.,M(λ) =V(ϕ)−1_{V(S)is real symmetric for alllÎ 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 symmetric matrix-valued function.

Corollary 2.5.Let Lm(h, H,Q)andL(h,˜ H,˜ Q)˜ be two real symmetric problems.

Sup-pose thatij(λ) =˜ij(λ)for(i, j) = (0, 0)or 1≤i≤j≤m,then_h=h˜,h=H˜ andQ=Q.

Proof. ForlÎℝ. bothM(λ)andM˜(λ)are real symmetric. Moreover,

Mji(λ) = 00(λ)−ij(λ)

00(λ)

= ˜00(λ)− ˜ij(λ) ˜

00(λ)

=M˜ji(λ), for 1≤i≤j≤m.

Hence,Mij(λ) =M˜ij(λ)forlÎℝand 1≤i,j≤m. This leads toij(λ) =˜ij(λ)for l Î ℝ. We conclude thatij(λ) =˜ij(λ)andMij(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 ,. . .,em

,

ij₌

0,. . .,(ith-column)ei ,. . .,(ith-column)ej ,. . ., 0

,

where _{ei= (0, 0,. . ., 0,}(ith-coordiante)_{1 , 0,. . ., 0)}t_.Hence,Γij+Γij=Im. LetΘij(l) be the characteristic function of the self-adjoint problem

associated with some boundary conditions

ijy(0,λ)−(ijh+ij)y(0,λ) = 0,

y(π,λ) +Hy(π,λ) = 0, (2:8)

then

ij(λ) = det[V(ϕ1),. . .,

(ith-column) V(Sj) ,. . .,

(jth-column)

V(Si) ,. . .,V(ϕm)],

where V(Lj) denotes thejth column of (V(L)) for am ×m matrixL. Similarly, we denote Ωij(l) the characteristic function of the real symmetric problem

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

ij(λ) = det ⎡ ⎢

⎣V(ϕ1),. . .,

(ith-column)

V(ϕi) +1

2V(Sj),. . .,

(jth-column)

V(ϕj) +1

2V(Si),. . .,V(ϕm) ⎤ ⎥ ⎦

= det[V(ϕ1),. . .,V(ϕi),. . .,V(ϕm)]

+ 1

2det[V(ϕ1),. . .,

(ith-column) V(Sj) ,. . .,

(jth-column)

V(ϕj) ,. . .,V(ϕm)]

+ 1

2det[V(ϕ1),. . .,

(ith-column) V(ϕi) ,. . .,

(jth-column)

V(Si) ,. . .,V(ϕm)]

+ det[V(ϕ1),. . .,

(ith-column) V(Sj) ,. . .,

(jth-column)

V(Si) ,. . .,V(ϕm)]

(2:9)

for 1≤i,j≤m. For simplicity, we write

00(λ) = det[V(ϕ1),. . .,V(ϕj),. . .,V(ϕm)].

Now, we are going to focus on self-adjoint problems. For a self-adjoint problemLm (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 ofLm(Q, h, H) then the characteristic function ofLm(Q,h, H) is

(λ) =C∞_i=1(1− λ

λi )mi

whereC 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)and Lm(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.,

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

ij(λ) = det[V(ϕ1),. . .,

(ith-column) V(ϕi) ,. . .,

(jth-column)

V(ϕj) +V(Si),. . .,V(ϕm)] = det[V(ϕ1),. . .,V(ϕj),. . .,V(ϕm)]

+ det[V(ϕ1),. . .,

(ith-column) V(ϕi) ,. . .,

(jth-column)

V(Si) ,. . .,V(ϕm)]

=00(λ) + det[V(ϕ1),. . .,

(ith-column) V(ϕi) ,. . .,

(jth-column)

V(Si) ,. . .,V(ϕm)] =00(λ)−00(λ)Mji(λ).

Similarly, ˜

ij(λ) =˜00(λ)− ˜00(λ)Mji(˜ λ).

Moreover, by the assumptions and Lemma 2.4, we haveMij(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 impliesLm(Q,h,H) =Lm(Q,˜ h,˜ H).˜

The authors want to emphasis that forn= 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 andH are diagonals thenLm(Q, h, H) is an uncoupled system. Hence, everything for the operatorLm(Q, h, H) can be obtained by applying inverse spectral theory for scalar Sturm-Liouville equation. Unfortunately, it is not true. We sayLm(Q,h,H) diagonal if allQ,handHare diagonals.

Corollary 2.7. Suppose Lm(Q, h, H) and Lm(Q,h,˜ H) are both diagonals. If

kk(λ) =kk(λ)for k =0, 1, ...,m, then_Q=Q,_h=h˜and_H=H.

Proof. SinceLm(Q,h,H) andLm(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 is_M(λ). Hence,

Mij(λ) = 0 fori=j, 1≤i,j≤m.

Moreover,

Mkk(λ) = − 1

00(λ)(ϕ

1(π,λ) +H1ϕ1(π,λ)· · ·

(k)

(S_k(π,λ) +HkSk(π,λ))· · ·

= −1

00(λ)(kk(λ)−00(λ))

= 00(λ)−kk(λ)

00(λ)

= 00(λ)−kk(λ)

for k = 1, 2, ..., m. This implies. _M(λ) =M(λ). Applying Theorem 2.1 again, we haveQ=Q,_h=h˜and _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 China2_{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

References

1. Borg, G: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math.78, 1–96 (1945)

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, vol. 189, pp. 85–92.American Mathematical Society, Providence, RI (1999) 3. Hochstadt, H: The inverse Sturm-Liouville problem. Commun Pure Appl Math.26, 75129 (1973)

4. Hochstadt, H, Lieberman, B: An inverse Sturm-Liouville problem with mixed given data. SIAM J Appl Math.34, 67680 (1978)

5. Kren, MG: Solution of the inverse Sturm-Liouville problem. Dokl Akad Nauk SSSR.76, 214 (1951) 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.19, 163 (1964). doi:10.1070/RM1964v019n03ABEH001151

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.267, 564–575 (2002). doi:10.1006/jmaa.2001.7792

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.167(1), 181–210 (2000). doi:10.1006/jdeq.1999.3758

14. Gesztesy, F, Kiselev, A, Makarov, K-A: Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-Type operators. Math Nachr.239-240(1), 103–145 (2002). doi:10.1002/1522-2616(200206)239:13.0.CO;2-F

15. Jodeit, M, Levitan, BM: The isospectrality problem for the classical Sturm-Liouville equation. Adv Differ Equ.22, 297_–318 (1997)

16. Jodeit, M, Levitan, BM: Isospectral vector-valued Sturm-Liouville problems. Lett Math Phys.43, 117–122 (1998). doi:10.1023/A:1007498010532

17. Shen, C-L: Some eigenvalue problems for the vectorial Hill’s equation. Inverse Probl.16(3), 749–783 (2000). doi:10.1088/ 0266-5611/16/3/313

18. Shen, C-L: Some inverse spectral problems for vectorial Sturm-Liouville equations. Inverse Probl.17(5), 1253–1294 (2001). doi:10.1088/0266-5611/17/5/303

19. Shieh, C-T: Isospectral sets and inverse problems for vector-valued Sturm-Liouville equations. Inverse Probl.23(6), 2457_–2468 (2007). doi:10.1088/0266-5611/23/6/011

20. Yurko, VA: Inverse problems for the matrix Sturm-Liouville equation on a finite intervl. Inverse Probl.22, 1139–1149 (2006). doi:10.1088/0266-5611/22/4/002

doi:10.1186/1687-2770-2011-40