RESEARC H Open Access
Uniqueness of the potential function for the
vectorial Sturm-Liouville equation on a finite
interval
Tsorng-Hwa Chang
1,2
and Chung-Tsun Shieh
1*
* 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
L
Q
= −
d
2
dx
2
+ Q(x
)
is conside red,
where Q(x) is an integrable m × m matrix-valued function defined on the interval
[0,π] The authors prove that m

,
(1:1)
on a finite interv al is devoted to deter mine the potential matrix Q(x) from the spec-
tral data of (1.1) with boundary conditions
U
(

y
)
:=

y

(
0
)
− h

y
(
0
)
=0, V
(

y
)
:=

y

(
C
)
and
Q(x)=[Q
ij
(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 m echanics 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 =2andQ(x) is a two-by-two real s ymme tric matrix- valu ed smooth func-
tions defined in the interval [0, π] Shen [18] showed that five spectral data can
Chang and Shieh Boundary Value Problems 2011, 2011:40
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© 2011 Chang and Shieh; licensee Springer. This is an Open Access art icle 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, provid ed 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

(0) − B
j

y(0) =

y(π )=

0
,
(1:4)
for j = 1, 2, 3, where
B
j
=

α
j
β
j
β
j
γ
j

is a real symmetric matrix and {(a
j
, b

(

Q
)
= σ
ND
(

Q
)
and
σ
j
(Q)=σ
j
(

Q
)
for j = 1, 2, 3, then
Q
(
x
)
=

Q
(
x
)

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
ij
]

+ ϕ
(
x, λ
)
M
(
λ
)
which satisfy the boundary conditions

U( )=

(0) − h(0) = I
m
,
V()=

(π)+H(π)=0
m
.
(2:1)
Then,
M
(
λ
)
= 
(
0, λ
)

Q,
˜
h,

H
)
separately. S uppose
M
(
λ
)
=

M
(
λ
)
, then
h
=
˜
h
,
h
=
˜
h
and
H
=

))
−
1
,
(2:2)
M
(
λ
)
= −
(
V
(
ϕ
))
−1
V
(
S
)
= ψ
(
0, λ
)(
U
(
ψ
))
−
1

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 o f
A. In the remaining of this section , we shall prove some uniqueness theorems for vec-
torial Sturm-Liouville equations. Let
B(i, j)=

b
rs

r
,
s=1
,
m
,
b

ij
be defined as above. Then

ij
(λ)=
00
(λ)+det(Augment[ϕ

1
(π,λ)+Hϕ
1
(π,λ), ,
(jth column)
S

i
(π,λ)+HS
i
(π,λ), , ϕ

m
(π,λ)+Hϕ
m
(π,λ)])
,
where 
k
(π, l) is the k th column of  (π, l) and S
k
(π, l) the kth colum n of S (π, l)


(π,λ)+Hϕ(π,λ)) +
(jth column)
[0, S

i
(π,λ)+HS
i
(π,λ)0])
=det(ϕ

(π,λ)+Hϕ(π,λ)) + det(ϕ

1
(π,λ)+Hϕ
1
(π,λ),
,
(jth column)
S

i
(π,λ)+HS
i
(π,λ), , ϕ

m
(π,λ)+Hϕ
m
(π,λ))

denotes the analogous object related
to
L
m
(

Q,
˜
h,

H
)
.
Theorem 2.3. Suppose that

i
j
(λ)=


i
j
(λ
)
for (i, j)=(0,0)or 1 ≤ i, j ≤ mthen
h
=
˜
h
,

)
+ ϕ
(
x, λ
)M(
λ
),
we have that
−
(
S

(
π,λ
)
+ HS
(
π,λ
))
e
i
=
(
ϕ

(
π,λ
)
+ Hϕ
(

(π,λ), ,
(jth column)
S

i
(π,λ)+HS
i
(π,λ), , ϕ

m
(π,λ)+Hϕ
m
(π,λ))
det(ϕ

(π,λ)+Hϕ(π ,λ))
=

00
(λ) − 
ij
(λ)

00
(λ)
=


00
(λ) −

Lemma 2.4. Suppose that h, H a re real symmetric matri ces and Q(x) is a real sym-
metric matrix-valuedfunction.Then,
M(
λ
)
= −V
(
ϕ
)
−1
V
(
S
)
is re al 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)
For l Î ℝ,
⎧

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
,
(

(x, λ)=

−(S)
∗
(x, λ)(S
∗
)

(x, λ)
ϕ
∗
(x, λ) −(ϕ
∗
)

(x, λ)

.
(2:6)
Chang and Shieh Boundary Value Problems 2011, 2011:40
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Page 4 of 8
Now let
U
2
(x, 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
,

1
V
(
S
)
is real symmetric for all l Î ℝ. □
Definition 2.1. We call L
m
(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 L
m
(h, H, Q) and
L
(
˜
h,
˜
H,
˜
Q
)
be two real symmetric problems. Sup-
pose that

i
j
(λ)=
˜


are real symmetric. Moreover,
M
ji
(λ)=

00
(λ) − ij(λ)

00
(λ)
=
˜

00
(λ) −
˜
ij(λ)
˜

00
(λ)
=
˜
M
j
i
(λ), for 1 ≤ i ≤ j ≤ m
.
Hence,
M

(λ
)
and
M
i
j
(λ)=
˜
M
i
j
(λ
)
for lÎℂ.Thiscom-
pletes the proof. □
From now on, we let L
m
(Q, h, H) be a real symm etric problem. We would like to
know that how many spectral data can determine the problem L
m
(Q, h, H)ifwe
require all spectral data come from real symmetric problems. Denote

ij
=

e
1
, ,
(ith-column)

1
,0, ,0
)
t
.
Hence, Γ
ij
+ Γ
ij
= I
m
.LetΘ
ij
(l)bethe
characteristic function of the self-adjoint problem
y

+
(
λI
m
− Q
(
x
))
y =0, 0< x <
π
(2:7)
Chang and Shieh Boundary Value Problems 2011, 2011:40
http://www.boundaryvalueproblems.com/content/2011/1/40

i
), , V(ϕ
m
)]
,
where V (L
j
) d enotes the jth column of (V( L)) for a m × m matrix L. Similarly, we
denote Ω
ij
(l) 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

ij
(λ)=det
⎡
⎢
⎣
V(ϕ
1
), ,
(ith-column)
V(ϕ

2
det[V(ϕ
1
), ,
(ith-column)
V(S
j
), ,
(jth-column)
V(ϕ
j
), , V(ϕ
m
)]
+
1
2
det[V(ϕ
1
), ,
(ith-column)
V(ϕ
i
), ,
(jth-column)
V(S
i
), , V(ϕ
m
)]

, m
i
)}
i =1,∞
the spectr al
data of L
m
(Q, h, H)wherem
i
is the multiplicity of the eigenvalue l
i
of L
m
(Q, h, H)
then the characteristic function of L
m
(Q, h, H)is
(λ)=C
∞
i=1
(1 −
λ
λ
i
)
m
i
where C is determined by {(l
i
, m

)
for (i, j) = (0, 0) or 1 ≤ i ≤ j ≤ m,
(2)

i
j
(λ)=


i
j
(λ
)
for 1 ≤ i<j≤ m.,
are satisfied, then
h
=
˜
h
,
H
=

H
and
Q
(
x
)
=

-co
l
umn
)
V(ϕ
j
)+V(S
i
), , V(ϕ
m
)]
=det[V(ϕ
1
), , V(ϕ
j
), , V(ϕ
m
)]
+det[V(ϕ
1
), ,
(ith-column)
V(ϕ
i
), ,
(jth-column)
V(S
i
), , V(ϕ
m

(λ)=
˜

00
(λ) −
˜

00
(λ)
˜
M
j
i
(λ)
.
Moreover, by the assumptions and Lemma 2.4, we have M
ij
(l)=M
ji
(l) Hence,
(1) Δ
ij
(l)=Δ
ji
(l) and
˜

i
j
(λ)=

λ
)
for i = 0, 1, , m,
(3)

i
j
(λ)=
i
j
(λ) − 
i
j
(λ)=


i
j
(λ) −


i
j
(λ)=


i
j
(λ
)

Corollary 2.7. Suppose L
m
(Q, h, H) and
L
m
(

Q,
˜
h,

H
)
are both diagonals. If

kk
(
λ
)
=


kk
(
λ
)
for k = 0, 1, , m, then
Q
=


det
(
ϕ

(
π,λ
)
+ Hϕ
(
π,λ
))
(S

(π,λ)+HS(π,λ)
)
is diagonal and so is

M
(
λ
)
. Hence,
M
i
j
(λ)=0fori = j,1≤ i, j ≤ m
.
Moreover,
M
kk

(λ) − 
00
(λ))
=

00
(λ) − 
kk
(λ)

00
(λ)
=


00
(λ) −


kk
(λ)


00
(λ)
=

M
kk
(

=

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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