EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 3, 2017, 535-543
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On an inverse problem for Sturm-Liouville Equation

Döne Karahan1,∗, Khanlar R. Mamedov2

1 Department of Mathematics, Harran University, Turkey
2 Department of Mathematics, Mersin University, Turkey

Abstract. In this study, the theorem on necessary and sufficient conditions for the solvability
of inverse problem for Sturm-Liouville operator with discontinuous coefficient is proved and the
algorithm of reconstruction of potential from spectral data (eigenvalues and normalizing numbers)
is given.

2010 Mathematics Subject Classifications: 34A55, 34B24

Key Words and Phrases: Sturm-Liouville operator, inverse problem, necessary and sufficient
conditions

1. Introduction

We consider the boundary value problem

−y′′ + q(x)y = λ2ρ(x)y, 0 ≤ x ≤ π, (1)

y′(0) = 0, y(π) = 0, (2)

where q (x) ∈ L2 (0, π) is a real-valued function, ρ(x) is a piecewise continuous function,
λ is a complex parameter. This spectral problem appears while solving wave or heat
equations for nonhomogeneous density of the material [1], [2]. Physical applications of
discontinuous Sturm-Liouville problem are given in [3]-[8].

For simplicity, we will assume that the density function has only one discontinuity
point such that

ρ(x) =

{
1, 0 ≤ x ≤ a,
α2, a < x ≤ π, (3)

where 0 < α 6= 1.
Direct problem of spectral analysis for Sturm-Liouville problem is investigated prop-

erties of eigenvalues and eigenfunctions, finding normalizing numbers, spectrum set of the
boundary value problem, scattering data and some other values. It is important to inves-
tigate these properties. Inverse problem of spectral analysis is to final the coefficient of

∗Corresponding author.

Email addresses: dkarahan@harran.edu.tr (D. Karahan), hanlar@mersin.edu.tr (Kh. R. Mamedov)

http://www.ejpam.com 535 c© 2017 EJPAM All rights reserved.



D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, 10 (3) (2017), 535-543 536

the equation for given spectral data. This has to be done uniquely, so that it gives the
uniqueness of the inverse problem. In the process of the solution of the inverse problem
giving an algorithm for constructing the potential is important. For ρ(x) ≡ 1, solutions of
inverse problem for equation (1) is given by [9]-[16]. For ρ(x) 6= 1, under different bound-
ary conditions similar problem is solved in [17]-[21]. When boundary conditions contain
spectral parameter, it is solved by [22], [23].

The inverse problem for this equation is to find necessary and sufficient conditions for
any data set to be spectral data. The main of this work is to find these conditions for (1),
(2) boundary value problem. Firstly spectral data is defined. Characteristic properties
of these values are investigated in [20] and also uniqueness of the solution of the inverse
problem is proved.

Consequently, in this work for (1), (2) spectral problem, solution of the inverse problem
is given with respect to the spectral data.

For (1), (2) boundary value problem in [20], it is shown that the real numbers
{
λ2n, αn

}
n≥1

satisfy the following

λn = λ0n +
dn
λ0n

+
kn
n
, αn = α0

n +
tn
n
, {kn}, {tn} ∈ l2, (4)

where λ0n are zeros of the function

∆0(λ) =
1

2
(1 +

1

α
) cosλµ+(π) +

1

2
(1− 1

α
) cosλµ−(π),

dn =
h+ sinλ0nµ

+(π) + h− sinλ0nµ
−(π)

1
2(1 + 1

α)µ+(π) sinλ0nµ
+(π) + 1

2(1− 1
α)µ−(π) sinλ0nµ

−(π)

is a bounded sequence.
In [18] it is proved, that the solution ϕ(x, λ) of the equation (1) with initial date

ϕ(0, λ) = 1, ϕ′(0, λ) = 0 can be represented as

ϕ(x, λ) = ϕ0(x, λ) +

∫ µ+(x)

0
A(x, t) cosλtdt, (5)

where A(x, t) belongs to the space L2(0, π) for each fixed x ∈ [0, π] and is related to the
coefficient q(x) of the equation (1) by the formula:

d

dx
A(x, µ+(x)) =

1

4
√
ρ(x)

(
1 +

1√
ρ(x)

)
q(x), (6)

ϕ0(x, λ) =
1

2

(
1 +

1√
ρ(x)

)
cosλµ+(x) +

1

2

(
1− 1√

ρ(x)

)
cosλµ−(x) (7)

is the solution of (1) when q(x) ≡ 0,

µ+(x) = ±x
√
ρ(x) + a

(
1∓

√
ρ(x)

)
. (8)



D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, 10 (3) (2017), 535-543 537

The characteristic function ∆(λ) of the problem (1), (2) is

∆(λ) :=< ϕ(x, λ), ψ(x, λ) >= ϕ(x, λ)ψ′(x, λ)− ϕ′(x, λ)ψ(x, λ)

where ∆(λ) is independent from x ∈ [0, π]. Substituting x = 0 and x = π into above the
equation, we get

∆(λ) = ϕ(π, λ) = ψ′(0, λ).

Theorem 1. For each fixed x ∈ [0, π] the kernel A(x, t) from the representation (5)satisfies
the following linear functional integral equation

2

1 +
√
ρ(t)

A
(
x, µ+(t)

)
+

1−
√
ρ(2a− t)

1 +
√
ρ(2a− t)

A (x, 2a− t) +

+F (x, t) +

∫ µ+(x)

0
A(x, ξ)F0(ξ, t)dξ = 0, 0 < t < x (9)

where

F0(x, t) =

∞∑
n=1

(
ϕ0(t, λn) cosλnx

αn
− ϕ0(t, λ

0
n) cosλ0nx

α0
n

)
(10)

F (x, t) =
1

2

(
1 +

1√
ρ(x)

)
F0(µ

+(x), t) +
1

2

(
1− 1√

ρ(x)

)
F0(µ

−(x), t) (11)

{
λ0n
}2

are eigenvalues and α0
n are norming constants of the boundary value problem (1),

(2) when q(x) ≡ 0.

Theorem 2. For each fixed x ∈ [0, π] main equation (9) has a unique solution A(x, .) ∈
L2,ρ (0, µ+(x)).

The proof of Theorem 1 and Theorem 2 is given in [21].

2. Sufficient conditions for solvability of the inverse problem

Assume that the real numbers
{
λ2n, αn

}
n≥1 is given by the formula (4). Now, let’s

construct F0(x, t) and F (x, t) functions by using the formulas (10), (11) and write the
integral equation (9).

We determine A(x, t) from the main equation (9). We shall construct the function
ϕ(x, λ) with the formula (5) i.e.

ϕ(x, λ) := ϕ0(x, λ) +

∫ µ+(x)

0
A(x, t) cosλtdt,

and the function q(x) with formula

q(x) :=
4ρ(x)√
ρ(x) + 1

d

dx
A
(
x, µ+(x)

)
. (12)



D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, 10 (3) (2017), 535-543 538

Denote

b(x) :=
∞∑
n=1

(
cosλnx

αnλ2n
− cosλ0nx

α0
nλ

0
n
2

)
.

Similar to Lemma 1.3.4 in [15], it is shown that b(x) ∈ W 1
2 (0, π). According to (4) and

(5) we have
F0tt(x, t) = ρ(t)F0xx(x, t), ρ(t)Fxx(x, t) = ρ(x)Ftt(x, t), (13)

F0(x, t)|x=0 = 0, F0(x, t)|t=0 = 0, (14)

∂

∂x
F0(µ

±(x), t) = ±
√
ρ (x)

∂

∂ξ
F0 (ξ, t)|ξ=µ±(x) . (15)

Using the main equation (9) it can be proved that

A(x, 0) = 0, (16)√
ρ(x)− 1√
ρ(x) + 1

d

dx
A(x, µ+(x)) =

d

dx

{
A(x, µ−(x) + 0)−A(x, µ−(x)− 0

}
. (17)

2.1. Derivation of the Differential Equation

Lemma 1. The following relations hold

−ϕ′′(x, λ) + q(x)ϕ(x, λ) = λ2ρ(x)ϕ(x, λ), (18)

ϕ(0, λ) = 1, ϕ′(0, λ) = 0. (19)

Proof. Assume that b(x) ∈W 2
2 (0, π) and

J(x, λ) :=
2

1 +
√
ρ(t)

A
(
x, µ+(t)

)
+

1−
√
ρ(2a− t)

1 +
√
ρ(2a− t)

A (x, 2a− t) +

+F (x, t) +

∫ µ+(x)

0
A(x, ξ)F0(ξ, t)dξ = 0, (20)

Differentiating (20) twice with respect to x and t we get

J ′′xx(x, t)− ρ(x)J ′′tt(x, t)− q(x)J(x, λ) ≡ 0.

Using the formulas (9), (12)-(15) and (17), we obtain the following homogeneous equa-
tion

2

1 +
√
ρ(t)

[
Axx

(
x, µ+(t)

)
− ρ(x)Att

(
x, µ+(t)

)
− q(x)A

(
x, µ+(t)

)]
+

+
1−

√
ρ(2a− t)

1 +
√
ρ(2a− t)

[Axx (x, 2a− t)− ρ(x)Att (x, 2a− t)− q(x)A (x, 2a− t)] +



D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, 10 (3) (2017), 535-543 539

+

∫ µ+(x)

0
[Axx(x, ξ)− ρ(x)Aξξ(x, ξ)− q(x)A(x, ξ)]F0(ξ, t)dξ = 0.

We know that from [21] this equation has only trivial solution:

Axx(x, t)− ρ(x)Att(x, t)− q(x)A(x, t) = 0, 0 < t < x. (21)

Differentiating (5) twice, integrating by parts twice and using (16) we obtain

ϕ′′(x, λ) + λ2ρ(x)ϕ(x, λ)− q(x)ϕ(x, λ) = ϕ′′0(x, λ) +

∫ µ+(x)

0
Axx(x, t) cosλtdt+

−λρ(x)A(x, µ+(x)) sinλµ+(x) +
√
ρ(x)Ax(x, µ+(x)) cosλµ+(x)+

+λρ(x) sinλµ−(x)
(
A
(
x, µ−(x) + 0

)
−A

(
x, µ−(x)− 0

))
+

+
√
ρ(x) cosλµ−(x)

d

dx

(
A
(
x, µ−(x) + 0

)
−A

(
x, µ−(x)− 0

))
+

+
√
ρ(x) cosλµ+(x)

∂A(x, t)

∂x

∣∣∣∣
t=µ+(x)

+

+
√
ρ(x) cosλµ−(x)

(
∂A(x, t)

∂x

∣∣∣∣
t=µ−(x)+0

− ∂A(x, t)

∂x

∣∣∣∣
t=µ−(x)−0

)
−

−ϕ′′0(x, λ) + λρ(x) sinλµ+(x)A(x, µ+(x)) + ρ(x) cosλµ+(x)
∂A(x, t)

∂t

∣∣∣∣
t=µ+(x)

−

−λρ(x) sinλµ−(x)
{
A
(
x, µ−(x) + 0

)
−A

(
x, µ−(x)− 0

)}
+

+ρ(x) cosλµ−(x)

[
∂A(x, t)

∂t

∣∣∣∣
t=µ−(x)−0

− ∂A(x, t)

∂t

∣∣∣∣
t=µ−(x)+0

]
−

−ρ(x)

∫ µ+(x)

0
A′′tt(x, t) cosλtdt−

−q(x)

[
1

2

(
1 +

1√
ρ(x)

)
cosλµ+(x)+

+
1

2

(
1− 1√

ρ(x)

)
cosλµ−(x) +

∫ µ+(x)

0
A(x, t) cosλtdt

]
.

Hence using (12), (17) and (21) we arrive at (18). The relations (19) follow from (5) for
x = 0. Lemma 1 is proved in the case b(x) ∈W 2

2 (0, π).
The proof of Lemma 1 in the case b(x) ∈W 1

2 (0, π) is carried out by a standard method
(see e.g. [8] p. 40).

As in the theory of Sturm-Liouville problems (see [15], Lemma 1.5.8 and Corollary
1.5.1) the following lemmas can be proved.



D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, 10 (3) (2017), 535-543 540

Lemma 2. For each function g(x) ∈ L2,ρ(0, π),∫ π

0
ρ(x)g2(x)dx =

∞∑
n=1

1

αn

(∫ π

0
ρ(t)g(t)ϕ(t, λn)dt

)2

. (22)

Corollary 1. For arbitrary functions f(x), g(x) ∈ L2,ρ(0, π),∫ π

0
ρ(x)f(x)g(x)dx =

∞∑
n=1

1

αn

∫ π

0
ρ(t)f(t)ϕ(t, λn)dt

∫ π

0
ρ(t)g(t)ϕ(t, λn)dt. (23)

Using the below lemmas the following lemma is proved with standard method.

Lemma 3. The following relation holds∫ π

0
ρ(x)ϕ(t, λn)ϕ(t, λk)dt =

{
0, n 6= k
αn, n = k.

(24)

2.2. Derivation of Boundary Condition

Lemma 4. For all n ≥ 1 the equality

ϕ(π, λn) = 0

holds.

Proof. Since
−ϕ′′(x, λn) + q(x)ϕ(x, λn) = λ2nρ(x)ϕ(x, λn),

−ϕ′′(x, λm) + q(x)ϕ(x, λm) = λ2mρ(x)ϕ(x, λm),

we get
d

dx

(
ϕ(x, λn)ϕ′(x, λm)− ϕ′(x, λn)ϕ(x, λm)

)
=

=
(
λ2n − λ2m

)
ρ(x)ϕ(x, λn)ϕ(x, λm) (25)

From (25) we have (
λ2n − λ2m

) ∫ π

0
ρ(x)ϕ(x, λn)ϕ(x, λm)dx =

= ϕ(π, λn)ϕ′(π, λm)− ϕ′(π, λn)ϕ(π, λm).

By (24) we get
ϕ(π, λn)ϕ′(π, λm)− ϕ′(π, λn)ϕ(π, λm) = 0. (26)

Clearly, ϕ′(π, λn) 6= 0, for all n ≥ 1. Indeed, if we suppose that ϕ′(π, λm) = 0 for a certain
m, then ϕ(π, λm) 6= 0, and in view of (26) ϕ′(π, λn) = 0 for all n.

On the other hand,

ϕ′(π, λn) = ϕ′0(π, λn) +O(e|Imλ|µ
+(x)), |λ| → ∞



D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, 10 (3) (2017), 535-543 541

i.e. for any n, ϕ′(π, λn) ≈ ϕ′0(π, λ
0
n) 6= 0 as n → ∞, that contradicts the condition

ϕ′(π, λn) = 0, n 6= m. Thus, ϕ′(π, λn) 6= 0, for all n ≥ 1 and from (26) we have

ϕ(π, λn)

ϕ′(π, λn)
=
ϕ(π, λm)

ϕ′(π, λm)
= H,

i.e. for any n, ϕ(π, λn) = Hϕ′(π, λn). Since ϕ(π, λn) = o(1) as n → ∞, we have H = 0
i.e. ϕ(π, λn) = 0.

Thus, we prove that the numbers
{
λ2n, αn

}
n≥1 are spectral data of the constructed

boundary value problem (1), (2). Then, the following theorem is proved.

Theorem 3. For the sequences
{
λ2n, αn

}
n≥1 , where λn 6= λm for n 6= m,αn > 0 for all

n to be spectral date of a problem L(q(x)) of the form (1)-(3) with q(x) ∈ L2(0, π), it is
necessary and sufficient to satisfy conditions

λn = λ0n +
dn
λ0n

+
kn
n
, αn = α0

n +
tn
n
, {kn} , {tn} ∈ l2

Here λ0n are the zeros of the function

∆0(λ) =
1

2

(
1 +

1

α

)
cosλµ+(π) +

1

2

(
1− 1

α

)
cosλµ−(π),

α0
n =

∫ π

0
ϕ2
0(x, λn)ρ(x)dx,

ϕ0(x, λ) =
1

2

(
1 +

1√
ρ(x)

)
cosλµ+(x) +

1

2

(
1− 1√

ρ(x)

)
cosλµ−(x),

µ±(x) = ±x
√
ρ(x) + a

(
1∓

√
ρ(x)

)
,

dn is a bounded sequence; {kn} , {tn} ∈ l2.

Algorithm of the construction of the function q(x) by spectral date
{
λ2n, αn

}
follows

from the proof of the Theorem 3:
1) By the given numbers

{
λ2n, αn

}
n≥1 the functions F0(x, t) and F (x, t) are constructed

by the formulas (10) and (11), respectively;
2) The function A(x, t) is found from equation (9);
3) q(x) is calculated by the formula (12).

Acknowledgements

This work is supported by the Scientific and Technological Research Council of Turkey
(TUBITAK).



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