Research

# Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition

Khanlar R Mamedov* and F Ayca Cetinkaya

Author Affiliations

Department of Mathematics, Science and Letters Faculty, Mersin University, Mersin, 33343, Turkey

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Boundary Value Problems 2013, 2013:183  doi:10.1186/1687-2770-2013-183

 Received: 5 April 2013 Accepted: 29 July 2013 Published: 14 August 2013

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.

### Abstract

This work aims to examine a Sturm-Liouville operator with a piece-wise continuous coefficient and a spectral parameter in boundary condition. The orthogonality of the eigenfunctions, realness and simplicity of the eigenvalues are investigated. The asymptotic formula of the eigenvalues is found, and the resolvent operator is constructed. It is shown that the eigenfunctions form a complete system and the expansion formula with respect to eigenfunctions is obtained. Also, the evolution of the Weyl solution and Weyl function is discussed. Uniqueness theorems for the solution of the inverse problem with Weyl function and spectral data are proved.

MSC: 34L10, 34L40, 34A55.

##### Keywords:
Sturm-Liouville operator; expansion formula; inverse problem; Weyl function

### 1 Introduction

In recent years, there has been a growing interest in physical applications of boundary value problems with a spectral parameter, contained in the boundary conditions. The relationship between diffusion processes and Sturm-Liouville problem with eigen-parameter in the boundary conditions has been shown in [1]. Another example of this relationship between the same problem and the wave equation has been examined in [2,3]. Sturm-Liouville problems with a discontinuous coefficient arise upon non-homogeneous material properties.

In a finite interval, inverse problems for the Sturm-Liouville operator with spectral parameter, contained in the boundary conditions, have been investigated, and the uniqueness of the solution of these problems has been shown in [4-9]. The inverse problem has been analyzed by zeros of the eigenfunctions in [6], by numerical methods in [9] and by two spectra, consisting of sequences of eigenvalues and the normed constants in [10]. In [11,12], eigenvalue-dependent inverse problem with the discontinuities inside the interval was examined by the Weyl function. In a finite interval, discontinuous and no eigenvalue parameter containing direct problem and inverse problem with the Weyl function were discussed in [13,14]. The similar problem was investigated in the half line by scattering data in [15,16].

We consider the boundary value problem

(1)

(2)

(3)

where is a real valued function, λ is a complex parameter, , , , are real numbers and

where .

### 2 Special solutions

Let and be the solutions of equation (1) satisfying the initial conditions

(4)

(5)

For the solution of equation (1), the following integral representation is obtained in [13] for all λ:

where .

The kernel has the partial derivative belonging to the space for every , and the properties below hold:

Moreover, if is differentiable, then the following are valid

Using the representation of the solution and formula

we obtain the integral representation of the solution

(6)

where . The kernel can be represented with the coefficients and

With the help of equation (6), we have a representation for the function

(7)

where is a real function.

Denote

(8)

which is independent of . Substituting and into (8), we get

is an entire function of λ and is called the characteristic function of the boundary value problem (1)-(3).

### 3 Some spectral properties

Lemma 1The square values of the rootsof the characteristic function coincide with the eigenvalues of the boundary value problem (1)-(3), and for every, there exists a sequencesuch that

(9)

whereandare the eigenfunctions of the boundary value problem (1)-(3), corresponding to the eigenvalue.

Proof The proof can be done in a similar way to [8]. Indeed, let us assume that is an eigenvalue of the function . Then

holds, i.e., the functions and are linearly dependent (), and they satisfy the boundary conditions (2), (3). Hence is an eigenvalue, and are eigenfunctions, related to this eigenvalue. Conversely, let be an eigenvalue of the operator A, and let , be the corresponding eigenfunctions. Then the boundary conditions (2), (3) hold both for the eigenfunctions and . Additionally, if the functions and satisfy the conditions , , then , . According to boundary conditions (2), (3), we have

Similarly, if we assume that , , then , . Again from the boundary conditions (2), (3), it is obvious that

Therefore, we have proved that for each eigenvalue , there exists only one (up to a multiplicative constant) eigenfunction. □

In the Hilbert space an inner product defined by

where

Let us define

with

where

The boundary value problem (1)-(3) is equivalent to the equation .

The eigenfunctions of operator A are in the form of

Lemma 2The eigenfunctionsand, corresponding to different eigenvalues, are orthogonal.

Proof Since and are the solutions of the boundary value problem (1)-(3), the equations below are valid

Multiplying the first equation by and the second equation by and adding together, we have

Integrating it from 0 to π, and using the boundary condition (3), we obtain

Since , the lemma is proved. □

Corollary 3The eigenvalues of the boundary value problem (1)-(3) are real.

The values

(10)

are called the norming constants of the boundary value problem (1)-(3).

Now, let us agree to denote differentiation with respect to λ with a dot ().

Lemma 4The following equality holds

(11)

Proof Since

we get

With the help of (2) and (3),

Taking into consideration (9) and (10), for , we arrive (11). □

Corollary 5All zeros ofare simple, i.e., .

### 4 Asymptotic formulas of eigenvalues

Let be the solution of equation (1) satisfying the initial conditions (4) when

(12)

The eigenvalues () of the boundary value problem (1)-(3) when can be found using the equation

from [17] (see also [18]) and can be represented in the following way

(13)

where .

Lemma 6Rootsof the functionare separated, i.e.,

Proof Assume the contrary. Then there are sequences , of zeros of functions such that

Since the eigenfunctions , are orthogonal, we have

where

Thus,

(14)

Let us prove that as . In fact, (12) implies the following estimate

Consequently, holds uniformly on . Now, passing to the limit in equality (14), as , we have . This is a contradiction, and it proves the validity of lemma’s statement. □

Lemma 7The eigenvalues of the boundary value problem (1)-(3) are in the form of

(15)

whereis a bounded sequence

and.

Proof From (6), it follows that

(16)

Lemma 1.3.1 in [19] and from [17], we get

(17)

Therefore, for sufficiently large n, on the contours

we have

By the Rouche theorem, we obtain that the number of zeros of the function inside the contour coincides with the number of zeros of the function . Furthermore, applying the Rouche theorem to the circle , we get that, for sufficiently large n there exists only one zero of the function in . Owing to the arbitrariness of we have

(18)

Substituting (18) into (16), we get

Hence, as , taking into account the equality and relations , , integrating by parts and using the properties of the kernel , we have

where

Let us show that . It is obvious that

can be reduced to

where . Now, take

It is clear from [19] (p.66) that . By virtue of this we have (see [20,21]). Therefore, as

the validity of can be seen directly. Lemma is proved. □

### 5 Expansion formula

Assume that is not a spectrum point of operator A. Then, there exists resolvent operator . Let us find the expression of the operator .

Lemma 8The resolventis the integral operator with the kernel

(19)

Proof To construct the resolvent operator of A, we need to solve the boundary value problem

(20)

(21)

(22)

where . By applying the method of variation of constants, we seek the solution of the problem (20)-(22) in the following form

(23)

and we get the coefficients and as

(24)

(25)

Substituting (24) and (25) into (23) and taking into account the boundary conditions (21) and (22), we have

(26)

where is as in (19). □

Theorem 9The eigenfunctionsof the boundary value problem (1)-(3) form a complete system in.

Proof With the help of (9) and (11), we can write

(27)

Using (19) and (26), we get

(28)

Now, let and assume

(29)

Then from (28), we have . Consequently, for fixed the function is entire with respect to λ.

Let us denote that

where δ is a sufficiently small positive number. (16) is valid from Theorem 12.4 in [22] for .

On the other hand, we can say from Lemma 1.3.1 in [19] that for every , the following relation holds

(30)

Also, for , the relations below hold

(31)

(32)

(33)

(34)

From the estimates (31)-(34), it is obvious that

(35)

From (26), it follows that for fixed and sufficiently large , we have

Using maximum principle for module of analytic functions and Liouville theorem, we get . From this and the expression of the boundary value problem (20)-(22), we obtain that a.e. on . Thus, we reach the completeness of the eigenfunctions in . □

Theorem 10If, then the expansion formula

(36)

is valid, where

and the series converges uniformly with respect to. For, the series converges in, moreover, the Parseval equality holds

Proof Since and are the solutions of the boundary value problem (1)-(3), we have

(37)

Integrating by parts and taking into account the boundary conditions (2), (3), we obtain

(38)

where

as . Let us consider the following contour integral

where is a contour oriented counter-clockwise, and n is a sufficiently large natural number. With the help of Residue theorem, we get

(39)

where

On the other hand, taking into account (38), we have

(40)

Comparing (39) and (40), we obtain

where

Thus, we obtain

(41)

Now, let us show that

(42)

From estimates (31)-(34) of solutions , and the inequality (35) for the function , it follows, for fixed and sufficiently large

(43)

Let us show that . If we suppose that , and by then integrate by parts the expression of , we obtain

Hence, similar to , we have

(44)

In general case, let us take an arbitrary fixed number and assume that , such that . Then we can find a that for and . Also, using the equation below,

and with the help of the estimates of functions , and , we get

as , . Hence we have

From the arbitrariness of ϵ, we reach

(45)

The validity of (42) can be easily seen from (43) and (44). Thus, we obtain

If we take

the last equation gives us the expansion formula

Since the system of is complete and orthogonal in , the Parseval equality

holds. Extension of the Parseval equality to an arbitrary vector-function of the class can be carried out by usual methods. □

### 6 Weyl solution, Weyl function

Let be the solution of equation (1) that satisfies the conditions

Denote by the solution of equation (1), which satisfies the initial conditions , . Then the solution can be represented as follows

or

(46)

Denote

(47)

It is clear that

(48)

The functions and are respectively called the Weyl solution and the Weyl function of the boundary value problem (1)-(3). The Weyl function is a meromorphic function having simple poles at points eigenvalues of boundary value problem (1)-(3). Relations (46), (48) yield

(49)

It can be shown that

(50)

Let us take into consideration a boundary value problem with the coefficient similar to (1)-(3) and assume that if an element α belongs to boundary value problem (1)-(3), then belongs to one with .

Validity of the equation below can be shown analogously to [8]

(51)

Theorem 11The boundary value problem (1)-(3) is identically denoted by the Weyl function. (If, then.)

Proof Let us identify the matrix as

(52)

From (50) and (52), we have

(53)

or

(54)

Taking equation (49) into consideration in (54), we get

(55)

Now, from the estimates

and

we have from equation (55)

(56)

for . Now, if we take consideration equation (48) into (53), we get

Therefore, if , then and are entire functions for every fixed x. It can easily be seen from equation (56) that and . Consequently, we get and for every x and λ. Hence, we arrive at . □

Theorem 12The spectral data identically define the boundary value problem (1)-(3).

Proof From (51), it is clear that the function can be constructed by . Since for every , from Theorem 10, we can say that . Then from Theorem 11, it is obvious that . □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

### Acknowledgements

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

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