# Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient

Khanlar R Mamedov and Ozge Akcay*

Author Affiliations

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

For all author emails, please log on.

Boundary Value Problems 2014, 2014:110  doi:10.1186/1687-2770-2014-110

 Received: 30 November 2013 Accepted: 25 April 2014 Published: 13 May 2014

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 credited.

### Abstract

In this paper, the inverse problem of recovering the coefficient of a Dirac operator is studied from the sequences of eigenvalues and normalizing numbers. The theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given.

MSC: 34A55, 34L40.

##### Keywords:
Dirac operator; inverse problem; necessary and sufficient condition

### 1 Introduction

In this paper, we consider the boundary value problem generated by the system of Dirac equations on the finite interval :

(1)

with boundary conditions

(2)

where

, are real valued functions, , , λ is a spectral parameter,

and .

The inverse problem for the Dirac operator with separable boundary conditions was completely solved by two spectra in [1,2]. The reconstruction of the potential from one spectrum and norming constants was investigated in [3]. For the Dirac operator, the inverse periodic and antiperiodic boundary value problems were given in [4-6]. Using the Weyl-Titschmarsh function, the direct and inverse problems for a Dirac type-system were developed in [7,8]. Uniqueness of the inverse problem for the Dirac operator with a discontinuous coefficient by the Weyl function was studied in [9] and discontinuity conditions inside an interval were worked out in [10,11]. The inverse problem for weighted Dirac equations was obtained in [12]. The reconstruction of the potential by the spectral function was given in [13]. For the Dirac operator with peculiarity, the inverse problem was found in [14]. Inverse nodal problems for the Dirac operator were examined in [15,16]. In the case of potentials that belong entrywise to , for some , the inverse spectral problem for the Dirac operator was studied in [17], and in this work, not only the Gelfand-Levitan-Marchenko method but also the Krein method [18] was used. In the positive half line, the inverse scattering problem for the Dirac operator with discontinuous coefficient was analyzed in [19]. Besides, in a finite interval, for Sturm-Liouville operator inverse problem has widely been developed (see [20-22]). The inverse problem of the Sturm-Liouville operator with discontinuous coefficient was worked out in [23,24] and discontinuous conditions inside an interval were obtained in [25]. In the mathematical and physical literature, the direct and inverse problems for the Dirac operator are widespread, so there are numerous investigations as regards the Dirac operator. Therefore, we can mention the studies concerned with a discontinuity, which is close to our topic, in the references list.

In this paper, our aim is to solve the inverse problem for the Dirac operator with a piecewise continuous coefficient on a finite interval. Let and () be, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2). The quantities () are called spectral data. We can state the inverse problem for a system of Dirac equations in the following way: knowing the spectral data () to indicate a method of determining the potential and to find necessary and sufficient conditions for () to be the spectral data of a problem (1), (2). In this paper, this problem is completely solved.

We give a brief account of the contents of this paper in the following section.

### 2 Preliminaries

Let be solution of the system (1) satisfying the initial conditions

The solution has an integral representation [26] as follows:

(3)

where

is a quadratic matrix function and is the solution of the problem

(4)

Equation (4) gives the relation between the kernel and the coefficient of (1). Let be solutions of the system (1) satisfying the initial conditions

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

(5)

where is the Wronskian of the solutions and and independent of . The zeros of the characteristic function coincide with the eigenvalues of the boundary value problem (1), (2). The functions and are eigenfunctions and there exists a sequence such that

(6)

Denote the normalizing numbers by

The following relation is valid:

(7)

where . In fact, since and are solutions of the problem (1), (2), we get

Multiplying the equations by , , , , respectively, adding them together, integrating from 0 to π and using the condition (2),

is found. From (6) as , we obtain

The following two theorems are obtained by Huseynov and Latifova in [27].

Theorem 1 (i) The boundary value problem (1), (2) has a countable set of simple eigenvalues () where

(8)

(ii) The eigen vector-functions of problem (1), (2) can be represented in the form

(iii) The normalizing numbers of problem (1), (2) have the form

(9)

Theorem 2 (i) The system of eigen vector-functions () of problem (1), (2) is complete in space.

(ii) Letbe an absolutely continuous vector-function on the segmentand. Then

(10)

moreover, the series converges uniformly with respect to.

(iii) Forseries (10) converges in; moreover, the Parseval equality holds:

(11)

From [27], the following inequality holds:

(12)

where is a positive number and this inequality is valid in the domain

where () are zeros of the function and δ is a sufficiently small number.

In Section 3, the fundamental equation

is derived by using the method by Gelfand-Levitan-Marchenko, where

and

In Section 4, we show that the fundamental equation has a unique solution and the boundary value problem (1), (2) can be uniquely determined from the spectral data. In Section 5, the result is obtained from Lemma 6 that the function defined by (3) satisfies the equation

where

where is the solution of the fundamental equation. In Lemma 7, using the fundamental equation, the Parseval equality

is found. We demonstrate by using Lemma 6, Lemma 9, and Lemma 10 that () are spectral data of the boundary value problem (1), (2). Then necessary and sufficient conditions for the solvability of problem (1), (2) are obtained in Theorem 11. Finally, we give an algorithm of the construction of the function by the spectral data ().

Note that throughout this paper, denotes the transposed matrix of ϕ.

### 3 Fundamental equation

Theorem 3For each fixedthe kernelfrom the representation (3) satisfies the following equation:

(13)

where

(14)

and

(15)

whereandare, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2) when.

Proof According to (3) we have

(16)

It follows from (3) and (16) that

and

Using the last two equalities, we obtain

or

(17)

where

It is easily found by using (14) and (15) that

(18)

Let . Then according to the expansion formula (10) in Theorem 2, we obtain uniformly on

(19)

From (18), we find

(20)

It follows from (3) that

(21)

Taking into account (21) and expansion formula (10) in Theorem 2, we get

Substituting , we obtain

(22)

Now, we calculate

(23)

Using (7) and the residue theorem, we get

(24)

where is oriented counter-clockwise, N is a sufficiently large number. Taking into account the asymptotic formulas as

and the relations ([20], Lemma 1.3.1)

it follows from (12) and (24) that

(25)

Thus, using (17), (19), (20), (22) (23), and (25), we find

Since can be chosen arbitrarily,

is obtained. □

### 4 Uniqueness

Lemma 4For each fixed, (13) has a unique solution.

Proof When , (13) can be rewritten as

where

(26)

Now, we shall prove that is invertible, i.e. has a bounded inverse in .

Consider the equation , . From this and (24),

We show that

In fact,

Thus, the operator is invertible in . Therefore the fundamental equation (13) is equivalent to

and is completely continuous in . Then it is sufficient to prove that the equation

(27)

has only the trivial solution . Let be a non-trivial solution of (27). Then

It follows from (14) that

Using (21), we get

Substituting into the last two integrals, we obtain

(28)

Using the Parseval equality,

it follows from (28) that

Since the system () is complete in , we have , i.e.. For invertible in , is obtained. □

Theorem 5Letandbe two boundary value problems and

Then

Proof According to (14) and (15), and . Then, from the fundamental equation (13), we have . It follows from (4) that a.e. on . □

### 5 Reconstruction by spectral data

Let the real numbers () of the form (8) and (9) be given. Using these numbers, we construct the functions and by (14) and (15) and determine from the fundamental equation (13).

Now, let us construct the function by (3) and the function by (4). From [2], and have a derivative in both variables and these derivatives belong to .

Lemma 6The following relations hold:

(29)

(30)

Proof Differentiating to x and y, (13), respectively, we get

(31)

(32)

It follows from (14) and (15) that

(33)

(34)

and using the fundamental equation (13), we obtain

(35)

Multiplying (31) on the left by B and we get

(36)

and multiplying (32) on the right by B and we have

(37)

Adding (36) and (37) and using (34), we find

(38)

From (33), we get

(39)

Integrating by parts and from (35)

(40)

is obtained. Substituting (40) into (38) and dividing by , we have

(41)

Multiplying (13) on the left by in the form of (4) and adding to (41)

(42)

is obtained. Setting

we can rewrite (42) as follows:

(43)

According to Lemma 4, the homogeneous equation (43) has only the trivial solution, i.e.

(44)

Differentiating (3) and multiplying on the left by B, we have

(45)

On the other hand, multiplying (3) on the left by and then integrating by parts and using (35), we find

(46)

It follows from (45) and (46) that

Taking into account (4) and (44),

is obtained. For , from (3) we get (30). □

Lemma 7For each function, the following relation holds:

(47)

Proof It follows from (3) and (21) that

(48)

Using the expression

the fundamental equation (13) is transformed into the following form:

(49)

From (48), we get

(50)

and for the kernel we have the identity

(51)

Denote

and using (48) it is transformed into the following form:

where

(52)

Similarly, in view of (50), we have

(53)

According to (52),

It follows from (49) and (51) that

(54)

From (18) and the Parseval equality we obtain

Taking into account (54), we have

whence, by (52) and (53),

is obtained, i.e., (47) is valid. □

Corollary 8For any functionand, the following relation holds:

(55)

Lemma 9The relation

(56)

is valid.

Proof (1) Let . Consider the series

(57)

where

(58)

Using Lemma 6 and integrating by parts, we get

Applying the asymptotic formulas in Theorem 1, is found. Consequently the series (57) converges absolutely and uniformly on . According to (55) and (58), we have

Since is arbitrary, is obtained, i.e.

(59)

(2) Fix and assume . Then, by virtue of (59),

where

The system is minimal in and consequently by (3), the system is minimal in . Hence and we obtain (56). □

Lemma 10For allthe equality

is valid.

Proof It is easily found that

According to (56), we get

(60)

We shall prove that for any n, . Assume the contrary, i.e. there exists m such that . Then for , it follows from (60) that . On the other hand, since as

. This contradicts the condition , . Hence, for any n. From (60), we have

Thus, we get , for any n. Since

we find , and then is obtained. □

Theorem 11For the sequences () to be the spectral data for a certain boundary value problemof the form (1), (2) with, it is necessary and sufficient that the relations (8) and (9) hold.

Proof Necessity of the problem is proved in [27]. Let us prove the sufficiency. Let the real numbers () of the form (8) and (9) be given. It follows from Lemma 6, Lemma 9, and Lemma 10 that the numbers () are spectral data for the constructed boundary value problem . The theorem is proved. □

The algorithm of the construction of the function by the spectral data () follows from the proof of the theorem:

(1) By the given numbers () the functions and are constructed, respectively, by (14) and (15).

(2) The function is found from (13).

(3) is calculated by (4).

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Acknowledgements

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

### References

1. Gasymov, MG, Levitan, BM: The inverse problem for the Dirac system. Dokl. Akad. Nauk SSSR. 167, 967–970 (1966)

2. Gasymov, MG, Dzabiev, TT: Solution of the inverse problem by two spectra for the Dirac equation on a finite interval. Dokl. Akad. Nauk Azerb. SSR. 22(7), 3–6 (1966)

3. Dzabiev, TT: The inverse problem for the Dirac equation with a singularity. Dokl. Akad. Nauk Azerb. SSR. 22(11), 8–12 (1966)

4. Misyura, TV: Characteristics of spectrums of periodical and antiperiodical boundary value problems generated by Dirac operation. II. Teoriya funktsiy, funk. analiz i ikh prilozheiniya, pp. 102–109. Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkov (1979)

5. Nabiev, IM: Solution of a class of inverse problems for the Dirac operator. Trans. Natl. Acad. Sci. Azerb.. 21(1), 146–157 (2001)

6. Nabiev, IM: Characteristic of spectral data of Dirac operators. Trans. Natl. Acad. Sci. Azerb.. 24(7), 161–166 (2004)

7. Sakhnovich, A: Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg-Marchenko theorems. Inverse Probl.. 22(6), 2083–2101 (2006). Publisher Full Text

8. Fritzsche, B, Kirstein, B, Roitberg, IY, Sakhnovich, A: Skew-self-adjoint Dirac system with a rectangular matrix potential: Weyl theory, direct and inverse problems. Integral Equ. Oper. Theory. 74(2), 163–187 (2012). Publisher Full Text

9. Latifova, AR: The inverse problem of one class of Dirac operators with discontinuous coefficients by the Weyl function. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb.. 22(30), 65–70 (2005)

10. Amirov, RK: On system of Dirac differential equations with discontinuity conditions inside an interval. Ukr. Math. J.. 57(5), 712–727 (2005). Publisher Full Text

11. Huseynov, HM, Latifova, AR: The main equation for the system of Dirac equation with discontinuity conditions interior to interval. Trans. Natl. Acad. Sci. Azerb.. 28(1), 63–76 (2008)

12. Watson, BA: Inverse spectral problems for weighted Dirac systems. Inverse Probl.. 15(3), 793–805 (1999). Publisher Full Text

13. Mamedov, SG: The inverse boundary value problem on a finite interval for Dirac’s system of equations. Azerb. Gos. Univ. Ucen. Zap. Ser. Fiz-Mat. Nauk. 5, 61–67 (1975)

14. Panakhov, ES: Some aspects inverse problem for Dirac operator with peculiarity. Trans. Natl. Acad. Sci. Azerb.. 3, 39–44 (1995)

15. Yang, CF, Huang, ZY: Reconstruction of the Dirac operator from nodal data. Integral Equ. Oper. Theory. 66, 539–551 (2010). Publisher Full Text

16. Yang, CF, Pivovarchik, VN: Inverse nodal problem for Dirac system with spectral parameter in boundary conditions. Complex Anal. Oper. Theory. 7, 1211–1230 (2013). Publisher Full Text

17. Albeverio, S, Hryniv, R, Mykytyuk, Y: Inverse spectral problems for Dirac operators with summable potentials. Russ. J. Math. Phys.. 12(14), 406–423 (2005)

18. Krein, MG: On integral equations generating differential equations of the second order. Dokl. Akad. Nauk SSSR. 97, 21–24 (1954)

19. Mamedov, KR, Çöl, A: On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition. Math. Methods Appl. Sci.. 35(14), 1712–1720 (2012). Publisher Full Text

20. Marchenko, VA: Sturm-Liouville Operators and Applications, Am. Math. Soc., Providence (2011)

21. Freiling, G, Yurko, V: Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, New York (2008)

22. Guliyev, NJ: Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Probl.. 21, 1315–1330 (2005). Publisher Full Text

23. Mamedov, KR, Cetinkaya, FA: Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition. Bound. Value Probl. (2013). BioMed Central Full Text

24. Akhmedova, EN, Huseynov, HM: On solution of the inverse Sturm-Liouville problem with discontinuous coefficients. Trans. Natl. Acad. Sci. Azerb.. 27(7), 33–44 (2007)

25. Yang, CF, Yang, XP: An interior inverse problem for the Sturm Liouville operator with discontinuous conditions. Appl. Math. Lett.. 22, 1315–1319 (2009). Publisher Full Text

26. Latifova, AR: On the representation of solution with initial conditions for Dirac equations system with discontinuous coefficients. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb.. 16(24), 64–68 (2002)

27. Huseynov, HM, Latifova, AR: On eigenvalues and eigenfunctions of one class of Dirac operators with discontinuous coefficients. Trans. Natl. Acad. Sci. Azerb.. 24(1), 103–112 (2004)