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
with boundary conditions
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 . 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  and discontinuity conditions inside an interval were worked out in [10,11]. The inverse problem for weighted Dirac equations was obtained in . The reconstruction of the potential by the spectral function was given in . For the Dirac operator with peculiarity, the inverse problem was found in . 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 , and in this work, not only the Gelfand-Levitan-Marchenko method but also the Krein method  was used. In the positive half line, the inverse scattering problem for the Dirac operator with discontinuous coefficient was analyzed in . 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 . 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.
The solution has an integral representation  as follows:
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
Denote the normalizing numbers by
The following relation is valid:
The following two theorems are obtained by Huseynov and Latifova in .
(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
From , the following inequality holds:
In Section 3, the fundamental equation
is derived by using the method by Gelfand-Levitan-Marchenko, where
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
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 ().
3 Fundamental equation
Proof According to (3) we have
It follows from (3) and (16) that
Using the last two equalities, we obtain
It is easily found by using (14) and (15) that
From (18), we find
It follows from (3) that
Taking into account (21) and expansion formula (10) in Theorem 2, we get
Now, we calculate
Using (7) and the residue theorem, we get
and the relations (, Lemma 1.3.1)
it follows from (12) and (24) that
Thus, using (17), (19), (20), (22) (23), and (25), we find
is obtained. □
We show that
It follows from (14) that
Using (21), we get
Using the Parseval equality,
it follows from (28) that
5 Reconstruction by spectral data
Now, let us construct the function by (3) and the function by (4). From , and have a derivative in both variables and these derivatives belong to .
Lemma 6The following relations hold:
Proof Differentiating to x and y, (13), respectively, we get
It follows from (14) and (15) that
and using the fundamental equation (13), we obtain
Adding (36) and (37) and using (34), we find
From (33), we get
Integrating by parts and from (35)
is obtained. Setting
we can rewrite (42) as follows:
According to Lemma 4, the homogeneous equation (43) has only the trivial solution, i.e.
Differentiating (3) and multiplying on the left by B, we have
It follows from (45) and (46) that
Taking into account (4) and (44),
Proof It follows from (3) and (21) that
Using the expression
the fundamental equation (13) is transformed into the following form:
From (48), we get
and using (48) it is transformed into the following form:
Similarly, in view of (50), we have
According to (52),
It follows from (49) and (51) that
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. □
Lemma 9The relation
Using Lemma 6 and integrating by parts, we get
Proof It is easily found that
According to (56), we get
Proof Necessity of the problem is proved in . 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 authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
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